Finite Group Databasebeta

← #29581404

Subgroups — #29581404

Order 15616.

480 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 85 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#21456215C211yesmaximal, characteristic
3#21444536C211yesmaximal, characteristic, complemented
4#21444421C211yesmaximal, characteristic, complemented
5#21444326C211yesmaximal, characteristic
6#21443712C211yesmaximal, characteristic, complemented
7C61×#2642C211yesmaximal, characteristic, complemented
8#16840312C2211yescharacteristic
9#16840172—21no—
10#16840128—21no—
11#16840124—22no—
12#16840114—21no—
13#16840110—21no—
14#16840080—21no—
15#16840075—21no—
16#16840059—22no—
17#16839819C2211yescharacteristic
18#16839819C2211yescharacteristic
19#16839761—22no—
20C4×#11758194C2211yescharacteristic
21#16839508—22no—
22#16839504—22no—
23#16839495—22no—
24C2×C4⋊Q8×C61C2211yescharacteristic
25C2×#125×C61C2211yescharacteristic
26C61×#400—22no—
27C61×#400—22no—
28C61×#391—21no—
29C61×#391—21no—
30C61×#376C2211yescharacteristic
31C2×#11758222—21no—
32C2×#11758194—21no—
33#14519777D411yescharacteristic
34#14519774D411yescharacteristic
35#14519765—21no—
36#14519751—21no—
37#14519751—21no—
38C4×D244—21no—
39C4×D244—21no—
40#14519710—22no—
41#14519708—22no—
42C4⋊#92418—21no—
43C4⋊#92418D412yes—
44#14519700—24no—
45#14519697—24no—
46#14519697—22no—
47#14519697—22no—
48C22⋊#92418—21no—
49C22⋊#92418—21no—
50#14519688—21no—
51#14519688—21no—
52#14519688D412yes—
53C4×#92418—21no—
54C22×Q8×C61—21no—
55C4⋊Q8×C61—21no—
56C4⋊Q8×C61—21no—
57C4⋊Q8×C61—21no—
58C4⋊Q8×C61—21no—
59#125×C61—21no—
60#125×C61—21no—
61#125×C61—21no—
62#125×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61—21no—
66C2×C4⋊C4×C61D412yes—
67C2×C4⋊C4×C61D411yescharacteristic
68C2×C4⋊C4×C61D411yescharacteristic
69C2×C42×C61—21no—
70C2×C42×C61C2311yescharacteristic
71#95×C61—24no—
72#95×C61—24no—
73#95×C61—21no—
74#95×C61D412yes—
75C2×Q8×C61—44no—
76C2×Q8×C61—41no—
77C2×Q8×C61—41no—
78C22×C4×C61—22no—
79C22×C4×C61—21no—
80C22×C4×C61—21no—
81C22×C4×C61—22no—
82C22×C4×C61C2×D411yescharacteristic
83C22×C4×C61C2×D411yescharacteristic
84C22×C4×C61C2×D411yescharacteristic
85C2×C22⋊D61—41no—
86#11758222—44no—
87C2×D244—21no—
88C2×D244—21no—
89C2×D244—21no—
90C2×C4×D61—41no—
91C2×Dic122—21no—
92C4⋊C4×C61—21no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—22no—
95C4⋊C4×C61—21no—
96C4⋊C4×C61—21no—
97C4⋊C4×C61—22no—
98C4⋊C4×C61—41no—
99C4⋊C4×C61—41no—
100C4⋊C4×C61—41no—
101C4⋊C4×C61—41no—
102C4⋊C4×C61—41no—
103C4⋊C4×C61—41no—
104C42×C61—42no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61#4111yescharacteristic
109#11758198—41no—
110#11758198—41no—
111C4⋊Dic61—21no—
112C4⋊Dic61—21no—
113#11758194—42no—
114C2×#92418—22no—
115C2×#92418—22no—
116C2×#92418—41no—
117C23×C61#12011yescharacteristic
118C22×D61—41no—
119Q8×C61—82no—
120Q8×C61—82no—
121C2×C4×C61—22no—
122C2×C4×C61—22no—
123C2×C4×C61—41no—
124C2×C4×C61—42no—
125C2×C4×C61—42no—
126C2×C4×C61—42no—
127C2×C4×C61—44no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—42no—
131C2×C4×C61—42no—
132C2×C4×C61—42no—
133C2×C4×C61#12011yescharacteristic
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12711yescharacteristic
139C22⋊D61—82no—
140C2×Dic61—41no—
141D244—42no—
142D244—42no—
143D244—81no—
144C4×D61—82no—
145Dic122—42no—
146#92418—42no—
147#92418—42no—
148#92418—81no—
149#10095—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61#39211yescharacteristic
153C22×C61#45412yes—
154C2×D61—81no—
155C2×D61—81no—
156C2×D61—81no—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—82no—
163C4×C61—81no—
164C4×C61—82no—
165C4×C61—81no—
166C4×C61#45412yes—
167Dic61—81no—
168#3234—611no—
169#3141—611no—
170#2642D6111yescharacteristic, complemented
171#1556—611no—
172#1528—611no—
173#1505—611no—
174#1300—611no—
175C2×C61—22no—
176C2×C61#200311yescharacteristic
177C2×C61#200311yescharacteristic
178C2×C61#200911yescharacteristic
179D61—161no—
180#532—611no—
181C2×C4⋊Q8C2×D6111yescharacteristic
182C2×#125C2×D6111yescharacteristic
183#501—1221no—
184#490—1221no—
185#489—1222no—
186#487—1221no—
187#486—1221no—
188#481—1221no—
189#480—1221no—
190#478—1222no—
191#432—611no—
192#432—611no—
193#417—1222no—
194C4×M16—611no—
195#400—22no—
196#400—22no—
197#391—21no—
198#391—21no—
199#376C2×D6111yescharacteristic
200#332—1222no—
201#330—1222no—
202#327—1222no—
203C61#1009511yescharacteristic, complemented, Sylow
204C2×#41—1221no—
205C22×Q8—21no—
206C2×M16—1221no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—611no—
212#127—611no—
213#125—21no—
214#125—21no—
215#125—21no—
216#125—21no—
217#124—1221no—
218#121—1221no—
219#121—1221no—
220C4×D4—1221no—
221C4×D4—1221no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4C22⋊D6111yescharacteristic
226C2×C4⋊C4C22⋊D6111yescharacteristic
227C2×C4⋊C4C22⋊D6112yes—
228C2×C42—21no—
229C2×C42C22×D6111yescharacteristic
230#107—1222no—
231#106—1222no—
232C4⋊C8—612no—
233C4⋊C8—1221no—
234#103—1224no—
235#102—1224no—
236#102—1222no—
237#102—1222no—
238C22⋊C8—1221no—
239C22⋊C8—1221no—
240#97—612no—
241#97—1221no—
242#97—1221no—
243C4×C8—1221no—
244#95—24no—
245#95—24no—
246#95—21no—
247#95C22⋊D6112yes—
248#41—2444no—
249C2×Q8—44no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—1221no—
253C2×D4—1221no—
254C2×D4—1221no—
255C2×D4—1221no—
256C2×D4—2441no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—21no—
260C22×C4—22no—
261C22×C4—2441no—
262C22×C4C2×C22⋊D6111yescharacteristic
263C22×C4C2×C22⋊D6111yescharacteristic
264C22×C4C2×C22⋊D6111yescharacteristic
265M16—2442no—
266C2×C8—1222no—
267C2×C8—1222no—
268C2×C8—2441no—
269C4⋊C4—22no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—1221no—
282C4⋊C4—1221no—
283C22⋊C4—2441no—
284C22⋊C4—2441no—
285C42—42no—
286C42D4×D6111yescharacteristic
287C42D4×D6111yescharacteristic
288C42D4×D6111yescharacteristic
289C42#1175822411yescharacteristic
290C23—2441no—
291C23#1451974611yescharacteristic
292Q8—82no—
293Q8—82no—
294Q8—2442no—
295D4—4881no—
296D4—4882no—
297D4—2442no—
298D4—2442no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—4882no—
312C2×C4—2441no—
313C2×C4#1451974511yescharacteristic
314C2×C4#1451974511yescharacteristic
315C2×C4#1451974811yescharacteristic
316C2×C4#1451974911yescharacteristic
317C2×C4#1451974911yescharacteristic
318C2×C4#1451977311yescharacteristic
319C8—4881no—
320C8—2442no—
321C8—2442no—
322C22—21no—
323C22—21no—
324C22—4881no—
325C22—4881no—
326C22—4881no—
327C22#1683967011yescharacteristic
328C22#1683993512yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—82no—
335C4—81no—
336C4—82no—
337C4—81no—
338C4—4881no—
339C4#1683993412yes—
340C2—22no—
341C2—9761no—
342C2#2144683911yescharacteristic
343C2#2144684111yescharacteristic
344C2#2144687911yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#2642C2
socleC22×C61#392
prime radical#16840312C22
hypercenter#2642D61
hypocenterC61#10095
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2642

Complements

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

NH ≅ G/Ntype
3. #21444536C2semidirect
4. #21444421C2semidirect
6. #21443712C2semidirect
7. C61×#2642C2semidirect
170. #2642D61semidirect
203. C61#10095semidirect

Maximal subgroups

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

subgroupindex
#214570092
#214562152
#214445362
#214444212
#214443262
#214437122
C61×#26422
#1009561