Finite Group Databasebeta

← #29835183

Subgroups — #29835183

Order 15616.

556 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456471C211yesmaximal, characteristic, complemented
2#21454671C211yesmaximal, characteristic
3#21448028C211yesmaximal, characteristic, complemented
4#21448024C211yesmaximal, characteristic, complemented
5#21447978C211yesmaximal, characteristic, complemented
6#21447974C211yesmaximal, characteristic, complemented
7#21447862C211yesmaximal, characteristic, complemented
8#21447862C211yesmaximal, characteristic, complemented
9#21447748C211yesmaximal, characteristic
10#21447684C211yesmaximal, characteristic, complemented
11#21447680C211yesmaximal, characteristic, complemented
12#21447674C211yesmaximal, characteristic, complemented
13SD32×Dic61C211yesmaximal, characteristic, complemented
14#21447623C211yesmaximal, characteristic, complemented
15C61×#2187C211yesmaximal, characteristic
16#16840372—21no—
17#16840237—21no—
18C2×#14519721C2211yescharacteristic
19C2×#14519721C2211yescharacteristic
20#16840163C2211yescharacteristic, complemented
21#16840140C2211yescharacteristic, complemented
22#16840129—21no—
23#16840018—21no—
24#16840003C2211yescharacteristic
25#16839874—21no—
26#16839874—21no—
27#16839866—21no—
28Q16×Dic61C2211yescharacteristic, complemented
29#16839856C2211yescharacteristic, complemented
30D8×Dic61C2211yescharacteristic
31D8×Dic61C2211yescharacteristic
32D8×Dic61C2211yescharacteristic
33D8×Dic61C2211yescharacteristic
34#16839800C2211yescharacteristic
35#16839767—21no—
36#16839717C2211yescharacteristic
37#16839590C2211yescharacteristic, complemented
38#16839589C2211yescharacteristic, complemented
39#16839588C2211yescharacteristic, complemented
40#16839587C2211yescharacteristic, complemented
41#16839580C2211yescharacteristic, complemented
42#16839580C2211yescharacteristic, complemented
43#16839567C2211yescharacteristic, complemented
44#16839567C2211yescharacteristic, complemented
45#16839566C2211yescharacteristic, complemented
46#16839566C2211yescharacteristic, complemented
47#16839563C2211yescharacteristic
48#16839563C2211yescharacteristic
49C22⋊#11758185C2211yescharacteristic
50#16839537C2211yescharacteristic, complemented
51C16×Dic61C2211yescharacteristic, complemented
52C22×D8×C61C2211yescharacteristic
53C2×SD32×C61C2211yescharacteristic
54C2×SD32×C61C2211yescharacteristic
55C61×#467C2211yescharacteristic
56C61×#357C2211yescharacteristic
57C61×#357C2211yescharacteristic
58C61×C22⋊C16C2211yescharacteristic
59C2×Dic244C2311yescharacteristic
60C2×#11758202—21no—
61C2×#11758194D411yescharacteristic
62#14519775—21no—
63#14519770—41no—
64#14519770—41no—
65#14519768—21no—
66#14519768—41no—
67#14519767—41no—
68#14519756—21no—
69Q8×Dic61—21no—
70#14519742—21no—
71D4×Dic61—21no—
72D4×Dic61—21no—
73D4×Dic61—21no—
74D4×Dic61—21no—
75D4×Dic61—41no—
76D4×Dic61—41no—
77#14519736—21no—
78#14519736D411yescharacteristic
79C2×C22⋊Dic61—41no—
80C2×C22⋊Dic61—41no—
81#14519721—22no—
82#14519721—22no—
83C2×#11758185C2311yescharacteristic
84C2×#11758185C2311yescharacteristic
85#14519711C2311yescharacteristic
86#14519711C2311yescharacteristic
87#14519711C2311yescharacteristic
88#14519709—21no—
89#14519708—21no—
90#14519702—21no—
91#14519701—21no—
92#14519700—21no—
93#14519698—21no—
94#14519698—21no—
95#14519698—21no—
96#14519698—21no—
97#14519687D411yescharacteristic
98#14519687D411yescharacteristic
99C8×Dic61C2311yescharacteristic
100C8×Dic61C2311yescharacteristic
101C22×D4×C61—21no—
102C2×Q16×C61C2311yescharacteristic
103C2×D8×C61—22no—
104C2×D8×C61—22no—
105C2×D8×C61—21no—
106C2×D8×C61C2311yescharacteristic
107C2×D8×C61C2311yescharacteristic
108C22×C8×C61C2311yescharacteristic
109C22⋊Q8×C61—21no—
110SD32×C61—22no—
111SD32×C61—22no—
112C2×C16×C61C2311yescharacteristic
113C2×C16×C61C2311yescharacteristic
114#107×C61C2311yescharacteristic
115#103×C61—21no—
116C24×C61—41no—
117C2×Q8×C61—21no—
118C2×D4×C61—24no—
119C2×D4×C61—21no—
120C2×D4×C61—21no—
121C2×D4×C61—42no—
122C2×D4×C61—41no—
123C22×C4×C61C2×D411yescharacteristic
124C22×Dic61—41no—
125C22×Dic61—41no—
126C22×Dic61—41no—
127C22×Dic61—41no—
128C2×Dic122—21no—
129Q16×C61—21no—
130Q16×C61—21no—
131D8×C61—24no—
132D8×C61—21no—
133D8×C61—21no—
134D8×C61#4112yes—
135D8×C61#4112yes—
136C2×C8×C61—21no—
137C2×C8×C61—21no—
138C2×C8×C61C2×D411yescharacteristic
139C2×C8×C61C2411yescharacteristic
140C4⋊C4×C61—21no—
141C4⋊C4×C61—41no—
142C22⋊C4×C61—41no—
143C22⋊Dic61—41no—
144C22⋊Dic61—41no—
145C22⋊Dic61—41no—
146C22⋊Dic61—41no—
147C22⋊Dic61—41no—
148C22⋊Dic61—41no—
149C22⋊Dic61D811yescharacteristic
150C22⋊Dic61D811yescharacteristic
151#11758202—42no—
152C4⋊Dic61—21no—
153C4⋊Dic61—21no—
154C4⋊Dic61—21no—
155C4⋊Dic61—41no—
156C4⋊Dic61—41no—
157C4⋊Dic61D811yescharacteristic
158C4⋊Dic61D811yescharacteristic
159#11758196—41no—
160#11758196—41no—
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—44no—
179C23×C61—41no—
180C23×C61—41no—
181C23×C61—161no—
182C23×C61—161no—
183C23×C61C2×D811yescharacteristic
184Q8×C61—41no—
185Q8×C61—41no—
186D4×C61—22no—
187D4×C61—22no—
188D4×C61—44no—
189D4×C61—41no—
190D4×C61—41no—
191C2×C4×C61—21no—
192C2×C4×C61—21no—
193C2×C4×C61—41no—
194C2×C4×C61—41no—
195C2×C4×C61C2×D811yescharacteristic
196C2×C4×C61C22×D411yescharacteristic
197C2×Dic61—41no—
198C2×Dic61—41no—
199C2×Dic61—41no—
200C2×Dic61—41no—
201C2×Dic61—161no—
202C2×Dic61—161no—
203C2×Dic61—161no—
204C2×Dic61—161no—
205C2×Dic61C2×D811yescharacteristic
206C2×Dic61C2×D811yescharacteristic
207C2×Dic61C2×D811yescharacteristic
208C2×Dic61C2×D811yescharacteristic
209Dic122—41no—
210Dic122—41no—
211C8×C61—21no—
212C8×C61C2×#4111yescharacteristic
213C8×C61C2×#4111yescharacteristic
214#92418—21no—
215#92418—21no—
216#33683—611nomaximal, Sylow
217C22×C61—21no—
218C22×C61—21no—
219C22×C61—21no—
220C22×C61—42no—
221C22×C61—42no—
222C22×C61—81no—
223C22×C61—81no—
224C22×C61—81no—
225C22×C61—81no—
226C22×C61—164no—
227C22×C61C22×D811yescharacteristic
228C4×C61—21no—
229C4×C61—81no—
230C4×C61—81no—
231C4×C61#52511yescharacteristic
232C4×C61#57511yescharacteristic
233Dic61—21no—
234Dic61—21no—
235Dic61—21no—
236Dic61—21no—
237Dic61—81no—
238Dic61—81no—
239Dic61—81no—
240Dic61—81no—
241#3177—611no—
242#2948—611no—
243#2237—611no—
244#2237—611no—
245#2228—611no—
246#2228—611no—
247#2208—611no—
248#2208—611no—
249#2187—611no—
250#2187D6111yescharacteristic
251#2175—611no—
252#2175—611no—
253C4×SD32—611no—
254C4×SD32—611no—
255#2164—611no—
256C2×C61—41no—
257C2×C61—82no—
258C2×C61—82no—
259C2×C61#310011yescharacteristic
260C2×C61#341611yescharacteristic
261C2×C61#341611yescharacteristic
262C22×D8C2×D6111yescharacteristic
263#541—1221no—
264#518—1221no—
265C2×SD32—611no—
266C2×SD32—611no—
267C2×SD32C2×D6111yescharacteristic
268C2×SD32C2×D6111yescharacteristic
269#499—611no—
270#494—611no—
271#491—1221no—
272#470—1221no—
273#467—611no—
274#467C2×D6111yescharacteristic
275#442—1221no—
276#442—1221no—
277#441—1221no—
278C4×Q16—611no—
279C4×Q16—611no—
280C4×D8—611no—
281C4×D8—611no—
282C4×D8—611no—
283C4×D8—611no—
284#427—611no—
285#418—1221no—
286#405—611no—
287#367—611no—
288#367—611no—
289#366—611no—
290#366—611no—
291C4⋊C16—611no—
292C4⋊C16—611no—
293#358—611no—
294#358—611no—
295#358—611no—
296#358—611no—
297#357—611no—
298#357—611no—
299#357C2×D6111yescharacteristic
300#357C2×D6111yescharacteristic
301C22⋊C16—611no—
302C22⋊C16C2×D6111yescharacteristic
303#346—611no—
304C4×C16—611no—
305C61#3368311yescharacteristic, complemented, Sylow
306C22×D4—21no—
307C2×Q16—611no—
308C2×Q16—1221no—
309C2×Q16C22×D6111yescharacteristic
310C2×D8—22no—
311C2×D8—22no—
312C2×D8—21no—
313C2×D8C22×D6111yescharacteristic
314C2×D8C22×D6111yescharacteristic
315C2×M16—611no—
316C22×C8C22×D6111yescharacteristic
317C4⋊Q8—1221no—
318#126—2441no—
319#126—2441no—
320#125—1221no—
321#125—2441no—
322#125—2441no—
323C22⋊Q8—21no—
324C22⋊Q8—1221no—
325C4×Q8—1221no—
326C4×Q8—1221no—
327C4×D4—1221no—
328C4×D4—1221no—
329C4×D4—1221no—
330C4×D4—1221no—
331C4×D4—2441no—
332C4×D4—2441no—
333#117—611no—
334#117—1221no—
335C2×C22⋊C4—2441no—
336C2×C22⋊C4—2441no—
337SD32—22no—
338SD32—22no—
339SD32—1222no—
340SD32—1222no—
341C2×C16—611no—
342C2×C16—611no—
343C2×C16C22×D6111yescharacteristic
344C2×C16C22×D6111yescharacteristic
345#107—611no—
346#107—611no—
347#107—611no—
348#107C22×D6111yescharacteristic
349#106—1221no—
350#106—1221no—
351#103—21no—
352#103—1221no—
353#103—1221no—
354#103—1221no—
355#102—1221no—
356#102—1221no—
357#102—1221no—
358#102—1221no—
359#97—611no—
360#97—611no—
361C4×C8—611no—
362C4×C8—611no—
363C24—41no—
364C2×Q8—21no—
365C2×Q8—1221no—
366C2×D4—24no—
367C2×D4—21no—
368C2×D4—21no—
369C2×D4—42no—
370C2×D4—41no—
371C22×C4—2441no—
372C22×C4—2441no—
373C22×C4—2441no—
374C22×C4—2441no—
375C22×C4D4×D6111yescharacteristic
376Q16—21no—
377Q16—21no—
378Q16—1221no—
379Q16—1221no—
380Q16—2442no—
381D8—24no—
382D8—21no—
383D8—21no—
384D8#1175822412yes—
385D8#1175822412yes—
386M16—2441no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—611no—
390C2×C8—611no—
391C2×C8D4×D6111yescharacteristic
392C2×C8C23×D6111yescharacteristic
393C4⋊C4—21no—
394C4⋊C4—41no—
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—
406C4⋊C4—2441no—
407C22⋊C4—41no—
408C22⋊C4—611no—
409C22⋊C4—611no—
410C22⋊C4—2441no—
411C22⋊C4—2441no—
412C22⋊C4—2441no—
413C22⋊C4—2441no—
414C22⋊C4—2441no—
415C22⋊C4—2441no—
416C42—611no—
417C42—611no—
418C42—2441no—
419C42—2441no—
420C42—2441no—
421C16—21no—
422C16—21no—
423C16—1221no—
424C16—1221no—
425C23—44no—
426C23—41no—
427C23—41no—
428C23—161no—
429C23—161no—
430C23D8×D6111yescharacteristic
431Q8—41no—
432Q8—41no—
433Q8—2441no—
434Q8—2441no—
435D4—22no—
436D4—22no—
437D4—44no—
438D4—41no—
439D4—41no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—611no—
445C2×C4—611no—
446C2×C4—611no—
447C2×C4—611no—
448C2×C4—9761no—
449C2×C4—9761no—
450C2×C4—9761no—
451C2×C4—9761no—
452C2×C4—2441no—
453C2×C4—2441no—
454C2×C4—2441no—
455C2×C4—2441no—
456C2×C4D8×D6111yescharacteristic
457C2×C4C2×D4×D6111yescharacteristic
458C8—21no—
459C8—1221no—
460C8—1221no—
461C8C2×#1175822411yescharacteristic
462C8C2×#1175822411yescharacteristic
463C22—21no—
464C22—21no—
465C22—21no—
466C22—42no—
467C22—42no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—164no—
473C22C2×D8×D6111yescharacteristic
474C4—21no—
475C4—81no—
476C4—81no—
477C4—1221no—
478C4—1221no—
479C4—1221no—
480C4—1221no—
481C4—4881no—
482C4—4881no—
483C4—4881no—
484C4—4881no—
485C4#1684026811yescharacteristic
486C4#1684055811yescharacteristic
487C2—41no—
488C2—82no—
489C2—82no—
490C2#2145581411yescharacteristic
491C2#2145896111yescharacteristic
492C2#2145896511yescharacteristic
subgroupG/H
centerC22C2×D8×D61
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
FittingC61×#2187C2
socleC22×C61C22×D8
prime radicalC22×D8×C61C22
hypercenter#2187D61
hypocenterC61#33683
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2187

Complements

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

NH ≅ G/Ntype
1. #21456471C2semidirect
3. #21448028C2semidirect
4. #21448024C2semidirect
5. #21447978C2semidirect
6. #21447974C2semidirect
7. #21447862C2semidirect
8. #21447862C2semidirect
10. #21447684C2semidirect
11. #21447680C2semidirect
12. #21447674C2semidirect
13. SD32×Dic61C2semidirect
14. #21447623C2semidirect
20. #16840163C22semidirect
21. #16840140C22semidirect
28. Q16×Dic61C22semidirect
29. #16839856C22semidirect
37. #16839590C22semidirect
38. #16839589C22semidirect
39. #16839588C22semidirect
40. #16839587C22semidirect
41. #16839580C22semidirect
42. #16839580C22semidirect
43. #16839567C22semidirect
44. #16839567C22semidirect
45. #16839566C22semidirect
46. #16839566C22semidirect
50. #16839537C22semidirect
51. C16×Dic61C22semidirect
305. C61#33683semidirect

Maximal subgroups

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

subgroupindex
#214564712
#214546712
#214480282
#214480242
#214479782
#214479742
#214478622
#214477482
#214476842
#214476802
#214476742
SD32×Dic612
#214476232
C61×#21872
#3368361