Finite Group Databasebeta

← #29713876

Subgroups — #29713876

Order 15616.

1122 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1122 conjugacy classes of subgroups — too many to draw. Only its 191 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#21454138C212yesmaximal, complemented
2#21448301C211yesmaximal, characteristic, complemented
3#21447171C212yesmaximal, complemented
4#21447067C212yesmaximal
5#21446278C212yesmaximal, complemented
6#21446100C214yesmaximal, complemented
7#21445896C211yesmaximal, characteristic, complemented
8C61×#3491C211yesmaximal, characteristic
9C2×#14519736C2211yescharacteristic
10C22×C4×Dic61C2211yescharacteristic
11C2×#14519692C2214yes—
12#16839658—22no—
13#16839650—24no—
14#16839643—24no—
15#16839642C2214yes—
16#16839634C2214yes—
17#16839634C2212yes—
18#16839634C2212yes—
19#16839625C2212yes—
20C4×#11758196—22no—
21#16839529C2212yes—
22#16839529C2214yes—
23#16839529C2212yes—
24C61×#552—24no—
25C61×#551—24no—
26C61×#543C2211yescharacteristic
27C61×#539—22no—
28C61×#534C2211yescharacteristic
29C61×#529—22no—
30C2×#123×C61C2212yes—
31C61×#518C2212yes—
32C2×#117×C61C2211yescharacteristic
33C23×Dic61C2312yes—
34#14519736—24no—
35C2×C4⋊Dic61—21no—
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758196—24no—
39C2×C22⋊Dic61—24no—
40C2×C22⋊Dic61C2314yes—
41C2×C22⋊Dic61C2312yes—
42C2×C4×Dic61—21no—
43C2×C4×Dic61—21no—
44C2×C4×Dic61—24no—
45C2×C4×Dic61—21no—
46#14519692—28no—
47#14519684—22no—
48#14519684—22no—
49#14519684—24no—
50#14519684—24no—
51#14519684—24no—
52C22×D4×C61C2312yes—
53C23×C4×C61C2311yescharacteristic
54#126×C61—24no—
55#126×C61—24no—
56#125×C61—24no—
57#125×C61—22no—
58#124×C61—22no—
59#123×C61—24no—
60#123×C61—22no—
61#123×C61—22no—
62#123×C61—24no—
63#123×C61—22no—
64C22⋊Q8×C61—24no—
65C22⋊Q8×C61—22no—
66C22⋊Q8×C61—24no—
67#121×C61—22no—
68#120×C61—22no—
69C4×Q8×C61—22no—
70C4×D4×C61—22no—
71C4×D4×C61—24no—
72C4×D4×C61—24no—
73#117×C61—24no—
74#117×C61—24no—
75#117×C61—22no—
76C2×C4⋊C4×C61D412yes—
77C2×C4⋊C4×C61D414yes—
78C2×C22⋊C4×C61—22no—
79C2×C22⋊C4×C61D414yes—
80C2×C22⋊C4×C61C2311yescharacteristic
81C2×C22⋊C4×C61C2312yes—
82C2×C22⋊C4×C61C2311yescharacteristic
83C2×C42×C61D412yes—
84C24×C61C2×D412yes—
85C24×C61C2411yescharacteristic
86C2×Q8×C61—22no—
87C2×D4×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—24no—
90C2×D4×C61—24no—
91C2×D4×C61—22no—
92C22×C4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—24no—
95C22×C4×C61C2×D412yes—
96C22×C4×C61C2×D411yescharacteristic
97C22×C4×C61C2×D414yes—
98C22×C4×C61C2×D411yescharacteristic
99C22×C4×C61C2×D412yes—
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D414yes—
102C22×Dic61—24no—
103C22×Dic61—22no—
104C22×Dic61—22no—
105C22×Dic61—24no—
106C22×Dic61—44no—
107C4⋊C4×C61—24no—
108C4⋊C4×C61—24no—
109C4⋊C4×C61—22no—
110C4⋊C4×C61—22no—
111C4⋊C4×C61—24no—
112C4⋊C4×C61—24no—
113C4⋊C4×C61—24no—
114C22⋊C4×C61—24no—
115C22⋊C4×C61—24no—
116C22⋊C4×C61—22no—
117C22⋊C4×C61—22no—
118C22⋊C4×C61—22no—
119C22⋊C4×C61—22no—
120C22⋊C4×C61—24no—
121C22⋊C4×C61—24no—
122C42×C61—24no—
123C42×C61—22no—
124C42×C61—22no—
125C22⋊Dic61—22no—
126C22⋊Dic61—24no—
127C22⋊Dic61—24no—
128C22⋊Dic61—48no—
129C22⋊Dic61#4114yes—
130C4⋊Dic61—24no—
131#11758196—44no—
132#11758196—44no—
133#11758196—48no—
134C4×Dic61—24no—
135C4×Dic61—42no—
136C4×Dic61—42no—
137C23×C61—22no—
138C23×C61—24no—
139C23×C61—42no—
140C23×C61—44no—
141C23×C61C22×D411yescharacteristic
142C23×C61C22×D412yes—
143C23×C61C2×#4112yes—
144C23×C612−1+412yes—
145Q8×C61—82no—
146D4×C61—42no—
147D4×C61—42no—
148D4×C61—44no—
149D4×C61—82no—
150C2×C4×C61—21no—
151C2×C4×C61—21no—
152C2×C4×C61—21no—
153C2×C4×C61—21no—
154C2×C4×C61—28no—
155C2×C4×C61—22no—
156C2×C4×C61—22no—
157C2×C4×C61—22no—
158C2×C4×C61—24no—
159C2×C4×C61—24no—
160C2×C4×C61—42no—
161C2×C4×C61—44no—
162C2×C4×C61—42no—
163C2×C4×C61—84no—
164C2×C4×C61#12012yes—
165C2×C4×C61#12012yes—
166C2×Dic61—24no—
167C2×Dic61—28no—
168C2×Dic61—24no—
169C2×Dic61—42no—
170C2×Dic61—42no—
171C2×Dic61—42no—
172C2×Dic61—44no—
173C2×Dic61—42no—
174C2×Dic61—44no—
175C2×Dic61—44no—
176C2×Dic61—44no—
177#24113—611nomaximal, Sylow
178C22×C61—22no—
179C22×C61—24no—
180C22×C61—42no—
181C22×C61—42no—
182C22×C61—42no—
183C22×C61—44no—
184C22×C61—84no—
185C22×C61C2×#12011yescharacteristic
186C22×C61#52511yescharacteristic
187C22×C61#53611yescharacteristic
188C22×C61#54714yes—
189C4×C61—22no—
190C4×C61—22no—
191C4×C61—42no—
192C4×C61—42no—
193C4×C61—44no—
194C4×C61—42no—
195Dic61—44no—
196Dic61—44no—
197Dic61—84no—
198#3491D6111yescharacteristic
199#2853—612no—
200#2296—611no—
201#2062—612no—
202#2044—612no—
203#1899—612no—
204#1867—614no—
205#1820—611no—
206C2×C61—21no—
207C2×C61—24no—
208C2×C61—21no—
209C2×C61—82no—
210C2×C61#240611yescharacteristic
211C2×C61#302512yes—
212#552—24no—
213#551—24no—
214#543C2×D6111yescharacteristic
215#539—22no—
216#534C2×D6111yescharacteristic
217#529—22no—
218C2×#123C2×D6112yes—
219#518C2×D6112yes—
220C2×#117—611no—
221C2×#117C2×D6111yescharacteristic
222C22×C42—611no—
223C2×#99—614no—
224#389—1222no—
225#388—1224no—
226#387—1224no—
227#386—614no—
228#385—614no—
229#385—612no—
230#385—612no—
231#381—612no—
232C4×C4⋊C4—1222no—
233#342—612no—
234#342—614no—
235#342—612no—
236C61#2411311yescharacteristic, complemented, Sylow
237C22×D4C22×D6112yes—
238C23×C4—612no—
239C23×C4C22×D6111yescharacteristic
240#126—24no—
241#126—24no—
242#125—22no—
243#125—24no—
244#124—22no—
245#123—24no—
246#123—22no—
247#123—22no—
248#123—24no—
249#123—22no—
250C22⋊Q8—24no—
251C22⋊Q8—22no—
252C22⋊Q8—24no—
253#121—22no—
254#120—22no—
255C4×Q8—22no—
256C4×D4—22no—
257C4×D4—24no—
258C4×D4—24no—
259#117—24no—
260#117—24no—
261#117—22no—
262#117—1224no—
263C2×C4⋊C4—1221no—
264C2×C4⋊C4—1222no—
265C2×C4⋊C4—1222no—
266C2×C4⋊C4—1224no—
267C2×C4⋊C4C22⋊D6112yes—
268C2×C4⋊C4C22⋊D6114yes—
269C2×C22⋊C4—22no—
270C2×C22⋊C4—612no—
271C2×C22⋊C4—614no—
272C2×C22⋊C4—1224no—
273C2×C22⋊C4C22⋊D6114yes—
274C2×C22⋊C4C22×D6111yescharacteristic
275C2×C22⋊C4C22×D6111yescharacteristic
276C2×C22⋊C4C22×D6112yes—
277C2×C42—1221no—
278C2×C42—1221no—
279C2×C42—1224no—
280C2×C42—1221no—
281C2×C42C22⋊D6112yes—
282#99—1228no—
283#95—1222no—
284#95—1222no—
285#95—1224no—
286#95—1224no—
287#95—1224no—
288C24C2×C22⋊D6112yes—
289C24C23×D6111yescharacteristic
290C2×Q8—22no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—24no—
294C2×D4—24no—
295C2×D4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—1224no—
300C22×C4—1222no—
301C22×C4—1222no—
302C22×C4—1224no—
303C22×C4—2444no—
304C22×C4C2×C22⋊D6112yes—
305C22×C4C2×C22⋊D6114yes—
306C22×C4C2×C22⋊D6111yescharacteristic
307C22×C4C2×C22⋊D6111yescharacteristic
308C22×C4C2×C22⋊D6112yes—
309C22×C4C2×C22⋊D6112yes—
310C22×C4C2×C22⋊D6114yes—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—1224no—
319C4⋊C4—2448no—
320C4⋊C4—2444no—
321C4⋊C4—2444no—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—22no—
325C22⋊C4—22no—
326C22⋊C4—22no—
327C22⋊C4—22no—
328C22⋊C4—24no—
329C22⋊C4—24no—
330C22⋊C4—614no—
331C22⋊C4—1222no—
332C22⋊C4—1224no—
333C22⋊C4—1224no—
334C22⋊C4—2448no—
335C42—24no—
336C42—22no—
337C42—22no—
338C42—1224no—
339C42—2442no—
340C42—2442no—
341C23—22no—
342C23—24no—
343C23—42no—
344C23—44no—
345C23C22×C22⋊D6111yescharacteristic
346C23C22×C22⋊D6112yes—
347C23#41×D6112yes—
348C23#1451982412yes—
349Q8—82no—
350D4—42no—
351D4—42no—
352D4—44no—
353D4—82no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—28no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—22no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—84no—
368C2×C4—1228no—
369C2×C4—1224no—
370C2×C4—1224no—
371C2×C4—2444no—
372C2×C4—2444no—
373C2×C4—2444no—
374C2×C4—2442no—
375C2×C4—2442no—
376C2×C4—2442no—
377C2×C4—2444no—
378C2×C4—2442no—
379C2×C4#1451974612yes—
380C2×C4#1451974612yes—
381C22—22no—
382C22—24no—
383C22—42no—
384C22—42no—
385C22—44no—
386C22—42no—
387C22—84no—
388C22C2×#1451974611yescharacteristic
389C22#1684026911yescharacteristic
390C22#1684033311yescharacteristic
391C22#1684041114yes—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—44no—
396C4—42no—
397C4—42no—
398C4—4884no—
399C4—2444no—
400C4—2444no—
401C2—24no—
402C2—21no—
403C2—21no—
404C2—82no—
405C2#2144915011yescharacteristic
406C2#2145521112yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3491C2
socleC22×C61C2×#120
prime radicalC61×#534C22
hypercenter#3491D61
hypocenterC61#24113
LCSG ▷ C24×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3491

Complements

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

NH ≅ G/Ntype
1. #21454138C2semidirect
2. #21448301C2semidirect
3. #21447171C2semidirect
5. #21446278C2semidirect
6. #21446100C2semidirect
7. #21445896C2semidirect
236. C61#24113semidirect

Maximal subgroups

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

subgroupindex
#214541382
#214483012
#214471712
#214470672
#214462782
#214461002
#214458962
C61×#34912
#2411361