Finite Group Databasebeta

← #29662452

Subgroups — #29662452

Order 15616.

726 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456398C211yesmaximal, characteristic, complemented
2#21455276C212yesmaximal, complemented
3C8×#11758222C211yesmaximal, characteristic, complemented
4#21445264C212yesmaximal, complemented
5#21445257C212yesmaximal, complemented
6#21445214C211yesmaximal, characteristic, complemented
7#21445209C211yesmaximal, characteristic, complemented
8#21445180C211yesmaximal, characteristic, complemented
9#21445175C211yesmaximal, characteristic, complemented
10#21444469C212yesmaximal, complemented
11C2×C61×#335C211yesmaximal, 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—
26C8×Dic122C2211yescharacteristic
27C8×Dic122C2211yescharacteristic
28C4×D488—21no—
29C4×D488—21no—
30C8×D244C2211yescharacteristic
31C8×D244C2211yescharacteristic
32C8×C22⋊D61—21no—
33C8×C22⋊D61—21no—
34#16839815C2211yescharacteristic
35#16839813—21no—
36#16839759—22no—
37C4×C8×D61—21no—
38#16839509C2212yes—
39#16839509C2212yes—
40#16839505C2212yes—
41#16839505C2212yes—
42C2×C4⋊C8×C61C2212yes—
43C2×C4×C8×C61C2211yescharacteristic
44C61×#335C2211yescharacteristic
45C61×#335C2211yescharacteristic
46C61×#335C2212yes—
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×C8×D61—21no—
57C2×C8×D61—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—
89C4×#92418—21no—
90C8×Dic61—21no—
91C8×Dic61—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—22no—
109#11758222—44no—
110#11758222—44no—
111#11758222D814yescomplemented
112C2×D244—22no—
113C2×D244—22no—
114C2×D244—22no—
115C2×D244C2×D411yescharacteristic
116C2×C4×D61—21no—
117C2×C4×D61—21no—
118C2×C4×D61—21no—
119C2×C4×D61—42no—
120C2×Dic122—22no—
121C2×Dic122C2×D411yescharacteristic
122C2×C8×C61—22no—
123C2×C8×C61—22no—
124C2×C8×C61—22no—
125C2×C8×C61—41no—
126C2×C8×C61—44no—
127C2×C8×C61C2×D411yescharacteristic
128C2×C8×C61C2×D411yescharacteristic
129C2×C8×C61C2×D411yescharacteristic
130C2×C8×C61C2×D411yescharacteristic
131C42×C61C2×D411yescharacteristic
132C42×C61C2×D411yescharacteristic
133C42×C61#4111yescharacteristic
134C42×C61C2411yescharacteristic
135C22⋊Dic61—21no—
136#11758198—21no—
137#11758198—21no—
138#11758198—42no—
139#11758198—42no—
140C4⋊Dic61—22no—
141C4⋊Dic61—22no—
142C4⋊Dic61C2×D411yescharacteristic
143C4⋊Dic61C2×D411yescharacteristic
144#11758196—21no—
145#11758196—21no—
146C4×Dic61—21no—
147C4×Dic61—21no—
148C4×Dic61—21no—
149C2×#92418—21no—
150C2×#92418—21no—
151Dic244—44no—
152D488—42no—
153D488—44no—
154D488—42no—
155#11758190—44no—
156#11758190—44no—
157C8×D61—42no—
158C8×D61—42no—
159C23×C61#12111yescharacteristic
160C22×D61—21no—
161C22×D61—42no—
162C2×C4×C61—21no—
163C2×C4×C61—21no—
164C2×C4×C61—41no—
165C2×C4×C61#12111yescharacteristic
166C2×C4×C61#12111yescharacteristic
167C2×C4×C61#12111yescharacteristic
168C2×C4×C61C2×D812yes—
169C2×C4×C61#13512yes—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—24no—
174C22⋊D61—84no—
175C2×Dic61—21no—
176C2×Dic61—21no—
177C2×Dic61—21no—
178C2×Dic61—42no—
179D244—21no—
180D244—42no—
181D244—42no—
182D244—42no—
183D244—42no—
184D244—82no—
185D244C2×D812yes—
186C4×D61—24no—
187C4×D61—42no—
188C4×D61—41no—
189C4×D61—41no—
190C4×D61—84no—
191Dic122—21no—
192Dic122—42no—
193Dic122—42no—
194Dic122C2×D812yes—
195C8×C61—21no—
196C8×C61—21no—
197C8×C61—21no—
198C8×C61—42no—
199C8×C61—42no—
200C8×C61#13512yescomplemented
201#92418—41no—
202#92418—41no—
203#19125—611nomaximal, Sylow
204C22×C61—21no—
205C22×C61—21no—
206C22×C61#46612yes—
207C22×C61C2×#12111yescharacteristic
208C2×D61—22no—
209C2×D61—41no—
210C2×D61—41no—
211C2×D61—82no—
212C2×D61—82no—
213C2×D61—82no—
214C4×C61—21no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61#46612yes—
218C4×C61C22×D811yescharacteristic
219C4×C61C2×#13511yescharacteristic
220C4×C61C2×#13511yescharacteristic
221C4×C61#57511yescharacteristic
222Dic61—22no—
223Dic61—41no—
224Dic61—41no—
225Dic61—82no—
226#3169—611no—
227#3035—612no—
228C8×#41—611no—
229#1682—612no—
230#1681—612no—
231#1674—611no—
232#1673—611no—
233#1668—611no—
234#1667—611no—
235C2×#335D6111yescharacteristic, complemented
236#1540—612no—
237C2×C61—22no—
238C2×C61#303511yescharacteristic
239C2×C61C2×#46611yescharacteristic
240C2×C61#306711yescharacteristic
241D61—42no—
242D61—162no—
243C2×#135—1222no—
244#532—612no—
245C4×#41—611no—
246#500—611no—
247#493—611no—
248#486—612no—
249#466—1221no—
250#466—1221no—
251#464—1222no—
252#463—1222no—
253#462—612no—
254#461—612no—
255#460—612no—
256#459—612no—
257C8×Q8—611no—
258C8×Q8—611no—
259C4×D8—1221no—
260C4×D8—1221no—
261C8×D4—611no—
262C8×D4—611no—
263C8×D4—1221no—
264C8×D4—1221no—
265#431—611no—
266#431—1221no—
267C2×C4⋊C8C2×D6112yes—
268#416—1222no—
269C2×C4×C8—1221no—
270C2×C4×C8C2×D6111yescharacteristic
271#335C2×D6111yescharacteristic
272#335C2×D6111yescharacteristic
273#335C2×D6112yes—
274#332—612no—
275#332—612no—
276#331—612no—
277#331—612no—
278C61#1912511yescharacteristic, complemented, Sylow
279C2×#41—611no—
280C2×#41—1222no—
281#135—2448no—
282C2×Q16—1222no—
283C2×SD16—1222no—
284C2×SD16—1222no—
285C2×D8—1222no—
286C2×D8—1221no—
287C2×D8—1221no—
288C22×C8—22no—
289C22×C8—1221no—
290C22×C8—1221no—
291C22×C8D24411yescharacteristic
292C22×C8D24411yescharacteristic
293C4⋊Q8—612no—
294#127—612no—
295#124—1222no—
296#121—1222no—
297#121—1222no—
298C4×Q8—611no—
299C4×Q8—611no—
300C4×D4—611no—
301C4×D4—611no—
302C4×D4—1221no—
303C4×D4—1221no—
304C4×D4—1222no—
305C4×D4—1222no—
306#117—611no—
307#117—1221no—
308C2×C42—1221no—
309C2×C42C22×D6111yescharacteristic
310#107—1221no—
311#107—1221no—
312C4⋊C8—22no—
313C4⋊C8—611no—
314C4⋊C8—611no—
315C4⋊C8—1221no—
316C4⋊C8—1221no—
317C4⋊C8C22×D6112yes—
318C4⋊C8C22×D6112yes—
319#103—1222no—
320#103—1222no—
321#102—1222no—
322#102—1222no—
323#102—1222no—
324#102—1222no—
325C22⋊C8—1221no—
326C22⋊C8—1221no—
327C22⋊C8—1221no—
328C4×C8—1221no—
329C4×C8—1221no—
330C4×C8—1221no—
331C4×C8D24411yescharacteristic
332C4×C8D24411yescharacteristic
333C4×C8C22×D6111yescharacteristic
334C4×C8C22×D6111yescharacteristic
335#41—614no—
336#41—1222no—
337#41—2444no—
338#41—2444no—
339C2×Q8—611no—
340C2×Q8—1222no—
341C2×D4—611no—
342C2×D4—1221no—
343C2×D4—1222no—
344C2×D4—1222no—
345C2×D4—1222no—
346C2×D4—2442no—
347C22×C4—1221no—
348C22×C4—1221no—
349C22×C4—1221no—
350C22×C4—2442no—
351C22×C4C2×D24411yescharacteristic
352C22×C4D4×D6111yescharacteristic
353C22×C4#1175822611yescharacteristic
354Q16—2444no—
355SD16—2444no—
356SD16—2444no—
357D8—2442no—
358D8—2444no—
359D8—2442no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—41no—
364C2×C8—44no—
365C2×C8—1221no—
366C2×C8—1221no—
367C2×C8—2442no—
368C2×C8—2442no—
369C2×C8C2×D24411yescharacteristic
370C2×C8C2×D24411yescharacteristic
371C2×C8C2×D24411yescharacteristic
372C2×C8C2×D24411yescharacteristic
373C4⋊C4—611no—
374C4⋊C4—611no—
375C4⋊C4—1221no—
376C4⋊C4—1221no—
377C4⋊C4—1222no—
378C4⋊C4—1222no—
379C22⋊C4—1221no—
380C22⋊C4—1221no—
381C22⋊C4—1221no—
382C22⋊C4—2442no—
383C22⋊C4—2442no—
384C42—1221no—
385C42—1221no—
386C42—1221no—
387C42C2×D24411yescharacteristic
388C42D4×D6111yescharacteristic
389C42#1175822611yescharacteristic
390C42C23×D6111yescharacteristic
391C23—1221no—
392C23—2442no—
393C23#1451975211yescharacteristic
394Q8—612no—
395Q8—1221no—
396Q8—2442no—
397Q8—2442no—
398D4—612no—
399D4—1224no—
400D4—1221no—
401D4—4882no—
402D4—4884no—
403D4—2442no—
404D4—2442no—
405D4—2442no—
406D4—2442no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—41no—
410C2×C4—1221no—
411C2×C4—1224no—
412C2×C4—1221no—
413C2×C4—1221no—
414C2×C4—4884no—
415C2×C4—2442no—
416C2×C4—2442no—
417C2×C4—2441no—
418C2×C4—2441no—
419C2×C4#1451975211yescharacteristic
420C2×C4#1451975211yescharacteristic
421C2×C4#1451975211yescharacteristic
422C2×C4D8×D6112yes—
423C2×C4#1451980012yes—
424C2×C4C2×D4×D6111yescharacteristic
425C2×C4C22×D24411yescharacteristic
426C2×C4C2×#1175822611yescharacteristic
427C8—21no—
428C8—21no—
429C8—21no—
430C8—42no—
431C8—42no—
432C8—2441no—
433C8—2441no—
434C8#1451979912yescomplemented
435C22—21no—
436C22—21no—
437C22—1222no—
438C22—4882no—
439C22—4882no—
440C22—4882no—
441C22—2441no—
442C22—2441no—
443C22#1683999912yes—
444C22C2×#1451975211yescharacteristic
445C4—21no—
446C4—21no—
447C4—21no—
448C4—1222no—
449C4—4882no—
450C4—2441no—
451C4—2441no—
452C4#1683999912yes—
453C4C2×D8×D6111yescharacteristic
454C4C2×#1451979911yescharacteristic
455C4C2×#1451980011yescharacteristic
456C4#1684056211yescharacteristic
457C2—22no—
458C2—9762no—
459C2—2442no—
460C2#2145527611yescharacteristic
461C2C2×#1683999911yescharacteristic
462C2#2145554011yescharacteristic
subgroupG/H
centerC2×C4#14519752
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC2×C61×#335C2
socleC22×C61C2×#121
hypercenterC2×#335D61
hypocenterC61#19125
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×#335

Complements

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

NH ≅ G/Ntype
1. #21456398C2semidirect
2. #21455276C2semidirect
3. C8×#11758222C2semidirect
4. #21445264C2semidirect
5. #21445257C2semidirect
6. #21445214C2semidirect
7. #21445209C2semidirect
8. #21445180C2semidirect
9. #21445175C2semidirect
10. #21444469C2semidirect
11. C2×C61×#335C2semidirect
111. #11758222D8semidirect
200. C8×C61#135semidirect
235. C2×#335D61semidirect
278. C61#19125semidirect
434. C8#14519799semidirect

Maximal subgroups

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

subgroupindex
#214563982
#214552762
C8×#117582222
#214452642
#214452572
#214452142
#214452092
#214451802
#214451752
#214444692
C2×C61×#3352
#1912561