Finite Group Databasebeta

← #29663836

Subgroups — #29663836

Order 15616.

694 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#21456401C211yesmaximal, characteristic, complemented
2#21455919C211yesmaximal, characteristic
3#21455914C211yesmaximal, characteristic
4C2×#16839827C211yesmaximal, characteristic
5#21445245C212yesmaximal, complemented
6#21445241C212yesmaximal, complemented
7#21445207C212yesmaximal, complemented
8#21445183C212yesmaximal, complemented
9#21444580C211yesmaximal, characteristic, complemented
10#21444517C211yesmaximal, characteristic, complemented
11C61×#1552C211yesmaximal, characteristic, complemented
12C22×C8×D61—21no—
13#16840319C2211yescharacteristic
14C2×C4×D244C2211yescharacteristic
15#16840165C2211yescharacteristic, complemented
16#16840141C2211yescharacteristic, complemented
17#16840114C2211yescharacteristic, complemented
18#16840112C2211yescharacteristic, complemented
19#16840098—21no—
20#16840095—21no—
21#16840043C2212yes—
22#16840039C2212yes—
23#16840001—21no—
24#16840001—21no—
25#16839968C2212yes—
26#16839964C2212yes—
27#16839915—21no—
28#16839912—21no—
29#16839864—22no—
30#16839860—22no—
31C4×#11758202—21no—
32C4×#11758202—21no—
33#16839827—21no—
34#16839827—21no—
35#16839827C2212yes—
36#16839827C2212yes—
37#16839819C2211yescharacteristic
38C2×#14519689—21no—
39C2×C4×#92418C2211yescharacteristic
40#16839514C2212yescomplemented
41#16839512C2212yescomplemented
42#16839509C2212yescomplemented
43#16839507C2212yescomplemented
44C61×#533C2211yescharacteristic
45C61×#432C2211yescharacteristic
46C2×C4⋊C8×C61C2211yescharacteristic
47C61×#332C2212yescomplemented
48C61×#332C2212yescomplemented
49C22×D244D411yescharacteristic
50C22×C4×D61—21no—
51C2×#11758202—21no—
52C2×#11758202—21no—
53C2×#11758190—22no—
54C2×#11758201—22no—
55C22×#92418D411yescharacteristic
56C22×#92418D411yescharacteristic
57C2×C8×D61—22no—
58C2×C8×D61—22no—
59C2×C8×D61—44no—
60#14519777C2311yescharacteristic
61#14519777C2311yescharacteristic
62#14519769—21no—
63#14519756—21no—
64#14519756—21no—
65C4×Dic122—21no—
66C4×Dic122—21no—
67C4×D244—21no—
68C4×D244—21no—
69C4×D244D412yes—
70C4×D244C2312yes—
71#14519736—21no—
72C2×C4⋊Dic61D411yescharacteristic
73C2×#11758198—21no—
74#14519711—22no—
75#14519710—22no—
76#14519710—21no—
77#14519710—21no—
78C4⋊#92418C2312yes—
79#14519702—22no—
80#14519701—22no—
81#14519701—21no—
82#14519701—21no—
83#14519700—21no—
84#14519700—21no—
85#14519699—22no—
86#14519697—22no—
87C22⋊#92418—21no—
88#14519689—22no—
89#14519689—22no—
90#14519688—21no—
91C4×#92418D411yescharacteristic
92C4×#92418D411yescharacteristic
93C4×#92418C2311yescharacteristic
94C4×#92418C2311yescharacteristic
95C22×C8×C61—21no—
96C4⋊Q8×C61C2311yescharacteristic
97C4⋊Q8×C61C2311yescharacteristic
98#125×C61—21no—
99C22⋊Q8×C61—21no—
100C22⋊Q8×C61—21no—
101C4×Q8×C61—21no—
102C4×Q8×C61—21no—
103#117×C61—21no—
104C2×C42×C61C2311yescharacteristic
105C4⋊C8×C61—21no—
106C4⋊C8×C61C2312yes—
107C4⋊C8×C61C2312yes—
108C22⋊C8×C61—21no—
109#97×C61—21no—
110C23×D61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C22×C4×C61C2×D411yescharacteristic
114C22×C4×C61C2×D411yescharacteristic
115C22×C4×C61#4111yescharacteristic
116C22×Dic61—21no—
117C2×D244—21no—
118C2×D244—24no—
119C2×D244C2×D412yes—
120C2×C4×D61—22no—
121C2×C4×D61—22no—
122C2×C4×D61—42no—
123C2×C4×D61—42no—
124C2×Dic122—21no—
125C2×Dic122—21no—
126C2×C8×C61—22no—
127C2×C8×C61—22no—
128C2×C8×C61—42no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C22⋊C4×C61—41no—
136C42×C61—41no—
137C42×C61C2×D411yescharacteristic
138C42×C61C2×D411yescharacteristic
139C42×C61#4111yescharacteristic
140C42×C61C2411yescharacteristic
141C22⋊Dic61—41no—
142#11758202—42no—
143#11758202—42no—
144#11758201—44no—
145#11758198—22no—
146#11758198—22no—
147C4⋊Dic61—21no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152C4⋊Dic61C2×D412yes—
153#11758196—41no—
154#11758196—41no—
155C4×Dic61—41no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—41no—
159C2×#92418C2×D411yescharacteristic
160C2×#92418C2×D411yescharacteristic
161C2×#92418C2×D411yescharacteristic
162C2×#92418C2×D411yescharacteristic
163#11758190—44no—
164C8×D61—44no—
165C8×D61—44no—
166C23×C61#12111yescharacteristic
167C22×D61—24no—
168C22×D61—22no—
169C22×D61—42no—
170C22×D61—42no—
171Q8×C61—42no—
172Q8×C61—41no—
173Q8×C61—41no—
174C2×C4×C61—22no—
175C2×C4×C61—22no—
176C2×C4×C61—21no—
177C2×C4×C61—21no—
178C2×C4×C61—41no—
179C2×C4×C61—41no—
180C2×C4×C61#12111yescharacteristic
181C2×C4×C61#12111yescharacteristic
182C2×C4×C61#12111yescharacteristic
183C2×C4×C61C22×D411yescharacteristic
184C2×C4×C61C22×D411yescharacteristic
185C2×C4×C61C2×#4111yescharacteristic
186C2×Dic61—22no—
187C2×Dic61—42no—
188C2×Dic61—41no—
189C2×Dic61—41no—
190D244—24no—
191D244—22no—
192D244#13514yes—
193C4×D61—44no—
194C4×D61—42no—
195C4×D61—42no—
196Dic122—42no—
197Dic122—41no—
198Dic122—41no—
199C8×C61—42no—
200C8×C61—42no—
201#92418—21no—
202#92418—21no—
203#92418—21no—
204#92418—42no—
205#92418AGL(1,ℤ/8)12yes—
206#19259—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61#47312yes—
210C22×C61C2×#12111yescharacteristic
211C2×D61—24no—
212C2×D61—48no—
213C2×D61—42no—
214C2×D61—42no—
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—81no—
220C4×C61—81no—
221C4×C61C2×#13512yes—
222C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
223C4×C61#57511yescharacteristic
224Dic61—42no—
225Dic61—81no—
226Dic61—81no—
227#3170—611no—
228#3110—611no—
229#3110—611no—
230C2×C8×D4—611no—
231#1679—612no—
232#1679—612no—
233#1673—612no—
234#1669—612no—
235#1568—611no—
236#1552—611no—
237#1552D6111yescharacteristic, complemented
238C2×C61—22no—
239C2×C61#303611yescharacteristic
240C2×C61#305011yescharacteristic
241C2×C61C2×#47311yescharacteristic
242D61—44no—
243C23×C8—1221no—
244#533—611no—
245#533C2×D6111yescharacteristic
246C2×C4×D4—611no—
247#500—611no—
248#494—611no—
249#487—611no—
250#487—611no—
251#484—1221no—
252#484—1221no—
253#474—612no—
254#474—612no—
255#467—1221no—
256#467—1221no—
257#461—612no—
258#461—612no—
259#450—1221no—
260#450—1221no—
261#440—1222no—
262#440—1222no—
263C4×Q16—1221no—
264C4×Q16—1221no—
265C8×D4—612no—
266C8×D4—612no—
267C8×D4—1221no—
268C8×D4—1221no—
269#432—611no—
270#432C2×D6111yescharacteristic
271C2×C4⋊C8C2×D6111yescharacteristic
272C2×C22⋊C8—1221no—
273C2×C4×C8—611no—
274#335—612no—
275#334—612no—
276#332—612no—
277#332—612no—
278#332C2×D6112yescomplemented
279#332C2×D6112yescomplemented
280C61#1925911yescharacteristic, complemented, Sylow
281C22×D4—611no—
282C23×C4—1221no—
283C2×Q16—1221no—
284C2×Q16—1221no—
285C2×SD16—1222no—
286C2×SD16—1222no—
287C22×C8—21no—
288C22×C8—611no—
289C22×C8—611no—
290C22×C8—1222no—
291C22×C8—1222no—
292C22×C8—2444no—
293C4⋊Q8—611no—
294C4⋊Q8—611no—
295C4⋊Q8C22×D6111yescharacteristic
296C4⋊Q8C22×D6111yescharacteristic
297#125—21no—
298#125—1221no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C22⋊Q8—1221no—
302C22⋊Q8—1221no—
303C4×Q8—21no—
304C4×Q8—21no—
305C4×Q8—1221no—
306C4×Q8—1221no—
307C4×D4—612no—
308C4×D4—612no—
309C4×D4—1221no—
310C4×D4—1221no—
311#117—21no—
312#117—1221no—
313C2×C4⋊C4—611no—
314C2×C22⋊C4—1221no—
315C2×C42C22×D6111yescharacteristic
316#107—1222no—
317#107—1221no—
318#107—1221no—
319#107—1222no—
320C4⋊C8—21no—
321C4⋊C8—612no—
322C4⋊C8C22×D6112yes—
323C4⋊C8C22×D6112yes—
324#103—1221no—
325#103—1221no—
326#103—1222no—
327#103—1222no—
328#103—1221no—
329#103—1221no—
330#102—1222no—
331#102—1222no—
332C22⋊C8—21no—
333C22⋊C8—1222no—
334C22⋊C8—1222no—
335C22⋊C8—1221no—
336#97—21no—
337#97—1221no—
338C4×C8—611no—
339C4×C8—611no—
340C4×C8—611no—
341C4×C8—611no—
342C24—1221no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×D4—612no—
348C2×D4—1224no—
349C2×D4—1221no—
350C22×C4—1221no—
351C22×C4—1222no—
352C22×C4—1222no—
353C22×C4—2442no—
354C22×C4—2442no—
355C22×C4#1175822211yescharacteristic
356C22×C4D4×D6111yescharacteristic
357C22×C4D4×D6111yescharacteristic
358Q16—2442no—
359Q16—2442no—
360SD16—2444no—
361SD16—2444no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—42no—
365C2×C8—611no—
366C2×C8—611no—
367C2×C8—611no—
368C2×C8—611no—
369C2×C8—1222no—
370C2×C8—1222no—
371C2×C8—2444no—
372C2×C8—2444no—
373C2×C8—2441no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—612no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—1221no—
386C4⋊C4—2441no—
387C4⋊C4—2441no—
388C22⋊C4—41no—
389C22⋊C4—1222no—
390C22⋊C4—1222no—
391C22⋊C4—2441no—
392C42—41no—
393C42—2441no—
394C42#1175822211yescharacteristic
395C42D4×D6111yescharacteristic
396C42D4×D6111yescharacteristic
397C42C23×D6111yescharacteristic
398C23—1224no—
399C23—1222no—
400C23—2442no—
401C23—2442no—
402C23#1451975011yescharacteristic
403Q8—42no—
404Q8—41no—
405Q8—41no—
406Q8—2442no—
407Q8—2441no—
408Q8—2441no—
409D4—614no—
410D4—1224no—
411D4—1222no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—1222no—
419C2×C4—2444no—
420C2×C4—2441no—
421C2×C4—2441no—
422C2×C4—2442no—
423C2×C4—2442no—
424C2×C4—2442no—
425C2×C4#1451975011yescharacteristic
426C2×C4#1451975011yescharacteristic
427C2×C4#1451975011yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C2×D4×D6111yescharacteristic
430C2×C4C2×#1175822211yescharacteristic
431C8—42no—
432C8—42no—
433C8—612no—
434C8—1221no—
435C8—1221no—
436C8—1221no—
437C8—2442no—
438C22—21no—
439C22—21no—
440C22—1224no—
441C22—2448no—
442C22—2442no—
443C22—2442no—
444C22#1684003612yes—
445C22C2×#1451975011yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—81no—
451C4—81no—
452C4—4881no—
453C4—4881no—
454C4—2442no—
455C4#135×D6112yes—
456C4AGL(1,ℤ/8)×D6111yescharacteristic
457C4#1684056111yescharacteristic
458C2—22no—
459C2—2444no—
460C2#2145528211yescharacteristic
461C2#2145539911yescharacteristic
462C2C2×#1684003611yescharacteristic
subgroupG/H
centerC22C2×#14519750
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1552C2
socleC22×C61C2×#121
prime radicalC22×D244D4
hypercenter#1552D61
hypocenterC61#19259
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1552

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
5. #21445245C2semidirect
6. #21445241C2semidirect
7. #21445207C2semidirect
8. #21445183C2semidirect
9. #21444580C2semidirect
10. #21444517C2semidirect
11. C61×#1552C2semidirect
15. #16840165C22semidirect
16. #16840141C22semidirect
17. #16840114C22semidirect
18. #16840112C22semidirect
40. #16839514C22semidirect
41. #16839512C22semidirect
42. #16839509C22semidirect
43. #16839507C22semidirect
47. C61×#332C22semidirect
48. C61×#332C22semidirect
237. #1552D61semidirect
278. #332C2×D61semidirect
279. #332C2×D61semidirect
280. C61#19259semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214559192
#214559142
C2×#168398272
#214452452
#214452412
#214452072
#214451832
#214445802
#214445172
C61×#15522
#1925961