Finite Group Databasebeta

← #29708664

Subgroups — #29708664

Order 15616.

1054 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839799C211yesmaximal, characteristic
2C2×#16839787C211yesmaximal, characteristic
3C2×#16839769C211yesmaximal, characteristic
4C2×#103×D61C211yesmaximal, characteristic
5#21448441C211yesmaximal, characteristic
6#21448402C211yesmaximal, characteristic
7#21447213C211yesmaximal, characteristic, complemented
8#21447212C211yesmaximal, characteristic, complemented
9#21447209C211yesmaximal, characteristic, complemented
10#21447201C211yesmaximal, characteristic, complemented
11#21447200C211yesmaximal, characteristic, complemented
12#21447197C211yesmaximal, characteristic, complemented
13#21447196C211yesmaximal, characteristic, complemented
14#21445537C211yesmaximal, characteristic
15C61×#2068C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×C8×D61C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×#14519734C2211yescharacteristic
22C2×C4⋊C4×D61C2211yescharacteristic
23#16839799—24no—
24C2×#14519711C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27#16839787—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31#16839769—24no—
32C2×#14519702C2211yescharacteristic
33C2×#14519702C2211yescharacteristic
34C2×#14519701C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519700C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38#103×D61—24no—
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839666—21no—
43#16839666—21no—
44#16839661C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839636—21no—
48#16839633—21no—
49#16839631—21no—
50#16839626—21no—
51#16839624—21no—
52#16839624—21no—
53#16839525C2211yescharacteristic
54#16839524C2211yescharacteristic
55#16839524C2211yescharacteristic
56C2×#107×C61C2211yescharacteristic
57C2×C4⋊C8×C61C2211yescharacteristic
58C2×#103×C61C2211yescharacteristic
59C2×#103×C61C2211yescharacteristic
60C61×#391C2211yescharacteristic
61C61×#389C2211yescharacteristic
62C61×#340C2211yescharacteristic
63C22×Dic122—21no—
64C22×Dic122C2311yescharacteristic
65C2×Q8×D61—22no—
66C2×Q8×D61—22no—
67C2×Q8×D61—22no—
68C2×Q8×D61—22no—
69C2×Q8×D61—22no—
70C2×Q8×D61Q814yes—
71C22×C4×D61—21no—
72C22×C4×D61—21no—
73C22×C4×D61C2311yescharacteristic
74C2×#11758189—22no—
75C2×#11758189—22no—
76C2×#11758189—22no—
77C22×#92418C2311yescharacteristic
78C22×#92418C2311yescharacteristic
79C2×C8×D61—22no—
80C2×C8×D61—22no—
81C2×C8×D61D414yes—
82#14519735—22no—
83#14519735—22no—
84#14519735—22no—
85#14519735D414yes—
86#14519734—22no—
87#14519734—22no—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C2×#11758196D411yescharacteristic
96C2×#11758196D411yescharacteristic
97C4⋊C4×D61—22no—
98C4⋊C4×D61—22no—
99C2×#11758198—21no—
100C2×#11758198—21no—
101C2×#11758198D411yescharacteristic
102C2×#11758198D411yescharacteristic
103C2×C4×Dic61—21no—
104C2×C4×Dic61—21no—
105C2×C4×Dic61C2311yescharacteristic
106#14519711—22no—
107#14519708—22no—
108#14519708—22no—
109C4⋊#92418—22no—
110#14519706—22no—
111#14519706—22no—
112#14519702—22no—
113#14519702—22no—
114#14519701—22no—
115#14519701—22no—
116#14519700—22no—
117#14519700—22no—
118#14519684—21no—
119#14519684—21no—
120#14519684—21no—
121#14519684—21no—
122#14519684—21no—
123#14519684—21no—
124#14519684—21no—
125#14519684—21no—
126#14519684—21no—
127C22×Q8×C61C2311yescharacteristic
128C22×C8×C61C2311yescharacteristic
129C22×C8×C61C2311yescharacteristic
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61C2311yescharacteristic
132C2×C4⋊C4×C61C2311yescharacteristic
133C2×C4⋊C4×C61C2311yescharacteristic
134C2×C42×C61C2311yescharacteristic
135#107×C61—22no—
136C4⋊C8×C61—22no—
137#103×C61—22no—
138#103×C61—22no—
139#95×C61—21no—
140#95×C61—21no—
141#95×C61—21no—
142C23×D61C2×D411yescharacteristic
143C2×Q8×C61—21no—
144C2×Q8×C61—21no—
145C2×Q8×C61—21no—
146C2×Q8×C61—21no—
147C2×Q8×C61—21no—
148C2×Q8×C61C2×Q812yes—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D411yescharacteristic
155C22×C4×C61C2411yescharacteristic
156C22×Dic61—21no—
157C22×Dic61—21no—
158C22×Dic61—21no—
159C22×Dic61—21no—
160C22×Dic61C2×D411yescharacteristic
161C22×Dic61C2×D411yescharacteristic
162C22×Dic61C2×D411yescharacteristic
163Q8×D61—28no—
164Q8×D61—28no—
165Q8×D61—44no—
166Q8×D61—44no—
167C2×C4×D61—24no—
168C2×C4×D61—42no—
169C2×C4×D61—42no—
170C2×C4×D61—42no—
171C2×C4×D61—81no—
172C2×C4×D61—81no—
173C2×C4×D61C2×D412yes—
174C2×C4×D61C2×D412yes—
175C2×C4×D61C2×Q812yes—
176C2×C4×D61#4112yes—
177C2×C4×D61#4112yes—
178C2×C4×D61#4112yes—
179C2×Dic122—21no—
180C2×Dic122—21no—
181C2×Dic122—21no—
182C2×Dic122—21no—
183C2×Dic122—21no—
184C2×Dic122—24no—
185C2×Dic122—42no—
186C2×Dic122—42no—
187C2×Dic122C2×Q812yes—
188C2×C8×C61—21no—
189C2×C8×C61—21no—
190C2×C8×C61—21no—
191C2×C8×C61—21no—
192C2×C8×C61—21no—
193C2×C8×C61C2×D412yes—
194C4⋊C4×C61—21no—
195C4⋊C4×C61—21no—
196C4⋊C4×C61—21no—
197C4⋊C4×C61—21no—
198C4⋊C4×C61—21no—
199C4⋊C4×C61—81no—
200C4⋊C4×C61C2×D412yes—
201C42×C61—21no—
202C42×C61—21no—
203#11758198—24no—
204#11758198—22no—
205#11758198—22no—
206#11758198—42no—
207C4⋊Dic61—21no—
208C4⋊Dic61—21no—
209C4⋊Dic61—21no—
210C4⋊Dic61—21no—
211C4⋊Dic61—21no—
212C4⋊Dic61—81no—
213C4⋊Dic61C2×D412yes—
214#11758196—22no—
215#11758196—22no—
216#11758196—42no—
217#11758196—81no—
218#11758196—81no—
219C4×Dic61—21no—
220C4×Dic61—21no—
221C4×Dic61—24no—
222C4×Dic61—42no—
223C2×#92418—21no—
224C2×#92418—21no—
225C2×#92418—21no—
226C2×#92418—21no—
227C2×#92418—21no—
228C2×#92418C2×D412yes—
229#11758189—44no—
230C8×D61—28no—
231C23×C61C22×D411yescharacteristic
232C22×D61—21no—
233C22×D61—21no—
234C22×D61—21no—
235C22×D61—21no—
236C22×D61#12112yes—
237C22×D61#12112yes—
238C22×D61C22⋊Q812yes—
239Q8×C61—22no—
240Q8×C61—22no—
241Q8×C61—41no—
242Q8×C61—41no—
243C2×C4×C61—22no—
244C2×C4×C61—21no—
245C2×C4×C61—21no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—21no—
251C2×C4×C61—41no—
252C2×C4×C61—41no—
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×C61C22×D411yescharacteristic
262C2×C4×C61C22×D411yescharacteristic
263C2×C4×C61C22×Q811yescharacteristic
264C2×C4×C61C2×#4111yescharacteristic
265C2×C4×C61C2×#4111yescharacteristic
266C2×C4×C61C2×#4111yescharacteristic
267C2×Dic61—21no—
268C2×Dic61—21no—
269C2×Dic61—22no—
270C2×Dic61—22no—
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—21no—
275C2×Dic61—41no—
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#12112yes—
287C2×Dic61C22⋊Q812yes—
288C4×D61—48no—
289C4×D61—84no—
290C4×D61#12114yes—
291C4×D61C22⋊Q814yes—
292C4×D61C22⋊Q814yes—
293C4×D61#12614yes—
294Dic122—22no—
295Dic122—22no—
296Dic122—41no—
297Dic122—41no—
298Dic122—48no—
299C8×C61—22no—
300C8×C61—41no—
301#92418—22no—
302#92418—41no—
303#23613—611nomaximal, Sylow
304C22×C61C2×#12111yescharacteristic
305C22×C61C2×#12111yescharacteristic
306C22×C61C2×C22⋊Q811yescharacteristic
307C22×C61#57711yescharacteristic
308C22×C61#57711yescharacteristic
309C22×C61#57811yescharacteristic
310C22×C61#57811yescharacteristic
311C2×D61—22no—
312C2×D61—22no—
313C2×D61—22no—
314C2×D61—22no—
315C2×D61—22no—
316C2×D61—22no—
317C2×D61#39614yes—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—41no—
321C4×C61—41no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61C2×#12111yescharacteristic
325C4×C61C2×C22⋊Q811yescharacteristic
326C4×C61C2×C22⋊Q811yescharacteristic
327C4×C61C2×#12611yescharacteristic
328Dic61—42no—
329Dic61—42no—
330Dic61—41no—
331Dic61—41no—
332Dic61—81no—
333Dic61—81no—
334Dic61#39614yes—
335C2×#427—611no—
336C2×#424—611no—
337C2×#419—611no—
338C22×#103—611no—
339#2317—611no—
340#2310—611no—
341#2068—611no—
342#2068—611no—
343#2068—611no—
344#2068D6111yescharacteristic, complemented
345#2066—611no—
346#2066—611no—
347#2066—611no—
348#2066—611no—
349#1736—611no—
350C2×C61C2×#39611yescharacteristic
351C2×C61#304611yescharacteristic
352C2×C61#304611yescharacteristic
353C2×C61#305711yescharacteristic
354C2×C61#305911yescharacteristic
355C2×C61#308211yescharacteristic
356C2×C61#308211yescharacteristic
357D61—24no—
358C23×Q8—611no—
359C22×M16—611no—
360C23×C8—611no—
361C2×#117—611no—
362C2×#117—611no—
363C2×#117—611no—
364C22×C4⋊C4—611no—
365#427—1224no—
366C2×#107—611no—
367C2×#107C2×D6111yescharacteristic
368C2×#106—611no—
369C2×#106—611no—
370#424—1224no—
371C2×C4⋊C8—611no—
372C2×C4⋊C8—611no—
373C2×C4⋊C8—611no—
374C2×C4⋊C8C2×D6111yescharacteristic
375#419—1224no—
376C2×#103—611no—
377C2×#103—611no—
378C2×#103—611no—
379C2×#103—611no—
380C2×#103—611no—
381C2×#103—611no—
382C2×#103—1224no—
383C2×#103C2×D6111yescharacteristic
384C2×#103C2×D6111yescharacteristic
385#391—611no—
386#391—611no—
387#391—611no—
388#391—1221no—
389#391—1221no—
390#391C2×D6111yescharacteristic
391#389—611no—
392#389—611no—
393#389—611no—
394#389C2×D6111yescharacteristic
395#385—1221no—
396#385—1221no—
397#384—1221no—
398#382—1221no—
399#381—1221no—
400#381—1221no—
401#340—611no—
402#340—611no—
403#340—611no—
404#340C2×D6111yescharacteristic
405C61#2361311yescharacteristic, complemented, Sylow
406C22×Q8—614no—
407C22×Q8—611no—
408C22×Q8—1221no—
409C22×Q8—1222no—
410C22×Q8—1222no—
411C22×Q8—1222no—
412C22×Q8—1222no—
413C22×Q8—1222no—
414C22×Q8C22×D6111yescharacteristic
415C23×C4—611no—
416C23×C4—1221no—
417C23×C4—1221no—
418C2×M16—1222no—
419C2×M16—1222no—
420C2×M16—1222no—
421C22×C8—611no—
422C22×C8—614no—
423C22×C8—611no—
424C22×C8—1222no—
425C22×C8—1222no—
426C22×C8C22×D6111yescharacteristic
427C22×C8C22×D6111yescharacteristic
428#117—614no—
429#117—1222no—
430#117—1222no—
431#117—1222no—
432#117—1222no—
433#117—1222no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—611no—
436C2×C4⋊C4—611no—
437C2×C4⋊C4—611no—
438C2×C4⋊C4—611no—
439C2×C4⋊C4—611no—
440C2×C4⋊C4—1222no—
441C2×C4⋊C4—1222no—
442C2×C4⋊C4—1221no—
443C2×C4⋊C4—1221no—
444C2×C4⋊C4—1221no—
445C2×C4⋊C4—1221no—
446C2×C4⋊C4C22×D6111yescharacteristic
447C2×C4⋊C4C22×D6111yescharacteristic
448C2×C4⋊C4C22×D6111yescharacteristic
449C2×C22⋊C4—611no—
450C2×C22⋊C4—611no—
451C2×C22⋊C4—1221no—
452C2×C22⋊C4—1221no—
453C2×C42—611no—
454C2×C42—1221no—
455C2×C42—1221no—
456C2×C42C22×D6111yescharacteristic
457#107—22no—
458#107—1222no—
459#106—1222no—
460#106—1222no—
461C4⋊C8—22no—
462C4⋊C8—1222no—
463C4⋊C8—1222no—
464C4⋊C8—1222no—
465#103—22no—
466#103—22no—
467#103—1222no—
468#103—1222no—
469#103—1222no—
470#103—1222no—
471#103—1222no—
472#103—1222no—
473#95—21no—
474#95—21no—
475#95—21no—
476#95—1221no—
477#95—1221no—
478#95—1221no—
479#95—1221no—
480#95—1221no—
481#95—1221no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485C24—611no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—612no—
492C2×Q8—1224no—
493C2×Q8—1228no—
494C2×Q8—1228no—
495C2×Q8—1221no—
496C2×Q8—1221no—
497C2×Q8—1221no—
498C2×Q8—1221no—
499C2×Q8—1221no—
500C2×Q8—2442no—
501C2×Q8—2442no—
502C2×Q8—2444no—
503C2×Q8—2444no—
504C2×Q8Q8×D6112yes—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—611no—
510C22×C4—612no—
511C22×C4—612no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—611no—
517C22×C4—611no—
518C22×C4—1224no—
519C22×C4—1221no—
520C22×C4—1221no—
521C22×C4—1221no—
522C22×C4—1221no—
523C22×C4—4881no—
524C22×C4—4881no—
525C22×C4—2442no—
526C22×C4—2442no—
527C22×C4—2442no—
528C22×C4D4×D6111yescharacteristic
529C22×C4D4×D6111yescharacteristic
530C22×C4C23×D6111yescharacteristic
531M16—2444no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—612no—
538C2×C8—1228no—
539C2×C8—1221no—
540C2×C8—1221no—
541C2×C8—1221no—
542C2×C8—1221no—
543C2×C8—1221no—
544C2×C8D4×D6112yes—
545C4⋊C4—21no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—21no—
549C4⋊C4—21no—
550C4⋊C4—81no—
551C4⋊C4—612no—
552C4⋊C4—1222no—
553C4⋊C4—1222no—
554C4⋊C4—1221no—
555C4⋊C4—1221no—
556C4⋊C4—1221no—
557C4⋊C4—1221no—
558C4⋊C4—1221no—
559C4⋊C4—4881no—
560C4⋊C4—4881no—
561C4⋊C4—4881no—
562C4⋊C4—2442no—
563C4⋊C4D4×D6112yes—
564C22⋊C4—1222no—
565C22⋊C4—1222no—
566C22⋊C4—1224no—
567C22⋊C4—2442no—
568C42—21no—
569C42—21no—
570C42—1224no—
571C42—1221no—
572C42—1221no—
573C42—2442no—
574C23—612no—
575C23—612no—
576C23—612no—
577C23—1221no—
578C23—1221no—
579C23—1221no—
580C23—1221no—
581C23C2×D4×D6111yescharacteristic
582Q8—22no—
583Q8—22no—
584Q8—41no—
585Q8—41no—
586Q8—1222no—
587Q8—1222no—
588Q8—2448no—
589Q8—2441no—
590Q8—2441no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—612no—
610C2×C4—612no—
611C2×C4—612no—
612C2×C4—614no—
613C2×C4—614no—
614C2×C4—614no—
615C2×C4—614no—
616C2×C4—1222no—
617C2×C4—1221no—
618C2×C4—1221no—
619C2×C4—1222no—
620C2×C4—1221no—
621C2×C4—1221no—
622C2×C4—1221no—
623C2×C4—1221no—
624C2×C4—4884no—
625C2×C4—2448no—
626C2×C4—2441no—
627C2×C4—2441no—
628C2×C4—2441no—
629C2×C4—2441no—
630C2×C4—2441no—
631C2×C4—2441no—
632C2×C4—2441no—
633C2×C4—2441no—
634C2×C4—2441no—
635C2×C4—2441no—
636C2×C4C2×D4×D6111yescharacteristic
637C2×C4C2×D4×D6111yescharacteristic
638C2×C4C2×Q8×D6111yescharacteristic
639C2×C4#41×D6111yescharacteristic
640C2×C4#41×D6111yescharacteristic
641C2×C4#41×D6111yescharacteristic
642C8—22no—
643C8—41no—
644C8—1222no—
645C8—2441no—
646C22—614no—
647C22—1222no—
648C22—1222no—
649C22—1222no—
650C22—1222no—
651C22—1222no—
652C22—1222no—
653C22#121×D6111yescharacteristic
654C22#121×D6111yescharacteristic
655C22C22⋊Q8×D6111yescharacteristic
656C22#1684057111yescharacteristic
657C22#1684057211yescharacteristic
658C22#1684057511yescharacteristic
659C22#1684057611yescharacteristic
660C4—42no—
661C4—42no—
662C4—41no—
663C4—41no—
664C4—81no—
665C4—81no—
666C4—614no—
667C4—4881no—
668C4—4881no—
669C4—2442no—
670C4—2442no—
671C4—2441no—
672C4—2441no—
673C4#121×D6111yescharacteristic
674C4C22⋊Q8×D6111yescharacteristic
675C4C22⋊Q8×D6111yescharacteristic
676C4#126×D6111yescharacteristic
677C2—1224no—
678C2#396×D6111yescharacteristic
679C2#2145536611yescharacteristic
680C2#2145537011yescharacteristic
681C2#2145545611yescharacteristic
682C2#2145546911yescharacteristic
683C2#2145565611yescharacteristic
684C2#2145566011yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2068C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2068D61
hypocenterC61#23613
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2068

Complements

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

NH ≅ G/Ntype
7. #21447213C2semidirect
8. #21447212C2semidirect
9. #21447209C2semidirect
10. #21447201C2semidirect
11. #21447200C2semidirect
12. #21447197C2semidirect
13. #21447196C2semidirect
15. C61×#2068C2semidirect
344. #2068D61semidirect
405. C61#23613semidirect

Maximal subgroups

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

subgroupindex
C2×#168397992
C2×#168397872
C2×#168397692
C2×#103×D612
#214484412
#214484022
#214472132
#214472122
#214472092
#214472012
#214472002
#214471972
#214471962
#214455372
C61×#20682
#2361361