Finite Group Databasebeta

← #29739941

Subgroups — #29739941

Order 15616.

854 conjugacy classes of subgroups.

The subgroup lattice has 682 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840100C211yesmaximal, characteristic, complemented
2C2×#16839802C211yesmaximal, characteristic, complemented
3#21448768C211yesmaximal, characteristic
4#21447362C211yesmaximal, characteristic, complemented
5#21447295C211yesmaximal, characteristic, complemented
6#21447255C211yesmaximal, characteristic, complemented
7#21447012C211yesmaximal, characteristic, complemented
8#21446457C211yesmaximal, characteristic, complemented
9#21446408C211yesmaximal, characteristic, complemented
10#21446253C211yesmaximal, characteristic, complemented
11#21446132C211yesmaximal, characteristic, complemented
12#21446117C211yesmaximal, characteristic
13#21446016C211yesmaximal, characteristic, complemented
14#21446012C211yesmaximal, characteristic, complemented
15C61×#2417C211yesmaximal, characteristic
16C22×#11758194C2211yescharacteristic
17C2×#14519736C2211yescharacteristic
18C2×#14519736C2211yescharacteristic
19#16840100—24no—
20#16839802—24no—
21C2×#14519710C2211yescharacteristic
22C2×#14519710C2211yescharacteristic
23C2×#14519708C2211yescharacteristic
24C2×#14519708C2211yescharacteristic
25C2×#14519708C2211yescharacteristic
26C2×#14519708C2211yescharacteristic
27C2×#14519701C2211yescharacteristic
28C2×#14519701C2211yescharacteristic
29C2×#14519701C2211yescharacteristic
30C2×#14519701C2211yescharacteristic
31C2×C22⋊#92418C2211yescharacteristic
32C2×C22⋊#92418C2211yescharacteristic
33C2×C22⋊#92418C2211yescharacteristic
34#16839665—21no—
35#16839655—21no—
36#16839655—21no—
37#16839632C2211yescharacteristic
38#16839622C2211yescharacteristic
39#16839622C2211yescharacteristic
40C4⋊C4×Dic61—21no—
41C4×C4⋊Dic61C2211yescharacteristic
42#16839524C2211yescharacteristic
43#16839524C2211yescharacteristic
44#16839524C2211yescharacteristic
45#16839524C2211yescharacteristic
46#16839524C2211yescharacteristic
47#16839524C2211yescharacteristic
48#16839517C2211yescharacteristic
49#16839517C2211yescharacteristic
50C2×C22⋊Q8×C61C2211yescharacteristic
51C2×C22⋊Q8×C61C2211yescharacteristic
52C2×C4×D4×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#398—21no—
55C61×#396—21no—
56C61×#391C2211yescharacteristic
57C61×#389C2211yescharacteristic
58C61×#386—21no—
59C61×#380—21no—
60C61×#380C2211yescharacteristic
61C2×#11758194—22no—
62C2×#11758194—22no—
63C2×#11758194—22no—
64C22×#92418C2311yescharacteristic
65C22×#92418C2311yescharacteristic
66C22×#92418C2311yescharacteristic
67C22×#92418C2311yescharacteristic
68#14519736—22no—
69#14519736—22no—
70#14519736—41no—
71#14519736—41no—
72C2×C4⋊Dic61—21no—
73C2×C4⋊Dic61—21no—
74C2×C4⋊Dic61C2311yescharacteristic
75C2×C4⋊Dic61C2311yescharacteristic
76C2×C4⋊Dic61C2311yescharacteristic
77C2×C4⋊Dic61C2311yescharacteristic
78C2×C22⋊Dic61—21no—
79C2×C22⋊Dic61—21no—
80C2×C22⋊Dic61—21no—
81C2×C22⋊Dic61—21no—
82C2×C4×Dic61—21no—
83C2×C4×Dic61—21no—
84C2×C4×Dic61—21no—
85#14519710—22no—
86#14519710D414yescomplemented
87#14519708—22no—
88#14519708—22no—
89#14519708—22no—
90#14519708—41no—
91#14519701—22no—
92#14519701—22no—
93#14519701—22no—
94#14519701—41no—
95C22⋊#92418—22no—
96C22⋊#92418—22no—
97C22⋊#92418—22no—
98#14519684—21no—
99#14519684—21no—
100#14519684—21no—
101C22×Q8×C61C2311yescharacteristic
102C22×D4×C61D411yescharacteristic
103C23×C4×C61—21no—
104C23×C4×C61C2311yescharacteristic
105C23×C4×C61C2311yescharacteristic
106C22⋊Q8×C61—22no—
107C22⋊Q8×C61—22no—
108C22⋊Q8×C61—22no—
109C22⋊Q8×C61—41no—
110C4×D4×C61—22no—
111C4×D4×C61—22no—
112C2×C4⋊C4×C61—22no—
113C2×C4⋊C4×C61—21no—
114C2×C4⋊C4×C61—21no—
115C2×C4⋊C4×C61—21no—
116C2×C4⋊C4×C61—21no—
117C2×C4⋊C4×C61—41no—
118C2×C4⋊C4×C61D411yescharacteristic
119C2×C4⋊C4×C61C2311yescharacteristic
120C2×C4⋊C4×C61C2311yescharacteristic
121C2×C4⋊C4×C61C2311yescharacteristic
122C2×C22⋊C4×C61—21no—
123C2×C22⋊C4×C61—21no—
124C2×C22⋊C4×C61—21no—
125C2×C22⋊C4×C61D411yescharacteristic
126C2×C22⋊C4×C61D411yescharacteristic
127C2×C42×C61C2311yescharacteristic
128#95×C61—21no—
129#95×C61—21no—
130#95×C61—21no—
131#95×C61—21no—
132C24×C61C2×D411yescharacteristic
133C24×C61C2×D411yescharacteristic
134C2×Q8×C61—21no—
135C2×Q8×C61—21no—
136C2×Q8×C61—21no—
137C2×Q8×C61—21no—
138C2×Q8×C61—21no—
139C2×Q8×C61—21no—
140C2×D4×C61—22no—
141C2×D4×C61—22no—
142C2×D4×C61—41no—
143C22×C4×C61—21no—
144C22×C4×C61—21no—
145C22×C4×C61—21no—
146C22×C4×C61—21no—
147C22×C4×C61—21no—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—21no—
151C22×C4×C61—21no—
152C22×C4×C61—21no—
153C22×C4×C61—21no—
154C22×C4×C61—21no—
155C22×C4×C61—21no—
156C22×C4×C61—21no—
157C22×C4×C61—42no—
158C22×C4×C61—42no—
159C22×C4×C61—81no—
160C22×C4×C61C2×D411yescharacteristic
161C22×C4×C61C2×D411yescharacteristic
162C22×C4×C61C2×D411yescharacteristic
163C22×C4×C61C2×D411yescharacteristic
164C22×C4×C61#4112yes—
165C22×C4×C61#4112yes—
166C22×C4×C61C2411yescharacteristic
167C22×Dic61—21no—
168C22×Dic61—21no—
169C22×Dic61—21no—
170C22×Dic61—21no—
171C4⋊C4×C61—22no—
172C4⋊C4×C61—21no—
173C4⋊C4×C61—21no—
174C4⋊C4×C61—21no—
175C4⋊C4×C61—21no—
176C4⋊C4×C61—21no—
177C4⋊C4×C61—42no—
178C4⋊C4×C61—42no—
179C4⋊C4×C61—42no—
180C4⋊C4×C61—81no—
181C4⋊C4×C61C2×D412yes—
182C22⋊C4×C61—22no—
183C22⋊C4×C61—22no—
184C22⋊C4×C61—42no—
185C22⋊C4×C61—42no—
186C22⋊C4×C61—81no—
187C42×C61—21no—
188C42×C61—21no—
189C22⋊Dic61—42no—
190C22⋊Dic61—42no—
191C22⋊Dic61—81no—
192C22⋊Dic61—81no—
193C4⋊Dic61—21no—
194C4⋊Dic61—21no—
195C4⋊Dic61—21no—
196C4⋊Dic61—21no—
197C4⋊Dic61—21no—
198C4⋊Dic61—21no—
199C4⋊Dic61—21no—
200C4⋊Dic61—42no—
201C4⋊Dic61—81no—
202C4⋊Dic61C2×D412yes—
203C4×Dic61—42no—
204C4×Dic61—42no—
205C4×Dic61—42no—
206#11758194—28no—
207C2×#92418—21no—
208C2×#92418—21no—
209C2×#92418—21no—
210C2×#92418—21no—
211C2×#92418—21no—
212C2×#92418—21no—
213C2×#92418—21no—
214C2×#92418—21no—
215C2×#92418—21no—
216C2×#92418—21no—
217C2×#92418C2×D412yes—
218C2×#92418#4112yes—
219C23×C61—21no—
220C23×C61—21no—
221C23×C61—21no—
222C23×C61—21no—
223C23×C61—21no—
224C23×C61—21no—
225C23×C61—21no—
226C23×C61—21no—
227C23×C61—21no—
228C23×C61—21no—
229C23×C61—21no—
230C23×C61—21no—
231C23×C61—21no—
232C23×C61#12312yes—
233C23×C61C22×D411yescharacteristic
234Q8×C61—41no—
235Q8×C61—41no—
236Q8×C61—41no—
237Q8×C61—41no—
238D4×C61—44no—
239C2×C4×C61—21no—
240C2×C4×C61—21no—
241C2×C4×C61—21no—
242C2×C4×C61—21no—
243C2×C4×C61—22no—
244C2×C4×C61—22no—
245C2×C4×C61—22no—
246C2×C4×C61—22no—
247C2×C4×C61—21no—
248C2×C4×C61—21no—
249C2×C4×C61—21no—
250C2×C4×C61—22no—
251C2×C4×C61—21no—
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—41no—
265C2×C4×C61—41no—
266C2×C4×C61—41no—
267C2×C4×C61—41no—
268C2×C4×C61—41no—
269C2×C4×C61—84no—
270C2×C4×C61#12312yes—
271C2×C4×C61#13712yes—
272C2×C4×C61C22×D411yescharacteristic
273C2×C4×C61C2×#4111yescharacteristic
274C2×C4×C61C2×#4111yescharacteristic
275C2×C4×C61C2×#4111yescharacteristic
276C2×C4×C61H(5,𝔽2)11yescharacteristic
277C2×C4×C612−1+411yescharacteristic
278C2×Dic61—22no—
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—41no—
289C2×Dic61—41no—
290#92418—22no—
291#92418—22no—
292#92418—41no—
293#92418—41no—
294#26155—611nomaximal, Sylow
295C22×C61—22no—
296C22×C61—22no—
297C22×C61—22no—
298C22×C61—41no—
299C22×C61—41no—
300C22×C61—41no—
301C22×C61—41no—
302C22×C61—41no—
303C22×C61—41no—
304C22×C61—41no—
305C22×C61—41no—
306C22×C61—41no—
307C22×C61—41no—
308C22×C61—41no—
309C22×C61C2×#12311yescharacteristic
310C22×C61D4211yescharacteristic
311C22×C61#54711yescharacteristic
312C22×C61C2×#13711yescharacteristic
313C22×C61#57511yescharacteristic
314C22×C61#57711yescharacteristic
315C22×C61#57711yescharacteristic
316C4×C61—22no—
317C4×C61—22no—
318C4×C61—41no—
319C4×C61—41no—
320C4×C61—42no—
321C4×C61—81no—
322C4×C61—81no—
323C4×C61—81no—
324C4×C61#52511yescharacteristic
325C4×C61#53911yescharacteristic
326C4×C61#54811yescharacteristic
327C4×C61#55311yescharacteristic
328Dic61—42no—
329Dic61—81no—
330Dic61—81no—
331Dic61—81no—
332C2×#485—611no—
333C2×#428—611no—
334#2417D6111yescharacteristic
335#2354—611no—
336#2096—611no—
337#2084—611no—
338#2077—611no—
339#2035—611no—
340#1928—611no—
341#1921—611no—
342#1894—611no—
343#1873—611no—
344#1870—611no—
345#1854—611no—
346#1853—611no—
347C2×C61—42no—
348C2×C61—81no—
349C2×C61#244311yescharacteristic
350C2×C61#309511yescharacteristic
351C2×C61#309911yescharacteristic
352C2×C61#326711yescharacteristic
353C2×C61#327811yescharacteristic
354C2×C61#334811yescharacteristic
355C2×C61#335611yescharacteristic
356C22×M16—611no—
357C2×C22⋊Q8C2×D6111yescharacteristic
358C2×C22⋊Q8C2×D6111yescharacteristic
359C2×C4×D4C2×D6111yescharacteristic
360C2×#117—611no—
361C2×#117—611no—
362C22×C4⋊C4C2×D6111yescharacteristic
363#485—1224no—
364#428—1224no—
365C2×#107—611no—
366C2×#107—611no—
367C2×#106—611no—
368C2×#106—611no—
369C2×#106—611no—
370C2×#106—611no—
371C2×#103—611no—
372C2×#103—611no—
373C2×#103—611no—
374C2×#103—611no—
375C2×C22⋊C8—611no—
376C2×C22⋊C8—611no—
377C2×C22⋊C8—611no—
378#398—21no—
379#396—21no—
380#391C2×D6111yescharacteristic
381#390—1221no—
382#389C2×D6111yescharacteristic
383#388—1221no—
384#388—1221no—
385#386—21no—
386#384—611no—
387#380—21no—
388#380—611no—
389#380—611no—
390#380C2×D6111yescharacteristic
391C4×C4⋊C4—611no—
392C4×C4⋊C4—1221no—
393#340—611no—
394#340—611no—
395#340—611no—
396#340—611no—
397#340—611no—
398#340—611no—
399#336—611no—
400#336—611no—
401C61#2615511yescharacteristic, complemented, Sylow
402C22×Q8C22×D6111yescharacteristic
403C22×D4C22⋊D6111yescharacteristic
404C23×C4—21no—
405C23×C4C22×D6111yescharacteristic
406C23×C4C22×D6111yescharacteristic
407C2×M16—1222no—
408C2×M16—1222no—
409C2×M16—1222no—
410C22×C8—611no—
411C22×C8—611no—
412C22×C8—611no—
413C22×C8—611no—
414C22⋊Q8—22no—
415C22⋊Q8—22no—
416C22⋊Q8—22no—
417C22⋊Q8—41no—
418C4×D4—22no—
419C4×D4—22no—
420#117—1222no—
421#117—1222no—
422#117—2441no—
423#117—2441no—
424C2×C4⋊C4—22no—
425C2×C4⋊C4—21no—
426C2×C4⋊C4—21no—
427C2×C4⋊C4—21no—
428C2×C4⋊C4—21no—
429C2×C4⋊C4—41no—
430C2×C4⋊C4—611no—
431C2×C4⋊C4—611no—
432C2×C4⋊C4—611no—
433C2×C4⋊C4—611no—
434C2×C4⋊C4—1221no—
435C2×C4⋊C4—1221no—
436C2×C4⋊C4C22⋊D6111yescharacteristic
437C2×C4⋊C4C22×D6111yescharacteristic
438C2×C4⋊C4C22×D6111yescharacteristic
439C2×C4⋊C4C22×D6111yescharacteristic
440C2×C22⋊C4—21no—
441C2×C22⋊C4—21no—
442C2×C22⋊C4—21no—
443C2×C22⋊C4—1221no—
444C2×C22⋊C4—1221no—
445C2×C22⋊C4—1221no—
446C2×C22⋊C4—1221no—
447C2×C22⋊C4C22⋊D6111yescharacteristic
448C2×C22⋊C4C22⋊D6111yescharacteristic
449C2×C42—1221no—
450C2×C42—1221no—
451C2×C42—1221no—
452C2×C42C22×D6111yescharacteristic
453#107—614no—
454#107—1222no—
455#106—1222no—
456#106—1222no—
457#106—1222no—
458#106—2441no—
459#103—1222no—
460#103—1222no—
461#103—1222no—
462#103—2441no—
463C22⋊C8—1222no—
464C22⋊C8—1222no—
465C22⋊C8—1222no—
466#95—21no—
467#95—21no—
468#95—21no—
469#95—21no—
470#95—1221no—
471#95—1221no—
472#95—1221no—
473C24C2×C22⋊D6111yescharacteristic
474C24C2×C22⋊D6111yescharacteristic
475C2×Q8—21no—
476C2×Q8—21no—
477C2×Q8—21no—
478C2×Q8—21no—
479C2×Q8—21no—
480C2×Q8—21no—
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—41no—
484C22×C4—21no—
485C22×C4—21no—
486C22×C4—21no—
487C22×C4—21no—
488C22×C4—21no—
489C22×C4—21no—
490C22×C4—21no—
491C22×C4—21no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—21no—
495C22×C4—21no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—42no—
499C22×C4—42no—
500C22×C4—81no—
501C22×C4—1221no—
502C22×C4—1221no—
503C22×C4—1221no—
504C22×C4—1221no—
505C22×C4#1175822412yes—
506C22×C4#1175822412yes—
507C22×C4C2×C22⋊D6111yescharacteristic
508C22×C4C2×C22⋊D6111yescharacteristic
509C22×C4C2×C22⋊D6111yescharacteristic
510C22×C4C2×C22⋊D6111yescharacteristic
511C22×C4C23×D6111yescharacteristic
512M16—1228no—
513C2×C8—612no—
514C2×C8—612no—
515C2×C8—1221no—
516C2×C8—1221no—
517C2×C8—1221no—
518C2×C8—1221no—
519C2×C8—1221no—
520C2×C8—1221no—
521C2×C8—1221no—
522C2×C8—1221no—
523C2×C8—1221no—
524C2×C8—1221no—
525C4⋊C4—21no—
526C4⋊C4—22no—
527C4⋊C4—21no—
528C4⋊C4—21no—
529C4⋊C4—21no—
530C4⋊C4—21no—
531C4⋊C4—42no—
532C4⋊C4—42no—
533C4⋊C4—42no—
534C4⋊C4—81no—
535C4⋊C4—612no—
536C4⋊C4—1221no—
537C4⋊C4—1221no—
538C4⋊C4—1221no—
539C4⋊C4—1221no—
540C4⋊C4—1221no—
541C4⋊C4—1221no—
542C4⋊C4—1221no—
543C4⋊C4—4881no—
544C4⋊C4—2442no—
545C4⋊C4D4×D6112yes—
546C22⋊C4—22no—
547C22⋊C4—22no—
548C22⋊C4—42no—
549C22⋊C4—42no—
550C22⋊C4—81no—
551C22⋊C4—4881no—
552C22⋊C4—4881no—
553C22⋊C4—2442no—
554C22⋊C4—2442no—
555C42—21no—
556C42—21no—
557C42—2442no—
558C42—2442no—
559C42—2442no—
560C23—21no—
561C23—21no—
562C23—21no—
563C23—21no—
564C23—21no—
565C23—21no—
566C23—21no—
567C23—21no—
568C23—21no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23#1451975812yes—
574C23C22×C22⋊D6111yescharacteristic
575Q8—41no—
576Q8—41no—
577Q8—41no—
578Q8—41no—
579D4—44no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—22no—
585C2×C4—22no—
586C2×C4—22no—
587C2×C4—22no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—21no—
595C2×C4—41no—
596C2×C4—41no—
597C2×C4—41no—
598C2×C4—41no—
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—41no—
610C2×C4—84no—
611C2×C4—1222no—
612C2×C4—2441no—
613C2×C4—2441no—
614C2×C4—2441no—
615C2×C4—2441no—
616C2×C4—2441no—
617C2×C4—2441no—
618C2×C4—2441no—
619C2×C4—2441no—
620C2×C4—2441no—
621C2×C4—2441no—
622C2×C4—2441no—
623C2×C4#1451975812yes—
624C2×C4#1451980812yes—
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4#41×D6111yescharacteristic
627C2×C4C2×#1175822411yescharacteristic
628C2×C4C2×#1175822411yescharacteristic
629C2×C4#1451982211yescharacteristic
630C2×C4#1451982411yescharacteristic
631C8—1222no—
632C8—1222no—
633C8—2441no—
634C8—2441no—
635C22—22no—
636C22—22no—
637C22—22no—
638C22—41no—
639C22—41no—
640C22—41no—
641C22—41no—
642C22—41no—
643C22—41no—
644C22—41no—
645C22—41no—
646C22—41no—
647C22—41no—
648C22—41no—
649C22C2×#1451975811yescharacteristic
650C22D4×C22⋊D6111yescharacteristic
651C22#1684041111yescharacteristic
652C22C2×#1451980811yescharacteristic
653C22#1684056011yescharacteristic
654C22#1684057011yescharacteristic
655C22#1684057311yescharacteristic
656C4—22no—
657C4—22no—
658C4—41no—
659C4—41no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—81no—
664C4—4881no—
665C4—4881no—
666C4—4881no—
667C4—2442no—
668C4#1684027211yescharacteristic
669C4#1684035711yescharacteristic
670C4#1684041811yescharacteristic
671C4#1684045511yescharacteristic
672C2—42no—
673C2—81no—
674C2#2144955011yescharacteristic
675C2#2145576311yescharacteristic
676C2#2145580411yescharacteristic
677C2#2145734011yescharacteristic
678C2#2145748111yescharacteristic
679C2#2145832811yescharacteristic
680C2#2145843411yescharacteristic
subgroupG/H
centerC23C22×C22⋊D61
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#2417C2
socleC23×C61C22×D4
prime radicalC22×D4×C61D4
hypercenter#2417D61
hypocenterC61#26155
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2417

Complements

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

NH ≅ G/Ntype
1. C2×#16840100C2semidirect
2. C2×#16839802C2semidirect
4. #21447362C2semidirect
5. #21447295C2semidirect
6. #21447255C2semidirect
7. #21447012C2semidirect
8. #21446457C2semidirect
9. #21446408C2semidirect
10. #21446253C2semidirect
11. #21446132C2semidirect
13. #21446016C2semidirect
14. #21446012C2semidirect
86. #14519710D4semidirect
401. C61#26155semidirect

Maximal subgroups

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

subgroupindex
C2×#168401002
C2×#168398022
#214487682
#214473622
#214472952
#214472552
#214470122
#214464572
#214464082
#214462532
#214461322
#214461172
#214460162
#214460122
C61×#24172
#2615561