Finite Group Databasebeta

← #29652874

Subgroups — #29652874

Order 15616.

902 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 183 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#21455945C212yesmaximal
2#21449110C211yesmaximal, characteristic
3#21444827C214yesmaximal
4#21444793C214yesmaximal
5#21444331C212yesmaximal
6#21444163C211yesmaximal, characteristic
7C61×#3502C211yesmaximal, characteristic
8C2×C4×Dic122C2211yescharacteristic
9#16840090C2214yes—
10#16840044—22no—
11#16839969—22no—
12#16839917C2214yes—
13#16839867—24no—
14#16839819C2212yes—
15#16839693—21no—
16#16839692C2212yes—
17#16839682C2212yes—
18#16839681—21no—
19C4×#11758196—22no—
20#16839608C2211yescharacteristic
21#16839499C2218yes—
22#16839488C2214yes—
23Q82×C61—22no—
24C61×#557—22no—
25C61×#557—24no—
26C61×#544—21no—
27C61×#544C2214yes—
28C61×#533—22no—
29C61×#533C2212yes—
30C61×#519—21no—
31C2×C4×Q8×C61C2211yescharacteristic
32C22×Dic122C2311yescharacteristic
33C2×#11758202—24no—
34C4×Dic122—22no—
35C4×Dic122—22no—
36C2×C4⋊Dic61C2311yescharacteristic
37C2×#11758196—22no—
38C2×#11758196—21no—
39C2×#11758196—22no—
40C2×C4×Dic61—21no—
41C2×C4×Dic61—21no—
42#14519708—24no—
43C4⋊#92418—22no—
44#14519701—24no—
45#14519700—24no—
46C22⋊#92418C2314yes—
47#14519688—22no—
48#14519684—22no—
49#14519684—21no—
50#14519684—22no—
51#14519684C2312yes—
52C22×Q8×C61D411yescharacteristic
53C4⋊Q8×C61—22no—
54C4⋊Q8×C61—22no—
55C4⋊Q8×C61—24no—
56C4⋊Q8×C61—28no—
57C4⋊Q8×C61D412yes—
58C4⋊Q8×C61D412yes—
59#125×C61—24no—
60#125×C61—24no—
61#125×C61—28no—
62#125×C61—22no—
63C22⋊Q8×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61C2314yes—
66C4×Q8×C61—24no—
67C4×Q8×C61—24no—
68C4×Q8×C61—24no—
69C4×Q8×C61—24no—
70C4×Q8×C61—22no—
71C4×Q8×C61—22no—
72C4×Q8×C61—22no—
73C4×Q8×C61D414yes—
74#117×C61—24no—
75#117×C61—22no—
76C2×C4⋊C4×C61D411yescharacteristic
77C2×C4⋊C4×C61C2312yes—
78C2×C42×C61D412yes—
79C2×C42×C61C2311yescharacteristic
80C2×Q8×C61—24no—
81C2×Q8×C61—22no—
82C2×Q8×C61—22no—
83C2×Q8×C61—24no—
84C2×Q8×C61C2×D414yes—
85C22×C4×C61C2×D411yescharacteristic
86C22×C4×C61C2×D411yescharacteristic
87C22×C4×C61C2×D412yes—
88C22×C4×C61C2×D412yes—
89C22×C4×C61C2411yescharacteristic
90C22×Dic61—21no—
91C22×Dic61—22no—
92C22×Dic61—21no—
93C2×Dic122—22no—
94C2×Dic122—42no—
95C4⋊C4×C61—24no—
96C4⋊C4×C61—24no—
97C4⋊C4×C61—24no—
98C4⋊C4×C61—28no—
99C4⋊C4×C61—22no—
100C4⋊C4×C61—22no—
101C4⋊C4×C61—22no—
102C4⋊C4×C61—22no—
103C4⋊C4×C61—24no—
104C4⋊C4×C61—24no—
105C4⋊C4×C61—24no—
106C4⋊C4×C61C2×D414yes—
107C22⋊C4×C61—24no—
108C42×C61—28no—
109C42×C61—22no—
110C42×C61—22no—
111C42×C61—24no—
112C42×C61C2×D412yes—
113C42×C61C2×D412yes—
114#11758202—48no—
115C4⋊Dic61—22no—
116#11758196—44no—
117#11758196—42no—
118#11758196—44no—
119C4×Dic61—24no—
120C4×Dic61—41no—
121C4×Dic61—41no—
122C2×#92418—24no—
123C23×C61C22×D411yescharacteristic
124Q8×C61—28no—
125Q8×C61—44no—
126Q8×C61—44no—
127Q8×C61—84no—
128C2×C4×C61—22no—
129C2×C4×C61—22no—
130C2×C4×C61—22no—
131C2×C4×C61—22no—
132C2×C4×C61—24no—
133C2×C4×C61—24no—
134C2×C4×C61—22no—
135C2×C4×C61—41no—
136C2×C4×C61—42no—
137C2×C4×C61#12011yescharacteristic
138C2×C4×C61#12012yes—
139C2×C4×C61#12011yescharacteristic
140C2×C4×C61#13714yes—
141C2×C4×C61C22×D412yes—
142C2×Dic61—24no—
143C2×Dic61—22no—
144C2×Dic61—42no—
145C2×Dic61—41no—
146C2×Dic61—41no—
147C2×Dic61—42no—
148C2×Dic61—41no—
149C2×Dic61—42no—
150Dic122—24no—
151Dic122—42no—
152#92418—44no—
153#18189—611nomaximal, Sylow
154C22×C61—21no—
155C22×C61—21no—
156C22×C61C2×#12011yescharacteristic
157C22×C61C2×#13712yes—
158C4×C61—22no—
159C4×C61—24no—
160C4×C61—21no—
161C4×C61—22no—
162C4×C61—44no—
163C4×C61—42no—
164C4×C61—44no—
165C4×C61#45612yes—
166Dic61—44no—
167Dic61—42no—
168Dic61—81no—
169#3502D6111yescharacteristic
170#3113—612no—
171#2399—611no—
172#1611—614no—
173#1605—614no—
174#1506—612no—
175#1467—611no—
176C2×C61—22no—
177C2×C61#300411yescharacteristic
178C2×C61C2×#45611yescharacteristic
179C2×C61#302511yescharacteristic
180Q82—22no—
181#557—22no—
182#557—24no—
183#544—21no—
184#544C2×D6114yes—
185#533—22no—
186#533C2×D6112yes—
187#519—21no—
188C2×C4×Q8—611no—
189C2×C4×Q8C2×D6111yescharacteristic
190#483—614no—
191#475—1222no—
192#462—1222no—
193#451—614no—
194#441—1224no—
195#432—612no—
196#398—612no—
197#398—1221no—
198#395—612no—
199#395—1221no—
200C4×C4⋊C4—1222no—
201#376—611no—
202#328—618no—
203#324—614no—
204C61#1818911yescharacteristic, complemented, Sylow
205C22×Q8—611no—
206C22×Q8C22⋊D6111yescharacteristic
207C2×Q16—1224no—
208C4⋊Q8—22no—
209C4⋊Q8—22no—
210C4⋊Q8—24no—
211C4⋊Q8—28no—
212C4⋊Q8C22⋊D6112yes—
213C4⋊Q8C22⋊D6112yes—
214#125—24no—
215#125—24no—
216#125—28no—
217#125—22no—
218C22⋊Q8—22no—
219C22⋊Q8—24no—
220C22⋊Q8C22×D6114yes—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—22no—
226C4×Q8—22no—
227C4×Q8—22no—
228C4×Q8—1222no—
229C4×Q8—1222no—
230C4×Q8C22⋊D6114yes—
231#117—24no—
232#117—22no—
233C2×C4⋊C4—611no—
234C2×C4⋊C4—1221no—
235C2×C4⋊C4—1222no—
236C2×C4⋊C4—1222no—
237C2×C4⋊C4C22⋊D6111yescharacteristic
238C2×C4⋊C4C22×D6112yes—
239C2×C42—1221no—
240C2×C42—1221no—
241C2×C42C22⋊D6112yes—
242C2×C42C22×D6111yescharacteristic
243#106—1224no—
244C4⋊C8—1222no—
245#103—1224no—
246#103—1224no—
247C22⋊C8—614no—
248#97—1222no—
249#95—612no—
250#95—1221no—
251#95—1222no—
252#95—1222no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—22no—
257C2×Q8—1222no—
258C2×Q8—2442no—
259C2×Q8C2×C22⋊D6114yes—
260C22×C4—1222no—
261C22×C4—1221no—
262C22×C4—1221no—
263C22×C4C2×C22⋊D6112yes—
264C22×C4C2×C22⋊D6111yescharacteristic
265C22×C4C2×C22⋊D6111yescharacteristic
266C22×C4C2×C22⋊D6112yes—
267C22×C4C23×D6111yescharacteristic
268Q16—2448no—
269C2×C8—1224no—
270C4⋊C4—24no—
271C4⋊C4—24no—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—24no—
281C4⋊C4—1222no—
282C4⋊C4—2444no—
283C4⋊C4—2444no—
284C4⋊C4—2442no—
285C4⋊C4C2×C22⋊D6114yes—
286C22⋊C4—24no—
287C42—28no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—1224no—
292C42—2441no—
293C42—2441no—
294C42C2×C22⋊D6112yes—
295C42C2×C22⋊D6112yes—
296C23C22×C22⋊D6111yescharacteristic
297Q8—28no—
298Q8—44no—
299Q8—44no—
300Q8—84no—
301Q8—1224no—
302Q8—2442no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—22no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—1224no—
313C2×C4—1222no—
314C2×C4—2441no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2441no—
318C2×C4—2441no—
319C2×C4—2442no—
320C2×C4#1451974611yescharacteristic
321C2×C4#1451974612yes—
322C2×C4#1451974611yescharacteristic
323C2×C4#1451980814yes—
324C2×C4C22×C22⋊D6112yes—
325C8—2444no—
326C22—21no—
327C22—21no—
328C22C2×#1451974611yescharacteristic
329C22C2×#1451980812yes—
330C4—21no—
331C4—22no—
332C4—24no—
333C4—22no—
334C4—44no—
335C4—42no—
336C4—44no—
337C4—4881no—
338C4—2444no—
339C4—2442no—
340C4#1683994512yes—
341C2—22no—
342C2#2145505311yescharacteristic
343C2C2×#1683994511yescharacteristic
344C2#2145521111yescharacteristic
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC22×C4×C61C24
FrattiniC22×C4C23×D61
FittingC61×#3502C2
socleC22×C61C2×#120
prime radicalC23×C61C22×D4
hypercenter#3502D61
hypocenterC61#18189
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
204. C61#18189semidirect

Maximal subgroups

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

subgroupindex
#214559452
#214491102
#214448272
#214447932
#214443312
#214441632
C61×#35022
#1818961