Finite Group Databasebeta

← #29835345

Subgroups — #29835345

Order 15616.

602 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456358C211yesmaximal, characteristic, complemented
2#21448060C212yesmaximal, complemented
3#21448055C212yesmaximal, complemented
4#21448050C212yesmaximal, complemented
5#21448042C212yesmaximal, complemented
6#21447535C211yesmaximal, characteristic, complemented
7#21447531C211yesmaximal, characteristic, complemented
8#21447526C211yesmaximal, characteristic, complemented
9C2×#16839563C211yesmaximal, characteristic, complemented
10#21447416C211yesmaximal, characteristic, complemented
11C61×#3155C211yesmaximal, characteristic
12#16840390—21no—
13C2×#14519717C2211yescharacteristic
14C22×#11758185C2211yescharacteristic
15#16840171—22no—
16#16840167C2212yescomplemented
17#16840164C2212yescomplemented
18#16839804—21no—
19#16839800C2211yescharacteristic
20C2×#14519711C2211yescharacteristic
21#16839566C2211yescharacteristic, complemented
22#16839566C2211yescharacteristic, complemented
23#16839566C2212yescomplemented
24#16839563C2212yes—
25#16839563C2212yes—
26#16839563C2212yes—
27#16839563C2212yes—
28#16839563C2211yescharacteristic, complemented
29#16839563C2211yescharacteristic, complemented
30#16839563C2212yescomplemented
31#16839538C2212yescomplemented
32C4×#11758185C2212yescomplemented
33C2×AGL(1,ℤ/8)×C61—21no—
34C2×#135×C61C2211yescharacteristic
35C22×D8×C61C2211yescharacteristic
36C61×#535—21no—
37C61×#496—22no—
38C61×#495C2212yes—
39C61×#493C2212yes—
40C61×#405C2211yescharacteristic
41#14519777—22no—
42#14519776—41no—
43#14519769—22no—
44#14519768—41no—
45#14519754—42no—
46#14519736—21no—
47C2×C4⋊Dic61—21no—
48#14519717—22no—
49C2×#11758185—22no—
50C2×#11758185C2311yescharacteristic
51C2×#11758185C2312yes—
52C2×#11758185C2311yescharacteristic
53#14519711—21no—
54#14519711—22no—
55#14519711C2312yes—
56#14519711C2311yescharacteristic
57#14519711C2311yescharacteristic
58#14519709—21no—
59#14519709—22no—
60C2×#41×C61—21no—
61C22×D4×C61—21no—
62AGL(1,ℤ/8)×C61—44no—
63#135×C61—22no—
64#135×C61—22no—
65C2×Q16×C61C2311yescharacteristic
66C2×SD16×C61—21no—
67C2×SD16×C61—22no—
68C2×D8×C61—21no—
69C2×D8×C61—22no—
70C2×D8×C61—22no—
71C2×D8×C61—22no—
72C2×D8×C61C2312yes—
73C2×D8×C61C2311yescharacteristic
74C2×M16×C61D411yescharacteristic
75C22×C8×C61C2311yescharacteristic
76#127×C61—22no—
77#124×C61—22no—
78#121×C61—41no—
79#121×C61—41no—
80#120×C61—42no—
81#117×C61D411yescharacteristic
82#97×C61D412yes—
83C4×C8×C61C2312yes—
84C24×C61—41no—
85#41×C61—42no—
86#41×C61—42no—
87C2×Q8×C61—21no—
88C2×D4×C61—22no—
89C2×D4×C61—21no—
90C2×D4×C61—41no—
91C2×D4×C61—42no—
92C2×D4×C61—41no—
93C2×D4×C61—42no—
94C2×D4×C61—42no—
95C2×D4×C61—42no—
96C22×C4×C61—41no—
97C22×C4×C61C2×D411yescharacteristic
98C22×Dic61—41no—
99C2×Dic122—42no—
100Q16×C61—21no—
101Q16×C61—21no—
102SD16×C61—42no—
103SD16×C61—44no—
104D8×C61—22no—
105D8×C61—22no—
106D8×C61—21no—
107D8×C61—21no—
108D8×C61—42no—
109D8×C61—44no—
110M16×C61SD1614yes—
111C2×C8×C61C2×D411yescharacteristic
112C2×C8×C61C2×D412yes—
113C2×C8×C61#4112yes—
114C2×C8×C61#4112yes—
115C2×C8×C61C2411yescharacteristic
116C4⋊C4×C61D811yescharacteristic
117C4⋊C4×C61D811yescharacteristic
118C22⋊C4×C61—42no—
119C22⋊C4×C61—42no—
120C22⋊C4×C61D812yes—
121C42×C61C2×D412yes—
122C22⋊Dic61—41no—
123C4⋊Dic61—21no—
124C4⋊Dic61—22no—
125C4⋊Dic61—21no—
126C4⋊Dic61—41no—
127C4⋊Dic61—42no—
128#11758196—42no—
129#11758196—42no—
130C4×Dic61—41no—
131#11758185—21no—
132#11758185—21no—
133#11758185—22no—
134C23×C61—42no—
135C23×C61—41no—
136C23×C61—82no—
137C23×C61—161no—
138C23×C61—161no—
139C23×C61C2×D811yescharacteristic
140Q8×C61—41no—
141Q8×C61—41no—
142D4×C61—42no—
143D4×C61—41no—
144D4×C61—41no—
145D4×C61—42no—
146D4×C61—82no—
147D4×C61—82no—
148D4×C61—162no—
149D4×C61—162no—
150C2×C4×C61—41no—
151C2×C4×C61—82no—
152C2×C4×C61#12412yes—
153C2×C4×C61C2×D812yes—
154C2×C4×C61C2×D812yes—
155C2×C4×C61C2×D811yescharacteristic
156C2×C4×C61C2×SD1612yes—
157C2×C4×C61C22×D411yescharacteristic
158C2×Dic61—41no—
159C2×Dic61—41no—
160C2×Dic61—42no—
161C2×Dic61—161no—
162Dic122—162no—
163C8×C61#12412yes—
164C8×C61C2×SD1614yes—
165C8×C61C2×#4111yescharacteristic
166C8×C61C2×#4111yescharacteristic
167#33702—611nomaximal, Sylow
168C22×C61—21no—
169C22×C61—21no—
170C22×C61—81no—
171C22×C61—82no—
172C22×C61—81no—
173C22×C61—82no—
174C22×C61—81no—
175C22×C61—82no—
176C22×C61—164no—
177C22×C61#48612yes—
178C22×C61C22×D811yescharacteristic
179C4×C61—22no—
180C4×C61—22no—
181C4×C61—81no—
182C4×C61#48612yes—
183C4×C61C2×#12411yescharacteristic
184C4×C61C22×SD1611yescharacteristic
185Dic61—81no—
186Dic61—82no—
187Dic61—81no—
188#3164—611no—
189#3155D6111yescharacteristic
190#2244—612no—
191#2243—612no—
192#2242—612no—
193#2240—612no—
194#2140—611no—
195#2139—611no—
196#2138—611no—
197C2×#357—611no—
198#2108—611no—
199C2×C61—22no—
200C2×C61—161no—
201C2×C61—162no—
202C2×C61C2×#48611yescharacteristic
203C2×C61#341511yescharacteristic
204C2×C61#341611yescharacteristic
205C2×AGL(1,ℤ/8)—21no—
206C2×#135C2×D6111yescharacteristic
207C22×D8C2×D6111yescharacteristic
208#544—1221no—
209#535—21no—
210C2×M32—611no—
211C22×C16—611no—
212#501—1222no—
213#500—612no—
214#499—612no—
215#496—22no—
216#495C2×D6112yes—
217#493C2×D6112yes—
218#428—1221no—
219#427—611no—
220C2×#107—611no—
221#405C2×D6111yescharacteristic
222#358—611no—
223#358—611no—
224#358—612no—
225#357—612no—
226#357—612no—
227#357—611no—
228#357—611no—
229#357—612no—
230#357—612no—
231#357—612no—
232#346—612no—
233C4×C16—612no—
234C61#3370211yescharacteristic, complemented, Sylow
235C2×#41—21no—
236C22×D4—21no—
237AGL(1,ℤ/8)—44no—
238#135—22no—
239#135—22no—
240C2×Q16C22×D6111yescharacteristic
241C2×SD16—21no—
242C2×SD16—22no—
243C2×D8—22no—
244C2×D8—22no—
245C2×D8—21no—
246C2×D8—22no—
247C2×D8C22×D6112yes—
248C2×D8C22×D6111yescharacteristic
249C2×M16C22⋊D6111yescharacteristic
250C22×C8C22×D6111yescharacteristic
251C4⋊Q8—1222no—
252C4⋊Q8—2441no—
253#127—22no—
254#125—1222no—
255#125—2441no—
256#124—22no—
257C22⋊Q8—2442no—
258#121—41no—
259#121—41no—
260#120—42no—
261#117—1221no—
262#117C22⋊D6111yescharacteristic
263C2×C4⋊C4—1221no—
264M32—1222no—
265C2×C16—611no—
266C2×C16—612no—
267C2×C16—611no—
268C2×C16—1222no—
269#107—612no—
270#107—611no—
271#107—611no—
272#107—1221no—
273#107—1222no—
274#106—1221no—
275#106—1222no—
276#97C22⋊D6112yes—
277C4×C8C22×D6112yes—
278C24—41no—
279#41—42no—
280#41—42no—
281C2×Q8—21no—
282C2×Q8—2442no—
283C2×D4—22no—
284C2×D4—21no—
285C2×D4—41no—
286C2×D4—42no—
287C2×D4—41no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—42no—
291C22×C4—41no—
292C22×C4—2441no—
293C22×C4C2×C22⋊D6111yescharacteristic
294Q16—21no—
295Q16—21no—
296SD16—42no—
297SD16—44no—
298D8—22no—
299D8—22no—
300D8—21no—
301D8—21no—
302D8—42no—
303D8—44no—
304M16#1175820014yes—
305C2×C8#1175822412yes—
306C2×C8#1175822412yes—
307C2×C8C2×C22⋊D6111yescharacteristic
308C2×C8C2×C22⋊D6112yes—
309C2×C8C23×D6111yescharacteristic
310C4⋊C4—1221no—
311C4⋊C4—1222no—
312C4⋊C4—1221no—
313C4⋊C4—2441no—
314C4⋊C4—2442no—
315C4⋊C4—2442no—
316C4⋊C4—2442no—
317C4⋊C4#1175819911yescharacteristic
318C4⋊C4#1175819911yescharacteristic
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—2441no—
322C22⋊C4#1175819912yes—
323C42—2441no—
324C42C2×C22⋊D6112yes—
325C16—1221no—
326C16—1221no—
327C16—1222no—
328C23—42no—
329C23—41no—
330C23—82no—
331C23—161no—
332C23—161no—
333C23C2×#1175819911yescharacteristic
334Q8—41no—
335Q8—41no—
336Q8—9762no—
337D4—42no—
338D4—41no—
339D4—41no—
340D4—42no—
341D4—82no—
342D4—82no—
343D4—162no—
344D4—162no—
345C2×C4—41no—
346C2×C4—82no—
347C2×C4—9761no—
348C2×C4—2441no—
349C2×C4—2441no—
350C2×C4—2442no—
351C2×C4#1451976412yes—
352C2×C4C2×#1175819912yes—
353C2×C4C2×#1175819912yes—
354C2×C4C2×#1175819911yescharacteristic
355C2×C4C2×#1175820012yes—
356C2×C4C22×C22⋊D6111yescharacteristic
357C8#1451976412yes—
358C8C2×#1175820014yes—
359C8C2×#1175822411yescharacteristic
360C8C2×#1175822411yescharacteristic
361C22—21no—
362C22—21no—
363C22—81no—
364C22—82no—
365C22—81no—
366C22—82no—
367C22—81no—
368C22—82no—
369C22—164no—
370C22#1684010812yes—
371C22C22×#1175819911yescharacteristic
372C4—22no—
373C4—22no—
374C4—81no—
375C4—4882no—
376C4—4881no—
377C4—4881no—
378C4#1684010812yes—
379C4C2×#1451976411yescharacteristic
380C4C22×#1175820011yescharacteristic
381C2—22no—
382C2—162no—
383C2—161no—
384C2C2×#1684010811yescharacteristic
385C2#2145895811yescharacteristic
386C2#2145896611yescharacteristic
subgroupG/H
centerC22C22×#11758199
commutator, cosocleC2×C8×C61C24
FrattiniC2×C8C23×D61
Fitting, prime radicalC61×#3155C2
socleC22×C61C22×D8
hypercenter#3155D61
hypocenterC61#33702
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3155

Complements

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

NH ≅ G/Ntype
1. #21456358C2semidirect
2. #21448060C2semidirect
3. #21448055C2semidirect
4. #21448050C2semidirect
5. #21448042C2semidirect
6. #21447535C2semidirect
7. #21447531C2semidirect
8. #21447526C2semidirect
9. C2×#16839563C2semidirect
10. #21447416C2semidirect
16. #16840167C22semidirect
17. #16840164C22semidirect
21. #16839566C22semidirect
22. #16839566C22semidirect
23. #16839566C22semidirect
28. #16839563C22semidirect
29. #16839563C22semidirect
30. #16839563C22semidirect
31. #16839538C22semidirect
32. C4×#11758185C22semidirect
234. C61#33702semidirect

Maximal subgroups

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

subgroupindex
#214563582
#214480602
#214480552
#214480502
#214480422
#214475352
#214475312
#214475262
C2×#168395632
#214474162
C61×#31552
#3370261