Finite Group Databasebeta

← #30595841

Subgroups — #30595841

Order 15616.

640 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 94 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#21461711C211yesmaximal, characteristic, complemented
2#21461546C211yesmaximal, characteristic, complemented
3#21460635C214yesmaximal, complemented
4#21454161C211yesmaximal, characteristic
5C2×#14519834—21no—
6C2×#14519834—21no—
7C2×#14519832—21no—
8#16840628—21no—
9#16840628C2212yes—
10#16840628C2212yes—
11C2×#14519748—21no—
12C2×#14519748—21no—
13C2×#14519748C412yes—
14C2×C42×D61C2211yescharacteristic
15#16839665—21no—
16#16839665—21no—
17#16839665C2212yes—
18#16839632C411yescharacteristic
19C2×#127×C61C411yescharacteristic
20C22×#92420—22no—
21C22×#92420—22no—
22#14519834—24no—
23#14519834—42no—
24#14519832—42no—
25C22×C22⋊D61—22no—
26C22×C4×D61C2×C411yescharacteristic
27C22×C4×D61D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29#14519748—44no—
30#14519748—44no—
31#14519748—44no—
32C2×C4⋊Dic61—21no—
33C2×C4⋊Dic61—21no—
34C2×C4⋊Dic61C2×C412yes—
35C2×C22⋊Dic61—22no—
36C2×C22⋊Dic61—22no—
37C2×C22⋊Dic61—22no—
38C2×C4×Dic61D411yescharacteristic
39C2×C4×Dic61D411yescharacteristic
40C2×C4×Dic61C2311yescharacteristic
41C42×D61—42no—
42C22×D4×C61—21no—
43C22×D4×C61—21no—
44C22×D4×C61C2×C412yes—
45#127×C61—42no—
46C2×C42×C61C2×C411yescharacteristic
47C24×C61—22no—
48C23×D61C22⋊C411yescharacteristic
49C2×D4×C61—24no—
50C2×D4×C61—42no—
51C2×D4×C61—42no—
52C2×D4×C61—42no—
53C2×D4×C61—42no—
54C2×D4×C61—42no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C2×C22⋊D61—44no—
62C2×C22⋊D61—44no—
63C2×C22⋊D61—44no—
64C22×Dic61—22no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C2×D411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—22no—
73C2×C4×D61—22no—
74C2×C4×D61—41no—
75C2×C4×D61—41no—
76C2×C4×D61—41no—
77C2×#92420—24no—
78C2×#92420—42no—
79C2×#92420—42no—
80C2×#92420—42no—
81C2×#92420—42no—
82C2×#92420—42no—
83C42×C61—22no—
84C22⋊Dic61—44no—
85C22⋊Dic61—44no—
86C22⋊Dic61—44no—
87C4⋊Dic61—42no—
88C4⋊Dic61—42no—
89C4⋊Dic61—42no—
90C4×Dic61—22no—
91C4×Dic61—41no—
92C4×Dic61—41no—
93C23×C61—42no—
94C23×C61—42no—
95C23×C61—42no—
96C23×C61—42no—
97C23×C61—42no—
98C23×C61—42no—
99C23×C61—42no—
100C23×C61C2×C22⋊C411yescharacteristic
101C22×D61—21no—
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61#10012yes—
108D4×C61—84no—
109D4×C61—84no—
110D4×C61—84no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61#9912yes—
118C2×C4×C61#10412yes—
119C2×C4×C61#13112yes—
120C22⋊D61—88no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61—42no—
131C2×Dic61AGL(1,ℤ/8)12yes—
132C2×Dic61AGL(1,ℤ/8)12yes—
133C4×D61—44no—
134C4×D61—44no—
135C4×D61—44no—
136#92420—44no—
137#92420—82no—
138#7704—611nomaximal, Sylow
139C22×C61—82no—
140C22×C61—82no—
141C22×C61—82no—
142C22×C61—82no—
143C22×C61—82no—
144C22×C61—82no—
145C22×C61—82no—
146C22×C61#40811yescharacteristic
147C22×C61C2×#9911yescharacteristic
148C22×C61C2×#10011yescharacteristic
149C22×C61#41811yescharacteristic
150C22×C61#41811yescharacteristic
151C22×C61C2×#10411yescharacteristic
152C22×C61#42111yescharacteristic
153C2×D61—22no—
154C2×D61—22no—
155C2×D61—22no—
156C2×D61—41no—
157C2×D61—41no—
158C2×D61—41no—
159C2×D61—41no—
160C4×C61—22no—
161C4×C61—22no—
162C4×C61—22no—
163Dic61—22no—
164Dic61—22no—
165Dic61—41no—
166Dic61—82no—
167#2856—611no—
168#1817—611no—
169#1749—611no—
170#1270—614no—
171C2×C61—162no—
172C2×C61#147111yescharacteristic
173C2×C61#148411yescharacteristic
174C2×C61#149611yescharacteristic
175C2×C61#150711yescharacteristic
176C2×C61#151011yescharacteristic
177C2×C61#153011yescharacteristic
178C2×C61#155511yescharacteristic
179D61—42no—
180C2×#127#692311yescharacteristic
181C2×#121—612no—
182C2×#121—1221no—
183C2×#121—1221no—
184C22×C42—611no—
185C2×C22⋊C8—1221no—
186C2×C22⋊C8—1221no—
187C2×#97—1221no—
188#390—612no—
189#390—1221no—
190#390—1221no—
191#384—611no—
192#336—612no—
193#336—612no—
194#336—1221no—
195C61#770411yescharacteristic, complemented, Sylow
196C22×D4—21no—
197C22×D4—21no—
198C22×D4—1222no—
199C22×D4C2×#692312yes—
200C23×C4—611no—
201C23×C4—611no—
202C23×C4—611no—
203C22×C8—1222no—
204C22×C8—1222no—
205#127—42no—
206#121—2444no—
207#121—2444no—
208#121—2444no—
209C2×C4⋊C4—612no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C22⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C42—611no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—2442no—
219C2×C42C2×#692311yescharacteristic
220C22⋊C8—1224no—
221C22⋊C8—2442no—
222#97—2442no—
223C24—22no—
224C24—611no—
225C2×D4—24no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—42no—
231C2×D4—42no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—2444no—
235C2×D4—2444no—
236C2×D4—2444no—
237C22×C4—611no—
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—1222no—
241C22×C4—1222no—
242C22×C4—1222no—
243C22×C4—1222no—
244C22×C4—1222no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—2441no—
248C22×C4—2441no—
249C22×C4—2441no—
250C22×C4C22⋊#692311yescharacteristic
251C22×C4C22⋊#692311yescharacteristic
252C22×C4C22×#692311yescharacteristic
253C2×C8—1224no—
254C2×C8—2442no—
255C2×C8—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C2×C8—2442no—
259C4⋊C4—2442no—
260C4⋊C4—2442no—
261C4⋊C4—2442no—
262C22⋊C4—2444no—
263C22⋊C4—2444no—
264C22⋊C4—2444no—
265C42—22no—
266C42—1222no—
267C42—2441no—
268C42—2441no—
269C23—42no—
270C23—42no—
271C23—42no—
272C23—42no—
273C23—42no—
274C23—42no—
275C23—42no—
276C23—612no—
277C23—1221no—
278C23—1221no—
279C23—1221no—
280C23—1221no—
281C23—1221no—
282C23—1221no—
283C23C2×C22⋊#692311yescharacteristic
284D4—84no—
285D4—84no—
286D4—84no—
287D4—4888no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—612no—
295C2×C4—612no—
296C2×C4—1221no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—1221no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—2444no—
304C2×C4—2444no—
305C2×C4—2444no—
306C2×C4—2442no—
307C2×C4—2442no—
308C2×C4—2442no—
309C2×C4#1451983512yes—
310C2×C4#1451984112yes—
311C2×C4#1451986412yes—
312C8—4882no—
313C8—2444no—
314C22—82no—
315C22—82no—
316C22—82no—
317C22—82no—
318C22—82no—
319C22—82no—
320C22—82no—
321C22—1222no—
322C22—1222no—
323C22—1222no—
324C22—2441no—
325C22—2441no—
326C22—2441no—
327C22—2441no—
328C22#1684073611yescharacteristic
329C22C2×#1451983511yescharacteristic
330C22D61⋊#10011yescharacteristic
331C22#1684075411yescharacteristic
332C22#1684075411yescharacteristic
333C22C2×#1451984111yescharacteristic
334C22#1684076011yescharacteristic
335C4—22no—
336C4—22no—
337C4—22no—
338C4—1222no—
339C4—1222no—
340C4—4882no—
341C4—2441no—
342C2—162no—
343C2—2442no—
344C2#2146109411yescharacteristic
345C2#2146113911yescharacteristic
346C2#2146117911yescharacteristic
347C2#2146121511yescharacteristic
348C2#2146122411yescharacteristic
349C2#2146128911yescharacteristic
350C2#2146134711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×#127×C61C4
socleC23×C61C2×C22⋊C4
prime radical#21454161C2
cosocleC2×C4×Dic61C23
hypercenterC2×#127#6923
hypocenterC61#7704
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. #21461711C2semidirect
2. #21461546C2semidirect
3. #21460635C2semidirect
195. C61#7704semidirect

Maximal subgroups

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

subgroupindex
#214617112
#214615462
#214606352
#214541612
#770461