Finite Group Databasebeta

← #29662724

Subgroups — #29662724

Order 15616.

704 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2#21455276C212yesmaximal, complemented
3#21454785C211yesmaximal, characteristic, complemented
4#21445388C212yesmaximal, complemented
5#21445381C212yesmaximal, complemented
6#21445338C211yesmaximal, characteristic, complemented
7#21445333C211yesmaximal, characteristic, complemented
8#21445304C211yesmaximal, characteristic, complemented
9#21445299C211yesmaximal, characteristic, complemented
10#21444469C212yesmaximal, complemented
11C2×C61×#334C211yesmaximal, characteristic, complemented
12C2×#14519799—22no—
13#16840312C2212yes—
14C4×#11758222C2211yescharacteristic
15#16840167C2211yescharacteristic
16#16840139C2211yescharacteristic
17#16840109C2212yes—
18#16839998—21no—
19#16839998—21no—
20#16839985—22no—
21#16839978—22no—
22#16839973C2212yes—
23#16839968C2212yes—
24#16839963C2212yes—
25#16839958C2212yes—
26#16839892C2211yescharacteristic
27#16839892C2211yescharacteristic
28C4×D488—21no—
29C4×D488—21no—
30#16839841C2211yescharacteristic
31#16839841C2211yescharacteristic
32#16839833—21no—
33#16839833—21no—
34#16839815C2211yescharacteristic
35#16839813—21no—
36#16839759—22no—
37C4×#11758189—21no—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C2×C4⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#334C2212yes—
45C61×#334C2211yescharacteristic
46C61×#334C2211yescharacteristic
47C2×#11758222—22no—
48C2×#11758222D411yescharacteristic
49#14519799—48no—
50C2×Dic244—22no—
51C2×#11758190—22no—
52C2×#11758190—22no—
53C2×D488—22no—
54C2×D488—21no—
55C2×D488—21no—
56C2×#11758189—21no—
57C2×#11758189—21no—
58#14519777C2312yes—
59#14519774C2312yes—
60#14519765—22no—
61#14519751—22no—
62#14519751—22no—
63C4×Dic122D411yescharacteristic
64C4×Dic122C2311yescharacteristic
65C4×D244—22no—
66C4×D244—22no—
67C4×D244D411yescharacteristic
68C4×D244C2311yescharacteristic
69C4×C22⋊D61—21no—
70C4×C22⋊D61—21no—
71#14519736D411yescharacteristic
72#14519734—21no—
73C42×D61—21no—
74#14519711—21no—
75#14519711—21no—
76C4⋊#92418C2311yescharacteristic
77C4⋊#92418C2311yescharacteristic
78#14519706—21no—
79#14519706—21no—
80#14519702—22no—
81#14519702—22no—
82#14519699—22no—
83#14519699—22no—
84#14519699—22no—
85#14519699—22no—
86C22⋊#92418—21no—
87#14519689—21no—
88#14519689—21no—
89#14519687—21no—
90#14519687—21no—
91C4×#92418—21no—
92C22×C8×C61—22no—
93C22×C8×C61D411yescharacteristic
94C22×C8×C61D411yescharacteristic
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61—22no—
97C4⋊C8×C61C2312yes—
98C4⋊C8×C61C2312yes—
99C4×C8×C61D411yescharacteristic
100C4×C8×C61D411yescharacteristic
101C4×C8×C61C2311yescharacteristic
102C4×C8×C61C2311yescharacteristic
103C22×C4×C61C2×D411yescharacteristic
104C22×C4×C61C2×D411yescharacteristic
105C22×C4×C61#4111yescharacteristic
106C2×C22⋊D61—21no—
107C2×C22⋊D61—42no—
108#11758222—24no—
109#11758222—44no—
110#11758222—44no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—22no—
114C2×D244C2×D411yescharacteristic
115C2×C4×D61—21no—
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—42no—
119C2×Dic122—22no—
120C2×Dic122C2×D411yescharacteristic
121C2×C8×C61—22no—
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61—41no—
125C2×C8×C61—44no—
126C2×C8×C61C2×D411yescharacteristic
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C42×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61#4111yescharacteristic
133C42×C61C2411yescharacteristic
134C22⋊Dic61—21no—
135#11758198—21no—
136#11758198—21no—
137#11758198—42no—
138#11758198—42no—
139C4⋊Dic61—22no—
140C4⋊Dic61—22no—
141C4⋊Dic61C2×D411yescharacteristic
142C4⋊Dic61C2×D411yescharacteristic
143#11758196—21no—
144#11758196—21no—
145C4×Dic61—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C2×#92418—21no—
149C2×#92418—21no—
150Dic244—44no—
151D488—42no—
152D488—44no—
153D488—42no—
154#11758190—44no—
155#11758190—44no—
156#11758189—42no—
157#11758189—42no—
158C23×C61#12111yescharacteristic
159C22×D61—21no—
160C22×D61—42no—
161C2×C4×C61—21no—
162C2×C4×C61—21no—
163C2×C4×C61—41no—
164C2×C4×C61#12111yescharacteristic
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61AGL(1,ℤ/8)14yes—
168C2×C4×C61C22×D411yescharacteristic
169C2×C4×C61C22×D411yescharacteristic
170C2×C4×C61C2×#4111yescharacteristic
171C22⋊D61—42no—
172C22⋊D61—84no—
173C2×Dic61—21no—
174C2×Dic61—21no—
175C2×Dic61—21no—
176C2×Dic61—42no—
177D244—22no—
178D244—42no—
179D244—42no—
180D244—42no—
181D244—42no—
182D244—82no—
183C4×D61—42no—
184C4×D61—41no—
185C4×D61—41no—
186C4×D61—42no—
187C4×D61—84no—
188Dic122—22no—
189Dic122—42no—
190Dic122—42no—
191C8×C61—21no—
192C8×C61—21no—
193C8×C61—21no—
194C8×C61—42no—
195C8×C61—42no—
196C8×C61#13512yescomplemented
197#92418—41no—
198#92418—41no—
199#19157—611nomaximal, Sylow
200C22×C61—21no—
201C22×C61—21no—
202C22×C61#46912yes—
203C22×C61C2×#12111yescharacteristic
204C2×D61—41no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61—82no—
208C2×D61—82no—
209C2×D61—82no—
210C4×C61—21no—
211C4×C61—21no—
212C4×C61—21no—
213C4×C61#46912yes—
214C4×C61C2×#13511yescharacteristic
215C4×C61C2×AGL(1,ℤ/8)12yes—
216C4×C61#57511yescharacteristic
217Dic61—41no—
218Dic61—41no—
219Dic61—41no—
220Dic61—82no—
221#3169—611no—
222#3035—612no—
223#2965—611no—
224#1702—612no—
225#1701—612no—
226#1694—611no—
227#1693—611no—
228#1688—611no—
229#1687—611no—
230C2×#334D6111yescharacteristic, complemented
231#1540—612no—
232C2×C61—22no—
233C2×C61#303511yescharacteristic
234C2×C61C2×#46911yescharacteristic
235C2×C61#306711yescharacteristic
236D61—81no—
237D61—162no—
238C2×#135—1222no—
239#532—612no—
240C4×#41—611no—
241#500—611no—
242#493—611no—
243#486—612no—
244#466—1221no—
245#466—1221no—
246#464—1222no—
247#463—1222no—
248#462—612no—
249#461—612no—
250#460—612no—
251#459—612no—
252#446—611no—
253#446—611no—
254C4×D8—1221no—
255C4×D8—1221no—
256#436—611no—
257#436—611no—
258#435—1221no—
259#435—1221no—
260#431—611no—
261#431—1221no—
262C2×C4⋊C8C2×D6112yes—
263#416—1222no—
264C4×M16—1221no—
265C2×C4×C8C2×D6111yescharacteristic
266#334C2×D6111yescharacteristic
267#334C2×D6111yescharacteristic
268#334C2×D6112yes—
269#332—612no—
270#332—612no—
271#331—612no—
272#331—612no—
273C61#1915711yescharacteristic, complemented, Sylow
274C2×#41—611no—
275C2×#41—1222no—
276#135—2448no—
277C2×Q16—1222no—
278C2×SD16—1222no—
279C2×SD16—1222no—
280C2×D8—1222no—
281C2×D8—1221no—
282C2×D8—1221no—
283C2×M16—1221no—
284C2×M16—1221no—
285C22×C8—22no—
286C22×C8D24411yescharacteristic
287C22×C8D24411yescharacteristic
288C4⋊Q8—612no—
289#127—612no—
290#124—1222no—
291#121—1222no—
292#121—1222no—
293C4×Q8—611no—
294C4×Q8—611no—
295C4×D4—611no—
296C4×D4—611no—
297C4×D4—1221no—
298C4×D4—1221no—
299C4×D4—1222no—
300C4×D4—1222no—
301#117—611no—
302#117—1221no—
303C2×C42—1221no—
304C2×C42C22×D6111yescharacteristic
305#107—1221no—
306#107—1221no—
307C4⋊C8—22no—
308C4⋊C8—611no—
309C4⋊C8—611no—
310C4⋊C8—1221no—
311C4⋊C8—1221no—
312C4⋊C8C22×D6112yes—
313C4⋊C8C22×D6112yes—
314#103—1222no—
315#103—1222no—
316#102—1222no—
317#102—1222no—
318#102—1222no—
319#102—1222no—
320C22⋊C8—1221no—
321C22⋊C8—1221no—
322C22⋊C8—1221no—
323#97—1221no—
324#97—1221no—
325C4×C8—1221no—
326C4×C8D24411yescharacteristic
327C4×C8D24411yescharacteristic
328C4×C8C22×D6111yescharacteristic
329C4×C8C22×D6111yescharacteristic
330#41—1224no—
331#41—2444no—
332#41—2444no—
333C2×Q8—611no—
334C2×Q8—1222no—
335C2×D4—611no—
336C2×D4—1221no—
337C2×D4—1222no—
338C2×D4—1222no—
339C2×D4—1222no—
340C2×D4—2442no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—2442no—
345C22×C4C2×D24411yescharacteristic
346C22×C4D4×D6111yescharacteristic
347C22×C4#1175822611yescharacteristic
348Q16—2444no—
349SD16—2444no—
350SD16—2444no—
351D8—2442no—
352D8—2444no—
353D8—2442no—
354M16—2442no—
355M16—2442no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—44no—
361C2×C8—1221no—
362C2×C8—1221no—
363C2×C8C2×D24411yescharacteristic
364C2×C8C2×D24411yescharacteristic
365C2×C8C2×D24411yescharacteristic
366C2×C8C2×D24411yescharacteristic
367C4⋊C4—611no—
368C4⋊C4—611no—
369C4⋊C4—1221no—
370C4⋊C4—1221no—
371C4⋊C4—1222no—
372C4⋊C4—1222no—
373C22⋊C4—1221no—
374C22⋊C4—1221no—
375C22⋊C4—1221no—
376C22⋊C4—2442no—
377C22⋊C4—2442no—
378C42—1221no—
379C42—1221no—
380C42—1221no—
381C42C2×D24411yescharacteristic
382C42D4×D6111yescharacteristic
383C42#1175822611yescharacteristic
384C42C23×D6111yescharacteristic
385C23—1221no—
386C23—2442no—
387C23#1451975211yescharacteristic
388Q8—1222no—
389Q8—2442no—
390Q8—2442no—
391D4—1222no—
392D4—4882no—
393D4—4884no—
394D4—2442no—
395D4—2442no—
396D4—2442no—
397D4—2442no—
398D4—2442no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—1221no—
403C2×C4—1221no—
404C2×C4—1221no—
405C2×C4—4884no—
406C2×C4—2442no—
407C2×C4—2442no—
408C2×C4—2442no—
409C2×C4—2441no—
410C2×C4—2441no—
411C2×C4#1451975211yescharacteristic
412C2×C4#1451975211yescharacteristic
413C2×C4#1451975211yescharacteristic
414C2×C4#1451980414yes—
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4C22×D24411yescharacteristic
417C2×C4C2×#1175822611yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—42no—
422C8—42no—
423C8—2441no—
424C8—2441no—
425C8#1451979912yescomplemented
426C22—21no—
427C22—21no—
428C22—4882no—
429C22—4882no—
430C22—4882no—
431C22—2441no—
432C22—2441no—
433C22—2441no—
434C22#1684001612yes—
435C22C2×#1451975211yescharacteristic
436C4—21no—
437C4—21no—
438C4—21no—
439C4—4882no—
440C4—2441no—
441C4—2441no—
442C4—2441no—
443C4#1684001612yes—
444C4C2×#1451979911yescharacteristic
445C4C2×#1451980412yes—
446C4#1684056211yescharacteristic
447C2—22no—
448C2—9762no—
449C2—4881no—
450C2#2145527611yescharacteristic
451C2C2×#1684001611yescharacteristic
452C2#2145554011yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#334C2
socleC22×C61C2×#121
hypercenterC2×#334D61
hypocenterC61#19157
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#334

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21455276C2semidirect
3. #21454785C2semidirect
4. #21445388C2semidirect
5. #21445381C2semidirect
6. #21445338C2semidirect
7. #21445333C2semidirect
8. #21445304C2semidirect
9. #21445299C2semidirect
10. #21444469C2semidirect
11. C2×C61×#334C2semidirect
196. C8×C61#135semidirect
230. C2×#334D61semidirect
273. C61#19157semidirect
425. C8#14519799semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214552762
#214547852
#214453882
#214453812
#214453382
#214453332
#214453042
#214452992
#214444692
C2×C61×#3342
#1915761