Finite Group Databasebeta

← #29593017

Subgroups — #29593017

Order 15616.

428 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 63 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#21447169C211yesmaximal, characteristic, complemented
2#21444826C211yesmaximal, characteristic, complemented
3#21444758C211yesmaximal, characteristic
4#21443825C211yesmaximal, characteristic, complemented
5#21443822C211yesmaximal, characteristic
6#21443763C211yesmaximal, characteristic, complemented
7C61×#2609C211yesmaximal, characteristic
8#16840393C2211yescharacteristic
9#16840092—21no—
10#16839913—21no—
11#16839802—21no—
12#16839531C2211yescharacteristic
13#16839522—21no—
14#16839498—21no—
15#16839498—21no—
16#16839488C2211yescharacteristic
17#16839488C2211yescharacteristic
18C2×#117×C61—21no—
19C61×#398C2211yescharacteristic
20C61×#396—21no—
21C61×#394C2211yescharacteristic
22C61×#388—21no—
23C61×#387—21no—
24C61×#386—21no—
25C61×#385—21no—
26C61×#382C2211yescharacteristic
27C2×#11758200—41no—
28C2×#11758194—21no—
29#14519776—41no—
30#14519768—41no—
31#14519756—21no—
32#14519756—21no—
33#14519736—21no—
34#14519710—41no—
35#14519708—41no—
36#14519708—41no—
37#14519700—41no—
38#14519698—41no—
39C22⋊#92418—21no—
40C22⋊#92418—21no—
41C22×D4×C61D411yescharacteristic
42C23×C4×C61—21no—
43#117×C61—42no—
44#117×C61—42no—
45C2×C4⋊C4×C61—21no—
46C2×C4⋊C4×C61—21no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61D411yescharacteristic
49C2×C4⋊C4×C61D411yescharacteristic
50C2×C4⋊C4×C61C2311yescharacteristic
51C2×C22⋊C4×C61—21no—
52C2×C22⋊C4×C61—21no—
53C2×C22⋊C4×C61—21no—
54C2×C22⋊C4×C61—21no—
55C2×C22⋊C4×C61—21no—
56C2×C42×C61—21no—
57C2×C42×C61D411yescharacteristic
58#95×C61—21no—
59#95×C61—21no—
60#95×C61—21no—
61#95×C61—21no—
62#95×C61D411yescharacteristic
63#95×C61D411yescharacteristic
64C24×C61—21no—
65C2×D4×C61—41no—
66C2×D4×C61—42no—
67C22×C4×C61—21no—
68C22×C4×C61—21no—
69C22×C4×C61—21no—
70C22×C4×C61—21no—
71C22×C4×C61—21no—
72C22×C4×C61—42no—
73C22×C4×C61—42no—
74C22×C4×C61—42no—
75C22×C4×C61C2×D411yescharacteristic
76C22×C4×C61C2×D411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C2×Dic122—21no—
79C2×Dic122—21no—
80C4⋊C4×C61—41no—
81C4⋊C4×C61—42no—
82C4⋊C4×C61—42no—
83C4⋊C4×C61—41no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C22⋊C4×C61—42no—
88C22⋊C4×C61—42no—
89C22⋊C4×C61—42no—
90C22⋊C4×C61—41no—
91C22⋊C4×C61—41no—
92C22⋊C4×C61—81no—
93C42×C61—21no—
94C42×C61—42no—
95C42×C61SD1612yescomplemented
96C22⋊Dic61—41no—
97#11758200—82no—
98C4⋊Dic61—21no—
99C4⋊Dic61—21no—
100C4⋊Dic61—21no—
101C4⋊Dic61—21no—
102#11758196—41no—
103#11758196—41no—
104C4×Dic61—41no—
105#11758194—22no—
106#11758194—41no—
107C2×#92418—41no—
108C2×#92418—41no—
109C2×#92418—41no—
110C23×C61—22no—
111C23×C61—41no—
112C23×C61—41no—
113C23×C61—41no—
114C23×C61—41no—
115C23×C61—41no—
116C23×C61—41no—
117C23×C61#12011yescharacteristic
118D4×C61—42no—
119D4×C61—81no—
120C2×C4×C61—21no—
121C2×C4×C61—21no—
122C2×C4×C61—22no—
123C2×C4×C61—22no—
124C2×C4×C61—21no—
125C2×C4×C61—21no—
126C2×C4×C61—41no—
127C2×C4×C61—41no—
128C2×C4×C61—41no—
129C2×C4×C61—48no—
130C2×C4×C61—41no—
131C2×C4×C61—41no—
132C2×C4×C61—41no—
133C2×C4×C61—41no—
134C2×C4×C61—41no—
135C2×C4×C61—41no—
136C2×C4×C61—41no—
137C2×C4×C61—41no—
138C2×C4×C61—41no—
139C2×C4×C61—41no—
140C2×C4×C61—41no—
141C2×C4×C61—41no—
142C2×C4×C61C2×SD1611yescharacteristic
143C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
144C2×Dic61—41no—
145C2×Dic61—41no—
146Dic122—81no—
147Dic122—81no—
148#92418—42no—
149#92418—81no—
150#92418—81no—
151#12248—611nomaximal, Sylow
152C22×C61—21no—
153C22×C61—21no—
154C22×C61—42no—
155C22×C61—42no—
156C22×C61—42no—
157C22×C61—81no—
158C22×C61—81no—
159C22×C61—81no—
160C22×C61—81no—
161C22×C61#45011yescharacteristic
162C22×C61#45611yescharacteristic
163C22×C61B(4,𝔽2)11yescharacteristic
164C4×C61—21no—
165C4×C61—21no—
166C4×C61—42no—
167C4×C61—42no—
168C4×C61—41no—
169C4×C61—41no—
170C4×C61—81no—
171C4×C61—81no—
172C4×C61—81no—
173Dic61—81no—
174Dic61—81no—
175#2609D6111yescharacteristic
176#2061—611no—
177#1611—611no—
178#1599—611no—
179#1346—611no—
180#1345—611no—
181#1324—611no—
182C2×C61—21no—
183C2×C61—21no—
184C2×C61—82no—
185C2×C61#160411yescharacteristic
186C2×C61#219711yescharacteristic
187C2×C61#220411yescharacteristic
188#544—611no—
189C2×#117—21no—
190#483—1221no—
191#450—1221no—
192#428—1221no—
193#398C2×D6111yescharacteristic
194#396—21no—
195#394C2×D6111yescharacteristic
196#388—21no—
197#387—21no—
198#386—21no—
199#385—21no—
200#382C2×D6111yescharacteristic
201#343—611no—
202#339—1221no—
203#328—1221no—
204#328—1221no—
205#324—611no—
206#324—611no—
207C61#1224811yescharacteristic, complemented, Sylow
208C22×D4C22⋊D6111yescharacteristic
209C23×C4—21no—
210C2×SD16—2441no—
211C2×M16—1221no—
212C4⋊Q8—2441no—
213#125—2441no—
214C22⋊Q8—1221no—
215C22⋊Q8—1221no—
216#117—42no—
217#117—42no—
218#117—1221no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4C22⋊D6111yescharacteristic
223C2×C4⋊C4C22⋊D6111yescharacteristic
224C2×C4⋊C4C22×D6111yescharacteristic
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42—21no—
231C2×C42C22⋊D6111yescharacteristic
232#107—2441no—
233#106—2441no—
234#106—2441no—
235#103—2441no—
236#102—2441no—
237C22⋊C8—1221no—
238C22⋊C8—1221no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—21no—
243#95C22⋊D6111yescharacteristic
244#95C22⋊D6111yescharacteristic
245C24—21no—
246C2×Q8—1221no—
247C2×Q8—1221no—
248C2×D4—41no—
249C2×D4—42no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4C2×C22⋊D6111yescharacteristic
259C22×C4C2×C22⋊D6111yescharacteristic
260C22×C4C2×C22⋊D6111yescharacteristic
261SD16—4882no—
262M16—1222no—
263M16—2441no—
264C2×C8—2441no—
265C2×C8—2441no—
266C2×C8—2441no—
267C4⋊C4—41no—
268C4⋊C4—41no—
269C4⋊C4—41no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C4⋊C4—1221no—
275C4⋊C4—1221no—
276C4⋊C4—1221no—
277C4⋊C4—1221no—
278C4⋊C4—2441no—
279C4⋊C4—2441no—
280C22⋊C4—42no—
281C22⋊C4—42no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C22⋊C4—41no—
285C22⋊C4—81no—
286C22⋊C4—2441no—
287C42—21no—
288C42—42no—
289C42—2441no—
290C42#1175820012yescomplemented
291C23—22no—
292C23—41no—
293C23—41no—
294C23—41no—
295C23—41no—
296C23—41no—
297C23—41no—
298C23#1451974611yescharacteristic
299Q8—4881no—
300Q8—4881no—
301D4—42no—
302D4—81no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—48no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—2441no—
326C2×C4—2441no—
327C2×C4C2×#1175820011yescharacteristic
328C2×C4#1451980311yescharacteristic
329C8—4881no—
330C8—4881no—
331C8—2442no—
332C22—21no—
333C22—21no—
334C22—42no—
335C22—42no—
336C22—42no—
337C22—81no—
338C22—81no—
339C22—81no—
340C22—81no—
341C22#1683991311yescharacteristic
342C22#1683994511yescharacteristic
343C22#1683994911yescharacteristic
344C4—21no—
345C4—21no—
346C4—42no—
347C4—42no—
348C4—41no—
349C4—41no—
350C4—81no—
351C4—81no—
352C4—81no—
353C4—4881no—
354C4—4881no—
355C2—21no—
356C2—21no—
357C2—82no—
358C2#2144478811yescharacteristic
359C2#2144780911yescharacteristic
360C2#2144784511yescharacteristic
subgroupG/H
centerC22#16839949
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2609C2
socleC22×C61B(4,𝔽2)
prime radicalC22×D4×C61D4
hypercenter#2609D61
hypocenterC61#12248
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2609

Complements

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

NH ≅ G/Ntype
1. #21447169C2semidirect
2. #21444826C2semidirect
4. #21443825C2semidirect
6. #21443763C2semidirect
95. C42×C61SD16semidirect
207. C61#12248semidirect
290. C42#11758200semidirect

Maximal subgroups

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

subgroupindex
#214471692
#214448262
#214447582
#214438252
#214438222
#214437632
C61×#26092
#1224861