Finite Group Databasebeta

← #29835192

Subgroups — #29835192

Order 15616.

546 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456451C211yesmaximal, characteristic, complemented
2#21454695C211yesmaximal, characteristic, complemented
3#21448022C211yesmaximal, characteristic, complemented
4#21447997C211yesmaximal, characteristic, complemented
5#21447969C211yesmaximal, characteristic, complemented
6#21447954C211yesmaximal, characteristic, complemented
7#21447879C211yesmaximal, characteristic, complemented
8#21447872C211yesmaximal, characteristic, complemented
9#21447759C211yesmaximal, characteristic, complemented
10#21447684C211yesmaximal, characteristic, complemented
11#21447680C211yesmaximal, characteristic, complemented
12#21447675C211yesmaximal, characteristic, complemented
13D16×Dic61C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2186C211yesmaximal, characteristic
16#16840379—21no—
17#16840245—21no—
18C2×#14519723C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20#16840166C2211yescharacteristic
21#16840145C2211yescharacteristic
22#16840120—21no—
23#16840006—21no—
24#16840003C2211yescharacteristic
25#16839863—21no—
26#16839863—21no—
27#16839859—21no—
28Q16×Dic61C2211yescharacteristic
29Q16×Dic61C2211yescharacteristic
30#16839856C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34C2×#14519711C2211yescharacteristic
35#16839763—21no—
36#16839717C2211yescharacteristic
37#16839588C2211yescharacteristic
38#16839588C2211yescharacteristic
39#16839587C2211yescharacteristic
40#16839587C2211yescharacteristic
41#16839580C2211yescharacteristic
42#16839580C2211yescharacteristic
43#16839567C2211yescharacteristic
44#16839567C2211yescharacteristic
45#16839566C2211yescharacteristic
46#16839563C2211yescharacteristic
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic
51C16×Dic61C2211yescharacteristic
52C2×#135×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×D16×C61C2211yescharacteristic
55C61×#466C2211yescharacteristic
56C61×#358C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758200—21no—
61C2×#11758194D411yescharacteristic
62#14519776—21no—
63#14519770—41no—
64#14519764—21no—
65#14519764—41no—
66#14519758—41no—
67#14519756—21no—
68#14519754—41no—
69Q8×Dic61—21no—
70Q8×Dic61—21no—
71#14519742—21no—
72D4×Dic61—21no—
73D4×Dic61—21no—
74D4×Dic61—21no—
75D4×Dic61—41no—
76D4×Dic61—41no—
77#14519736—41no—
78#14519736D411yescharacteristic
79C2×C4⋊Dic61—21no—
80C2×C4⋊Dic61—41no—
81#14519723—22no—
82#14519721—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711—21no—
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519711C2311yescharacteristic
89#14519710—21no—
90#14519702—21no—
91#14519701—21no—
92#14519701—21no—
93#14519700—21no—
94#14519698—21no—
95#14519698—21no—
96#14519698—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C2×#41×C61—21no—
102#135×C61—22no—
103#135×C61—22no—
104C2×Q16×C61C2311yescharacteristic
105C2×SD16×C61—21no—
106C2×D8×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109#121×C61—21no—
110SD32×C61—22no—
111D16×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#102×C61—21no—
116#41×C61—24no—
117#41×C61—42no—
118C2×Q8×C61—21no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—41no—
122C2×D4×C61—41no—
123C22×C4×C61—41no—
124C22×C4×C61C2×D411yescharacteristic
125C22×Dic61—41no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C22×Dic61—41no—
129C2×Dic122—21no—
130C2×Dic122—41no—
131Q16×C61—21no—
132Q16×C61#4112yes—
133SD16×C61—24no—
134D8×C61—21no—
135D8×C61—21no—
136D8×C61—21no—
137D8×C61#4112yes—
138C2×C8×C61—21no—
139C2×C8×C61—21no—
140C2×C8×C61C2×D411yescharacteristic
141C2×C8×C61C2411yescharacteristic
142C4⋊C4×C61—21no—
143C22⋊C4×C61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149C22⋊Dic61—41no—
150C22⋊Dic61D811yescharacteristic
151C22⋊Dic61D811yescharacteristic
152#11758200—42no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—21no—
156C4⋊Dic61—41no—
157C4⋊Dic61—41no—
158C4⋊Dic61—41no—
159C4⋊Dic61D811yescharacteristic
160C4⋊Dic61D811yescharacteristic
161#11758196—41no—
162#11758196—41no—
163#11758196—41no—
164C4×Dic61—41no—
165C4×Dic61—41no—
166C4×Dic61—41no—
167C4×Dic61C2×D411yescharacteristic
168C4×Dic61C2×D411yescharacteristic
169#11758194—41no—
170C2×#92418C2×D411yescharacteristic
171C2×#92418C2×D411yescharacteristic
172Dic244—21no—
173Dic244—21no—
174C16×C61—21no—
175C16×C61—21no—
176#11758185—21no—
177#11758185—21no—
178C23×C61—41no—
179C23×C61—41no—
180C23×C61C2×D811yescharacteristic
181Q8×C61—22no—
182Q8×C61—41no—
183D4×C61—22no—
184D4×C61—44no—
185D4×C61—41no—
186D4×C61—41no—
187D4×C61—41no—
188D4×C61—161no—
189C2×C4×C61—21no—
190C2×C4×C61—21no—
191C2×C4×C61—44no—
192C2×C4×C61—41no—
193C2×C4×C61—41no—
194C2×C4×C61C2×D811yescharacteristic
195C2×C4×C61C22×D411yescharacteristic
196C2×Dic61—41no—
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—161no—
201C2×Dic61—161no—
202C2×Dic61—161no—
203C2×Dic61—161no—
204C2×Dic61C2×D811yescharacteristic
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208Dic122—41no—
209Dic122—41no—
210Dic122—161no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33684—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—42no—
221C22×C61—81no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61C22×D811yescharacteristic
227C4×C61—21no—
228C4×C61—42no—
229C4×C61—81no—
230C4×C61#52511yescharacteristic
231C4×C61#57511yescharacteristic
232Dic61—21no—
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—81no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240#3175—611no—
241#2951—611no—
242#2236—611no—
243#2232—611no—
244#2227—611no—
245#2224—611no—
246#2211—611no—
247#2210—611no—
248#2189—611no—
249#2186D6111yescharacteristic
250#2175—611no—
251#2175—611no—
252C4×Q32—611no—
253C4×D16—611no—
254#2164—611no—
255C2×C61—41no—
256C2×C61—82no—
257C2×C61—161no—
258C2×C61#310011yescharacteristic
259C2×C61#341511yescharacteristic
260C2×C61#341711yescharacteristic
261C2×#135C2×D6111yescharacteristic
262#542—1221no—
263#520—1221no—
264C2×Q32—611no—
265C2×SD32—611no—
266C2×SD32C2×D6111yescharacteristic
267C2×D16C2×D6111yescharacteristic
268#500—611no—
269#495—611no—
270#489—1221no—
271#468—1221no—
272#467—611no—
273#466C2×D6111yescharacteristic
274#440—1221no—
275#440—1221no—
276#440—1221no—
277C4×Q16—611no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×D8—611no—
281C4×D8—611no—
282C4×D8—611no—
283C2×#107—611no—
284#417—1221no—
285#405—611no—
286#366—611no—
287#366—611no—
288#366—611no—
289#366—611no—
290C4⋊C16—611no—
291C4⋊C16—611no—
292#358—611no—
293#358—611no—
294#358—611no—
295#358C2×D6111yescharacteristic
296#357—611no—
297#357—611no—
298#357—611no—
299#357C2×D6111yescharacteristic
300C22⋊C16—611no—
301C22⋊C16C2×D6111yescharacteristic
302#346—611no—
303C4×C16—611no—
304C61#3368411yescharacteristic, complemented, Sylow
305C2×#41—21no—
306#135—22no—
307#135—22no—
308C2×Q16—611no—
309C2×Q16C22×D6111yescharacteristic
310C2×SD16—21no—
311C2×SD16—1221no—
312C2×D8C22×D6111yescharacteristic
313C2×D8C22×D6111yescharacteristic
314C2×M16—611no—
315C22×C8C22×D6111yescharacteristic
316C4⋊Q8—1221no—
317#126—2441no—
318#124—1221no—
319#124—2441no—
320#123—2441no—
321C22⋊Q8—1221no—
322C22⋊Q8—2441no—
323#121—21no—
324C4×Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—2441no—
331C4×D4—2441no—
332#117—611no—
333#117—2441no—
334C2×C4⋊C4—1221no—
335C2×C4⋊C4—2441no—
336Q32—1222no—
337SD32—22no—
338SD32—1222no—
339D16—22no—
340C2×C16—611no—
341C2×C16—611no—
342C2×C16C22×D6111yescharacteristic
343C2×C16C22×D6111yescharacteristic
344#107—611no—
345#107—611no—
346#107—611no—
347#107—1221no—
348#107—1221no—
349#107C22×D6111yescharacteristic
350#103—1221no—
351#103—1221no—
352#103—1221no—
353#103—1221no—
354#102—21no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#97—611no—
359#97—611no—
360C4×C8—611no—
361C4×C8—611no—
362#41—24no—
363#41—42no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×Q8—2441no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—41no—
370C2×D4—41no—
371C22×C4—41no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4—2441no—
376C22×C4D4×D6111yescharacteristic
377Q16—21no—
378Q16—1221no—
379Q16—1221no—
380Q16#1175822412yes—
381SD16—24no—
382SD16—2442no—
383D8—21no—
384D8—21no—
385D8—21no—
386D8#1175822412yes—
387M16—2441no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—611no—
391C2×C8—611no—
392C2×C8D4×D6111yescharacteristic
393C2×C8C23×D6111yescharacteristic
394C4⋊C4—21no—
395C4⋊C4—611no—
396C4⋊C4—611no—
397C4⋊C4—1221no—
398C4⋊C4—1221no—
399C4⋊C4—1221no—
400C4⋊C4—2441no—
401C4⋊C4—2441no—
402C4⋊C4—2441no—
403C4⋊C4—2441no—
404C4⋊C4—2441no—
405C4⋊C4—2441no—
406C22⋊C4—41no—
407C22⋊C4—611no—
408C22⋊C4—611no—
409C22⋊C4—2441no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C42—611no—
416C42—611no—
417C42—2441no—
418C42—2441no—
419C42—2441no—
420C16—21no—
421C16—21no—
422C16—1221no—
423C16—1221no—
424C23—41no—
425C23—41no—
426C23D8×D6111yescharacteristic
427Q8—22no—
428Q8—41no—
429Q8—9761no—
430Q8—2441no—
431Q8—2441no—
432D4—22no—
433D4—44no—
434D4—41no—
435D4—41no—
436D4—41no—
437D4—161no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—44no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—611no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—9761no—
448C2×C4—9761no—
449C2×C4—9761no—
450C2×C4—9761no—
451C2×C4—2441no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4D8×D6111yescharacteristic
456C2×C4C2×D4×D6111yescharacteristic
457C8—21no—
458C8—1221no—
459C8—1221no—
460C8C2×#1175822411yescharacteristic
461C8C2×#1175822411yescharacteristic
462C22—21no—
463C22—21no—
464C22—21no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22C2×D8×D6111yescharacteristic
472C4—21no—
473C4—42no—
474C4—81no—
475C4—1221no—
476C4—1221no—
477C4—1221no—
478C4—1221no—
479C4—4881no—
480C4—4881no—
481C4—4881no—
482C4—4881no—
483C4#1684026811yescharacteristic
484C4#1684055811yescharacteristic
485C2—41no—
486C2—82no—
487C2—161no—
488C2#2145581411yescharacteristic
489C2#2145895911yescharacteristic
490C2#2145896911yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#2186C2
socleC22×C61C22×D8
hypercenter#2186D61
hypocenterC61#33684
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2186

Complements

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

NH ≅ G/Ntype
1. #21456451C2semidirect
2. #21454695C2semidirect
3. #21448022C2semidirect
4. #21447997C2semidirect
5. #21447969C2semidirect
6. #21447954C2semidirect
7. #21447879C2semidirect
8. #21447872C2semidirect
9. #21447759C2semidirect
10. #21447684C2semidirect
11. #21447680C2semidirect
12. #21447675C2semidirect
13. D16×Dic61C2semidirect
14. #21447623C2semidirect
304. C61#33684semidirect

Maximal subgroups

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

subgroupindex
#214564512
#214546952
#214480222
#214479972
#214479692
#214479542
#214478792
#214478722
#214477592
#214476842
#214476802
#214476752
D16×Dic612
#214476232
C61×#21862
#3368461