Finite Group Databasebeta

← #29708826

Subgroups — #29708826

Order 15616.

1010 conjugacy classes of subgroups.

The subgroup lattice has 702 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839762C211yesmaximal, characteristic
2C2×#16839757C211yesmaximal, characteristic
3#21448444C211yesmaximal, characteristic, complemented
4#21448410C211yesmaximal, characteristic
5#21447240C211yesmaximal, characteristic, complemented
6#21447236C211yesmaximal, characteristic, complemented
7#21447233C211yesmaximal, characteristic, complemented
8#21447230C211yesmaximal, characteristic, complemented
9#21447227C211yesmaximal, characteristic, complemented
10#21447226C211yesmaximal, characteristic, complemented
11#21447224C211yesmaximal, characteristic, complemented
12#21445537C211yesmaximal, characteristic, complemented
13#340×D61C211yesmaximal, characteristic, complemented
14#21445514C211yesmaximal, characteristic, complemented
15C61×#2074C211yesmaximal, characteristic, complemented
16C22×#11758226C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519734C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839762—24no—
24#16839757—24no—
25C2×#14519701C2211yescharacteristic
26C2×#14519701C2211yescharacteristic
27C2×#14519699C2211yescharacteristic
28C2×#14519699C2211yescharacteristic
29C2×#14519697C2211yescharacteristic
30C2×#14519697C2211yescharacteristic
31#16839669C2211yescharacteristic
32#16839664C2211yescharacteristic
33#16839663C2211yescharacteristic
34#16839659—21no—
35#16839659—21no—
36#16839640—21no—
37#16839640—21no—
38#16839636—21no—
39#16839636—21no—
40#16839632C2211yescharacteristic
41#16839631C2211yescharacteristic
42#16839627C2211yescharacteristic
43#16839621—21no—
44#16839621—21no—
45#16839525C2211yescharacteristic
46#16839525C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839524C2211yescharacteristic
49#16839524C2211yescharacteristic
50#16839524C2211yescharacteristic
51#16839517C2211yescharacteristic
52#16839516C2211yescharacteristic
53#16839516C2211yescharacteristic
54C2×#103×C61C2211yescharacteristic
55C2×#103×C61C2211yescharacteristic
56C61×#391C2211yescharacteristic
57C61×#382C2211yescharacteristic
58C61×#340C2211yescharacteristic
59C61×#340C2211yescharacteristic
60C61×#336C2211yescharacteristic
61C2×#11758226—22no—
62C2×#11758226—22no—
63C2×#11758226—22no—
64C2×#11758226—22no—
65C2×#11758226—22no—
66C2×#11758226D414yes—
67C22×D244—21no—
68C22×D244C2311yescharacteristic
69C22×C4×D61—21no—
70C22×C4×D61—21no—
71C22×C4×D61—21no—
72C22×C4×D61C2311yescharacteristic
73C2×#11758189—22no—
74C2×#11758189—22no—
75C2×#11758189—22no—
76C22×#92418C2311yescharacteristic
77C22×#92418C2311yescharacteristic
78C2×C8×D61—22no—
79C2×C8×D61—22no—
80C2×C8×D61—22no—
81#14519735—22no—
82#14519735—22no—
83#14519734—22no—
84#14519734—22no—
85C2×C4⋊Dic61—21no—
86C2×C4⋊Dic61—21no—
87C2×C4⋊Dic61C2311yescharacteristic
88C2×C4⋊Dic61C2311yescharacteristic
89C2×C4⋊Dic61C2311yescharacteristic
90C2×#11758196—21no—
91C2×#11758196—21no—
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196D411yescharacteristic
95C2×#11758196D411yescharacteristic
96C4⋊C4×D61—22no—
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C4⋊C4×D61—22no—
100C2×#11758198—21no—
101C2×#11758198—21no—
102C2×#11758198—21no—
103C2×#11758198—21no—
104C2×#11758198—21no—
105C2×#11758198D411yescharacteristic
106C2×#11758198D411yescharacteristic
107C2×C4×Dic61—21no—
108C2×C4×Dic61C2311yescharacteristic
109#14519701—22no—
110#14519701—22no—
111#14519699—22no—
112#14519699—22no—
113#14519697—22no—
114#14519697—22no—
115#14519684—21no—
116#14519684—21no—
117#14519684—21no—
118#14519684—21no—
119C22×Q8×C61C2311yescharacteristic
120C22×C8×C61C2311yescharacteristic
121C22×C8×C61C2311yescharacteristic
122C2×C4⋊C4×C61C2311yescharacteristic
123C2×C4⋊C4×C61C2311yescharacteristic
124C2×C4⋊C4×C61C2311yescharacteristic
125C2×C42×C61C2311yescharacteristic
126#103×C61—22no—
127#103×C61—22no—
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C23×D61—21no—
133C23×D61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61C2×D412yes—
140C22×C4×C61—21no—
141C22×C4×C61—21no—
142C22×C4×C61—21no—
143C22×C4×C61—21no—
144C22×C4×C61C2×D411yescharacteristic
145C22×C4×C61C2×D411yescharacteristic
146C22×C4×C61C2411yescharacteristic
147C22×Dic61—21no—
148C22×Dic61—21no—
149C22×Dic61—21no—
150C22×Dic61C2×D411yescharacteristic
151C22×Dic61C2×D411yescharacteristic
152C22×Dic61C2×D411yescharacteristic
153#11758226—28no—
154#11758226—28no—
155#11758226—44no—
156#11758226—44no—
157C2×D244—21no—
158C2×D244—21no—
159C2×D244—21no—
160C2×D244—21no—
161C2×D244—24no—
162C2×D244—21no—
163C2×D244—42no—
164C2×D244—42no—
165C2×D244C2×D412yes—
166C2×C4×D61—24no—
167C2×C4×D61—42no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C2×C4×D61—81no—
174C2×C4×D61—81no—
175C2×C4×D61C2×D412yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4112yes—
178C2×C4×D61#4112yes—
179C2×C4×D61#4112yes—
180C2×C4×D61#4112yes—
181C2×C8×C61—21no—
182C2×C8×C61—21no—
183C2×C8×C61—21no—
184C2×C8×C61—21no—
185C2×C8×C61—21no—
186C2×C8×C61—21no—
187C4⋊C4×C61—21no—
188C4⋊C4×C61—21no—
189C4⋊C4×C61—21no—
190C4⋊C4×C61—21no—
191C4⋊C4×C61—21no—
192C4⋊C4×C61—21no—
193C42×C61—21no—
194C42×C61—21no—
195#11758198—22no—
196#11758198—22no—
197#11758198—42no—
198#11758198—81no—
199#11758198—81no—
200#11758198—81no—
201#11758198—81no—
202C4⋊Dic61—21no—
203C4⋊Dic61—21no—
204C4⋊Dic61—21no—
205C4⋊Dic61—21no—
206C4⋊Dic61—21no—
207C4⋊Dic61—21no—
208C4⋊Dic61—81no—
209C4⋊Dic61—81no—
210#11758196—22no—
211#11758196—22no—
212#11758196—42no—
213#11758196—42no—
214#11758196—81no—
215#11758196—81no—
216C4×Dic61—21no—
217C4×Dic61—21no—
218C4×Dic61—42no—
219C2×#92418—21no—
220C2×#92418—21no—
221C2×#92418—21no—
222C2×#92418—21no—
223C2×#92418—21no—
224C2×#92418—21no—
225#11758189—44no—
226C8×D61—44no—
227C23×C61C22×D411yescharacteristic
228C22×D61—21no—
229C22×D61—21no—
230C22×D61—21no—
231C22×D61—21no—
232C22×D61—22no—
233C22×D61—41no—
234C22×D61—41no—
235C22×D61—41no—
236C22×D61—41no—
237C22×D61—41no—
238C22×D61—41no—
239C22×D61#12112yes—
240C22×D61#12312yes—
241C22×D61#12412yes—
242Q8×C61—22no—
243Q8×C61—22no—
244Q8×C61—41no—
245Q8×C61—41no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—21no—
252C2×C4×C61—21no—
253C2×C4×C61—41no—
254C2×C4×C61—41no—
255C2×C4×C61—41no—
256C2×C4×C61—41no—
257C2×C4×C61—41no—
258C2×C4×C61—41no—
259C2×C4×C61—41no—
260C2×C4×C61—41no—
261C2×C4×C61—41no—
262C2×C4×C61—41no—
263C2×C4×C61—41no—
264C2×C4×C61C22×D411yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×C4×C61C2×#4111yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×Dic61—21no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—21no—
276C2×Dic61—41no—
277C2×Dic61—41no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61#12112yes—
286C2×Dic61#12312yes—
287C2×Dic61#12412yes—
288D244—22no—
289D244—22no—
290D244—48no—
291D244—41no—
292D244—41no—
293C4×D61—48no—
294C4×D61—84no—
295C4×D61—84no—
296C4×D61#12314yes—
297C4×D61#12314yes—
298C4×D61#12614yes—
299C4×D61#12614yes—
300C8×C61—41no—
301C8×C61—41no—
302#92418—41no—
303#92418—41no—
304#23634—611nomaximal, Sylow
305C22×C61C2×#12111yescharacteristic
306C22×C61C2×#12311yescharacteristic
307C22×C61C2×#12411yescharacteristic
308C22×C61#57611yescharacteristic
309C22×C61#57711yescharacteristic
310C22×C61#57711yescharacteristic
311C22×C61#57811yescharacteristic
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—22no—
317C2×D61—22no—
318C2×D61—42no—
319C2×D61—42no—
320C2×D61—42no—
321C2×D61—81no—
322C2×D61—81no—
323C2×D61—81no—
324C2×D61—81no—
325C2×D61#39714yes—
326C4×C61—41no—
327C4×C61—41no—
328C4×C61—42no—
329C4×C61—81no—
330C4×C61—81no—
331C4×C61—81no—
332C4×C61C2×#12311yescharacteristic
333C4×C61C2×#12311yescharacteristic
334C4×C61C2×#12611yescharacteristic
335C4×C61C2×#12611yescharacteristic
336Dic61—41no—
337Dic61—41no—
338Dic61—81no—
339Dic61—81no—
340Dic61—81no—
341Dic61#39714yes—
342C2×#417—611no—
343C2×#416—611no—
344#2317—611no—
345#2311—611no—
346#2074—611no—
347#2074D6111yescharacteristic, complemented
348#2073—611no—
349#2073—611no—
350#2072—611no—
351#2072—611no—
352#2071—611no—
353#2071—611no—
354#1736—611no—
355C2×#340—611no—
356#1729—611no—
357C2×C61C2×#39711yescharacteristic
358C2×C61#304511yescharacteristic
359C2×C61#304611yescharacteristic
360C2×C61#309811yescharacteristic
361C2×C61#309911yescharacteristic
362C2×C61#313611yescharacteristic
363C2×C61#313711yescharacteristic
364D61—24no—
365D61—82no—
366C22×#41—611no—
367C22×M16—611no—
368C23×C8—611no—
369C2×#117—611no—
370C2×#117—611no—
371C22×C4⋊C4—611no—
372C22×C4⋊C4—611no—
373#417—1224no—
374#416—1224no—
375C2×#103—611no—
376C2×#103—611no—
377C2×#103C2×D6111yescharacteristic
378C2×#103C2×D6111yescharacteristic
379C2×#102—611no—
380C2×#102—611no—
381C2×#102—611no—
382C2×#102—611no—
383#391—611no—
384#391C2×D6111yescharacteristic
385#390—611no—
386#390—611no—
387#389—1221no—
388#389—1221no—
389#386—1221no—
390#386—1221no—
391#385—1221no—
392#385—1221no—
393#384—611no—
394#384—611no—
395#382—611no—
396#382C2×D6111yescharacteristic
397#380—1221no—
398#380—1221no—
399#340—611no—
400#340—611no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340—611no—
405#340C2×D6111yescharacteristic
406#340C2×D6111yescharacteristic
407#336—611no—
408#336—611no—
409#336—611no—
410#336C2×D6111yescharacteristic
411C61#2363411yescharacteristic, complemented, Sylow
412C2×#41—614no—
413C2×#41—1222no—
414C2×#41—1222no—
415C2×#41—1222no—
416C2×#41—1222no—
417C2×#41—1222no—
418C22×Q8C22×D6111yescharacteristic
419C22×D4—611no—
420C22×D4—1221no—
421C23×C4—611no—
422C23×C4—1221no—
423C23×C4—1221no—
424C23×C4—1221no—
425C2×M16—1222no—
426C2×M16—1222no—
427C2×M16—1222no—
428C22×C8—611no—
429C22×C8—611no—
430C22×C8—1222no—
431C22×C8—1222no—
432C22×C8—1222no—
433C22×C8C22×D6111yescharacteristic
434C22×C8C22×D6111yescharacteristic
435#117—1222no—
436#117—1222no—
437#117—1222no—
438#117—1222no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—611no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1222no—
447C2×C4⋊C4—1222no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4—1221no—
452C2×C4⋊C4—1221no—
453C2×C4⋊C4—1221no—
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C4⋊C4C22×D6111yescharacteristic
456C2×C4⋊C4C22×D6111yescharacteristic
457C2×C22⋊C4—611no—
458C2×C22⋊C4—611no—
459C2×C22⋊C4—1221no—
460C2×C22⋊C4—1221no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C22⋊C4—1221no—
464C2×C42—611no—
465C2×C42—1221no—
466C2×C42C22×D6111yescharacteristic
467#103—22no—
468#103—22no—
469#103—1222no—
470#103—1222no—
471#102—1222no—
472#102—1222no—
473#102—1222no—
474#102—1222no—
475#95—21no—
476#95—21no—
477#95—21no—
478#95—21no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483C24—611no—
484C24—1221no—
485#41—1228no—
486#41—1228no—
487#41—2444no—
488#41—2444no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8D4×D6112yes—
495C2×D4—612no—
496C2×D4—1224no—
497C2×D4—1221no—
498C2×D4—1221no—
499C2×D4—1221no—
500C2×D4—1221no—
501C2×D4—1221no—
502C2×D4—2442no—
503C2×D4—2442no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—611no—
509C22×C4—612no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—611no—
516C22×C4—611no—
517C22×C4—1224no—
518C22×C4—1221no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—4881no—
522C22×C4—4881no—
523C22×C4—4881no—
524C22×C4—4881no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4—2442no—
529C22×C4D4×D6111yescharacteristic
530C22×C4D4×D6111yescharacteristic
531C22×C4C23×D6111yescharacteristic
532M16—2444no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8—1221no—
545C2×C8—2444no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—21no—
551C4⋊C4—21no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—1221no—
560C4⋊C4—4881no—
561C4⋊C4—4881no—
562C4⋊C4—4881no—
563C4⋊C4—4881no—
564C4⋊C4—2442no—
565C4⋊C4—2442no—
566C22⋊C4—1222no—
567C22⋊C4—1222no—
568C22⋊C4—4881no—
569C22⋊C4—4881no—
570C22⋊C4—4881no—
571C22⋊C4—4881no—
572C22⋊C4—2442no—
573C42—21no—
574C42—21no—
575C42—1221no—
576C42—1221no—
577C42—2442no—
578C23—612no—
579C23—612no—
580C23—612no—
581C23—1221no—
582C23—1221no—
583C23—1221no—
584C23—1221no—
585C23—1222no—
586C23—2441no—
587C23—2441no—
588C23—2441no—
589C23—2441no—
590C23—2441no—
591C23—2441no—
592C23C2×D4×D6111yescharacteristic
593Q8—22no—
594Q8—22no—
595Q8—41no—
596Q8—41no—
597D4—1222no—
598D4—1222no—
599D4—2448no—
600D4—2441no—
601D4—2441no—
602C2×C4—22no—
603C2×C4—21no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—612no—
621C2×C4—612no—
622C2×C4—612no—
623C2×C4—614no—
624C2×C4—614no—
625C2×C4—614no—
626C2×C4—614no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1221no—
630C2×C4—1221no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—4884no—
634C2×C4—4884no—
635C2×C4—2448no—
636C2×C4—2441no—
637C2×C4—2441no—
638C2×C4—2441no—
639C2×C4—2441no—
640C2×C4—2441no—
641C2×C4—2441no—
642C2×C4—2441no—
643C2×C4—2441no—
644C2×C4—2441no—
645C2×C4C2×D4×D6111yescharacteristic
646C2×C4#41×D6111yescharacteristic
647C2×C4#41×D6111yescharacteristic
648C2×C4#41×D6111yescharacteristic
649C2×C4#41×D6111yescharacteristic
650C2×C4#41×D6111yescharacteristic
651C8—41no—
652C8—41no—
653C8—2441no—
654C8—2441no—
655C22—614no—
656C22—1222no—
657C22—1222no—
658C22—1222no—
659C22—1222no—
660C22—1222no—
661C22—1222no—
662C22—4881no—
663C22—4881no—
664C22—4881no—
665C22—4881no—
666C22—2442no—
667C22—2442no—
668C22—2442no—
669C22#121×D6111yescharacteristic
670C22#123×D6111yescharacteristic
671C22#124×D6111yescharacteristic
672C22#1684056411yescharacteristic
673C22#1684056911yescharacteristic
674C22#1684057111yescharacteristic
675C22#1684057811yescharacteristic
676C4—42no—
677C4—41no—
678C4—41no—
679C4—81no—
680C4—81no—
681C4—81no—
682C4—614no—
683C4—4881no—
684C4—4881no—
685C4—4881no—
686C4—2441no—
687C4—2441no—
688C4#123×D6111yescharacteristic
689C4#123×D6111yescharacteristic
690C4#126×D6111yescharacteristic
691C4#126×D6111yescharacteristic
692C2—1224no—
693C2—4882no—
694C2#397×D6111yescharacteristic
695C2#2145536111yescharacteristic
696C2#2145537211yescharacteristic
697C2#2145580011yescharacteristic
698C2#2145580511yescharacteristic
699C2#2145617211yescharacteristic
700C2#2145618511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2074C2
socleC23×C61C22×D4
prime radicalC22×#11758226C22
hypercenter#2074D61
hypocenterC61#23634
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2074

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
3. #21448444C2semidirect
5. #21447240C2semidirect
6. #21447236C2semidirect
7. #21447233C2semidirect
8. #21447230C2semidirect
9. #21447227C2semidirect
10. #21447226C2semidirect
11. #21447224C2semidirect
12. #21445537C2semidirect
13. #340×D61C2semidirect
14. #21445514C2semidirect
15. C61×#2074C2semidirect
347. #2074D61semidirect
411. C61#23634semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#168397622
C2×#168397572
#214484442
#214484102
#214472402
#214472362
#214472332
#214472302
#214472272
#214472262
#214472242
#214455372
#340×D612
#214455142
C61×#20742
#2363461