Finite Group Databasebeta

← #30601806

Subgroups — #30601806

Order 15616.

716 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 130 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
2C2×#16840615C214yesmaximal
3C4×(C2×#6923)⋊C4C211yesmaximal, characteristic
4#21454172C211yesmaximal, characteristic
5C22×C4×#6923—21no—
6C2×C4⋊#92420—21no—
7C2×#14519834C2212yes—
8C2×(C2×#6923)⋊C4—21no—
9C2×(C2×#6923)⋊C4C2212yes—
10#16840628—21no—
11#16840615—28no—
12C2×#14519777C411yescharacteristic, complemented
13C2×#14519757C2212yes—
14C2×C42×D61C2211yescharacteristic
15#16839667C412yescomplemented
16#16839661—21no—
17#16839661—21no—
18#16839619—21no—
19#16839619—21no—
20C61×#384C411yescharacteristic, complemented
21C23×#6923—22no—
22C22×#92420—22no—
23C2×C4×#6923—24no—
24C2×C4×#6923—42no—
25C2×C4×#6923—42no—
26C4⋊#92420—42no—
27#14519834—24no—
28(C2×#6923)⋊C4—24no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4D414yes—
31C22×Dic122C2×C412yes—
32C22×C4×D61D411yescharacteristic
33C22×C4×D61D411yescharacteristic
34C22×C4×D61C2311yescharacteristic
35#14519777—42no—
36#14519757—24no—
37#14519757—24no—
38C2×C4⋊Dic61—21no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—22no—
41C2×#11758198—22no—
42C2×C4×Dic61C2×C411yescharacteristic
43C2×C4×Dic61D411yescharacteristic
44C2×C4×Dic61D411yescharacteristic
45C42×D61—42no—
46#14519684—22no—
47#14519684—22no—
48C2×C4⋊C4×C61—21no—
49C2×C4⋊C4×C61—21no—
50C2×C4⋊C4×C61C2×C412yes—
51C2×C42×C61C2×C411yescharacteristic
52C23×D61C2×D411yescharacteristic
53C22×#6923—24no—
54C22×#6923—24no—
55C22×#6923—24no—
56C22×#6923—42no—
57C22×#6923—42no—
58C22×#6923—42no—
59C22×#6923—42no—
60C22×C4×C61—22no—
61C22×C4×C61C22⋊C411yescharacteristic
62C22×C4×C61C22⋊C411yescharacteristic
63C22×C4×C61C22×C411yescharacteristic
64C22×Dic61—22no—
65C22×Dic61C22⋊C411yescharacteristic
66C22×Dic61C22⋊C411yescharacteristic
67C22×Dic61C2×D411yescharacteristic
68C2×C4×D61—22no—
69C2×C4×D61—22no—
70C2×C4×D61—22no—
71C2×C4×D61—22no—
72C2×C4×D61—41no—
73C2×C4×D61—41no—
74C2×C4×D61—41no—
75C2×C4×D61SD1612yes—
76C2×C4×D61Q1612yes—
77C2×C4×D61C2×D412yes—
78C2×C4×D61#4112yes—
79C2×Dic122—24no—
80C2×Dic122—42no—
81C2×Dic122—42no—
82C2×#92420—42no—
83C2×#92420—42no—
84C2×#92420—42no—
85C4×#6923—48no—
86C4⋊C4×C61—24no—
87C4⋊C4×C61—42no—
88C4⋊C4×C61—42no—
89C42×C61—22no—
90#11758198—44no—
91C4⋊Dic61—42no—
92C4⋊Dic61—42no—
93C4⋊Dic61—42no—
94C4⋊Dic61—42no—
95C4×Dic61—22no—
96C4×Dic61—41no—
97C4×Dic61—41no—
98C23×C61C2×C22⋊C411yescharacteristic
99C22×D61—21no—
100C22×D61—21no—
101C22×D61—21no—
102C22×D61—21no—
103C22×D61#12112yes—
104C22×D61C2×SD1611yescharacteristic
105C22×D61C2×Q1611yescharacteristic
106C22×D61#13712yes—
107C2×#6923—28no—
108C2×#6923—44no—
109C2×#6923—44no—
110C2×#6923—44no—
111C2×#6923—44no—
112C2×#6923—44no—
113C2×#6923—44no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—21no—
118C2×C4×C61—42no—
119C2×C4×C61—42no—
120C2×C4×C61—42no—
121C2×C4×C61#9912yes—
122C2×C4×C61#10312yes—
123C2×C4×C61#10412yes—
124C2×C4×C61#10412yes—
125C2×C4×C61C4×D412yes—
126C2×Dic61—21no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—21no—
132C2×Dic61—21no—
133C2×Dic61—42no—
134C2×Dic61—42no—
135C2×Dic61—42no—
136C2×Dic61#10312yes—
137C2×Dic61#12411yescharacteristic
138C2×Dic61#12711yescharacteristic
139C4×D61—24no—
140C4×D61—24no—
141C4×D61—42no—
142C4×D61—42no—
143C4×D61—44no—
144Dic122—84no—
145#92420—82no—
146#9711—611nomaximal, Sylow
147C22×C61#39011yescharacteristic
148C22×C61C2×#9911yescharacteristic
149C22×C61#41011yescharacteristic
150C22×C61C2×#10311yescharacteristic
151C22×C61#41911yescharacteristic
152C22×C61C2×#10411yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C2×D61—22no—
155C2×D61—22no—
156C2×D61—21no—
157C2×D61—21no—
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—22no—
161C2×D61—22no—
162C2×D61#46112yes—
163C2×D61#46212yes—
164#6923—48no—
165C4×C61—22no—
166C4×C61—22no—
167C4×C61—82no—
168C4×C61#32814yescomplemented
169Dic61—21no—
170Dic61—21no—
171Dic61—22no—
172Dic61—41no—
173Dic61—82no—
174#2857—611no—
175#1797—611no—
176C2×#328—614no—
177C4×#95—611no—
178C2×C61C2×#32811yescharacteristic
179C2×C61#150311yescharacteristic
180C2×C61#150611yescharacteristic
181C2×C61#150811yescharacteristic
182C2×C61#196211yescharacteristic
183C2×C61#196411yescharacteristic
184C2×C61#198111yescharacteristic
185D61—22no—
186D61—22no—
187C2×C4⋊Q8—611no—
188C2×C22⋊Q8—612no—
189C22×C42—611no—
190C22×C42—1221no—
191C2×C4⋊C8—1221no—
192C2×C22⋊C8—612no—
193#391—612no—
194#389—1221no—
195#389—1221no—
196#384#692311yescharacteristic, complemented
197#380—1221no—
198#380—1221no—
199C2×#95—612no—
200C2×#95—1221no—
201#336—1221no—
202#328—1228no—
203C61#971111yescharacteristic, complemented, Sylow
204C22×Q8—612no—
205C23×C4—611no—
206C23×C4—611no—
207C23×C4—611no—
208C23×C4—1222no—
209C22×C8—1222no—
210C4⋊Q8—2442no—
211C22⋊Q8—1224no—
212C22⋊Q8—1224no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—1222no—
216C2×C4⋊C4—1221no—
217C2×C4⋊C4—1221no—
218C2×C4⋊C4C2×#692312yes—
219C2×C22⋊C4—1222no—
220C2×C42—611no—
221C2×C42—611no—
222C2×C42—611no—
223C2×C42—1224no—
224C2×C42—2442no—
225C2×C42—2442no—
226C2×C42—2442no—
227C2×C42C2×#692311yescharacteristic
228C4⋊C8—2442no—
229C22⋊C8—1224no—
230#95—614no—
231#95—1222no—
232#95—1222no—
233#95—1222no—
234#95—1224no—
235C24—611no—
236C2×Q8—1224no—
237C2×Q8—2442no—
238C2×Q8—2442no—
239C22×C4—22no—
240C22×C4—611no—
241C22×C4—611no—
242C22×C4—612no—
243C22×C4—612no—
244C22×C4—612no—
245C22×C4—612no—
246C22×C4—611no—
247C22×C4—1222no—
248C22×C4—1222no—
249C22×C4—1222no—
250C22×C4—1224no—
251C22×C4—1224no—
252C22×C4—1224no—
253C22×C4—1222no—
254C22×C4—1222no—
255C22×C4—2441no—
256C22×C4—2441no—
257C22×C4—2441no—
258C22×C4—2442no—
259C22×C4—2442no—
260C22×C4—2442no—
261C22×C4—2442no—
262C22×C4C22⋊#692311yescharacteristic
263C22×C4C22⋊#692311yescharacteristic
264C22×C4C22×#692311yescharacteristic
265C2×C8—2442no—
266C2×C8—2442no—
267C2×C8—2442no—
268C4⋊C4—24no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—2442no—
272C4⋊C4—2442no—
273C4⋊C4—2442no—
274C4⋊C4—2442no—
275C22⋊C4—2444no—
276C42—22no—
277C42—1222no—
278C42—2441no—
279C42—2441no—
280C42—2448no—
281C23—612no—
282C23—611no—
283C23—611no—
284C23—612no—
285C23—1221no—
286C23—1221no—
287C23—1221no—
288C23—1221no—
289C23C2×C22⋊#692311yescharacteristic
290Q8—4884no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—611no—
299C2×C4—611no—
300C2×C4—612no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—1221no—
305C2×C4—1224no—
306C2×C4—1224no—
307C2×C4—1228no—
308C2×C4—1221no—
309C2×C4—1221no—
310C2×C4—1221no—
311C2×C4—2442no—
312C2×C4—2442no—
313C2×C4—2444no—
314C2×C4—2442no—
315C2×C4—2442no—
316C2×C4—2442no—
317C2×C4—2444no—
318C2×C4—2444no—
319C2×C4—2444no—
320C2×C4—2444no—
321C2×C4—2444no—
322C2×C4—2444no—
323C2×C4#1451983612yes—
324C2×C4#1451984012yes—
325C2×C4#1451984112yes—
326C2×C4#1451984112yes—
327C2×C4D4×#692312yes—
328C8—4882no—
329C22—612no—
330C22—612no—
331C22—1222no—
332C22—1221no—
333C22—1221no—
334C22—1222no—
335C22—1221no—
336C22—1221no—
337C22—1222no—
338C22—1222no—
339C22#1684071411yescharacteristic
340C22C2×#1451983611yescharacteristic
341C22#1684074311yescharacteristic
342C22C2×#1451984011yescharacteristic
343C22#1684075511yescharacteristic
344C22C2×#1451984111yescharacteristic
345C22C2×#1451984111yescharacteristic
346C4—22no—
347C4—22no—
348C4—82no—
349C4—1222no—
350C4—1221no—
351C4—1221no—
352C4—4882no—
353C4—2448no—
354C4—2441no—
355C4#1684061514yescomplemented
356C2—1222no—
357C2—1222no—
358C2C2×#1684061511yescharacteristic
359C2#2146120211yescharacteristic
360C2#2146121111yescharacteristic
361C2#2146121811yescharacteristic
362C2#2146195311yescharacteristic
363C2#2146195911yescharacteristic
364C2#2146198511yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6923
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#384C4
socleC23×C61C2×C22⋊C4
prime radicalC23×D61C2×D4
cosocleC22×C4×D61C23
hypercenter#384#6923
hypocenterC61#9711
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
12. C2×#14519777C4semidirect
15. #16839667C4semidirect
20. C61×#384C4semidirect
168. C4×C61#328semidirect
196. #384#6923semidirect
203. C61#9711semidirect
355. C4#16840615semidirect

Maximal subgroups

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

subgroupindex
#214616742
C2×#168406152
C4×(C2×#6923)⋊C42
#214541722
#971161