Finite Group Databasebeta

← #29712502

Subgroups — #29712502

Order 15616.

1210 conjugacy classes of subgroups; only the 191 normal subgroups are shown.

normal subgroups only

This group has 1210 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#21451540C212yesmaximal, complemented
2#21448403C211yesmaximal, characteristic, complemented
3#21446985C212yesmaximal, complemented
4#21446948C212yesmaximal
5#21446278C212yesmaximal, complemented
6#21446232C214yesmaximal, complemented
7#21445896C211yesmaximal, characteristic, complemented
8C61×#3491C211yesmaximal, characteristic
9C23×Dic122C2211yescharacteristic
10C2×#14519758C2214yes—
11C2×#14519736C2211yescharacteristic
12C2×#14519692C2214yes—
13#16839682—24no—
14#16839674—24no—
15#16839668—22no—
16#16839667—22no—
17#16839625C2212yes—
18#16839623C2214yes—
19#16839623C2212yes—
20#16839623C2212yes—
21#16839529C2212yes—
22#16839529C2212yes—
23#16839529C2214yes—
24C61×#552—24no—
25C61×#551—24no—
26C61×#543C2211yescharacteristic
27C61×#539—22no—
28C61×#534C2211yescharacteristic
29C61×#529—22no—
30C2×#123×C61C2212yes—
31C61×#518C2212yes—
32C2×#117×C61C2211yescharacteristic
33C22×Dic122—21no—
34C22×Dic122—21no—
35C22×Dic122—24no—
36C22×Dic122—24no—
37C22×Dic122—42no—
38C23×Dic61C2312yes—
39#14519758—28no—
40#14519758—44no—
41#14519736—42no—
42C2×C4⋊Dic61—21no—
43C2×#11758196—24no—
44C2×C22⋊Dic61—24no—
45C2×C22⋊Dic61C2312yes—
46C2×C22⋊Dic61C2314yes—
47C2×C4×Dic61—21no—
48#14519692—44no—
49#14519684—28no—
50#14519684—24no—
51#14519684—24no—
52#14519684—24no—
53C22×D4×C61C2312yes—
54C23×C4×C61C2311yescharacteristic
55#126×C61—24no—
56#126×C61—24no—
57#125×C61—24no—
58#125×C61—22no—
59#124×C61—22no—
60#123×C61—24no—
61#123×C61—22no—
62#123×C61—22no—
63#123×C61—24no—
64#123×C61—22no—
65C22⋊Q8×C61—24no—
66C22⋊Q8×C61—22no—
67C22⋊Q8×C61—24no—
68#121×C61—22no—
69#120×C61—22no—
70C4×Q8×C61—22no—
71C4×D4×C61—22no—
72C4×D4×C61—24no—
73C4×D4×C61—24no—
74#117×C61—24no—
75#117×C61—24no—
76#117×C61—22no—
77C2×C4⋊C4×C61D414yes—
78C2×C4⋊C4×C61D412yes—
79C2×C22⋊C4×C61—22no—
80C2×C22⋊C4×C61D414yes—
81C2×C22⋊C4×C61C2311yescharacteristic
82C2×C22⋊C4×C61C2312yes—
83C2×C22⋊C4×C61C2311yescharacteristic
84C2×C42×C61D412yes—
85C24×C61C2×D412yes—
86C24×C61C2411yescharacteristic
87C2×Q8×C61—22no—
88C2×D4×C61—22no—
89C2×D4×C61—22no—
90C2×D4×C61—24no—
91C2×D4×C61—24no—
92C2×D4×C61—22no—
93C22×C4×C61—22no—
94C22×C4×C61—22no—
95C22×C4×C61—24no—
96C22×C4×C61C2×D412yes—
97C22×C4×C61C2×D411yescharacteristic
98C22×C4×C61C2×D414yes—
99C22×C4×C61C2×D411yescharacteristic
100C22×C4×C61C2×D412yes—
101C22×C4×C61C2×D412yes—
102C22×C4×C61C2×D414yes—
103C22×Dic61—22no—
104C22×Dic61—24no—
105C22×Dic61—22no—
106C22×Dic61—24no—
107C22×Dic61—22no—
108C22×Dic61—24no—
109C22×Dic61#4114yes—
110C2×Dic122—24no—
111C2×Dic122—24no—
112C2×Dic122—416no—
113C2×Dic122—42no—
114C2×Dic122—42no—
115C2×Dic122—42no—
116C2×Dic122—42no—
117C4⋊C4×C61—24no—
118C4⋊C4×C61—24no—
119C4⋊C4×C61—22no—
120C4⋊C4×C61—22no—
121C4⋊C4×C61—24no—
122C4⋊C4×C61—24no—
123C4⋊C4×C61—24no—
124C22⋊C4×C61—24no—
125C22⋊C4×C61—24no—
126C22⋊C4×C61—22no—
127C22⋊C4×C61—22no—
128C22⋊C4×C61—22no—
129C22⋊C4×C61—22no—
130C22⋊C4×C61—24no—
131C22⋊C4×C61—24no—
132C42×C61—24no—
133C42×C61—22no—
134C42×C61—22no—
135C22⋊Dic61—24no—
136C22⋊Dic61—24no—
137C22⋊Dic61—24no—
138C22⋊Dic61—48no—
139C4⋊Dic61—42no—
140#11758196—48no—
141C4×Dic61—42no—
142C23×C61—22no—
143C23×C61—24no—
144C23×C61—42no—
145C23×C61—44no—
146C23×C61C22×D411yescharacteristic
147C23×C61C22×D412yes—
148C23×C61C2×#4112yes—
149C23×C61H(5,𝔽2)12yes—
150Q8×C61—82no—
151D4×C61—42no—
152D4×C61—42no—
153D4×C61—44no—
154D4×C61—82no—
155C2×C4×C61—21no—
156C2×C4×C61—21no—
157C2×C4×C61—21no—
158C2×C4×C61—21no—
159C2×C4×C61—28no—
160C2×C4×C61—22no—
161C2×C4×C61—22no—
162C2×C4×C61—22no—
163C2×C4×C61—24no—
164C2×C4×C61—24no—
165C2×C4×C61—42no—
166C2×C4×C61—44no—
167C2×C4×C61—42no—
168C2×C4×C61—84no—
169C2×C4×C61#12012yes—
170C2×C4×C61#12012yes—
171C2×Dic61—24no—
172C2×Dic61—28no—
173C2×Dic61—42no—
174C2×Dic61—42no—
175C2×Dic61—44no—
176C2×Dic61—44no—
177C2×Dic61—48no—
178C2×Dic61—42no—
179C2×Dic61—44no—
180Dic122—48no—
181Dic122—48no—
182#23989—611nomaximal, Sylow
183C22×C61—22no—
184C22×C61—24no—
185C22×C61—42no—
186C22×C61—42no—
187C22×C61—42no—
188C22×C61—44no—
189C22×C61—84no—
190C22×C61C2×#12011yescharacteristic
191C22×C61#52511yescharacteristic
192C22×C61#53511yescharacteristic
193C22×C61#54614yes—
194C4×C61—22no—
195C4×C61—22no—
196C4×C61—42no—
197C4×C61—42no—
198C4×C61—44no—
199C4×C61—42no—
200Dic61—44no—
201Dic61—82no—
202Dic61—84no—
203#3491D6111yescharacteristic
204#2614—612no—
205#2310—611no—
206#2030—612no—
207#2022—612no—
208#1899—612no—
209#1890—614no—
210#1820—611no—
211C2×C61—21no—
212C2×C61—24no—
213C2×C61—21no—
214C2×C61—82no—
215C2×C61#240411yescharacteristic
216C2×C61#302812yes—
217C23×Q8—611no—
218#552—24no—
219#551—24no—
220#543C2×D6111yescharacteristic
221#539—22no—
222#534C2×D6111yescharacteristic
223#529—22no—
224C2×#123—614no—
225C2×#123C2×D6112yes—
226#518C2×D6112yes—
227C2×#117—611no—
228C2×#117C2×D6111yescharacteristic
229C2×#99—614no—
230#395—1224no—
231#393—1224no—
232#391—1222no—
233#391—1222no—
234#381—612no—
235#381—612no—
236#381—612no—
237#381—614no—
238#342—612no—
239#342—612no—
240#342—614no—
241C61#2398911yescharacteristic, complemented, Sylow
242C22×Q8—1224no—
243C22×Q8—1221no—
244C22×Q8—1221no—
245C22×Q8—1224no—
246C22×Q8—2442no—
247C22×D4C22×D6112yes—
248C23×C4—612no—
249C23×C4C22×D6111yescharacteristic
250#126—24no—
251#126—24no—
252#125—22no—
253#125—24no—
254#124—22no—
255#123—24no—
256#123—22no—
257#123—22no—
258#123—24no—
259#123—22no—
260#123—1228no—
261#123—2444no—
262C22⋊Q8—24no—
263C22⋊Q8—22no—
264C22⋊Q8—24no—
265#121—22no—
266#120—22no—
267C4×Q8—22no—
268C4×D4—22no—
269C4×D4—24no—
270C4×D4—24no—
271#117—24no—
272#117—24no—
273#117—22no—
274#117—2442no—
275C2×C4⋊C4—1221no—
276C2×C4⋊C4—1224no—
277C2×C4⋊C4C22⋊D6112yes—
278C2×C4⋊C4C22⋊D6114yes—
279C2×C22⋊C4—22no—
280C2×C22⋊C4—612no—
281C2×C22⋊C4—614no—
282C2×C22⋊C4—1224no—
283C2×C22⋊C4C22⋊D6114yes—
284C2×C22⋊C4C22×D6111yescharacteristic
285C2×C22⋊C4C22×D6111yescharacteristic
286C2×C22⋊C4C22×D6112yes—
287C2×C42—1221no—
288C2×C42C22⋊D6112yes—
289#99—2444no—
290#95—1224no—
291#95—1224no—
292#95—1228no—
293#95—1224no—
294C24C2×C22⋊D6112yes—
295C24C23×D6111yescharacteristic
296C2×Q8—22no—
297C2×Q8—1224no—
298C2×Q8—1224no—
299C2×Q8—24416no—
300C2×Q8—2442no—
301C2×Q8—2442no—
302C2×Q8—2442no—
303C2×Q8—2442no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—24no—
307C2×D4—24no—
308C2×D4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—24no—
312C22×C4—614no—
313C22×C4—1222no—
314C22×C4—1224no—
315C22×C4—1222no—
316C22×C4—1224no—
317C22×C4—1222no—
318C22×C4—1224no—
319C22×C4C2×C22⋊D6112yes—
320C22×C4C2×C22⋊D6114yes—
321C22×C4C2×C22⋊D6111yescharacteristic
322C22×C4C2×C22⋊D6111yescharacteristic
323C22×C4C2×C22⋊D6112yes—
324C22×C4C2×C22⋊D6112yes—
325C22×C4C2×C22⋊D6114yes—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—24no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—2448no—
334C4⋊C4—2442no—
335C22⋊C4—24no—
336C22⋊C4—24no—
337C22⋊C4—22no—
338C22⋊C4—22no—
339C22⋊C4—22no—
340C22⋊C4—22no—
341C22⋊C4—24no—
342C22⋊C4—24no—
343C22⋊C4—1224no—
344C22⋊C4—1224no—
345C22⋊C4—1224no—
346C22⋊C4—2448no—
347C42—24no—
348C42—22no—
349C42—22no—
350C42—2442no—
351C23—22no—
352C23—24no—
353C23—42no—
354C23—44no—
355C23C22×C22⋊D6111yescharacteristic
356C23C22×C22⋊D6112yes—
357C23#41×D6112yes—
358C23#1451982312yes—
359Q8—82no—
360Q8—2448no—
361Q8—2448no—
362D4—42no—
363D4—42no—
364D4—44no—
365D4—82no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—28no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—42no—
377C2×C4—44no—
378C2×C4—42no—
379C2×C4—84no—
380C2×C4—1224no—
381C2×C4—1228no—
382C2×C4—2444no—
383C2×C4—2448no—
384C2×C4—2442no—
385C2×C4—2444no—
386C2×C4—2442no—
387C2×C4—2442no—
388C2×C4—2444no—
389C2×C4#1451974612yes—
390C2×C4#1451974612yes—
391C22—22no—
392C22—24no—
393C22—42no—
394C22—42no—
395C22—44no—
396C22—42no—
397C22—84no—
398C22C2×#1451974611yescharacteristic
399C22#1684026911yescharacteristic
400C22#1684032611yescharacteristic
401C22#1684040014yes—
402C4—22no—
403C4—22no—
404C4—42no—
405C4—44no—
406C4—42no—
407C4—42no—
408C4—4882no—
409C4—4884no—
410C4—2444no—
411C2—21no—
412C2—24no—
413C2—21no—
414C2—82no—
415C2#2144913411yescharacteristic
416C2#2145522512yes—
subgroupG/H
centerC22C2×#14519746
commutator, cosocleC24×C61C24
FrattiniC24C23×D61
FittingC61×#3491C2
socleC22×C61C2×#120
prime radicalC61×#534C22
hypercenter#3491D61
hypocenterC61#23989
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. #21451540C2semidirect
2. #21448403C2semidirect
3. #21446985C2semidirect
5. #21446278C2semidirect
6. #21446232C2semidirect
7. #21445896C2semidirect
241. C61#23989semidirect

Maximal subgroups

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

subgroupindex
#214515402
#214484032
#214469852
#214469482
#214462782
#214462322
#214458962
C61×#34912
#2398961