Finite Group Databasebeta

← #29580910

Subgroups — #29580910

Order 15616.

492 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 81 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21456222C211yesmaximal, characteristic
2#21455941C211yesmaximal, characteristic
3#21444446C211yesmaximal, characteristic, complemented
4#21444365C211yesmaximal, characteristic
5#21444266C211yesmaximal, characteristic, complemented
6#21443683C211yesmaximal, characteristic, complemented
7C61×#2634C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840132—21no—
10#16840128—21no—
11#16840119—22no—
12#16840075—21no—
13#16840057—21no—
14#16839962—21no—
15#16839901—21no—
16#16839860—22no—
17#16839819C2211yescharacteristic
18#16839819C2211yescharacteristic
19#16839761—21no—
20#16839761—21no—
21C4×#11758194C2211yescharacteristic
22#16839504—22no—
23#16839495—22no—
24Q8⋊#92418—22no—
25C2×#125×C61C2211yescharacteristic
26C2×C4×Q8×C61C2211yescharacteristic
27C61×#400—21no—
28C61×#400—21no—
29C61×#398—21no—
30C61×#395—21no—
31C61×#387—22no—
32C61×#383C2211yescharacteristic
33C2×#11758222—21no—
34C2×#11758201—22no—
35C2×#11758194—21no—
36#14519777D411yescharacteristic
37#14519774D411yescharacteristic
38#14519765—21no—
39#14519751—21no—
40#14519751—21no—
41C4×D244—21no—
42C4×D244—21no—
43#14519710—22no—
44C4⋊#92418—21no—
45C4⋊#92418—21no—
46#14519701—22no—
47#14519700—22no—
48#14519700—22no—
49#14519697—22no—
50#14519697—22no—
51#14519697—22no—
52C22⋊#92418—21no—
53C22⋊#92418—21no—
54#14519688—21no—
55#14519688—21no—
56#14519688D412yes—
57C4×#92418D412yes—
58C22×Q8×C61D411yescharacteristic
59#125×C61—21no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63C4×Q8×C61—21no—
64C4×Q8×C61—21no—
65C4×Q8×C61—21no—
66C4×Q8×C61—21no—
67C2×C4⋊C4×C61—22no—
68C2×C4⋊C4×C61—21no—
69C2×C4⋊C4×C61—21no—
70C2×C4⋊C4×C61—21no—
71C2×C4⋊C4×C61D411yescharacteristic
72C2×C4⋊C4×C61D411yescharacteristic
73C2×C4⋊C4×C61D411yescharacteristic
74C2×C42×C61—21no—
75C2×C42×C61C2311yescharacteristic
76#95×C61—22no—
77#95×C61—22no—
78#95×C61—22no—
79#95×C61—21no—
80#95×C61—21no—
81#95×C61D412yes—
82C2×Q8×C61—21no—
83C2×Q8×C61—21no—
84C2×Q8×C61—42no—
85C22×C4×C61—21no—
86C22×C4×C61—22no—
87C22×C4×C61—21no—
88C22×C4×C61—21no—
89C22×C4×C61—21no—
90C22×C4×C61C2×D411yescharacteristic
91C22×C4×C61C2×D411yescharacteristic
92C22×C4×C61C2×D411yescharacteristic
93C2×C22⋊D61—41no—
94#11758222—44no—
95C2×D244—21no—
96C2×D244—21no—
97C2×D244—21no—
98C2×C4×D61—41no—
99C2×Dic122—21no—
100C4⋊C4×C61—21no—
101C4⋊C4×C61—21no—
102C4⋊C4×C61—21no—
103C4⋊C4×C61—21no—
104C4⋊C4×C61—21no—
105C4⋊C4×C61—21no—
106C4⋊C4×C61—44no—
107C4⋊C4×C61—41no—
108C4⋊C4×C61—41no—
109C4⋊C4×C61—41no—
110C4⋊C4×C61—41no—
111C4⋊C4×C61—41no—
112C4⋊C4×C61—41no—
113C42×C61—41no—
114C42×C61—41no—
115C42×C61C2×D411yescharacteristic
116C42×C61C2×D411yescharacteristic
117C42×C61C2×D411yescharacteristic
118C42×C61#4111yescharacteristic
119#11758201—44no—
120#11758198—41no—
121#11758198—41no—
122C4⋊Dic61—21no—
123C4⋊Dic61—21no—
124#11758194—42no—
125C2×#92418—22no—
126C2×#92418—22no—
127C2×#92418—41no—
128C23×C61#12011yescharacteristic
129C22×D61—41no—
130Q8×C61—22no—
131Q8×C61—22no—
132Q8×C61—41no—
133Q8×C61—41no—
134C2×C4×C61—22no—
135C2×C4×C61—22no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—42no—
141C2×C4×C61—42no—
142C2×C4×C61—42no—
143C2×C4×C61—42no—
144C2×C4×C61—41no—
145C2×C4×C61—41no—
146C2×C4×C61—41no—
147C2×C4×C61—42no—
148C2×C4×C61—42no—
149C2×C4×C61#12011yescharacteristic
150C2×C4×C61#12011yescharacteristic
151C2×C4×C61#12111yescharacteristic
152C2×C4×C61#12111yescharacteristic
153C2×C4×C61#12111yescharacteristic
154C2×C4×C61#12711yescharacteristic
155C22⋊D61—82no—
156C2×Dic61—41no—
157D244—42no—
158D244—42no—
159D244—81no—
160C4×D61—82no—
161Dic122—42no—
162#92418—42no—
163#92418—42no—
164#92418—81no—
165#10021—611nomaximal, Sylow
166C22×C61—21no—
167C22×C61—21no—
168C22×C61#39211yescharacteristic
169C22×C61#45412yes—
170C2×D61—81no—
171C2×D61—81no—
172C2×D61—81no—
173C4×C61—21no—
174C4×C61—21no—
175C4×C61—21no—
176C4×C61—21no—
177C4×C61—21no—
178C4×C61—21no—
179C4×C61—42no—
180C4×C61—82no—
181C4×C61—81no—
182C4×C61—81no—
183C4×C61—81no—
184Dic61—81no—
185#3142—611no—
186#3112—611no—
187#2634D6111yescharacteristic, complemented
188#1534—611no—
189#1514—611no—
190#1491—611no—
191#1288—611no—
192C2×C61—22no—
193C2×C61#200311yescharacteristic
194C2×C61#200911yescharacteristic
195C2×C61#201011yescharacteristic
196D61—161no—
197#532—611no—
198C2×#125C2×D6111yescharacteristic
199C2×C4×Q8C2×D6111yescharacteristic
200#491—1221no—
201#490—1221no—
202#488—1222no—
203#480—1221no—
204#477—1221no—
205#460—1221no—
206#448—1221no—
207#440—1222no—
208#432—611no—
209#432—611no—
210#417—1221no—
211#417—1221no—
212C4×M16—611no—
213#400—21no—
214#400—21no—
215#398—21no—
216#395—21no—
217#387—22no—
218#383C2×D6111yescharacteristic
219#330—1222no—
220#327—1222no—
221Q8⋊C8—1222no—
222C61#1002111yescharacteristic, complemented, Sylow
223C2×#41—1221no—
224C22×Q8C22⋊D6111yescharacteristic
225C2×SD16—1222no—
226C2×M16—1221no—
227C4⋊Q8—611no—
228#127—611no—
229#125—21no—
230#125—21no—
231#125—21no—
232#125—21no—
233#124—1221no—
234#121—1221no—
235#121—1221no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×D4—1221no—
241C4×D4—1221no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4C22⋊D6111yescharacteristic
247C2×C4⋊C4C22⋊D6111yescharacteristic
248C2×C4⋊C4C22⋊D6111yescharacteristic
249C2×C42—21no—
250C2×C42C22×D6111yescharacteristic
251#107—1222no—
252C4⋊C8—1221no—
253C4⋊C8—1221no—
254#103—1222no—
255#103—1222no—
256#103—1222no—
257#102—1222no—
258#102—1222no—
259#102—1222no—
260C22⋊C8—1221no—
261C22⋊C8—1221no—
262#97—612no—
263#97—1221no—
264#97—1221no—
265C4×C8—612no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—21no—
271#95C22⋊D6112yes—
272#41—2444no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—42no—
276C2×Q8—1221no—
277C2×D4—1221no—
278C2×D4—1221no—
279C2×D4—1221no—
280C2×D4—2441no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—2441no—
287C22×C4C2×C22⋊D6111yescharacteristic
288C22×C4C2×C22⋊D6111yescharacteristic
289C22×C4C2×C22⋊D6111yescharacteristic
290SD16—2444no—
291M16—2442no—
292C2×C8—1222no—
293C2×C8—1222no—
294C2×C8—2441no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—44no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—41no—
308C4⋊C4—1221no—
309C4⋊C4—1221no—
310C22⋊C4—2441no—
311C22⋊C4—2441no—
312C42—41no—
313C42—41no—
314C42D4×D6111yescharacteristic
315C42D4×D6111yescharacteristic
316C42D4×D6111yescharacteristic
317C42#1175822411yescharacteristic
318C23—2441no—
319C23#1451974611yescharacteristic
320Q8—22no—
321Q8—22no—
322Q8—41no—
323Q8—41no—
324Q8—2442no—
325D4—4881no—
326D4—4882no—
327D4—2442no—
328D4—2442no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—4882no—
345C2×C4—2441no—
346C2×C4#1451974511yescharacteristic
347C2×C4#1451974511yescharacteristic
348C2×C4#1451974811yescharacteristic
349C2×C4#1451974911yescharacteristic
350C2×C4#1451974911yescharacteristic
351C2×C4#1451977311yescharacteristic
352C8—4881no—
353C8—2442no—
354C8—2442no—
355C22—21no—
356C22—21no—
357C22—4881no—
358C22—4881no—
359C22—4881no—
360C22#1683967011yescharacteristic
361C22#1683993512yes—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—82no—
370C4—81no—
371C4—81no—
372C4—81no—
373C4—4881no—
374C2—22no—
375C2—9761no—
376C2#2144683911yescharacteristic
377C2#2144687911yescharacteristic
378C2#2144688411yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2634C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#2634D61
hypocenterC61#10021
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2634

Complements

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

NH ≅ G/Ntype
3. #21444446C2semidirect
5. #21444266C2semidirect
6. #21443683C2semidirect
7. C61×#2634C2semidirect
187. #2634D61semidirect
222. C61#10021semidirect

Maximal subgroups

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

subgroupindex
#214562222
#214559412
#214444462
#214443652
#214442662
#214436832
C61×#26342
#1002161