Finite Group Databasebeta

← #29708724

Subgroups — #29708724

Order 15616.

1102 conjugacy classes of subgroups.

The subgroup lattice has 698 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16839803C211yesmaximal, characteristic
2C2×#16839784C211yesmaximal, characteristic
3C2×#16839769C211yesmaximal, characteristic
4C2×#16839769C211yesmaximal, characteristic
5#21448300C211yesmaximal, characteristic
6#391×D61C211yesmaximal, characteristic
7#21447211C211yesmaximal, characteristic, complemented
8#21447210C211yesmaximal, characteristic, complemented
9#21447209C211yesmaximal, characteristic, complemented
10#21447201C211yesmaximal, characteristic, complemented
11#21447200C211yesmaximal, characteristic, complemented
12#21447199C211yesmaximal, characteristic, complemented
13#21447198C211yesmaximal, characteristic, complemented
14#21445537C211yesmaximal, characteristic
15C61×#2068C211yesmaximal, characteristic, complemented
16C22×Q8×D61C2211yescharacteristic
17C22×#11758189C2211yescharacteristic
18C22×#11758189C2211yescharacteristic
19C2×#14519735C2211yescharacteristic
20C2×#14519735C2211yescharacteristic
21C2×C4⋊C4×D61C2211yescharacteristic
22C2×C42×D61C2211yescharacteristic
23#16839803—24no—
24C2×#14519710C2211yescharacteristic
25C2×#14519709C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27#16839784—24no—
28C2×C4⋊#92418C2211yescharacteristic
29C2×#14519706C2211yescharacteristic
30C2×#14519706C2211yescharacteristic
31#16839769—24no—
32#16839769—24no—
33C2×#14519702C2211yescharacteristic
34C2×#14519702C2211yescharacteristic
35C2×#14519701C2211yescharacteristic
36C2×#14519701C2211yescharacteristic
37C2×#14519700C2211yescharacteristic
38C2×#14519700C2211yescharacteristic
39#16839669C2211yescharacteristic
40#16839668C2211yescharacteristic
41#16839667C2211yescharacteristic
42#16839666—21no—
43#16839666—21no—
44#16839661C2211yescharacteristic
45#16839660C2211yescharacteristic
46#16839658C2211yescharacteristic
47#16839636—21no—
48#16839633—21no—
49C4⋊C4×Dic61—21no—
50#16839613—21no—
51#95×D61—21no—
52#95×D61—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×D61D411yescharacteristic
74C22×C4×D61D411yescharacteristic
75C22×C4×D61C2311yescharacteristic
76C2×#11758189—22no—
77C2×#11758189—22no—
78C2×#11758189—22no—
79C2×#11758189—22no—
80C2×#11758189—22no—
81C2×#11758189D414yes—
82C22×#92418C2311yescharacteristic
83C22×#92418C2311yescharacteristic
84#14519735—22no—
85#14519735—22no—
86#14519735—22no—
87#14519735D414yes—
88C2×C4⋊Dic61—21no—
89C2×C4⋊Dic61C2311yescharacteristic
90C2×C4⋊Dic61C2311yescharacteristic
91C2×C4⋊Dic61C2311yescharacteristic
92C2×#11758196—21no—
93C2×#11758196—21no—
94C2×#11758196—21no—
95C4⋊C4×D61—22no—
96C4⋊C4×D61—22no—
97C2×#11758198—21no—
98C2×#11758198—21no—
99C2×C4×Dic61—21no—
100C2×C4×Dic61—21no—
101C2×C4×Dic61D411yescharacteristic
102C2×C4×Dic61D411yescharacteristic
103C2×C4×Dic61C2311yescharacteristic
104C42×D61—22no—
105C42×D61—22no—
106#14519710—22no—
107#14519709—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—22no—
168C2×C4×D61—22no—
169C2×C4×D61—24no—
170C2×C4×D61—22no—
171C2×C4×D61—22no—
172C2×C4×D61—41no—
173C2×C4×D61—41no—
174C2×C4×D61—42no—
175C2×C4×D61—42no—
176C2×C4×D61—42no—
177C2×C4×D61—42no—
178C2×C4×D61—42no—
179C2×C4×D61C2×D412yes—
180C2×C4×D61C2×D412yes—
181C2×C4×D61C2×Q812yes—
182C2×C4×D61#4112yes—
183C2×C4×D61#4112yes—
184C2×C4×D61#4112yes—
185C2×Dic122—21no—
186C2×Dic122—21no—
187C2×Dic122—21no—
188C2×Dic122—21no—
189C2×Dic122—21no—
190C2×Dic122—24no—
191C2×Dic122—42no—
192C2×Dic122—42no—
193C2×Dic122C2×Q812yes—
194C2×C8×C61—21no—
195C2×C8×C61—21no—
196C2×C8×C61—21no—
197C2×C8×C61—21no—
198C2×C8×C61—21no—
199C2×C8×C61C2×D412yes—
200C4⋊C4×C61—21no—
201C4⋊C4×C61—21no—
202C4⋊C4×C61—21no—
203C4⋊C4×C61—21no—
204C4⋊C4×C61—21no—
205C4⋊C4×C61—42no—
206C4⋊C4×C61C2×D412yes—
207C42×C61—21no—
208C42×C61—21no—
209#11758198—24no—
210#11758198—42no—
211C4⋊Dic61—21no—
212C4⋊Dic61—21no—
213C4⋊Dic61—21no—
214C4⋊Dic61—21no—
215C4⋊Dic61—21no—
216C4⋊Dic61—42no—
217C4⋊Dic61C2×D412yes—
218#11758196—42no—
219#11758196—42no—
220#11758196—42no—
221C4×Dic61—21no—
222C4×Dic61—21no—
223C4×Dic61—22no—
224C4×Dic61—22no—
225C4×Dic61—24no—
226C4×Dic61—42no—
227C2×#92418—21no—
228C2×#92418—21no—
229C2×#92418—21no—
230C2×#92418—21no—
231C2×#92418—21no—
232C2×#92418C2×D412yes—
233#11758189—28no—
234#11758189—44no—
235C23×C61C22×D411yescharacteristic
236C22×D61—21no—
237C22×D61—21no—
238C22×D61—21no—
239C22×D61—21no—
240C22×D61#12112yes—
241C22×D61#12112yes—
242C22×D61C22⋊Q812yes—
243Q8×C61—22no—
244Q8×C61—22no—
245Q8×C61—41no—
246Q8×C61—41no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—22no—
250C2×C4×C61—21no—
251C2×C4×C61—22no—
252C2×C4×C61—21no—
253C2×C4×C61—21no—
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×C61#13712yes—
265C2×C4×C61C22×D411yescharacteristic
266C2×C4×C61C22×D411yescharacteristic
267C2×C4×C61C22×Q811yescharacteristic
268C2×C4×C61C2×#4111yescharacteristic
269C2×C4×C61C2×#4111yescharacteristic
270C2×C4×C61C2×#4111yescharacteristic
271C2×Dic61—21no—
272C2×Dic61—21no—
273C2×Dic61—21no—
274C2×Dic61—22no—
275C2×Dic61—22no—
276C2×Dic61—21no—
277C2×Dic61—21no—
278C2×Dic61—41no—
279C2×Dic61—41no—
280C2×Dic61—41no—
281C2×Dic61—41no—
282C2×Dic61—41no—
283C2×Dic61—41no—
284C2×Dic61—41no—
285C2×Dic61—41no—
286C2×Dic61—41no—
287C2×Dic61—41no—
288C2×Dic61#12112yes—
289C2×Dic61#12112yes—
290C2×Dic61C22⋊Q812yes—
291C2×Dic61#13712yes—
292C4×D61—44no—
293C4×D61—44no—
294C4×D61—48no—
295C4×D61—84no—
296C4×D61#12114yes—
297C4×D61C22⋊Q814yes—
298C4×D61C22⋊Q814yes—
299C4×D61#12614yes—
300Dic122—22no—
301Dic122—22no—
302Dic122—41no—
303Dic122—41no—
304Dic122—48no—
305C8×C61—22no—
306C8×C61—41no—
307#92418—22no—
308#92418—41no—
309#23619—611nomaximal, Sylow
310C22×C61C2×#12111yescharacteristic
311C22×C61C2×#12111yescharacteristic
312C22×C61C2×C22⋊Q811yescharacteristic
313C22×C61C2×#13711yescharacteristic
314C22×C61C2×#13711yescharacteristic
315C22×C61#57511yescharacteristic
316C22×C61#57511yescharacteristic
317C2×D61—22no—
318C2×D61—22no—
319C2×D61—22no—
320C2×D61—22no—
321C2×D61—22no—
322C2×D61—22no—
323C2×D61#39614yes—
324C4×C61—22no—
325C4×C61—42no—
326C4×C61—42no—
327C4×C61—41no—
328C4×C61—81no—
329C4×C61—81no—
330C4×C61C2×#12111yescharacteristic
331C4×C61C2×C22⋊Q811yescharacteristic
332C4×C61C2×C22⋊Q811yescharacteristic
333C4×C61C2×#12611yescharacteristic
334Dic61—22no—
335Dic61—42no—
336Dic61—42no—
337Dic61—41no—
338Dic61—81no—
339Dic61—81no—
340Dic61#39614yes—
341C2×#428—611no—
342C2×#423—611no—
343C2×#419—611no—
344C2×#419—611no—
345#2296—611no—
346C2×#391—611no—
347#2068—611no—
348#2068—611no—
349#2068—611no—
350#2068D6111yescharacteristic, complemented
351#2066—611no—
352#2066—611no—
353#2066—611no—
354#2066—611no—
355#1736—611no—
356C2×C61C2×#39611yescharacteristic
357C2×C61#303911yescharacteristic
358C2×C61#304011yescharacteristic
359C2×C61#305611yescharacteristic
360C2×C61#305611yescharacteristic
361C2×C61#307811yescharacteristic
362C2×C61#307811yescharacteristic
363D61—24no—
364C23×Q8—611no—
365C22×M16—611no—
366C22×M16—611no—
367C2×#117—611no—
368C2×#117—611no—
369C22×C4⋊C4—611no—
370C22×C42—611no—
371#428—1224no—
372C2×#107—611no—
373C2×#107C2×D6111yescharacteristic
374C2×#106—611no—
375C2×#106—611no—
376#423—1224no—
377C2×C4⋊C8—611no—
378C2×C4⋊C8—611no—
379C2×C4⋊C8—611no—
380C2×C4⋊C8C2×D6111yescharacteristic
381#419—1224no—
382#419—1224no—
383C2×#103—611no—
384C2×#103—611no—
385C2×#103—611no—
386C2×#103—611no—
387C2×#103—611no—
388C2×#103—611no—
389C2×#103C2×D6111yescharacteristic
390C2×#103C2×D6111yescharacteristic
391#391—611no—
392#391—611no—
393#391—611no—
394#391—1221no—
395#391—1221no—
396#391C2×D6111yescharacteristic
397#389—611no—
398#389—611no—
399#389—611no—
400#389C2×D6111yescharacteristic
401#385—1221no—
402#385—1221no—
403C4×C4⋊C4—1221no—
404C4×C4⋊C4—1221no—
405C2×#95—1221no—
406C2×#95—1221no—
407#340—611no—
408#340—611no—
409#340—611no—
410#340C2×D6111yescharacteristic
411C61#2361911yescharacteristic, complemented, Sylow
412C22×Q8—614no—
413C22×Q8—611no—
414C22×Q8—1221no—
415C22×Q8—1222no—
416C22×Q8—1222no—
417C22×Q8—1222no—
418C22×Q8—1222no—
419C22×Q8—1222no—
420C22×Q8C22×D6111yescharacteristic
421C23×C4—611no—
422C23×C4—611no—
423C23×C4—611no—
424C23×C4—1221no—
425C23×C4—1221no—
426C2×M16—614no—
427C2×M16—1222no—
428C2×M16—1222no—
429C2×M16—1222no—
430C2×M16—1222no—
431C2×M16—1222no—
432C22×C8—611no—
433C22×C8—611no—
434C22×C8C22×D6111yescharacteristic
435C22×C8C22×D6111yescharacteristic
436#117—614no—
437#117—1222no—
438#117—1222no—
439#117—1222no—
440C2×C4⋊C4—21no—
441C2×C4⋊C4—611no—
442C2×C4⋊C4—611no—
443C2×C4⋊C4—611no—
444C2×C4⋊C4—1222no—
445C2×C4⋊C4—1222no—
446C2×C4⋊C4—1221no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1221no—
450C2×C4⋊C4C22×D6111yescharacteristic
451C2×C4⋊C4C22×D6111yescharacteristic
452C2×C4⋊C4C22×D6111yescharacteristic
453C2×C22⋊C4—1221no—
454C2×C22⋊C4—1221no—
455C2×C42—611no—
456C2×C42—611no—
457C2×C42—611no—
458C2×C42—1222no—
459C2×C42—1222no—
460C2×C42—1221no—
461C2×C42—1221no—
462C2×C42C22×D6111yescharacteristic
463#107—22no—
464#107—1222no—
465#106—1222no—
466#106—1222no—
467C4⋊C8—22no—
468C4⋊C8—1222no—
469C4⋊C8—1222no—
470C4⋊C8—1222no—
471#103—22no—
472#103—22no—
473#103—1222no—
474#103—1222no—
475#103—1222no—
476#103—1222no—
477#103—1222no—
478#103—1222no—
479#95—21no—
480#95—21no—
481#95—21no—
482#95—1221no—
483#95—1221no—
484#95—1221no—
485#95—1221no—
486#95—1221no—
487#95—1221no—
488#95—1221no—
489#95—1221no—
490#95—1221no—
491C24—611no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—21no—
497C2×Q8—612no—
498C2×Q8—1224no—
499C2×Q8—1228no—
500C2×Q8—1228no—
501C2×Q8—1221no—
502C2×Q8—1221no—
503C2×Q8—1221no—
504C2×Q8—1221no—
505C2×Q8—1221no—
506C2×Q8—2442no—
507C2×Q8—2442no—
508C2×Q8—2444no—
509C2×Q8—2444no—
510C2×Q8Q8×D6112yes—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—611no—
516C22×C4—612no—
517C22×C4—612no—
518C22×C4—612no—
519C22×C4—612no—
520C22×C4—612no—
521C22×C4—612no—
522C22×C4—611no—
523C22×C4—611no—
524C22×C4—1222no—
525C22×C4—1222no—
526C22×C4—1222no—
527C22×C4—1224no—
528C22×C4—1221no—
529C22×C4—1221no—
530C22×C4—1222no—
531C22×C4—1221no—
532C22×C4—1221no—
533C22×C4—2441no—
534C22×C4—2441no—
535C22×C4—2442no—
536C22×C4—2442no—
537C22×C4—2442no—
538C22×C4—2442no—
539C22×C4—2442no—
540C22×C4D4×D6111yescharacteristic
541C22×C4D4×D6111yescharacteristic
542C22×C4C23×D6111yescharacteristic
543M16—1228no—
544M16—2444no—
545C2×C8—21no—
546C2×C8—21no—
547C2×C8—21no—
548C2×C8—21no—
549C2×C8—21no—
550C2×C8—612no—
551C2×C8—1221no—
552C2×C8—1221no—
553C2×C8—1221no—
554C2×C8—1221no—
555C2×C8—1221no—
556C2×C8D4×D6112yes—
557C4⋊C4—21no—
558C4⋊C4—21no—
559C4⋊C4—21no—
560C4⋊C4—21no—
561C4⋊C4—21no—
562C4⋊C4—42no—
563C4⋊C4—612no—
564C4⋊C4—1221no—
565C4⋊C4—1221no—
566C4⋊C4—1221no—
567C4⋊C4—1221no—
568C4⋊C4—1221no—
569C4⋊C4—2442no—
570C4⋊C4—2442no—
571C4⋊C4—2442no—
572C4⋊C4—2442no—
573C4⋊C4D4×D6112yes—
574C22⋊C4—1224no—
575C22⋊C4—2442no—
576C42—21no—
577C42—21no—
578C42—1222no—
579C42—1222no—
580C42—1224no—
581C42—1221no—
582C42—1221no—
583C42—2442no—
584C23—612no—
585C23—612no—
586C23—612no—
587C23—1221no—
588C23—1221no—
589C23—1221no—
590C23—1221no—
591C23C2×D4×D6111yescharacteristic
592Q8—22no—
593Q8—22no—
594Q8—41no—
595Q8—41no—
596Q8—1222no—
597Q8—1222no—
598Q8—2448no—
599Q8—2441no—
600Q8—2441no—
601C2×C4—21no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—22no—
605C2×C4—22no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—41no—
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—612no—
619C2×C4—612no—
620C2×C4—612no—
621C2×C4—614no—
622C2×C4—614no—
623C2×C4—614no—
624C2×C4—614no—
625C2×C4—612no—
626C2×C4—1221no—
627C2×C4—1221no—
628C2×C4—1221no—
629C2×C4—1222no—
630C2×C4—1222no—
631C2×C4—1221no—
632C2×C4—1221no—
633C2×C4—4884no—
634C2×C4—2444no—
635C2×C4—2444no—
636C2×C4—2448no—
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×C4—2441no—
646C2×C4—2441no—
647C2×C4#1451980912yes—
648C2×C4C2×D4×D6111yescharacteristic
649C2×C4C2×D4×D6111yescharacteristic
650C2×C4C2×Q8×D6111yescharacteristic
651C2×C4#41×D6111yescharacteristic
652C2×C4#41×D6111yescharacteristic
653C2×C4#41×D6111yescharacteristic
654C8—22no—
655C8—41no—
656C8—1222no—
657C8—2441no—
658C22—614no—
659C22—1222no—
660C22—1222no—
661C22—1222no—
662C22—1222no—
663C22—1222no—
664C22—1222no—
665C22#121×D6111yescharacteristic
666C22#121×D6111yescharacteristic
667C22C22⋊Q8×D6111yescharacteristic
668C22#137×D6111yescharacteristic
669C22C2×#1451980911yescharacteristic
670C22#1684055811yescharacteristic
671C22#1684055911yescharacteristic
672C4—22no—
673C4—42no—
674C4—41no—
675C4—42no—
676C4—81no—
677C4—81no—
678C4—614no—
679C4—1222no—
680C4—4881no—
681C4—4881no—
682C4—2442no—
683C4—2441no—
684C4—2442no—
685C4#121×D6111yescharacteristic
686C4C22⋊Q8×D6111yescharacteristic
687C4C22⋊Q8×D6111yescharacteristic
688C4#126×D6111yescharacteristic
689C2—1224no—
690C2#396×D6111yescharacteristic
691C2#2145530311yescharacteristic
692C2#2145530811yescharacteristic
693C2#2145544411yescharacteristic
694C2#2145544611yescharacteristic
695C2#2145562211yescharacteristic
696C2#2145562511yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2068C2
socleC23×C61C22×D4
prime radicalC23×D61C2×D4
hypercenter#2068D61
hypocenterC61#23619
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. #21447211C2semidirect
8. #21447210C2semidirect
9. #21447209C2semidirect
10. #21447201C2semidirect
11. #21447200C2semidirect
12. #21447199C2semidirect
13. #21447198C2semidirect
15. C61×#2068C2semidirect
350. #2068D61semidirect
411. C61#23619semidirect

Maximal subgroups

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

subgroupindex
C2×#168398032
C2×#168397842
C2×#168397692
#214483002
#391×D612
#214472112
#214472102
#214472092
#214472012
#214472002
#214471992
#214471982
#214455372
C61×#20682
#2361961