Finite Group Databasebeta

← #30604870

Subgroups — #30604870

Order 15616.

436 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 436 conjugacy classes of subgroups — too many to draw. Only its 68 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
1C4×#14519835C211yesmaximal, characteristic
2#21461058C211yesmaximal, characteristic
3#21460814C212yesmaximal, complemented
4#21460799C212yesmaximal, complemented
5#21451737C211yesmaximal, characteristic
6#16840779—21no—
7C2×#14519835—21no—
8C4×C22⋊#6923—21no—
9#16840640C2212yes—
10#16840606C2212yes—
11#16840605—21no—
12C2×C4×C22⋊D61C2211yescharacteristic
13#16839693C411yescharacteristic, complemented
14#16839690—21no—
15#16839688C2211yescharacteristic
16#16839681C411yescharacteristic, complemented
17#16839677—21no—
18#16839673C2211yescharacteristic
19#16839643—21no—
20#16839633—21no—
21#16839625C411yescharacteristic, complemented
22C2×C22⋊Q8×C61C411yescharacteristic, complemented
23C2×C22⋊#6923—22no—
24C2×C4×#6923—41no—
25#14519844—41no—
26#14519835—24no—
27C22⋊#92420—22no—
28C4×#92420—41no—
29(C2×#6923)⋊C4—41no—
30C22×C22⋊D61D411yescharacteristic
31C22×C4×D61D411yescharacteristic
32C4×C22⋊D61—41no—
33C4×C22⋊D61—41no—
34C2×#11758196—21no—
35C2×#11758196—21no—
36C2×#11758196C2×C411yescharacteristic
37C2×C22⋊Dic61—21no—
38C2×C22⋊Dic61D411yescharacteristic
39C2×#11758198—21no—
40C2×#11758198—21no—
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61D411yescharacteristic
43#14519684—21no—
44#14519684—21no—
45#14519684—21no—
46#14519684—21no—
47#14519684—21no—
48#14519684C2×C411yescharacteristic
49#14519684C2×C411yescharacteristic
50C22×Q8×C61C2×C411yescharacteristic
51C23×C4×C61C2×C411yescharacteristic
52C22⋊Q8×C61—41no—
53C22⋊Q8×C61—41no—
54C2×C4⋊C4×C61—21no—
55C2×C4⋊C4×C61C2×C411yescharacteristic
56C2×C22⋊C4×C61—21no—
57C24×C61C22⋊C411yescharacteristic
58C23×D61C2×D411yescharacteristic
59C22×#6923—42no—
60C2×Q8×C61—41no—
61C2×Q8×C61—81no—
62C22×C4×C61—21no—
63C22×C4×C61—21no—
64C22×C4×C61—21no—
65C22×C4×C61—21no—
66C22×C4×C61—81no—
67C22×C4×C61C22⋊C411yescharacteristic
68C22×C4×C61C22×C411yescharacteristic
69C2×C22⋊D61—22no—
70C2×C22⋊D61—22no—
71C2×C22⋊D61—41no—
72C22×Dic61—21no—
73C22×Dic61—21no—
74C22×Dic61C22⋊C411yescharacteristic
75C22×Dic61C22⋊C411yescharacteristic
76C22×Dic61C2×D411yescharacteristic
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—81no—
80C22⋊#6923—24no—
81C22⋊#6923—42no—
82C2×#92420—42no—
83C4×#6923—82no—
84C4⋊C4×C61—41no—
85C4⋊C4×C61—41no—
86C4⋊C4×C61—41no—
87C22⋊C4×C61—42no—
88C22⋊Dic61—41no—
89C22⋊Dic61—81no—
90#11758198—41no—
91#11758198—42no—
92#11758198—81no—
93#11758196—41no—
94#11758196—42no—
95#11758196—81no—
96C4×Dic61—21no—
97C4×Dic61—21no—
98C23×C61—21no—
99C23×C61—21no—
100C23×C61—21no—
101C23×C61—41no—
102C23×C61—41no—
103C23×C61C2×C22⋊C411yescharacteristic
104C22×D61—21no—
105C22×D61—21no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61#13512yes—
109C2×#6923—44no—
110C2×#6923—82no—
111C2×#6923—82no—
112Q8×C61—82no—
113C2×C4×C61—21no—
114C2×C4×C61—41no—
115C2×C4×C61—41no—
116C2×C4×C61—41no—
117C2×C4×C61—41no—
118C2×C4×C61—41no—
119C2×C4×C61—41no—
120C2×C4×C61—41no—
121C2×C4×C61—41no—
122C2×C4×C61—81no—
123C2×C4×C61—81no—
124C2×C4×C61#9911yescharacteristic
125C2×C4×C61#9911yescharacteristic
126C22⋊D61—44no—
127C2×Dic61—21no—
128C2×Dic61—21no—
129C2×Dic61—21no—
130C2×Dic61—21no—
131C2×Dic61—41no—
132C2×Dic61—41no—
133C2×Dic61—41no—
134C2×Dic61—41no—
135C2×Dic61—41no—
136C2×Dic61—41no—
137C2×Dic61—41no—
138C2×Dic61—41no—
139C2×Dic61#10412yes—
140C4×D61—42no—
141C4×D61—81no—
142#92420—82no—
143#11731—611nomaximal, Sylow
144C22×C61—21no—
145C22×C61—21no—
146C22×C61—41no—
147C22×C61—41no—
148C22×C61—41no—
149C22×C61—81no—
150C22×C61—81no—
151C22×C61C2×#9911yescharacteristic
152C22×C61#41611yescharacteristic
153C22×C61C2×#10411yescharacteristic
154C2×D61—22no—
155C2×D61—41no—
156C2×D61—41no—
157C2×D61—42no—
158C2×D61—81no—
159#6923—84no—
160C4×C61—21no—
161C4×C61—21no—
162C4×C61—41no—
163C4×C61—81no—
164C4×C61—81no—
165Dic61—22no—
166Dic61—41no—
167Dic61—41no—
168Dic61—81no—
169Dic61—81no—
170#2638—611no—
171C4×#99—611no—
172#1460—611no—
173#1351—612no—
174#1344—612no—
175C2×C61—21no—
176C2×C61—21no—
177C2×C61—81no—
178C2×C61#150111yescharacteristic
179C2×C61#212111yescharacteristic
180C2×C61#212911yescharacteristic
181D61—42no—
182C2×C22⋊Q8#692311yescharacteristic, complemented
183C2×C4×D4—611no—
184#431—1221no—
185C2×#99—1221no—
186#398—611no—
187#397—611no—
188#397—1221no—
189#395—611no—
190#394—1221no—
191#393—611no—
192#387—1221no—
193#385—1221no—
194#381—611no—
195C4×C22⋊C4—1221no—
196#342—612no—
197#323—612no—
198#323—1221no—
199C61#1173111yescharacteristic, complemented, Sylow
200C22×Q8C2×#692311yescharacteristic
201C22×D4—611no—
202C23×C4—611no—
203C23×C4C2×#692311yescharacteristic
204C22⋊Q8—41no—
205C22⋊Q8—41no—
206C4×D4—2441no—
207C4×D4—2441no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1221no—
212C2×C4⋊C4C2×#692311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—611no—
215C2×C22⋊C4—611no—
216C2×C22⋊C4—1222no—
217C2×C22⋊C4—1221no—
218C2×C22⋊C4—1221no—
219C2×C22⋊C4—1221no—
220C2×C42—611no—
221C2×C42—2441no—
222C4⋊C8—2441no—
223#99—1224no—
224C22⋊C8—1222no—
225C4×C8—2441no—
226#95—611no—
227#95—611no—
228#95—1221no—
229#95—1221no—
230#95—1221no—
231#95—1221no—
232#95—1221no—
233#95—2441no—
234C24—611no—
235C24C22⋊#692311yescharacteristic
236C2×Q8—41no—
237C2×Q8—81no—
238C2×D4—1222no—
239C2×D4—1222no—
240C2×D4—2441no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—81no—
246C22×C4—611no—
247C22×C4—611no—
248C22×C4—611no—
249C22×C4—1221no—
250C22×C4—1221no—
251C22×C4—1221no—
252C22×C4—1221no—
253C22×C4—4881no—
254C22×C4—2442no—
255C22×C4C22⋊#692311yescharacteristic
256C22×C4C22×#692311yescharacteristic
257C2×C8—2442no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—4881no—
262C4⋊C4—2441no—
263C4⋊C4—2442no—
264C22⋊C4—42no—
265C22⋊C4—1224no—
266C22⋊C4—4881no—
267C22⋊C4—4881no—
268C22⋊C4—2441no—
269C22⋊C4—2442no—
270C22⋊C4—2441no—
271C22⋊C4—2442no—
272C42—1221no—
273C42—1221no—
274C42—4882no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—41no—
279C23—41no—
280C23—612no—
281C23—1221no—
282C23—1221no—
283C23—2441no—
284C23—2441no—
285C23C2×C22⋊#692311yescharacteristic
286Q8—82no—
287D4—2444no—
288C2×C4—21no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—612no—
300C2×C4—1221no—
301C2×C4—1221no—
302C2×C4—1221no—
303C2×C4—1221no—
304C2×C4—4881no—
305C2×C4—4882no—
306C2×C4—4882no—
307C2×C4—2442no—
308C2×C4—2441no—
309C2×C4—2441no—
310C2×C4—2441no—
311C2×C4—2441no—
312C2×C4—2444no—
313C2×C4—2441no—
314C2×C4—2441no—
315C2×C4—2441no—
316C2×C4—2441no—
317C2×C4#1451983511yescharacteristic
318C2×C4#1451983511yescharacteristic
319C8—4882no—
320C22—21no—
321C22—21no—
322C22—41no—
323C22—41no—
324C22—41no—
325C22—81no—
326C22—81no—
327C22—1222no—
328C22—4881no—
329C22—2442no—
330C22—2441no—
331C22—2441no—
332C22C2×#1451983511yescharacteristic
333C22#1684075211yescharacteristic
334C22D61⋊#10411yescharacteristic
335C4—21no—
336C4—21no—
337C4—41no—
338C4—81no—
339C4—81no—
340C4—1222no—
341C4—4881no—
342C4—4881no—
343C4—4884no—
344C4—2441no—
345C4—2441no—
346C2—21no—
347C2—21no—
348C2—81no—
349C2—2442no—
350C2#2146119411yescharacteristic
351C2#2146207411yescharacteristic
352C2#2146210111yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C22⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×C22⋊D61D4
cosocleC2×#11758198C23
hypercenterC2×C22⋊Q8#6923
hypocenterC61#11731
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460814C2semidirect
4. #21460799C2semidirect
13. #16839693C4semidirect
16. #16839681C4semidirect
21. #16839625C4semidirect
22. C2×C22⋊Q8×C61C4semidirect
182. C2×C22⋊Q8#6923semidirect
199. C61#11731semidirect

Maximal subgroups

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

subgroupindex
C4×#145198352
#214610582
#214608142
#214607992
#214517372
#1173161