Finite Group Databasebeta

← #30598170

Subgroups — #30598170

Order 15616.

836 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 836 conjugacy classes of subgroups — too many to draw. Only its 146 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#21461674C211yesmaximal, characteristic
2C4×#14519834C211yesmaximal, characteristic
3C2×#16840612C214yesmaximal, complemented
4C2×#16840317C211yesmaximal, characteristic
5C2×C4⋊#92420C2212yes—
6C2×#14519834—21no—
7C2×#14519834—21no—
8C2×C4×#92420C2212yes—
9#16840628—21no—
10#16840628—21no—
11#16840612—28no—
12#16840317—28no—
13C2×#14519777C411yescharacteristic
14C2×#14519768—21no—
15C2×#14519757C412yes—
16C2×#14519753—21no—
17C2×Q8×Dic61—21no—
18C2×#14519742C2212yes—
19C2×#14519735—21no—
20C2×C42×D61C2211yescharacteristic
21C2×C4⋊Q8×C61C411yescharacteristic
22C22×#92420—22no—
23C22×#92420—22no—
24C4⋊#92420—24no—
25#14519834—24no—
26#14519834—42no—
27C4×#92420—24no—
28C22×Dic122C2×C412yes—
29C22×C4×D61C2×C411yescharacteristic
30C22×C4×D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32#14519777—24no—
33#14519768—28no—
34#14519757—24no—
35#14519757—24no—
36#14519753—28no—
37Q8×Dic61—28no—
38#14519742—24no—
39#14519742D418yes—
40#14519735—28no—
41C2×C4⋊Dic61—21no—
42C2×C4⋊Dic61—21no—
43C2×C4⋊Dic61—22no—
44C2×#11758196—22no—
45C2×#11758198—22no—
46C2×C4×Dic61—22no—
47C2×C4×Dic61D411yescharacteristic
48C2×C4×Dic61D411yescharacteristic
49C2×C4×Dic61C2311yescharacteristic
50C42×D61—24no—
51C22×Q8×C61—21no—
52C4⋊Q8×C61—24no—
53C2×C4⋊C4×C61—21no—
54C2×C4⋊C4×C61C2×C412yes—
55C2×C42×C61C2×C411yescharacteristic
56C23×D61C22⋊C411yescharacteristic
57C2×Q8×C61—24no—
58C2×Q8×C61—42no—
59C2×Q8×C61—42no—
60C22×C4×C61—22no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
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—22no—
75C2×C4×D61—41no—
76C2×C4×D61SD1612yes—
77C2×C4×D61Q1612yes—
78C2×Dic122—24no—
79C2×Dic122—24no—
80C2×Dic122—22no—
81C2×Dic122C22⋊C414yes—
82C2×#92420—24no—
83C2×#92420—42no—
84C2×#92420—42no—
85C2×#92420—42no—
86C2×#92420—42no—
87C2×#92420—42no—
88C4⋊C4×C61—24no—
89C4⋊C4×C61—22no—
90C4⋊C4×C61C22⋊C414yes—
91C42×C61—22no—
92#11758198—28no—
93C4⋊Dic61—24no—
94C4⋊Dic61—24no—
95C4⋊Dic61—28no—
96#11758196—28no—
97C4×Dic61—22no—
98C4×Dic61—22no—
99C4×Dic61—28no—
100C4×Dic61C2×D414yes—
101C23×C61C2×C22⋊C411yescharacteristic
102C22×D61—21no—
103C22×D61—21no—
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—21no—
107C22×D61—21no—
108C22×D61#10312yes—
109Q8×C61—48no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—24no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61#10312yes—
118C2×C4×C61#10412yes—
119C2×C4×C61C2×C22⋊C412yes—
120C2×C4×C61#13112yes—
121C2×C4×C61#13112yes—
122C2×Dic61—21no—
123C2×Dic61—21no—
124C2×Dic61—21no—
125C2×Dic61—24no—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—42no—
129C2×Dic61—42no—
130C2×Dic61#12012yes—
131C2×Dic61#12012yes—
132C2×Dic61C2×SD1611yescharacteristic
133C2×Dic61C2×Q1611yescharacteristic
134C2×Dic61#13712yes—
135C4×D61—24no—
136C4×D61—24no—
137C4×D61—44no—
138C4×D61—44no—
139Dic122—28no—
140Dic122—44no—
141#92420—44no—
142#92420—82no—
143#8314—611nomaximal, Sylow
144C22×C61#37911yescharacteristic
145C22×C61#40811yescharacteristic
146C22×C61#40811yescharacteristic
147C22×C61C2×#10311yescharacteristic
148C22×C61#41911yescharacteristic
149C22×C61C2×#10411yescharacteristic
150C22×C61#42111yescharacteristic
151C2×D61—22no—
152C2×D61—22no—
153C2×D61—22no—
154C2×D61—41no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—41no—
158C4×C61—22no—
159C4×C61—22no—
160C4×C61—44no—
161C4×C61#40814yes—
162Dic61—22no—
163Dic61—44no—
164Dic61—41no—
165Dic61#44812yes—
166Dic61#45112yes—
167C2×#533—611no—
168#1797—611no—
169C4×C22⋊C8—611no—
170C2×Q8⋊C8—614no—
171C2×C61#148111yescharacteristic
172C2×C61#148511yescharacteristic
173C2×C61#149311yescharacteristic
174C2×C61#149711yescharacteristic
175C2×C61#178011yescharacteristic
176C2×C61#178711yescharacteristic
177C2×C61#178911yescharacteristic
178D61—42no—
179#533—1228no—
180C2×C4⋊Q8—611no—
181C2×C4⋊Q8#692311yescharacteristic
182C2×#125—1221no—
183C2×C22⋊Q8—612no—
184C2×C22⋊Q8—1221no—
185C2×C4×Q8—612no—
186C2×C4×Q8—1221no—
187C2×#117—1221no—
188C22×C42—611no—
189C2×C4⋊C8—612no—
190C2×C22⋊C8—1221no—
191C2×C22⋊C8—1221no—
192C2×C4×C8—612no—
193#336—1221no—
194#336—1221no—
195Q8⋊C8—1228no—
196C61#831411yescharacteristic, complemented, Sylow
197C22×Q8—21no—
198C22×Q8—612no—
199C23×C4—611no—
200C23×C4—611no—
201C23×C4—611no—
202C22×C8—1222no—
203C22×C8—1222no—
204C4⋊Q8—24no—
205C4⋊Q8—1224no—
206#125—1228no—
207C22⋊Q8—1224no—
208C22⋊Q8—1228no—
209C22⋊Q8—1224no—
210C4×Q8—618no—
211C4×Q8—1224no—
212C4×Q8—1228no—
213#117—1228no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—1222no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4—1221no—
219C2×C4⋊C4C2×#692312yes—
220C2×C22⋊C4—1222no—
221C2×C42—611no—
222C2×C42—611no—
223C2×C42—611no—
224C2×C42—1224no—
225C2×C42—1222no—
226C2×C42C2×#692311yescharacteristic
227C4⋊C8—1224no—
228C22⋊C8—1224no—
229C22⋊C8—2442no—
230C4×C8—1224no—
231C24—611no—
232C2×Q8—24no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—614no—
236C2×Q8—1222no—
237C2×Q8—1224no—
238C2×Q8—1224no—
239C22×C4—22no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—611no—
245C22×C4—1222no—
246C22×C4—1222no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—1222no—
250C22×C4—1222no—
251C22×C4—1222no—
252C22×C4—1222no—
253C22×C4—2441no—
254C22×C4C22⋊#692311yescharacteristic
255C22×C4C22⋊#692311yescharacteristic
256C22×C4C22×#692311yescharacteristic
257C2×C8—1224no—
258C2×C8—2442no—
259C2×C8—2442no—
260C2×C8—2442no—
261C2×C8—2442no—
262C2×C8—2442no—
263C4⋊C4—24no—
264C4⋊C4—22no—
265C4⋊C4—1228no—
266C4⋊C4—1224no—
267C4⋊C4—1228no—
268C4⋊C4—1224no—
269C4⋊C4C22⋊#692314yes—
270C22⋊C4—1228no—
271C42—22no—
272C42—614no—
273C42—1222no—
274C42—1222no—
275C42—1228no—
276C23—612no—
277C23—1221no—
278C23—1221no—
279C23—1221no—
280C23—1221no—
281C23—1221no—
282C23—1221no—
283C23C2×C22⋊#692311yescharacteristic
284Q8—48no—
285Q8—1228no—
286Q8—2444no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—24no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—611no—
295C2×C4—611no—
296C2×C4—612no—
297C2×C4—612no—
298C2×C4—612no—
299C2×C4—1224no—
300C2×C4—1224no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1221no—
306C2×C4—1224no—
307C2×C4—2444no—
308C2×C4—2442no—
309C2×C4—2442no—
310C2×C4—2444no—
311C2×C4#1451984012yes—
312C2×C4#1451984112yes—
313C2×C4C2×C22⋊#692312yes—
314C2×C4#1451986512yes—
315C2×C4#1451986512yes—
316C8—4882no—
317C8—2444no—
318C22—1222no—
319C22—1222no—
320C22—1222no—
321C22—2441no—
322C22—2441no—
323C22—2441no—
324C22—2441no—
325C22#1684070311yescharacteristic
326C22#1684073711yescharacteristic
327C22#1684073711yescharacteristic
328C22C2×#1451984011yescharacteristic
329C22#1684075511yescharacteristic
330C22C2×#1451984111yescharacteristic
331C22#1684076011yescharacteristic
332C4—22no—
333C4—22no—
334C4—44no—
335C4—612no—
336C4—612no—
337C4—1222no—
338C4—2441no—
339C4—2444no—
340C4#1684073714yes—
341C2—2442no—
342C2#2146112911yescharacteristic
343C2#2146114311yescharacteristic
344C2#2146116911yescharacteristic
345C2#2146118311yescharacteristic
346C2#2146164411yescharacteristic
347C2#2146165811yescharacteristic
348C2#2146166011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C22⋊C4
cosocleC2×C4×Dic61C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#8314
LCSG ▷ C22×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. C2×#16840612C2semidirect
196. C61#8314semidirect

Maximal subgroups

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

subgroupindex
#214616742
C4×#145198342
C2×#168406122
C2×#168403172
#831461