Finite Group Databasebeta

← #29664032

Subgroups — #29664032

Order 15616.

658 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#21456403C211yesmaximal, characteristic, complemented
2#21455611C212yesmaximal
3#436×D61C211yesmaximal, characteristic
4#21445393C212yesmaximal, complemented
5#21445392C212yesmaximal, complemented
6#21445343C211yesmaximal, characteristic, complemented
7#21445342C211yesmaximal, characteristic, complemented
8#21445305C211yesmaximal, characteristic, complemented
9#21444573C211yesmaximal, characteristic, complemented
10#21444508C212yesmaximal, complemented
11C61×#1689C211yesmaximal, characteristic, complemented
12C2×M16×D61—21no—
13#16840317C2212yes—
14C4×D4×D61C2211yescharacteristic
15#16840167C2211yescharacteristic, complemented
16#16840112C2212yescomplemented
17#16840065—22no—
18#16840064—22no—
19#16840053C2212yes—
20#16840052C2212yes—
21#16840039C2212yes—
22#16840000—21no—
23#16840000—21no—
24Q16×Dic61—21no—
25Q16×Dic61—21no—
26#16839841C2211yescharacteristic
27#16839840C2211yescharacteristic
28#16839839C2211yescharacteristic
29#16839838—21no—
30#16839837—21no—
31#16839803—22no—
32#16839784C2212yes—
33C4⋊C8×D61C2211yescharacteristic
34#16839765—22no—
35C22⋊C8×D61—21no—
36C4×C8×D61C2211yescharacteristic
37#16839513C2211yescharacteristic, complemented
38#16839512C2212yescomplemented
39#16839510C2212yescomplemented
40#16839510C2212yescomplemented
41#16839508C2212yescomplemented
42C61×#494C2211yescharacteristic, complemented
43C61×#474C2212yes—
44C61×#436C2211yescharacteristic
45C61×#334C2211yescharacteristic, complemented
46C61×#332C2212yescomplemented
47C2×D4×D61D411yescharacteristic
48C22×C4×D61—21no—
49C2×#11758194—21no—
50C2×#11758189—22no—
51C2×#11758189—21no—
52C2×#11758189—21no—
53M16×D61—44no—
54C2×C8×D61—21no—
55C2×C8×D61D411yescharacteristic
56C2×C8×D61D411yescharacteristic
57#14519777C2312yes—
58#14519768—22no—
59#14519757—22no—
60#14519753—22no—
61Q8×Dic61—22no—
62#14519742—22no—
63C4×D244C2311yescharacteristic
64D4×Dic61D411yescharacteristic
65#14519739D411yescharacteristic
66C4×C22⋊D61—21no—
67#14519737—21no—
68#14519735—22no—
69C4⋊C4×D61D411yescharacteristic
70C22⋊C4×D61—21no—
71C42×D61C2311yescharacteristic
72#14519711—21no—
73#14519711—21no—
74#14519710—22no—
75#14519709—22no—
76#14519708—22no—
77C4⋊#92418C2311yescharacteristic
78C4⋊#92418C2312yes—
79#14519706—21no—
80#14519706—22no—
81#14519701—22no—
82#14519701—22no—
83#14519699—22no—
84#14519698—22no—
85#14519697—22no—
86C22⋊#92418—21no—
87#14519689—21no—
88#14519689—21no—
89C4×#92418C2311yescharacteristic
90C8×Dic61D411yescharacteristic
91C8×Dic61D411yescharacteristic
92C2×Q16×C61—21no—
93C2×Q16×C61—21no—
94C2×M16×C61—21no—
95C4⋊Q8×C61C2312yes—
96C4×D4×C61C2311yescharacteristic
97#107×C61—22no—
98C4⋊C8×C61C2311yescharacteristic
99C4⋊C8×C61C2312yes—
100#102×C61—22no—
101C22⋊C8×C61—21no—
102C4×C8×C61C2311yescharacteristic
103C23×D61—21no—
104C2×Q8×C61—22no—
105C2×D4×C61C2×D411yescharacteristic
106C22×C4×C61—21no—
107C2×C22⋊D61—21no—
108C22×Dic61—21no—
109D4×D61—24no—
110C2×D244C2×D411yescharacteristic
111C2×C4×D61—21no—
112C2×C4×D61—21no—
113C2×C4×D61—21no—
114C2×C4×D61—42no—
115C2×C4×D61—42no—
116C2×C4×D61C2×D411yescharacteristic
117C2×C4×D61C2×D411yescharacteristic
118C2×C4×D61#4111yescharacteristic
119C2×Dic122—22no—
120Q16×C61—42no—
121Q16×C61—42no—
122M16×C61—42no—
123C2×C8×C61—21no—
124C2×C8×C61—22no—
125C2×C8×C61C2×D411yescharacteristic
126C2×C8×C61C2×D411yescharacteristic
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61C2×D411yescharacteristic
130C22⋊C4×C61—21no—
131C42×C61C2411yescharacteristic
132C22⋊Dic61—21no—
133#11758198—21no—
134#11758198—21no—
135#11758198—42no—
136C4⋊Dic61—22no—
137C4⋊Dic61—22no—
138C4⋊Dic61—42no—
139C4⋊Dic61C2×D411yescharacteristic
140#11758196—21no—
141#11758196—42no—
142C4×Dic61—42no—
143C4×Dic61C2×D411yescharacteristic
144C4×Dic61C2×D411yescharacteristic
145C4×Dic61#4111yescharacteristic
146#11758194—42no—
147C2×#92418—21no—
148C2×#92418—22no—
149C2×#92418C2×D411yescharacteristic
150C2×#92418C2×D411yescharacteristic
151#11758189—42no—
152#11758189—44no—
153#11758189—42no—
154C8×D61—22no—
155C8×D61—41no—
156C8×D61—42no—
157C23×C61—21no—
158C22×D61—21no—
159C22×D61—42no—
160C22×D61—42no—
161C22×D61—42no—
162C22×D61#12111yescharacteristic
163Q8×C61—42no—
164Q8×C61—42no—
165D4×C61—22no—
166C2×C4×C61—21no—
167C2×C4×C61—41no—
168C2×C4×C61—41no—
169C2×C4×C61—42no—
170C2×C4×C61C22×D411yescharacteristic
171C2×C4×C61C22×D411yescharacteristic
172C2×C4×C61C2×#4111yescharacteristic
173C22⋊D61—42no—
174C2×Dic61—21no—
175C2×Dic61—42no—
176C2×Dic61—42no—
177C2×Dic61#12111yescharacteristic
178C2×Dic61#12111yescharacteristic
179C2×Dic61#12111yescharacteristic
180D244—22no—
181C4×D61—21no—
182C4×D61—21no—
183C4×D61—41no—
184C4×D61—42no—
185C4×D61—42no—
186C4×D61—41no—
187C4×D61—41no—
188C4×D61#13714yes—
189Dic122—82no—
190C8×C61—21no—
191C8×C61—41no—
192C8×C61—42no—
193C8×C61AGL(1,ℤ/8)12yes—
194#92418—21no—
195#92418—21no—
196#92418—41no—
197#92418—42no—
198#19281—611nomaximal, Sylow
199C22×C61—41no—
200C22×C61—41no—
201C22×C61—41no—
202C22×C61C2×#12111yescharacteristic
203C2×D61—21no—
204C2×D61—21no—
205C2×D61—41no—
206C2×D61—41no—
207C2×D61—41no—
208C2×D61—84no—
209C2×D61#47012yes—
210C4×C61—21no—
211C4×C61—41no—
212C4×C61—82no—
213C4×C61C2×AGL(1,ℤ/8)11yescharacteristic
214C4×C61C2×#13712yes—
215C4×C61#57511yescharacteristic
216Dic61—21no—
217Dic61—21no—
218Dic61—41no—
219Dic61—82no—
220Dic61#47012yes—
221#3170—611no—
222#3077—612no—
223C2×#436—611no—
224#1703—612no—
225#1703—612no—
226#1695—611no—
227#1695—611no—
228#1689—611no—
229#1689D6111yescharacteristic, complemented
230#1566—611no—
231#1550—612no—
232C2×C61—81no—
233C2×C61#303611yescharacteristic
234C2×C61C2×#47011yescharacteristic
235C2×C61#306711yescharacteristic
236D61—22no—
237D61—81no—
238C22×M16—1221no—
239#533—612no—
240C2×C4×D4—611no—
241#500—611no—
242#494C2×D6111yescharacteristic, complemented
243#487—612no—
244#478—1222no—
245#478—1222no—
246#476—612no—
247#476—612no—
248#474—612no—
249#474C2×D6112yes—
250#467—1221no—
251#467—1221no—
252C4×Q16—1221no—
253C4×Q16—1221no—
254#436—611no—
255#436—611no—
256#436—611no—
257#436—1221no—
258#436—1221no—
259#436C2×D6111yescharacteristic
260#428—1222no—
261#423—612no—
262C2×C4⋊C8—611no—
263#418—1222no—
264C2×C22⋊C8—1221no—
265C2×C4×C8—611no—
266#334—612no—
267#334—611no—
268#334C2×D6111yescharacteristic, complemented
269#333—612no—
270#333—612no—
271#332—612no—
272#332C2×D6112yescomplemented
273C61#1928111yescharacteristic, complemented, Sylow
274C22×D4—611no—
275C23×C4—1221no—
276C2×Q16—21no—
277C2×Q16—21no—
278C2×M16—21no—
279C2×M16—1221no—
280C2×M16—1222no—
281C2×M16—1221no—
282C2×M16—1221no—
283C2×M16—2444no—
284C22×C8—611no—
285C22×C8—611no—
286C22×C8—1221no—
287C4⋊Q8—612no—
288C4⋊Q8C22×D6112yes—
289#125—1222no—
290C22⋊Q8—1222no—
291C22⋊Q8—1222no—
292C4×Q8—1222no—
293C4×Q8—1222no—
294C4×D4—611no—
295C4×D4—611no—
296C4×D4—611no—
297C4×D4—1221no—
298C4×D4—1221no—
299C4×D4C22×D6111yescharacteristic
300#117—1222no—
301C2×C4⋊C4—611no—
302C2×C22⋊C4—1221no—
303C2×C42—611no—
304#107—22no—
305#107—1222no—
306#107—1221no—
307#107—1221no—
308#106—1222no—
309#106—1222no—
310C4⋊C8—611no—
311C4⋊C8—612no—
312C4⋊C8—1221no—
313C4⋊C8—1222no—
314C4⋊C8C22×D6111yescharacteristic
315C4⋊C8C22×D6112yes—
316#103—1222no—
317#103—1222no—
318#102—22no—
319#102—1222no—
320#102—1222no—
321#102—1222no—
322C22⋊C8—21no—
323C22⋊C8—1221no—
324C22⋊C8—1221no—
325C22⋊C8—1221no—
326C4×C8—611no—
327C4×C8—611no—
328C4×C8—611no—
329C4×C8C22×D6111yescharacteristic
330C24—1221no—
331C2×Q8—22no—
332C2×Q8—1222no—
333C2×D4—611no—
334C2×D4—1224no—
335C2×D4—1221no—
336C2×D4D4×D6111yescharacteristic
337C22×C4—21no—
338C22×C4—611no—
339C22×C4—611no—
340C22×C4—611no—
341C22×C4—1221no—
342C22×C4—1221no—
343C22×C4—1221no—
344C22×C4—1221no—
345C22×C4—2442no—
346C22×C4—2442no—
347Q16—42no—
348Q16—42no—
349M16—42no—
350M16—2442no—
351M16—2444no—
352M16—2442no—
353M16—2442no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—611no—
357C2×C8—611no—
358C2×C8—1222no—
359C2×C8—1221no—
360C2×C8—1222no—
361C2×C8—2441no—
362C2×C8—2442no—
363C2×C8D4×D6111yescharacteristic
364C2×C8D4×D6111yescharacteristic
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—611no—
368C4⋊C4—1221no—
369C4⋊C4—1222no—
370C4⋊C4—1222no—
371C4⋊C4—2442no—
372C4⋊C4—2442no—
373C4⋊C4D4×D6111yescharacteristic
374C22⋊C4—21no—
375C22⋊C4—1221no—
376C22⋊C4—1221no—
377C22⋊C4—1221no—
378C22⋊C4—2442no—
379C42—611no—
380C42—611no—
381C42—611no—
382C42—2442no—
383C42C23×D6111yescharacteristic
384C23—21no—
385C23—611no—
386C23—1221no—
387C23—2442no—
388C23—2442no—
389C23—2442no—
390Q8—42no—
391Q8—42no—
392Q8—4882no—
393D4—22no—
394D4—1222no—
395D4—2442no—
396C2×C4—21no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—611no—
401C2×C4—614no—
402C2×C4—611no—
403C2×C4—611no—
404C2×C4—1221no—
405C2×C4—1221no—
406C2×C4—1221no—
407C2×C4—2441no—
408C2×C4—2442no—
409C2×C4—2442no—
410C2×C4—2442no—
411C2×C4—2442no—
412C2×C4—2441no—
413C2×C4—2441no—
414C2×C4C2×D4×D6111yescharacteristic
415C2×C4C2×D4×D6111yescharacteristic
416C2×C4#41×D6111yescharacteristic
417C8—21no—
418C8—41no—
419C8—42no—
420C8—1221no—
421C8—1221no—
422C8—2441no—
423C8—2442no—
424C8#1451980112yes—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—612no—
429C22—1221no—
430C22—1221no—
431C22—4884no—
432C22—2441no—
433C22—2441no—
434C22—2441no—
435C22#121×D6111yescharacteristic
436C4—21no—
437C4—41no—
438C4—82no—
439C4—612no—
440C4—1221no—
441C4—1221no—
442C4—4882no—
443C4—2441no—
444C4C2×#1451980111yescharacteristic
445C4#137×D6112yes—
446C4#1684055811yescharacteristic
447C2—81no—
448C2—1222no—
449C2—4881no—
450C2#2145527811yescharacteristic
451C2#470×D6111yescharacteristic
452C2#2145554111yescharacteristic
subgroupG/H
centerC22#121×D61
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#1689C2
socleC22×C61C2×#121
prime radicalC2×D4×D61D4
hypercenter#1689D61
hypocenterC61#19281
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #1689

Complements

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

NH ≅ G/Ntype
1. #21456403C2semidirect
4. #21445393C2semidirect
5. #21445392C2semidirect
6. #21445343C2semidirect
7. #21445342C2semidirect
8. #21445305C2semidirect
9. #21444573C2semidirect
10. #21444508C2semidirect
11. C61×#1689C2semidirect
15. #16840167C22semidirect
16. #16840112C22semidirect
37. #16839513C22semidirect
38. #16839512C22semidirect
39. #16839510C22semidirect
40. #16839510C22semidirect
41. #16839508C22semidirect
42. C61×#494C22semidirect
45. C61×#334C22semidirect
46. C61×#332C22semidirect
229. #1689D61semidirect
242. #494C2×D61semidirect
268. #334C2×D61semidirect
272. #332C2×D61semidirect
273. C61#19281semidirect

Maximal subgroups

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

subgroupindex
#214564032
#214556112
#436×D612
#214453932
#214453922
#214453432
#214453422
#214453052
#214445732
#214445082
C61×#16892
#1928161