Finite Group Databasebeta

← #29721477

Subgroups — #29721477

Order 15616.

706 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21455628C211yesmaximal, characteristic, complemented
2C4×#14519803C211yesmaximal, characteristic
3#21454484C211yesmaximal, characteristic, complemented
4#21447265C212yesmaximal, complemented
5#21447112C212yesmaximal
6#21446809C211yesmaximal, characteristic, complemented
7#21446597C212yesmaximal, complemented
8#21446595C211yesmaximal, characteristic, complemented
9#21446478C212yesmaximal, complemented
10#21445811C211yesmaximal, characteristic, complemented
11C2×C61×#476C211yesmaximal, characteristic, complemented
12C2×#14519803C2211yescharacteristic
13C4×#11758222C2211yescharacteristic
14#16840054—21no—
15#16840049—21no—
16#16840044—21no—
17#16840039—21no—
18#16839872—21no—
19#16839862—21no—
20C4×#11758200—21no—
21C4×#11758199—21no—
22#16839822—22no—
23#16839818—21no—
24#16839814—21no—
25C2×#14519713C2212yes—
26#16839802C2212yes—
27#16839802C2211yescharacteristic
28#16839786C2211yescharacteristic
29#16839786C2211yescharacteristic
30#16839767C2211yescharacteristic
31#16839761C2212yes—
32#16839761C2211yescharacteristic
33C2×#14519694C2212yes—
34#16839728C2212yes—
35#16839717C2212yes—
36C4×#11758194C2211yescharacteristic
37#16839531C2212yes—
38#16839531C2212yes—
39#16839531C2212yes—
40#16839527C2212yes—
41C2×C4⋊Q8×C61C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#476—21no—
44C61×#476—21no—
45C61×#476—22no—
46C2×#106×C61C2212yes—
47C2×C4⋊C8×C61C2211yescharacteristic
48C2×#102×C61C2212yes—
49C2×#11758222C2311yescharacteristic
50#14519803—22no—
51#14519803—22no—
52C2×#11758200—21no—
53C2×#11758199—21no—
54C2×#11758194C2311yescharacteristic
55C2×#11758194C2312yes—
56C2×#11758194C2312yes—
57C2×#11758194C2311yescharacteristic
58C4×Dic122D411yescharacteristic
59C4×Dic122D411yescharacteristic
60C4×D244D411yescharacteristic
61C4×D244D411yescharacteristic
62C4×C22⋊D61—21no—
63C4×C22⋊D61—21no—
64#14519736C2311yescharacteristic
65#14519734—21no—
66C42×D61—21no—
67#14519713—24no—
68#14519710—22no—
69#14519710—21no—
70#14519708—22no—
71#14519708—21no—
72C4⋊#92418—21no—
73C4⋊#92418—21no—
74C4⋊#92418—21no—
75C4⋊#92418—21no—
76#14519700—22no—
77#14519700—21no—
78#14519698—21no—
79#14519698—21no—
80#14519697—22no—
81#14519697—21no—
82#14519694—24no—
83#14519689—22no—
84#14519689—22no—
85#14519688—21no—
86#14519687—22no—
87C4×#92418—21no—
88C8×Dic61—22no—
89C22×Q8×C61—21no—
90C22×D4×C61C2311yescharacteristic
91C23×C4×C61—21no—
92C22×C8×C61C2312yes—
93C4⋊Q8×C61—21no—
94C4⋊Q8×C61—21no—
95C4⋊Q8×C61—21no—
96C4⋊Q8×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61—21no—
99C4×D4×C61—21no—
100C4×D4×C61—21no—
101C2×C4⋊C4×C61—21no—
102C2×C4⋊C4×C61C2312yes—
103C2×C4⋊C4×C61C2311yescharacteristic
104C2×C22⋊C4×C61—21no—
105C2×C42×C61C2311yescharacteristic
106#106×C61—22no—
107#106×C61—22no—
108C4⋊C8×C61D411yescharacteristic
109C4⋊C8×C61D411yescharacteristic
110C4⋊C8×C61D412yes—
111#102×C61—22no—
112#102×C61—22no—
113C24×C61—21no—
114C2×Q8×C61—41no—
115C2×Q8×C61—41no—
116C2×Q8×C61—82no—
117C2×D4×C61—21no—
118C2×D4×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—21no—
122C2×D4×C61#4112yes—
123C22×C4×C61—21no—
124C22×C4×C61—22no—
125C22×C4×C61—41no—
126C22×C4×C61—41no—
127C22×C4×C61—82no—
128C22×C4×C61#4111yescharacteristic
129C22×C4×C61#4111yescharacteristic
130C22×C4×C61C2411yescharacteristic
131C2×C22⋊D61—21no—
132#11758222—21no—
133#11758222—21no—
134#11758222—21no—
135#11758222—21no—
136C2×D244C2×D411yescharacteristic
137C2×C4×D61—21no—
138C2×C4×D61—21no—
139C2×C4×D61—21no—
140C2×Dic122C2×D411yescharacteristic
141C2×C8×C61—22no—
142C2×C8×C61—22no—
143C2×C8×C61C2×D412yes—
144C2×C8×C61C2×D412yes—
145C4⋊C4×C61—22no—
146C4⋊C4×C61—21no—
147C4⋊C4×C61—21no—
148C4⋊C4×C61—22no—
149C4⋊C4×C61—41no—
150C4⋊C4×C61—41no—
151C22⋊C4×C61—41no—
152C22⋊C4×C61—41no—
153C42×C61C2×D411yescharacteristic
154C42×C61C2×D411yescharacteristic
155C42×C61C2×D411yescharacteristic
156C42×C61C2×D411yescharacteristic
157C22⋊Dic61—21no—
158#11758200—24no—
159#11758199—24no—
160#11758198—21no—
161#11758198—21no—
162C4⋊Dic61C2×D411yescharacteristic
163C4⋊Dic61C2×D411yescharacteristic
164#11758196—21no—
165#11758196—21no—
166C4×Dic61—21no—
167C4×Dic61—21no—
168C4×Dic61—21no—
169#11758194—21no—
170#11758194—22no—
171#11758194—22no—
172#11758194—22no—
173#11758194—21no—
174#11758194—21no—
175#11758194—22no—
176#11758194#4112yes—
177C2×#92418—21no—
178C2×#92418—22no—
179C2×#92418—21no—
180C2×#92418—22no—
181C23×C61—22no—
182C23×C61—41no—
183C23×C61—41no—
184C23×C61—81no—
185C23×C61—81no—
186C23×C61C2×#4111yescharacteristic
187C22×D61—21no—
188Q8×C61—82no—
189Q8×C61—82no—
190D4×C61—22no—
191D4×C61—22no—
192D4×C61—41no—
193D4×C61—41no—
194C2×C4×C61—22no—
195C2×C4×C61—41no—
196C2×C4×C61—41no—
197C2×C4×C61—42no—
198C2×C4×C61—82no—
199C2×C4×C61—82no—
200C2×C4×C61—81no—
201C2×C4×C61—82no—
202C2×C4×C61#12111yescharacteristic
203C2×C4×C61#12111yescharacteristic
204C2×C4×C61#12111yescharacteristic
205C2×C4×C61#12111yescharacteristic
206C2×C4×C61C22×D411yescharacteristic
207C2×C4×C61C22×D411yescharacteristic
208C2×C4×C61C2×#4111yescharacteristic
209C2×C4×C61C2×#4111yescharacteristic
210C2×C4×C612−1+411yescharacteristic
211C2×C4×C612−1+411yescharacteristic
212C22⋊D61—41no—
213C22⋊D61—41no—
214C2×Dic61—21no—
215C2×Dic61—21no—
216C2×Dic61—21no—
217D244—21no—
218D244—21no—
219C4×D61—41no—
220C4×D61—41no—
221C4×D61—42no—
222C4×D61—81no—
223Dic122—21no—
224Dic122—21no—
225C8×C61—22no—
226C8×C61—22no—
227#92418—22no—
228#92418—42no—
229#92418—41no—
230#92418—42no—
231#24721—611nomaximal, Sylow
232C22×C61—21no—
233C22×C61—21no—
234C22×C61—42no—
235C22×C61—82no—
236C22×C61—82no—
237C22×C61—81no—
238C22×C61—81no—
239C22×C61C2×#12111yescharacteristic
240C22×C61#55211yescharacteristic
241C22×C61#55611yescharacteristic
242C2×D61—41no—
243C2×D61—41no—
244C2×D61—41no—
245C4×C61—21no—
246C4×C61—21no—
247C4×C61—21no—
248C4×C61—81no—
249C4×C61—82no—
250C4×C61#47112yes—
251C4×C61#52511yescharacteristic
252C4×C61#53611yescharacteristic
253C4×C61#54711yescharacteristic
254C4×C61#54811yescharacteristic
255Dic61—41no—
256Dic61—41no—
257Dic61—41no—
258#3079—611no—
259C2×#476D6111yescharacteristic, complemented
260C4×AGL(1,ℤ/8)—611no—
261#2925—611no—
262#2079—612no—
263#2051—612no—
264#1997—611no—
265#1951—612no—
266#1950—611no—
267#1931—612no—
268#1800—611no—
269C2×C61—21no—
270C2×C61—21no—
271C2×C61—82no—
272C2×C61#241511yescharacteristic
273C2×C61C2×#47111yescharacteristic
274C2×C61#306811yescharacteristic
275D61—81no—
276C2×AGL(1,ℤ/8)—611no—
277C2×C4⋊Q8C2×D6111yescharacteristic
278C4×#41—611no—
279C2×C4×D4C2×D6111yescharacteristic
280#477—1221no—
281#476—21no—
282#476—21no—
283#476—22no—
284#476—1221no—
285#475—1221no—
286#474—1221no—
287#442—1221no—
288#440—1221no—
289C4×SD16—1221no—
290C4×D8—1221no—
291#433—1222no—
292#432—1221no—
293#431—1221no—
294C2×#108—612no—
295#428—612no—
296#428—611no—
297C2×#106C2×D6112yes—
298#423—611no—
299#423—611no—
300C2×C4⋊C8C2×D6111yescharacteristic
301#418—611no—
302#417—612no—
303#417—611no—
304C2×#102C2×D6112yes—
305C2×#100—612no—
306#408—612no—
307#405—612no—
308C4×M16—611no—
309#343—612no—
310#343—612no—
311#343—612no—
312#341—612no—
313C61#2472111yescharacteristic, complemented, Sylow
314C2×#41—611no—
315C22×Q8—21no—
316C22×D4C22×D6111yescharacteristic
317C23×C4—21no—
318AGL(1,ℤ/8)—1222no—
319AGL(1,ℤ/8)—1222no—
320C2×SD16—1221no—
321C2×D8—1221no—
322C2×M16—611no—
323C2×M16—611no—
324C2×M16—612no—
325C2×M16—612no—
326C22×C8C22×D6112yes—
327C4⋊Q8—21no—
328C4⋊Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—21no—
331C4×Q8—611no—
332C4×Q8—611no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4—21no—
336C4×D4—21no—
337C4×D4—611no—
338C4×D4—611no—
339C4×D4—1221no—
340C4×D4—1221no—
341#117—611no—
342#117—1221no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4C22×D6111yescharacteristic
345C2×C4⋊C4C22×D6112yes—
346C2×C22⋊C4—21no—
347C2×C42—1221no—
348C2×C42C22×D6111yescharacteristic
349#108—1224no—
350#107—1221no—
351#107—1222no—
352#106—22no—
353#106—22no—
354#106—1221no—
355#106—1222no—
356C4⋊C8—1221no—
357C4⋊C8—1221no—
358C4⋊C8—1221no—
359C4⋊C8—1221no—
360C4⋊C8C22⋊D6111yescharacteristic
361C4⋊C8C22⋊D6111yescharacteristic
362C4⋊C8C22⋊D6112yes—
363#103—1221no—
364#103—1222no—
365#102—22no—
366#102—22no—
367#102—1221no—
368#102—1222no—
369#102—1221no—
370#102—1221no—
371#100—1224no—
372C22⋊C8—1222no—
373C22⋊C8—1222no—
374#97—1222no—
375#97—1221no—
376C4×C8—1222no—
377C4×C8—1221no—
378C24—21no—
379#41—1221no—
380#41—1221no—
381#41—1221no—
382#41—1221no—
383C2×Q8—41no—
384C2×Q8—41no—
385C2×Q8—82no—
386C2×Q8—611no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—21no—
392C2×D4—611no—
393C2×D4—1221no—
394C2×D4#1175822612yes—
395C22×C4—21no—
396C22×C4—22no—
397C22×C4—41no—
398C22×C4—41no—
399C22×C4—82no—
400C22×C4—1221no—
401C22×C4—1221no—
402C22×C4—1221no—
403C22×C4#1175822411yescharacteristic
404C22×C4#1175822411yescharacteristic
405C22×C4C23×D6111yescharacteristic
406SD16—1224no—
407D8—1224no—
408M16—612no—
409M16—1221no—
410M16—1222no—
411M16—1222no—
412M16—1222no—
413M16—1221no—
414M16—1221no—
415M16—1222no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—1221no—
419C2×C8—1221no—
420C2×C8—1222no—
421C2×C8—1222no—
422C2×C8C2×C22⋊D6112yes—
423C2×C8C2×C22⋊D6112yes—
424C4⋊C4—21no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—21no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—611no—
431C4⋊C4—611no—
432C4⋊C4—1221no—
433C4⋊C4—1221no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—1221no—
437C22⋊C4—1221no—
438C22⋊C4—1221no—
439C42—1221no—
440C42—1221no—
441C42—1221no—
442C42D4×D6111yescharacteristic
443C42D4×D6111yescharacteristic
444C42C2×C22⋊D6111yescharacteristic
445C42C2×C22⋊D6111yescharacteristic
446C23—22no—
447C23—41no—
448C23—41no—
449C23—81no—
450C23—81no—
451C23—1221no—
452C23C2×#1175822411yescharacteristic
453Q8—82no—
454Q8—82no—
455Q8—1221no—
456Q8—1221no—
457D4—22no—
458D4—22no—
459D4—41no—
460D4—41no—
461D4—1221no—
462D4—1221no—
463D4—2441no—
464D4—2441no—
465C2×C4—22no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—42no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—1221no—
474C2×C4—1221no—
475C2×C4—1221no—
476C2×C4—4881no—
477C2×C4—2442no—
478C2×C4—2441no—
479C2×C4—2441no—
480C2×C4#1451974811yescharacteristic
481C2×C4#1451974811yescharacteristic
482C2×C4#1451974811yescharacteristic
483C2×C4#1451974811yescharacteristic
484C2×C4C2×D4×D6111yescharacteristic
485C2×C4C22×C22⋊D6111yescharacteristic
486C2×C4#41×D6111yescharacteristic
487C2×C4C2×#1175822611yescharacteristic
488C2×C4#1451982411yescharacteristic
489C2×C4#1451982511yescharacteristic
490C8—22no—
491C8—22no—
492C8—1222no—
493C8—2442no—
494C8—2441no—
495C8—2442no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—82no—
500C22—82no—
501C22—81no—
502C22—81no—
503C22—2441no—
504C22—2441no—
505C22—2441no—
506C22C2×#1451974811yescharacteristic
507C22#1684045411yescharacteristic
508C22#1684048311yescharacteristic
509C4—21no—
510C4—21no—
511C4—21no—
512C4—81no—
513C4—82no—
514C4—2441no—
515C4—2441no—
516C4—2441no—
517C4#1684002212yes—
518C4#1684027511yescharacteristic
519C4#1684033511yescharacteristic
520C4#1684041111yescharacteristic
521C4#1684042311yescharacteristic
522C2—21no—
523C2—21no—
524C2—82no—
525C2—4881no—
526C2#2144924411yescharacteristic
527C2C2×#1684002211yescharacteristic
528C2#2145554311yescharacteristic
subgroupG/H
centerC22C2×#14519748
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC2×C61×#476C2
socleC22×C61C2×#121
prime radicalC2×#14519803C22
hypercenterC2×#476D61
hypocenterC61#24721
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. #21455628C2semidirect
3. #21454484C2semidirect
4. #21447265C2semidirect
6. #21446809C2semidirect
7. #21446597C2semidirect
8. #21446595C2semidirect
9. #21446478C2semidirect
10. #21445811C2semidirect
11. C2×C61×#476C2semidirect
259. C2×#476D61semidirect
313. C61#24721semidirect

Maximal subgroups

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

subgroupindex
#214556282
C4×#145198032
#214544842
#214472652
#214471122
#214468092
#214465972
#214465952
#214464782
#214458112
C2×C61×#4762
#2472161