Finite Group Databasebeta

← #29581392

Subgroups — #29581392

Order 15616.

496 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 83 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#21457009C211yesmaximal, characteristic
2#21456213C211yesmaximal, characteristic
3#21451940C211yesmaximal, characteristic, complemented
4#21444537C211yesmaximal, characteristic, complemented
5#21444328C211yesmaximal, characteristic
6#21443713C211yesmaximal, characteristic, complemented
7C61×#1528C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9C2×#14519777C2211yescharacteristic
10#16840172—21no—
11#16840128—21no—
12#16840122—22no—
13#16840113—21no—
14#16840109—21no—
15#16840080—21no—
16#16840075—21no—
17#16840059—22no—
18#16839819C2211yescharacteristic
19#16839764—22no—
20#16839692—22no—
21#16839682—22no—
22#16839631—21no—
23#16839627—21no—
24#16839608C2211yescharacteristic
25#16839509—22no—
26#16839497—22no—
27C2×#125×C61C2211yescharacteristic
28C61×#432C2211yescharacteristic
29C4×M16×C61C2211yescharacteristic
30C61×#330—22no—
31C2×#11758222—21no—
32C22×Dic122—21no—
33#14519777—21no—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37#14519777D411yescharacteristic
38#14519774D411yescharacteristic
39#14519765—21no—
40#14519751—21no—
41#14519751—21no—
42C4×D244—21no—
43C4×D244—21no—
44C2×C4⋊Dic61—21no—
45C2×C4⋊Dic61D412yes—
46C2×#11758196—24no—
47C2×C4×Dic61—21no—
48#14519710—22no—
49#14519708—22no—
50C4⋊#92418D412yes—
51#14519702—24no—
52#14519699—24no—
53#14519697—22no—
54#14519697—22no—
55C22⋊#92418—21no—
56#14519688—21no—
57#14519684—24no—
58#14519684—21no—
59#14519684D412yes—
60C2×M16×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63#125×C61—21no—
64#125×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61—21no—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C42×C61C2311yescharacteristic
70C4⋊C8×C61—21no—
71C22⋊C8×C61—21no—
72#97×C61—21no—
73#97×C61D412yes—
74C4×C8×C61—21no—
75C22×C4×C61—21no—
76C22×C4×C61—21no—
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×C4×C61C2×D411yescharacteristic
80C2×C22⋊D61—41no—
81C22×Dic61—22no—
82C22×Dic61—22no—
83#11758222—44no—
84C2×D244—21no—
85C2×D244—21no—
86C2×D244—21no—
87C2×C4×D61—41no—
88C2×Dic122—21no—
89C2×Dic122—41no—
90C2×Dic122—41no—
91C2×Dic122—82no—
92M16×C61—42no—
93C2×C8×C61—22no—
94C2×C8×C61—41no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—21no—
98C4⋊C4×C61—21no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C42×C61C2×D411yescharacteristic
104C42×C61C2×D411yescharacteristic
105C42×C61C2×D411yescharacteristic
106C42×C61#4111yescharacteristic
107#11758198—41no—
108#11758198—41no—
109C4⋊Dic61—22no—
110C4⋊Dic61—21no—
111C4⋊Dic61—22no—
112C4⋊Dic61—21no—
113C4⋊Dic61—41no—
114C4⋊Dic61—41no—
115#11758196—84no—
116C4×Dic61—24no—
117C2×#92418—22no—
118C23×C61#12011yescharacteristic
119C22×D61—41no—
120C2×C4×C61—22no—
121C2×C4×C61—22no—
122C2×C4×C61—41no—
123C2×C4×C61—42no—
124C2×C4×C61—41no—
125C2×C4×C61—41no—
126C2×C4×C61—42no—
127C2×C4×C61#12011yescharacteristic
128C2×C4×C61#12011yescharacteristic
129C2×C4×C61#12111yescharacteristic
130C2×C4×C61#12111yescharacteristic
131C2×C4×C61#12111yescharacteristic
132C2×C4×C61#12711yescharacteristic
133C22⋊D61—82no—
134C2×Dic61—24no—
135C2×Dic61—42no—
136C2×Dic61—42no—
137C2×Dic61—44no—
138C2×Dic61—42no—
139C2×Dic61—41no—
140D244—42no—
141D244—42no—
142D244—81no—
143C4×D61—82no—
144Dic122—42no—
145Dic122—82no—
146Dic122—82no—
147C8×C61—42no—
148C8×C61—81no—
149#92418—42no—
150#10093—611nomaximal, Sylow
151C22×C61—21no—
152C22×C61—21no—
153C22×C61#39211yescharacteristic
154C22×C61#45412yes—
155C2×D61—81no—
156C2×D61—81no—
157C2×D61—81no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—21no—
163C4×C61—81no—
164C4×C61—81no—
165C4×C61#45612yes—
166Dic61—44no—
167Dic61—81no—
168Dic61—82no—
169#3234—611no—
170#3141—611no—
171#2660—611no—
172#1556—611no—
173#1528D6111yescharacteristic, complemented
174#1505—611no—
175#1300—611no—
176C2×C61—22no—
177C2×C61#200311yescharacteristic
178C2×C61#200511yescharacteristic
179C2×C61#200811yescharacteristic
180D61—161no—
181#532—611no—
182C2×C4⋊Q8—611no—
183C2×#125C2×D6111yescharacteristic
184#501—1221no—
185#490—1221no—
186#489—1222no—
187#487—1221no—
188#486—1221no—
189#481—1221no—
190#480—1221no—
191#478—1222no—
192#432—611no—
193#432C2×D6111yescharacteristic
194#417—1222no—
195C4×M16C2×D6111yescharacteristic
196#398—1222no—
197#395—1222no—
198#384—1221no—
199#382—1221no—
200#376—611no—
201#332—1222no—
202#330—22no—
203#327—1222no—
204C61#1009311yescharacteristic, complemented, Sylow
205C2×#41—1221no—
206C22×Q8—1221no—
207C2×M16—21no—
208C4⋊Q8—611no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8—1221no—
212C4⋊Q8—1221no—
213#127—611no—
214#125—21no—
215#125—21no—
216#125—21no—
217#125—21no—
218#124—1221no—
219#121—1221no—
220#121—1221no—
221C4×D4—1221no—
222C4×D4—1221no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—612no—
226C2×C4⋊C4—1224no—
227C2×C4⋊C4—1221no—
228C2×C4⋊C4D24411yescharacteristic
229C2×C4⋊C4D24411yescharacteristic
230C2×C42—1221no—
231C2×C42C22×D6111yescharacteristic
232#107—1222no—
233#106—1222no—
234C4⋊C8—21no—
235C4⋊C8—612no—
236#103—1224no—
237#102—1222no—
238#102—1222no—
239#102—1224no—
240C22⋊C8—21no—
241C22⋊C8—1221no—
242#97—21no—
243#97—1221no—
244#97C22⋊D6112yes—
245C4×C8—21no—
246#95—612no—
247#95—1224no—
248#95—1221no—
249#41—2444no—
250C2×Q8—1221no—
251C2×Q8—4882no—
252C2×Q8—2441no—
253C2×Q8—2441no—
254C2×D4—1221no—
255C2×D4—1221no—
256C2×D4—1221no—
257C2×D4—2441no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4—2441no—
263C22×C4C2×D24411yescharacteristic
264C22×C4D4×D6111yescharacteristic
265C22×C4D4×D6111yescharacteristic
266M16—42no—
267C2×C8—22no—
268C2×C8—41no—
269C2×C8—1222no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—1222no—
279C4⋊C4—1221no—
280C4⋊C4—1222no—
281C4⋊C4—1221no—
282C4⋊C4—4884no—
283C4⋊C4—2441no—
284C4⋊C4—2441no—
285C22⋊C4—2441no—
286C22⋊C4—2441no—
287C42—1224no—
288C42#1175822211yescharacteristic
289C42D4×D6111yescharacteristic
290C42D4×D6111yescharacteristic
291C42C2×C22⋊D6111yescharacteristic
292C23—2441no—
293C23#1451974711yescharacteristic
294Q8—4882no—
295Q8—4882no—
296Q8—2442no—
297D4—4881no—
298D4—4882no—
299D4—2442no—
300D4—2442no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—1224no—
309C2×C4—4882no—
310C2×C4—2444no—
311C2×C4—2441no—
312C2×C4—2442no—
313C2×C4—2442no—
314C2×C4—2442no—
315C2×C4#1451974511yescharacteristic
316C2×C4#1451974711yescharacteristic
317C2×C4#1451975011yescharacteristic
318C2×C4#1451975011yescharacteristic
319C2×C4#1451975111yescharacteristic
320C2×C4#1451977311yescharacteristic
321C8—42no—
322C8—81no—
323C8—2442no—
324C22—21no—
325C22—21no—
326C22—4881no—
327C22—4881no—
328C22—4881no—
329C22#1683967111yescharacteristic
330C22#1683993612yes—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—81no—
337C4—81no—
338C4—4881no—
339C4—4882no—
340C4—2444no—
341C4#1683994712yes—
342C2—22no—
343C2—9761no—
344C2#2144684011yescharacteristic
345C2#2144685711yescharacteristic
346C2#2144687311yescharacteristic
subgroupG/H
centerC22#16839671
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#1528C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#1528D61
hypocenterC61#10093
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1528

Complements

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

NH ≅ G/Ntype
3. #21451940C2semidirect
4. #21444537C2semidirect
6. #21443713C2semidirect
7. C61×#1528C2semidirect
173. #1528D61semidirect
204. C61#10093semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214562132
#214519402
#214445372
#214443282
#214437132
C61×#15282
#1009361