Finite Group Databasebeta

← #29663625

Subgroups — #29663625

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#21457161C211yesmaximal, characteristic, complemented
2#21456193C211yesmaximal, characteristic
3#21454768C211yesmaximal, characteristic, complemented
4#21445385C212yesmaximal, complemented
5#21445357C212yesmaximal, complemented
6#21445261C212yesmaximal, complemented
7#21445233C212yesmaximal, complemented
8#21444572C211yesmaximal, characteristic, complemented
9#21444474C211yesmaximal, characteristic, complemented
10C2×#16839506C211yesmaximal, characteristic
11C61×#3208C211yesmaximal, characteristic
12#16840319C2211yescharacteristic
13#16840126—22no—
14#16840108C2212yes—
15#16840108C2212yes—
16#16840057C2212yescomplemented
17#16840054—21no—
18#16840001—22no—
19#16839972C2212yescomplemented
20#16839969—21no—
21#16839893—22no—
22#16839893C2212yescomplemented
23Q8×#92418C2212yescomplemented
24#16839789—21no—
25C2×C4⋊#92418C2211yescharacteristic
26C2×C4⋊#92418C2211yescharacteristic
27#16839767—21no—
28#16839767—21no—
29#16839718—21no—
30C4×#11758194C2211yescharacteristic
31#16839514C2212yescomplemented
32#16839512C2212yescomplemented
33#16839510C2211yescharacteristic, complemented
34#16839510C2211yescharacteristic, complemented
35#16839509C2211yescharacteristic, complemented
36#16839509C2211yescharacteristic, complemented
37#16839506C2212yes—
38#16839506C2212yes—
39C61×#533C2211yescharacteristic
40C2×#127×C61C2211yescharacteristic
41C61×#492—21no—
42C61×#486—21no—
43C61×#460C2212yes—
44C61×#460C2212yes—
45C61×#450—22no—
46C61×#431C2211yescharacteristic
47C2×#11758202—22no—
48C2×#11758194D411yescharacteristic
49C2×#11758194D411yescharacteristic
50C22×#92418—21no—
51C22×#92418—21no—
52#14519777C2311yescharacteristic
53#14519777C2311yescharacteristic
54#14519769—21no—
55#14519756—21no—
56#14519756—21no—
57C4×Dic122—21no—
58C4×Dic122—21no—
59#14519736—21no—
60#14519710—22no—
61C4⋊#92418—21no—
62C4⋊#92418—21no—
63C4⋊#92418—22no—
64C4⋊#92418—22no—
65C4⋊#92418C2312yes—
66C4⋊#92418C2311yescharacteristic
67C4⋊#92418C2311yescharacteristic
68#14519701—22no—
69#14519700—22no—
70#14519698—22no—
71#14519698—22no—
72#14519698—22no—
73#14519698—22no—
74#14519688—22no—
75#14519688D412yes—
76C4×#92418—22no—
77C4×#92418C2312yes—
78C22×D4×C61—21no—
79C22×D4×C61—21no—
80C22×D4×C61—21no—
81C2×SD16×C61—22no—
82C2×SD16×C61—22no—
83C4⋊Q8×C61D411yescharacteristic
84C4⋊Q8×C61D411yescharacteristic
85#127×C61—21no—
86#127×C61—21no—
87#127×C61—21no—
88#127×C61C2312yes—
89#125×C61—21no—
90C22⋊Q8×C61—21no—
91C22⋊Q8×C61D412yes—
92C4×Q8×C61—21no—
93C4×Q8×C61C2312yes—
94#117×C61—21no—
95C2×C42×C61C2311yescharacteristic
96C4⋊C8×C61C2311yescharacteristic
97C4⋊C8×C61C2311yescharacteristic
98#102×C61—22no—
99#102×C61—22no—
100C22⋊C8×C61—21no—
101C4×C8×C61—21no—
102C24×C61—41no—
103C2×Q8×C61—21no—
104C2×Q8×C61C2×D412yes—
105C2×D4×C61—22no—
106C2×D4×C61—22no—
107C2×D4×C61—22no—
108C2×D4×C61—41no—
109C2×D4×C61—41no—
110C2×D4×C61—41no—
111C2×D4×C61—42no—
112C2×D4×C61—42no—
113C2×D4×C61—81no—
114C2×D4×C61—81no—
115C2×D4×C61—82no—
116C22×C4×C61C2×D411yescharacteristic
117C22×C4×C61C2×D411yescharacteristic
118C22×C4×C61#4111yescharacteristic
119C2×Dic122—21no—
120C2×Dic122—21no—
121SD16×C61—82no—
122SD16×C61—82no—
123C2×C8×C61—21no—
124C2×C8×C61—21no—
125C4⋊C4×C61—21no—
126C4⋊C4×C61—21no—
127C4⋊C4×C61—21no—
128C4⋊C4×C61—22no—
129C4⋊C4×C61—22no—
130C4⋊C4×C61C2×D412yes—
131C22⋊C4×C61—22no—
132C42×C61—22no—
133C42×C61C2×D411yescharacteristic
134C42×C61C2×D411yescharacteristic
135C42×C61#4111yescharacteristic
136C42×C61C2411yescharacteristic
137C22⋊Dic61—41no—
138#11758202—82no—
139C4⋊Dic61—21no—
140C4⋊Dic61—21no—
141C4⋊Dic61—21no—
142C4⋊Dic61—21no—
143#11758196—41no—
144#11758196—41no—
145C4×Dic61—41no—
146#11758194—22no—
147#11758194—22no—
148C2×#92418—21no—
149C2×#92418—21no—
150C2×#92418—22no—
151C2×#92418—42no—
152C2×#92418—81no—
153C2×#92418C2×D412yes—
154C2×#92418C2×D412yes—
155C23×C61—42no—
156C23×C61—82no—
157C23×C61—82no—
158C23×C61—82no—
159C23×C61#12111yescharacteristic
160Q8×C61—22no—
161Q8×C61—22no—
162Q8×C61—81no—
163D4×C61—42no—
164D4×C61—42no—
165D4×C61—42no—
166D4×C61—42no—
167D4×C61—82no—
168D4×C61—82no—
169D4×C61—82no—
170D4×C61—82no—
171C2×C4×C61—21no—
172C2×C4×C61—21no—
173C2×C4×C61—22no—
174C2×C4×C61—22no—
175C2×C4×C61—41no—
176C2×C4×C61#12111yescharacteristic
177C2×C4×C61#12111yescharacteristic
178C2×C4×C61#12111yescharacteristic
179C2×C4×C61#13512yes—
180C2×C4×C61#13712yes—
181C2×C4×C61C22×D411yescharacteristic
182C2×C4×C61C22×D411yescharacteristic
183C2×C4×C61C2×#4111yescharacteristic
184C2×Dic61—41no—
185C2×Dic61—41no—
186Dic122—41no—
187Dic122—41no—
188Dic122—81no—
189C8×C61—41no—
190C8×C61—41no—
191#92418—22no—
192#92418—22no—
193#92418—41no—
194#92418—41no—
195#92418—42no—
196#19242—611nomaximal, Sylow
197C22×C61—21no—
198C22×C61—21no—
199C22×C61—82no—
200C22×C61—82no—
201C22×C61—82no—
202C22×C61—164no—
203C22×C61#46412yes—
204C22×C61C2×#12111yescharacteristic
205C4×C61—21no—
206C4×C61—21no—
207C4×C61—21no—
208C4×C61—42no—
209C4×C61—42no—
210C4×C61#46412yes—
211C4×C61C2×#13511yescharacteristic
212C4×C61C2×#13711yescharacteristic
213C4×C61#57611yescharacteristic
214C4×C61#57711yescharacteristic
215Dic61—81no—
216Dic61—81no—
217#3249—611no—
218#3208D6111yescharacteristic
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×C61—162no—
230C2×C61C2×#46411yescharacteristic
231C2×C61#306211yescharacteristic
232C2×C61#306811yescharacteristic
233#533—611no—
234#533C2×D6111yescharacteristic
235C2×#127C2×D6111yescharacteristic
236#492—21no—
237#490—1222no—
238#486—21no—
239#486—612no—
240#486—612no—
241#477—612no—
242#477—1221no—
243#467—1222no—
244#462—612no—
245#462—1221no—
246#460C2×D6112yes—
247#460C2×D6112yes—
248#450—22no—
249#446—612no—
250#446—1222no—
251C8×Q8—612no—
252#431C2×D6111yescharacteristic
253#424—1221no—
254C2×C4⋊C8—611no—
255C2×C4⋊C8—611no—
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—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—1222no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×M16—611no—
276C2×M16—611no—
277C22×C8—1221no—
278C22×C8—1221no—
279C4⋊Q8—611no—
280C4⋊Q8—611no—
281C4⋊Q8C22⋊D6111yescharacteristic
282C4⋊Q8C22⋊D6111yescharacteristic
283#127—21no—
284#127—21no—
285#127—21no—
286#127C22×D6112yes—
287#125—21no—
288#125—1221no—
289C22⋊Q8—21no—
290C22⋊Q8—1221no—
291C22⋊Q8—1221no—
292C22⋊Q8C22⋊D6112yes—
293C4×Q8—21no—
294C4×Q8—1221no—
295C4×Q8—1221no—
296C4×Q8C22×D6112yes—
297#117—21no—
298#117—1221no—
299C2×C42C22×D6111yescharacteristic
300#107—1222no—
301C4⋊C8—611no—
302C4⋊C8—611no—
303C4⋊C8—612no—
304C4⋊C8—1222no—
305C4⋊C8—1222no—
306C4⋊C8—1221no—
307C4⋊C8—1221no—
308C4⋊C8C22×D6111yescharacteristic
309C4⋊C8C22×D6111yescharacteristic
310#103—1222no—
311#103—1222no—
312#102—22no—
313#102—22no—
314#102—1222no—
315#102—1222no—
316#102—1222no—
317#102—1222no—
318C22⋊C8—21no—
319#97—612no—
320#97—1222no—
321C4×C8—21no—
322C4×C8—612no—
323C4×C8—1222no—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—1221no—
327C2×Q8—1221no—
328C2×Q8C2×C22⋊D6112yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—42no—
336C2×D4—42no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4D4×D6111yescharacteristic
341C22×C4#1175822411yescharacteristic
342C22×C4C2×C22⋊D6111yescharacteristic
343Q16—4882no—
344SD16—82no—
345SD16—82no—
346M16—1222no—
347M16—1222no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—612no—
351C2×C8—612no—
352C2×C8—1221no—
353C2×C8—1221no—
354C2×C8—1222no—
355C2×C8—4881no—
356C2×C8—2442no—
357C4⋊C4—22no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—22no—
362C4⋊C4—1221no—
363C4⋊C4—1221no—
364C4⋊C4—1221no—
365C4⋊C4—1221no—
366C4⋊C4—2441no—
367C4⋊C4—2441no—
368C4⋊C4C2×C22⋊D6112yes—
369C22⋊C4—22no—
370C22⋊C4—2441no—
371C42—22no—
372C42—2441no—
373C42D4×D6111yescharacteristic
374C42#1175822411yescharacteristic
375C42C2×C22⋊D6111yescharacteristic
376C42C23×D6111yescharacteristic
377C23—42no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23#1451974911yescharacteristic
382Q8—22no—
383Q8—22no—
384Q8—81no—
385Q8—4881no—
386Q8—2441no—
387Q8—2441no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—41no—
401C2×C4—2441no—
402C2×C4—2441no—
403C2×C4#1451974911yescharacteristic
404C2×C4#1451974911yescharacteristic
405C2×C4#1451974911yescharacteristic
406C2×C4#1451979812yes—
407C2×C4#1451980612yes—
408C2×C4C2×D4×D6111yescharacteristic
409C2×C4C22×C22⋊D6111yescharacteristic
410C2×C4C2×#1175822411yescharacteristic
411C8—41no—
412C8—41no—
413C8—1222no—
414C8—1222no—
415C8—2441no—
416C8—2441no—
417C8—2442no—
418C22—21no—
419C22—21no—
420C22—82no—
421C22—82no—
422C22—82no—
423C22—164no—
424C22#1683998212yes—
425C22C2×#1451974911yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—4881no—
432C4—4881no—
433C4#1683998212yes—
434C4C2×#1451979811yescharacteristic
435C4C2×#1451980611yescharacteristic
436C4#1684056511yescharacteristic
437C4#1684057311yescharacteristic
438C2—22no—
439C2—162no—
440C2C2×#1683998211yescharacteristic
441C2#2145549511yescharacteristic
442C2#2145554411yescharacteristic
subgroupG/H
centerC22C2×#14519749
commutator, cosocleC42×C61C24
FrattiniC42C23×D61
FittingC61×#3208C2
socleC22×C61C2×#121
prime radicalC2×#127×C61C22
hypercenter#3208D61
hypocenterC61#19242
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3208

Complements

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

NH ≅ G/Ntype
1. #21457161C2semidirect
3. #21454768C2semidirect
4. #21445385C2semidirect
5. #21445357C2semidirect
6. #21445261C2semidirect
7. #21445233C2semidirect
8. #21444572C2semidirect
9. #21444474C2semidirect
16. #16840057C22semidirect
19. #16839972C22semidirect
22. #16839893C22semidirect
23. Q8×#92418C22semidirect
31. #16839514C22semidirect
32. #16839512C22semidirect
33. #16839510C22semidirect
34. #16839510C22semidirect
35. #16839509C22semidirect
36. #16839509C22semidirect
268. C61#19242semidirect

Maximal subgroups

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

subgroupindex
#214571612
#214561932
#214547682
#214453852
#214453572
#214452612
#214452332
#214445722
#214444742
C2×#168395062
C61×#32082
#1924261