Finite Group Databasebeta

← #29581402

Subgroups — #29581402

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#21456212C211yesmaximal, characteristic
2#21451770C211yesmaximal, characteristic, complemented
3#21444537C211yesmaximal, characteristic, complemented
4#21444421C211yesmaximal, characteristic, complemented
5#21444327C211yesmaximal, characteristic
6#21443711C211yesmaximal, characteristic, complemented
7C61×#3234C211yesmaximal, characteristic
8C2×#14519777C2211yescharacteristic
9#16840121—22no—
10#16840112—21no—
11#16840108—21no—
12#16839819C2211yescharacteristic
13#16839763—22no—
14C4×#11758194C2211yescharacteristic
15#16839701—22no—
16#16839701—22no—
17#16839667—21no—
18#16839667—21no—
19#16839608C2211yescharacteristic
20#16839509—22no—
21#16839504—22no—
22#16839496—22no—
23C61×#532C2211yescharacteristic
24C2×#125×C61C2211yescharacteristic
25C61×#501—21no—
26C61×#490—21no—
27C61×#481—21no—
28C61×#480—21no—
29C61×#478—22no—
30C61×#432C2211yescharacteristic
31C22×Dic122—21no—
32C2×#11758194—21no—
33#14519777—21no—
34#14519777—21no—
35#14519777—21no—
36#14519777—21no—
37C2×C4⋊Dic61—21no—
38C2×C4⋊Dic61D412yes—
39C2×C4×Dic61—21no—
40C4⋊#92418—21no—
41#14519701—24no—
42#14519698—24no—
43C22⋊#92418—21no—
44#14519688—21no—
45#14519688D412yes—
46C4×#92418—21no—
47#14519684—24no—
48#14519684—24no—
49#14519684—21no—
50#14519684D412yes—
51C2×#41×C61—21no—
52C4⋊Q8×C61D411yescharacteristic
53#127×C61D411yescharacteristic
54#125×C61—21no—
55#125×C61—21no—
56#125×C61—21no—
57#125×C61—21no—
58#124×C61—21no—
59#121×C61—21no—
60#121×C61—21no—
61C4×D4×C61—21no—
62C4×D4×C61—21no—
63C2×C4⋊C4×C61—21no—
64C2×C4⋊C4×C61—21no—
65C2×C4⋊C4×C61D411yescharacteristic
66C2×C4⋊C4×C61D411yescharacteristic
67C2×C42×C61C2311yescharacteristic
68#107×C61—22no—
69#106×C61—22no—
70C4⋊C8×C61D412yes—
71#102×C61—22no—
72#102×C61—22no—
73C22⋊C8×C61—21no—
74#97×C61—21no—
75#41×C61—44no—
76C2×Q8×C61—21no—
77C2×D4×C61—21no—
78C2×D4×C61—21no—
79C2×D4×C61—21no—
80C2×D4×C61—41no—
81C22×C4×C61—21no—
82C22×C4×C61—21no—
83C22×C4×C61—41no—
84C22×C4×C61C2×D411yescharacteristic
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×Dic61—22no—
88C22×Dic61—22no—
89C2×Dic122—44no—
90C2×Dic122—41no—
91C2×Dic122—41no—
92C2×C8×C61—22no—
93C4⋊C4×C61—21no—
94C4⋊C4×C61—21no—
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—
103C22⋊C4×C61—41no—
104C22⋊C4×C61—41no—
105C42×C61C2×D411yescharacteristic
106C42×C61C2×D411yescharacteristic
107C42×C61C2×D411yescharacteristic
108C42×C61#4111yescharacteristic
109C4⋊Dic61—22no—
110C4⋊Dic61—22no—
111C4⋊Dic61—41no—
112C4⋊Dic61—41no—
113C4×Dic61—42no—
114#11758194—42no—
115C2×#92418—22no—
116C2×#92418—41no—
117C23×C61—41no—
118C23×C61#12011yescharacteristic
119Q8×C61—42no—
120D4×C61—42no—
121D4×C61—42no—
122D4×C61—81no—
123D4×C61—82no—
124C2×C4×C61—22no—
125C2×C4×C61—22no—
126C2×C4×C61—41no—
127C2×C4×C61—42no—
128C2×C4×C61—41no—
129C2×C4×C61—41no—
130C2×C4×C61—41no—
131C2×C4×C61—42no—
132C2×C4×C61—82no—
133C2×C4×C61#12011yescharacteristic
134C2×C4×C61#12011yescharacteristic
135C2×C4×C61#12111yescharacteristic
136C2×C4×C61#12111yescharacteristic
137C2×C4×C61#12111yescharacteristic
138C2×C4×C61#12711yescharacteristic
139C2×Dic61—42no—
140C2×Dic61—44no—
141C2×Dic61—42no—
142C2×Dic61—42no—
143C2×Dic61—42no—
144Dic122—82no—
145Dic122—82no—
146C8×C61—42no—
147#92418—42no—
148#92418—81no—
149#10095—611nomaximal, Sylow
150C22×C61—21no—
151C22×C61—21no—
152C22×C61—81no—
153C22×C61—81no—
154C22×C61—81no—
155C22×C61#39211yescharacteristic
156C22×C61#45412yes—
157C4×C61—21no—
158C4×C61—21no—
159C4×C61—21no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—81no—
163C4×C61—81no—
164C4×C61—81no—
165C4×C61#45412yes—
166Dic61—82no—
167Dic61—82no—
168#3234D6111yescharacteristic
169#3141—611no—
170#2642—611no—
171#1556—611no—
172#1528—611no—
173#1505—611no—
174#1300—611no—
175C2×C61—22no—
176C2×C61—161no—
177C2×C61#200311yescharacteristic
178C2×C61#200311yescharacteristic
179C2×C61#200911yescharacteristic
180#532C2×D6111yescharacteristic
181C2×C4⋊Q8—611no—
182C2×#125C2×D6111yescharacteristic
183#501—21no—
184#490—21no—
185#489—1222no—
186#487—1221no—
187#486—1221no—
188#481—21no—
189#480—21no—
190#478—22no—
191#432—611no—
192#432C2×D6111yescharacteristic
193#417—1222no—
194C4×M16—611no—
195#400—1222no—
196#400—1222no—
197#391—1221no—
198#391—1221no—
199#376—611no—
200#332—1222no—
201#330—1222no—
202#327—1222no—
203C61#1009511yescharacteristic, complemented, Sylow
204C2×#41—21no—
205C22×Q8—1221no—
206C2×M16—1221no—
207C4⋊Q8—1221no—
208C4⋊Q8—1221no—
209C4⋊Q8—1221no—
210C4⋊Q8—1221no—
211C4⋊Q8C22⋊D6111yescharacteristic
212#127C22⋊D6111yescharacteristic
213#125—21no—
214#125—21no—
215#125—21no—
216#125—21no—
217#124—21no—
218#121—21no—
219#121—21no—
220C4×D4—21no—
221C4×D4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—612no—
225C2×C4⋊C4—1221no—
226C2×C4⋊C4C22⋊D6111yescharacteristic
227C2×C4⋊C4C22⋊D6111yescharacteristic
228C2×C42—1221no—
229C2×C42C22×D6111yescharacteristic
230#107—22no—
231#106—22no—
232C4⋊C8—1221no—
233C4⋊C8C22⋊D6112yes—
234#103—1224no—
235#102—22no—
236#102—22no—
237#102—1224no—
238C22⋊C8—21no—
239C22⋊C8—1221no—
240#97—21no—
241#97—612no—
242#97—1221no—
243C4×C8—1221no—
244#95—612no—
245#95—1224no—
246#95—1224no—
247#95—1221no—
248#41—44no—
249C2×Q8—21no—
250C2×Q8—2444no—
251C2×Q8—2441no—
252C2×Q8—2441no—
253C2×D4—21no—
254C2×D4—21no—
255C2×D4—21no—
256C2×D4—41no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—41no—
260C22×C4—1222no—
261C22×C4—1222no—
262C22×C4D4×D6111yescharacteristic
263C22×C4D4×D6111yescharacteristic
264C22×C4C2×C22⋊D6111yescharacteristic
265M16—2442no—
266C2×C8—22no—
267C2×C8—1222no—
268C2×C8—2441no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—1222no—
280C4⋊C4—1222no—
281C4⋊C4—2441no—
282C4⋊C4—2441no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C42—2442no—
286C42D4×D6111yescharacteristic
287C42#1175822411yescharacteristic
288C42C2×C22⋊D6111yescharacteristic
289C42C2×C22⋊D6111yescharacteristic
290C23—41no—
291C23#1451974511yescharacteristic
292Q8—42no—
293Q8—4882no—
294Q8—4882no—
295D4—42no—
296D4—42no—
297D4—81no—
298D4—82no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2444no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4#1451974511yescharacteristic
314C2×C4#1451974611yescharacteristic
315C2×C4#1451974811yescharacteristic
316C2×C4#1451974911yescharacteristic
317C2×C4#1451974911yescharacteristic
318C2×C4#1451977311yescharacteristic
319C8—42no—
320C8—4881no—
321C8—2442no—
322C22—21no—
323C22—21no—
324C22—81no—
325C22—81no—
326C22—81no—
327C22#1683967011yescharacteristic
328C22#1683993312yes—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—81no—
335C4—81no—
336C4—81no—
337C4—4882no—
338C4—4882no—
339C4#1683993512yes—
340C2—22no—
341C2—161no—
342C2#2144683811yescharacteristic
343C2#2144683911yescharacteristic
344C2#2144687611yescharacteristic
subgroupG/H
centerC22#16839670
commutator, cosocleC2×C42×C61C23
FrattiniC2×C42C22×D61
FittingC61×#3234C2
socleC22×C61#392
prime radicalC61×#532C22
hypercenter#3234D61
hypocenterC61#10095
LCSG ▷ C2×C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3234

Complements

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

NH ≅ G/Ntype
2. #21451770C2semidirect
3. #21444537C2semidirect
4. #21444421C2semidirect
6. #21443711C2semidirect
203. C61#10095semidirect

Maximal subgroups

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

subgroupindex
#214562122
#214517702
#214445372
#214444212
#214443272
#214437112
C61×#32342
#1009561