Finite Group Databasebeta

← #29663834

Subgroups — #29663834

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#21455918C211yesmaximal, characteristic
3C2×D4×#92418C211yesmaximal, characteristic
4#21445244C212yesmaximal, complemented
5#21445240C212yesmaximal, complemented
6#21445206C212yesmaximal, complemented
7#21445182C212yesmaximal, complemented
8#21444580C211yesmaximal, characteristic, complemented
9#21444520C211yesmaximal, characteristic, complemented
10#21444519C211yesmaximal, characteristic, complemented
11C61×#3110C211yesmaximal, characteristic
12C23×#92418—21no—
13#16840319C2211yescharacteristic
14#16840165C2211yescharacteristic, complemented
15#16840141C2211yescharacteristic, complemented
16#16840114C2211yescharacteristic, complemented
17#16840112C2211yescharacteristic, complemented
18#16840097—21no—
19#16840042C2212yes—
20#16840001—21no—
21#16840001—21no—
22#16839967C2212yes—
23#16839913—21no—
24#16839862—22no—
25C4×#11758202—21no—
26C4×#11758202—21no—
27D4×#92418—21no—
28D4×#92418—21no—
29D4×#92418C2212yes—
30D4×#92418C2212yes—
31#16839819C2211yescharacteristic
32C2×C4⋊#92418C2211yescharacteristic
33C2×C22⋊#92418—21no—
34C2×C4×#92418C2211yescharacteristic
35#16839514C2212yescomplemented
36#16839512C2212yescomplemented
37#16839509C2212yescomplemented
38#16839508C2212yescomplemented
39#16839508C2212yescomplemented
40#16839507C2212yescomplemented
41C61×#533C2211yescharacteristic
42C2×C4×D4×C61C2211yescharacteristic
43C61×#484—21no—
44C61×#474C2212yes—
45C61×#461C2212yes—
46C61×#450—21no—
47C61×#440—22no—
48C61×#432C2211yescharacteristic
49C2×#11758202—21no—
50C2×#11758202—21no—
51C2×#11758200—22no—
52C22×#92418—21no—
53C22×#92418—22no—
54C22×#92418—22no—
55C22×#92418—44no—
56C22×#92418D411yescharacteristic
57C22×#92418D411yescharacteristic
58#14519777C2311yescharacteristic
59#14519777C2311yescharacteristic
60#14519769—21no—
61#14519756—21no—
62#14519756—21no—
63C4×Dic122—21no—
64C4×Dic122—21no—
65#14519736—21no—
66#14519710—22no—
67#14519710—21no—
68#14519710—21no—
69C4⋊#92418—21no—
70C4⋊#92418C2312yes—
71C4⋊#92418C2312yes—
72#14519701—21no—
73#14519701—21no—
74#14519700—22no—
75#14519700—21no—
76#14519700—21no—
77#14519698—22no—
78C22⋊#92418—21no—
79C22⋊#92418—22no—
80C22⋊#92418—22no—
81#14519688—21no—
82C4×#92418D411yescharacteristic
83C4×#92418D411yescharacteristic
84C4×#92418C2311yescharacteristic
85C4×#92418C2311yescharacteristic
86C22×D4×C61D411yescharacteristic
87C23×C4×C61—21no—
88C2×SD16×C61—22no—
89C4⋊Q8×C61C2311yescharacteristic
90C4⋊Q8×C61C2311yescharacteristic
91#125×C61—21no—
92C22⋊Q8×C61—21no—
93C22⋊Q8×C61—21no—
94C4×Q8×C61—21no—
95C4×Q8×C61—21no—
96C4×D4×C61—21no—
97C4×D4×C61—21no—
98C4×D4×C61D412yes—
99C4×D4×C61C2312yes—
100#117×C61—21no—
101C2×C4⋊C4×C61D411yescharacteristic
102C2×C22⋊C4×C61—21no—
103C2×C42×C61C2311yescharacteristic
104#107×C61—22no—
105C4⋊C8×C61C2312yes—
106#103×C61—22no—
107#102×C61—22no—
108C22⋊C8×C61—21no—
109#97×C61—21no—
110C24×C61—21no—
111C2×Q8×C61—21no—
112C2×Q8×C61—21no—
113C2×D4×C61—24no—
114C2×D4×C61—21no—
115C2×D4×C61C2×D412yes—
116C22×C4×C61—21no—
117C22×C4×C61—22no—
118C22×C4×C61—22no—
119C22×C4×C61—42no—
120C22×C4×C61—42no—
121C22×C4×C61C2×D411yescharacteristic
122C22×C4×C61C2×D411yescharacteristic
123C22×C4×C61#4111yescharacteristic
124C2×Dic122—21no—
125C2×Dic122—21no—
126SD16×C61—44no—
127C2×C8×C61—22no—
128C4⋊C4×C61—21no—
129C4⋊C4×C61—21no—
130C4⋊C4×C61—21no—
131C4⋊C4×C61—21no—
132C4⋊C4×C61—21no—
133C4⋊C4×C61—41no—
134C4⋊C4×C61—41no—
135C4⋊C4×C61C2×D412yes—
136C22⋊C4×C61—22no—
137C22⋊C4×C61—22no—
138C22⋊C4×C61—41no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144C22⋊Dic61—41no—
145#11758202—42no—
146#11758202—42no—
147#11758200—44no—
148C4⋊Dic61—21no—
149C4⋊Dic61—21no—
150C4⋊Dic61—21no—
151C4⋊Dic61—21no—
152#11758196—41no—
153#11758196—41no—
154C4×Dic61—41no—
155C2×#92418—22no—
156C2×#92418—22no—
157C2×#92418—22no—
158C2×#92418—41no—
159C2×#92418—44no—
160C2×#92418—42no—
161C2×#92418—44no—
162C2×#92418C2×D411yescharacteristic
163C2×#92418C2×D411yescharacteristic
164C2×#92418C2×D411yescharacteristic
165C2×#92418C2×D411yescharacteristic
166C23×C61—24no—
167C23×C61—22no—
168C23×C61—42no—
169C23×C61—42no—
170C23×C61#12111yescharacteristic
171Q8×C61—42no—
172Q8×C61—41no—
173Q8×C61—41no—
174D4×C61—24no—
175D4×C61—22no—
176D4×C61#13514yes—
177C2×C4×C61—22no—
178C2×C4×C61—22no—
179C2×C4×C61—21no—
180C2×C4×C61—21no—
181C2×C4×C61—22no—
182C2×C4×C61—44no—
183C2×C4×C61—42no—
184C2×C4×C61—42no—
185C2×C4×C61—42no—
186C2×C4×C61—41no—
187C2×C4×C61—41no—
188C2×C4×C61#12111yescharacteristic
189C2×C4×C61#12111yescharacteristic
190C2×C4×C61#12111yescharacteristic
191C2×C4×C61C22×D411yescharacteristic
192C2×C4×C61C22×D411yescharacteristic
193C2×C4×C61C2×#4111yescharacteristic
194C2×Dic61—41no—
195C2×Dic61—41no—
196Dic122—42no—
197Dic122—41no—
198Dic122—41no—
199C8×C61—42no—
200#92418—21no—
201#92418—21no—
202#92418—21no—
203#92418—42no—
204#92418—42no—
205#92418AGL(1,ℤ/8)12yes—
206#19259—611nomaximal, Sylow
207C22×C61—21no—
208C22×C61—21no—
209C22×C61—24no—
210C22×C61—48no—
211C22×C61—42no—
212C22×C61—42no—
213C22×C61#47312yes—
214C22×C61C2×#12111yescharacteristic
215C4×C61—21no—
216C4×C61—21no—
217C4×C61—21no—
218C4×C61—21no—
219C4×C61—42no—
220C4×C61—81no—
221C4×C61—81no—
222C4×C61C2×#13512yes—
223C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
224C4×C61#57511yescharacteristic
225Dic61—81no—
226Dic61—81no—
227#3170—611no—
228#3110—611no—
229#3110D6111yescharacteristic
230C2×C8×D4—611no—
231#1679—612no—
232#1679—612no—
233#1673—612no—
234#1669—612no—
235#1568—611no—
236#1552—611no—
237#1552—611no—
238C2×C61—22no—
239C2×C61—44no—
240C2×C61#303611yescharacteristic
241C2×C61#305011yescharacteristic
242C2×C61C2×#47311yescharacteristic
243C23×C8—1221no—
244#533—611no—
245#533C2×D6111yescharacteristic
246C2×C4×D4C2×D6111yescharacteristic
247#500—611no—
248#494—611no—
249#487—611no—
250#487—611no—
251#484—21no—
252#484—1221no—
253#474—612no—
254#474C2×D6112yes—
255#467—1221no—
256#467—1221no—
257#461—612no—
258#461C2×D6112yes—
259#450—21no—
260#450—1221no—
261#440—22no—
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⋊C8—611no—
272C2×C22⋊C8—1221no—
273C2×C4×C8—611no—
274#335—612no—
275#334—612no—
276#332—612no—
277#332—612no—
278#332—612no—
279#332—612no—
280C61#1925911yescharacteristic, complemented, Sylow
281C22×D4C22⋊D6111yescharacteristic
282C23×C4—21no—
283C2×Q16—1221no—
284C2×Q16—1221no—
285C2×SD16—22no—
286C2×SD16—1222no—
287C22×C8—611no—
288C22×C8—611no—
289C22×C8—1222no—
290C22×C8—1222no—
291C22×C8—1221no—
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—21no—
308C4×D4—21no—
309C4×D4C22⋊D6112yes—
310C4×D4C22×D6112yes—
311#117—21no—
312#117—1221no—
313C2×C4⋊C4C22⋊D6111yescharacteristic
314C2×C22⋊C4—21no—
315C2×C42C22×D6111yescharacteristic
316#107—22no—
317#107—1221no—
318#107—1221no—
319#107—1222no—
320C4⋊C8—612no—
321C4⋊C8—612no—
322C4⋊C8—1221no—
323C4⋊C8C22×D6112yes—
324#103—22no—
325#103—1221no—
326#103—1221no—
327#103—1221no—
328#103—1221no—
329#103—1222no—
330#102—22no—
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—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—1221no—
346C2×Q8—1221no—
347C2×D4—24no—
348C2×D4—21no—
349C2×D4C2×C22⋊D6112yes—
350C22×C4—21no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4D4×D6111yescharacteristic
356C22×C4#1175822411yescharacteristic
357C22×C4C2×C22⋊D6111yescharacteristic
358Q16—2442no—
359Q16—2442no—
360SD16—44no—
361SD16—2444no—
362C2×C8—22no—
363C2×C8—611no—
364C2×C8—611no—
365C2×C8—611no—
366C2×C8—611no—
367C2×C8—1222no—
368C2×C8—1222no—
369C2×C8—1222no—
370C2×C8—2444no—
371C2×C8—2441no—
372C2×C8—2444no—
373C2×C8—2442no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—1221no—
382C4⋊C4—1221no—
383C4⋊C4—1221no—
384C4⋊C4—1221no—
385C4⋊C4—2441no—
386C4⋊C4—2441no—
387C4⋊C4C2×C22⋊D6112yes—
388C22⋊C4—22no—
389C22⋊C4—22no—
390C22⋊C4—41no—
391C22⋊C4—2441no—
392C42—41no—
393C42—2441no—
394C42D4×D6111yescharacteristic
395C42#1175822411yescharacteristic
396C42C2×C22⋊D6111yescharacteristic
397C42C23×D6111yescharacteristic
398C23—24no—
399C23—22no—
400C23—42no—
401C23—42no—
402C23#1451974911yescharacteristic
403Q8—42no—
404Q8—41no—
405Q8—41no—
406Q8—2442no—
407Q8—2441no—
408Q8—2441no—
409D4—24no—
410D4—22no—
411D4#1451979714yes—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—22no—
417C2×C4—44no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—2441no—
424C2×C4—2441no—
425C2×C4#1451974911yescharacteristic
426C2×C4#1451974911yescharacteristic
427C2×C4#1451974911yescharacteristic
428C2×C4C2×D4×D6111yescharacteristic
429C2×C4C22×C22⋊D6111yescharacteristic
430C2×C4C2×#1175822411yescharacteristic
431C8—42no—
432C8—612no—
433C8—1221no—
434C8—1221no—
435C8—1221no—
436C8—2442no—
437C8—2442no—
438C22—21no—
439C22—21no—
440C22—24no—
441C22—48no—
442C22—42no—
443C22—42no—
444C22#1684003512yes—
445C22C2×#1451974911yescharacteristic
446C4—21no—
447C4—21no—
448C4—21no—
449C4—21no—
450C4—42no—
451C4—81no—
452C4—81no—
453C4—4881no—
454C4—4881no—
455C4C2×#1451979712yes—
456C4AGL(1,ℤ/8)×D6111yescharacteristic
457C4#1684055811yescharacteristic
458C2—22no—
459C2—44no—
460C2#2145528011yescharacteristic
461C2#2145539711yescharacteristic
462C2C2×#1684003511yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3110C2
socleC22×C61C2×#121
prime radicalC22×D4×C61D4
hypercenter#3110D61
hypocenterC61#19259
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3110

Complements

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

NH ≅ G/Ntype
1. #21456401C2semidirect
4. #21445244C2semidirect
5. #21445240C2semidirect
6. #21445206C2semidirect
7. #21445182C2semidirect
8. #21444580C2semidirect
9. #21444520C2semidirect
10. #21444519C2semidirect
14. #16840165C22semidirect
15. #16840141C22semidirect
16. #16840114C22semidirect
17. #16840112C22semidirect
35. #16839514C22semidirect
36. #16839512C22semidirect
37. #16839509C22semidirect
38. #16839508C22semidirect
39. #16839508C22semidirect
40. #16839507C22semidirect
280. C61#19259semidirect

Maximal subgroups

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

subgroupindex
#214564012
#214559182
C2×D4×#924182
#214452442
#214452402
#214452062
#214451822
#214445802
#214445202
#214445192
C61×#31102
#1925961