Finite Group Databasebeta

← #29663631

Subgroups — #29663631

Order 15616.

636 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456757C211yesmaximal, characteristic
2#21456195C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21445389C212yesmaximal, complemented
5#21445361C212yesmaximal, complemented
6#21445265C212yesmaximal, complemented
7#21445237C212yesmaximal, complemented
8#21444569C211yesmaximal, characteristic, complemented
9#21444476C211yesmaximal, characteristic, complemented
10C2×#16839505C211yesmaximal, characteristic
11C61×#3249C211yesmaximal, characteristic, complemented
12C2×#14519774C2211yescharacteristic
13#16840136—21no—
14#16840128—22no—
15#16840110—21no—
16#16840110C2212yes—
17#16840110C2212yes—
18#16839962C2212yes—
19#16839962C2212yes—
20#16839912—22no—
21#16839893—22no—
22#16839893C2212yescomplemented
23Q8×#92418C2212yescomplemented
24#16839815C2211yescharacteristic
25#16839789—21no—
26C2×C4⋊#92418C2211yescharacteristic
27#16839766—21no—
28#16839766—21no—
29#16839718—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839511C2211yescharacteristic, complemented
34#16839511C2211yescharacteristic, complemented
35#16839508C2211yescharacteristic, complemented
36#16839508C2211yescharacteristic, complemented
37#16839505C2212yes—
38#16839505C2212yes—
39C61×#533C2211yescharacteristic
40C61×#533C2211yescharacteristic
41C61×#477—21no—
42C61×#477C2212yescomplemented
43C61×#467—22no—
44C61×#462—21no—
45C61×#462C2212yescomplemented
46C2×C4⋊C8×C61C2211yescharacteristic
47C22×D244—21no—
48C22×D244—21no—
49C22×D244—21no—
50C2×#11758201—22no—
51C2×#11758201—22no—
52C2×#11758194D411yescharacteristic
53C2×#11758194D411yescharacteristic
54C22×#92418—21no—
55#14519774—21no—
56#14519774—21no—
57#14519774—21no—
58#14519774C2312yes—
59C4⋊#92418—21no—
60C4⋊#92418—22no—
61C4⋊#92418—22no—
62C4⋊#92418C2311yescharacteristic
63C4⋊#92418C2311yescharacteristic
64C4⋊#92418C2311yescharacteristic
65C4⋊#92418C2311yescharacteristic
66#14519697—22no—
67#14519697—22no—
68#14519697—22no—
69#14519697—22no—
70#14519697—22no—
71#14519697—22no—
72C22⋊#92418—21no—
73#14519688—22no—
74#14519688D412yes—
75C4×#92418—21no—
76C4×#92418—22no—
77C4×#92418C2312yes—
78C2×Q16×C61—22no—
79C22×C8×C61—21no—
80C4⋊Q8×C61D411yescharacteristic
81C4⋊Q8×C61D411yescharacteristic
82C4⋊Q8×C61C2311yescharacteristic
83C4⋊Q8×C61C2311yescharacteristic
84#125×C61—21no—
85#125×C61—21no—
86C22⋊Q8×C61—21no—
87C22⋊Q8×C61—21no—
88C22⋊Q8×C61—21no—
89C22⋊Q8×C61D412yes—
90C4×Q8×C61—21no—
91C4×Q8×C61—21no—
92C4×Q8×C61—21no—
93C4×Q8×C61C2312yes—
94#117×C61—21no—
95#117×C61—21no—
96C2×C42×C61C2311yescharacteristic
97#107×C61—22no—
98C4⋊C8×C61—21no—
99C4⋊C8×C61C2312yes—
100#103×C61—22no—
101#103×C61—22no—
102C23×D61—41no—
103C2×Q8×C61—21no—
104C2×Q8×C61—21no—
105C2×Q8×C61—21no—
106C2×Q8×C61C2×D412yes—
107C22×C4×C61C2×D411yescharacteristic
108C22×C4×C61C2×D411yescharacteristic
109C22×C4×C61#4111yescharacteristic
110C2×D244—22no—
111C2×D244—22no—
112C2×D244—22no—
113C2×D244—42no—
114C2×D244—42no—
115C2×D244—41no—
116C2×D244—41no—
117C2×D244—41no—
118C2×D244—81no—
119C2×D244—81no—
120C2×D244—82no—
121Q16×C61—82no—
122C2×C8×C61—22no—
123C2×C8×C61—81no—
124C4⋊C4×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—22no—
128C4⋊C4×C61—22no—
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—41no—
138C42×C61—22no—
139C42×C61—41no—
140C42×C61C2×D411yescharacteristic
141C42×C61C2×D411yescharacteristic
142C42×C61#4111yescharacteristic
143C42×C61C2411yescharacteristic
144#11758201—82no—
145#11758201—82no—
146#11758194—22no—
147#11758194—22no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—21no—
151C2×#92418—21no—
152C2×#92418—42no—
153C2×#92418C2×D412yes—
154C2×#92418C2×D412yes—
155C23×C61#12111yescharacteristic
156C22×D61—42no—
157C22×D61—82no—
158C22×D61—82no—
159C22×D61—82no—
160Q8×C61—22no—
161Q8×C61—22no—
162Q8×C61—41no—
163Q8×C61—41no—
164Q8×C61—81no—
165Q8×C61—81no—
166C2×C4×C61—21no—
167C2×C4×C61—21no—
168C2×C4×C61—22no—
169C2×C4×C61—22no—
170C2×C4×C61—41no—
171C2×C4×C61—41no—
172C2×C4×C61—41no—
173C2×C4×C61#12111yescharacteristic
174C2×C4×C61#12111yescharacteristic
175C2×C4×C61#12111yescharacteristic
176C2×C4×C61#13512yes—
177C2×C4×C61#13712yes—
178C2×C4×C61C22×D411yescharacteristic
179C2×C4×C61C22×D411yescharacteristic
180C2×C4×C61C2×#4111yescharacteristic
181D244—42no—
182D244—42no—
183D244—42no—
184D244—42no—
185D244—82no—
186D244—82no—
187D244—82no—
188D244—82no—
189C8×C61—42no—
190#92418—22no—
191#92418—22no—
192#92418—41no—
193#92418—41no—
194#92418—41no—
195#92418—41no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61#46412yes—
200C22×C61C2×#12111yescharacteristic
201C2×D61—82no—
202C2×D61—82no—
203C2×D61—82no—
204C2×D61—164no—
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—42no—
209C4×C61—42no—
210C4×C61—81no—
211C4×C61—81no—
212C4×C61#46412yes—
213C4×C61C2×#13511yescharacteristic
214C4×C61C2×#13711yescharacteristic
215C4×C61#57611yescharacteristic
216C4×C61#57711yescharacteristic
217#3249D6111yescharacteristic, complemented
218#3208—611no—
219#3138—611no—
220#2962—611no—
221#1702—612no—
222#1698—612no—
223#1682—612no—
224#1678—612no—
225#1565—611no—
226#1542—611no—
227C2×#331—611no—
228C2×C61—22no—
229C2×C61C2×#46411yescharacteristic
230C2×C61#306211yescharacteristic
231C2×C61#306811yescharacteristic
232D61—162no—
233#533C2×D6111yescharacteristic
234#533C2×D6111yescharacteristic
235C2×#127—611no—
236#492—1221no—
237#490—1222no—
238#486—612no—
239#486—612no—
240#486—1221no—
241#477—21no—
242#477C2×D6112yescomplemented
243#467—22no—
244#462—21no—
245#462C2×D6112yescomplemented
246#460—612no—
247#460—612no—
248#450—1222no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431—611no—
253#424—1221no—
254C2×C4⋊C8—611no—
255C2×C4⋊C8C2×D6111yescharacteristic
256#418—1221no—
257#418—1221no—
258#405—1221no—
259C4×M16—611no—
260#335—612no—
261#334—612no—
262#333—611no—
263#333—611no—
264#332—611no—
265#332—611no—
266#331—612no—
267#331—612no—
268C61#1924211yescharacteristic, complemented, Sylow
269C22×D4—1221no—
270C22×D4—1221no—
271C22×D4—1221no—
272C2×Q16—22no—
273C2×SD16—1222no—
274C2×SD16—1222no—
275C2×M16—611no—
276C2×M16—611no—
277C22×C8—21no—
278C22×C8—1221no—
279C4⋊Q8C22⋊D6111yescharacteristic
280C4⋊Q8C22⋊D6111yescharacteristic
281C4⋊Q8C22×D6111yescharacteristic
282C4⋊Q8C22×D6111yescharacteristic
283#127—612no—
284#127—1221no—
285#127—1221no—
286#127—1221no—
287#125—21no—
288#125—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8—21no—
292C22⋊Q8C22⋊D6112yes—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8—21no—
296C4×Q8C22×D6112yes—
297#117—21no—
298#117—21no—
299C2×C42C22×D6111yescharacteristic
300#107—22no—
301C4⋊C8—21no—
302C4⋊C8—611no—
303C4⋊C8—611no—
304C4⋊C8—611no—
305C4⋊C8—611no—
306C4⋊C8—1222no—
307C4⋊C8—1222no—
308C4⋊C8—1221no—
309C4⋊C8C22×D6112yes—
310#103—22no—
311#103—22no—
312#102—1222no—
313#102—1222no—
314#102—1222no—
315#102—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—1221no—
319#97—612no—
320#97—1222no—
321C4×C8—612no—
322C4×C8—1222no—
323C4×C8—1221no—
324C24—2441no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×C22⋊D6112yes—
329C2×D4—1222no—
330C2×D4—1222no—
331C2×D4—1222no—
332C2×D4—4881no—
333C2×D4—4881no—
334C2×D4—4882no—
335C2×D4—2442no—
336C2×D4—2441no—
337C2×D4—2441no—
338C2×D4—2441no—
339C2×D4—2442no—
340C22×C4D4×D6111yescharacteristic
341C22×C4#1175822411yescharacteristic
342C22×C4C2×C22⋊D6111yescharacteristic
343Q16—82no—
344SD16—4882no—
345SD16—4882no—
346M16—1222no—
347M16—1222no—
348C2×C8—22no—
349C2×C8—81no—
350C2×C8—612no—
351C2×C8—612no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—1221no—
355C2×C8—1221no—
356C2×C8—2442no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4C2×C22⋊D6112yes—
369C22⋊C4—22no—
370C22⋊C4—41no—
371C42—22no—
372C42—41no—
373C42D4×D6111yescharacteristic
374C42#1175822411yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—4882no—
378C23—4882no—
379C23—4882no—
380C23—2442no—
381C23#1451974911yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—41no—
385Q8—41no—
386Q8—81no—
387Q8—81no—
388D4—4882no—
389D4—4882no—
390D4—4882no—
391D4—4882no—
392D4—2442no—
393D4—2442no—
394D4—2442no—
395D4—2442no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4#1451974911yescharacteristic
404C2×C4#1451974911yescharacteristic
405C2×C4#1451974911yescharacteristic
406C2×C4#1451980012yes—
407C2×C4#1451980912yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C22×C22⋊D6111yescharacteristic
410C2×C4C2×#1175822411yescharacteristic
411C8—42no—
412C8—1222no—
413C8—1222no—
414C8—2441no—
415C8—2441no—
416C8—2441no—
417C8—2441no—
418C22—21no—
419C22—21no—
420C22—9764no—
421C22—4882no—
422C22—4882no—
423C22—4882no—
424C22#1683998612yes—
425C22C2×#1451974911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4#1683998612yes—
434C4C2×#1451980011yescharacteristic
435C4C2×#1451980911yescharacteristic
436C4#1684056511yescharacteristic
437C4#1684057011yescharacteristic
438C2—22no—
439C2—9762no—
440C2C2×#1683998611yescharacteristic
441C2#2145550111yescharacteristic
442C2#2145554911yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3249C2
socleC22×C61C2×#121
prime radicalC2×#14519774C22
hypercenter#3249D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3249

Complements

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

NH ≅ G/Ntype
3. #21454768C2semidirect
4. #21445389C2semidirect
5. #21445361C2semidirect
6. #21445265C2semidirect
7. #21445237C2semidirect
8. #21444569C2semidirect
9. #21444476C2semidirect
11. C61×#3249C2semidirect
22. #16839893C22semidirect
23. Q8×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839511C22semidirect
34. #16839511C22semidirect
35. #16839508C22semidirect
36. #16839508C22semidirect
42. C61×#477C22semidirect
45. C61×#462C22semidirect
217. #3249D61semidirect
242. #477C2×D61semidirect
245. #462C2×D61semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214567572
#214561952
#214547682
#214453892
#214453612
#214452652
#214452372
#214445692
#214444762
C2×#168395052
C61×#32492
#1924261