Finite Group Databasebeta

← #30605477

Subgroups — #30605477

Order 15616.

508 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 80 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#21461680C211yesmaximal, characteristic
2#21461056C211yesmaximal, characteristic
3#21460816C214yesmaximal, complemented
4#21450773C211yesmaximal, characteristic
5C2×#14519836—21no—
6D61⋊(C22⋊C8)—21no—
7#16840704—21no—
8#16840640C2212yes—
9#16840607—21no—
10#16840606C2212yes—
11C2×#14519753—21no—
12C2×#14519753—21no—
13C2×C4×D244C2211yescharacteristic
14#16839697C412yescomplemented
15#16839697C2212yes—
16#16839663—21no—
17#16839631—21no—
18C4×C4⋊Dic61C411yescharacteristic, complemented
19C2×C4⋊Q8×C61C411yescharacteristic, complemented
20C2×D61⋊C8—41no—
21C2×C4⋊#6923—41no—
22C2×C22⋊#6923—22no—
23#14519836—42no—
24#14519834—41no—
25C22⋊#92420—22no—
26(C2×#6923)⋊C4—41no—
27C22×D244D411yescharacteristic
28C22×C4×D61D411yescharacteristic
29C22×C4×D61D411yescharacteristic
30#14519753—44no—
31#14519753—44no—
32C4×D244—41no—
33C4×D244—41no—
34C2×C4⋊Dic61—21no—
35C2×C4⋊Dic61D411yescharacteristic
36C2×#11758196—22no—
37C2×#11758196—22no—
38C2×#11758198—22no—
39C2×#11758198—22no—
40C2×#11758198C2×C411yescharacteristic
41C2×#11758198C2311yescharacteristic
42C2×C4×Dic61—21no—
43#14519684C2×C412yes—
44C22×Q8×C61—21no—
45C4⋊Q8×C61—41no—
46C4⋊Q8×C61—41no—
47C2×C4⋊C4×C61—21no—
48C2×C4⋊C4×C61C2×C412yes—
49C2×C42×C61C2×C411yescharacteristic
50C23×D61C22⋊C411yescharacteristic
51C23×D61C2×D411yescharacteristic
52C22×#6923—42no—
53C2×Q8×C61—42no—
54C2×Q8×C61—42no—
55C2×Q8×C61—41no—
56C2×Q8×C61—41no—
57C22×C4×C61—22no—
58C22×C4×C61C22⋊C411yescharacteristic
59C22×C4×C61C22⋊C411yescharacteristic
60C22×C4×C61C22×C411yescharacteristic
61C22×Dic61—22no—
62C22×Dic61C22⋊C411yescharacteristic
63C22×Dic61C2×D411yescharacteristic
64C2×D244—41no—
65C2×D244—41no—
66C2×D244SD1612yes—
67C2×D244Q1612yes—
68C2×C4×D61—21no—
69C2×C4×D61—21no—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61—42no—
73C2×C4×D61—42no—
74C22⋊#6923—42no—
75C22⋊#6923—42no—
76C2×#92420—42no—
77C4⋊#6923—82no—
78D61⋊C8—82no—
79C4⋊C4×C61—42no—
80C4⋊C4×C61—42no—
81C42×C61—21no—
82C42×C61—21no—
83#11758198—41no—
84#11758198—41no—
85#11758198—44no—
86#11758198—44no—
87C4⋊Dic61—24no—
88C4⋊Dic61—41no—
89#11758196—44no—
90#11758196—44no—
91C4×Dic61—42no—
92C23×C61C2×C22⋊C411yescharacteristic
93C22×D61—21no—
94C22×D61—21no—
95C22×D61—21no—
96C22×D61—21no—
97C22×D61—41no—
98C22×D61—41no—
99C22×D61—41no—
100C22×D61—41no—
101C22×D61#10312yes—
102C22×D61C2×SD1611yescharacteristic
103C22×D61C2×Q1611yescharacteristic
104C2×#6923—82no—
105C2×#6923—82no—
106C2×#6923—82no—
107Q8×C61—82no—
108Q8×C61—82no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—22no—
112C2×C4×C61—21no—
113C2×C4×C61—41no—
114C2×C4×C61—42no—
115C2×C4×C61—42no—
116C2×C4×C61—42no—
117C2×C4×C61#9911yescharacteristic
118C2×C4×C61#9911yescharacteristic
119C2×C4×C61#10312yes—
120C2×Dic61—24no—
121C2×Dic61—21no—
122C2×Dic61—21no—
123C2×Dic61—42no—
124C2×Dic61—42no—
125C2×Dic61—41no—
126C2×Dic61—41no—
127D244—42no—
128D244—42no—
129C4×D61—42no—
130C4×D61—42no—
131C4×D61—81no—
132C4×D61—81no—
133#92420—82no—
134#11901—611nomaximal, Sylow
135C22×C61—21no—
136C22×C61—21no—
137C22×C61C2×#9911yescharacteristic
138C22×C61C2×#10311yescharacteristic
139C22×C61#42111yescharacteristic
140C2×D61—21no—
141C2×D61—22no—
142C2×D61—21no—
143C2×D61—41no—
144C2×D61—41no—
145C2×D61—41no—
146C2×D61—41no—
147C2×D61—42no—
148C2×D61—41no—
149C2×D61—41no—
150C2×D61—81no—
151C2×D61—81no—
152#6923—162no—
153C4×C61—21no—
154C4×C61—22no—
155C4×C61—41no—
156C4×C61—82no—
157C4×C61#35312yes—
158Dic61—44no—
159Dic61—41no—
160Dic61—41no—
161#2542—611no—
162#1799—611no—
163#1459—611no—
164#1352—614no—
165C2×C61—21no—
166C2×C61—21no—
167C2×C61#152011yescharacteristic
168C2×C61#212311yescharacteristic
169C2×C61C2×#35311yescharacteristic
170D61—41no—
171D61—42no—
172D61—41no—
173C2×C4⋊Q8#692311yescharacteristic, complemented
174C2×C22⋊Q8—1221no—
175C2×C22⋊Q8—1221no—
176C2×C4×D4—611no—
177C2×#99—1221no—
178C2×C22⋊C8—1221no—
179#399—612no—
180#399—612no—
181#390—1221no—
182#384—1221no—
183#380—1221no—
184C4×C4⋊C4—611no—
185#342—612no—
186#324—1221no—
187#323—612no—
188C61#1190111yescharacteristic, complemented, Sylow
189C22×Q8—21no—
190C22×D4—611no—
191C23×C4—611no—
192C23×C4—611no—
193C22×C8—2441no—
194C4⋊Q8—41no—
195C4⋊Q8—41no—
196C22⋊Q8—2444no—
197C22⋊Q8—2444no—
198C4×D4—2441no—
199C4×D4—2441no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—611no—
202C2×C4⋊C4—1221no—
203C2×C4⋊C4—1222no—
204C2×C4⋊C4—1222no—
205C2×C4⋊C4—2441no—
206C2×C4⋊C4C2×#692312yes—
207C2×C22⋊C4—611no—
208C2×C22⋊C4—611no—
209C2×C22⋊C4—1222no—
210C2×C22⋊C4—1222no—
211C2×C22⋊C4—1222no—
212C2×C42—1221no—
213C2×C42C2×#692311yescharacteristic
214#99—2442no—
215C22⋊C8—1222no—
216C22⋊C8—2441no—
217#95—612no—
218#95—2441no—
219C24—611no—
220C24—611no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×D4—612no—
226C2×D4—612no—
227C2×D4—2441no—
228C2×D4—2441no—
229C22×C4—22no—
230C22×C4—611no—
231C22×C4—611no—
232C22×C4—1222no—
233C22×C4—1221no—
234C22×C4—1221no—
235C22×C4—1221no—
236C22×C4—1221no—
237C22×C4—2442no—
238C22×C4—2442no—
239C22×C4—2442no—
240C22×C4C22⋊#692311yescharacteristic
241C22×C4C22⋊#692311yescharacteristic
242C22×C4C22×#692311yescharacteristic
243C2×C8—4882no—
244C2×C8—2442no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—1224no—
248C4⋊C4—4882no—
249C4⋊C4—2441no—
250C4⋊C4—2444no—
251C4⋊C4—2444no—
252C22⋊C4—2441no—
253C22⋊C4—2441no—
254C22⋊C4—2444no—
255C22⋊C4—2444no—
256C22⋊C4—2442no—
257C22⋊C4—2442no—
258C42—21no—
259C42—21no—
260C42—2442no—
261C23—612no—
262C23—611no—
263C23—611no—
264C23—1221no—
265C23—1221no—
266C23—1221no—
267C23—1221no—
268C23—2441no—
269C23—2441no—
270C23—2441no—
271C23—2441no—
272C23C2×C22⋊#692311yescharacteristic
273Q8—82no—
274Q8—82no—
275D4—2442no—
276D4—2442no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—21no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—1224no—
286C2×C4—1221no—
287C2×C4—1221no—
288C2×C4—4881no—
289C2×C4—4881no—
290C2×C4—4882no—
291C2×C4—4882no—
292C2×C4—4882no—
293C2×C4—2442no—
294C2×C4—2442no—
295C2×C4—2442no—
296C2×C4—2442no—
297C2×C4—2441no—
298C2×C4—2441no—
299C2×C4#1451983511yescharacteristic
300C2×C4#1451983511yescharacteristic
301C2×C4#1451984012yes—
302C8—4882no—
303C22—21no—
304C22—21no—
305C22—1221no—
306C22—1222no—
307C22—1221no—
308C22—4881no—
309C22—4881no—
310C22—2441no—
311C22—2442no—
312C22—2441no—
313C22—2441no—
314C22—2441no—
315C22—2441no—
316C22—2441no—
317C22C2×#1451983511yescharacteristic
318C22C2×#1451984011yescharacteristic
319C22#1684076111yescharacteristic
320C4—22no—
321C4—21no—
322C4—41no—
323C4—82no—
324C4—9762no—
325C4—2444no—
326C4—2441no—
327C4—2441no—
328C4#1684066612yes—
329C2—21no—
330C2—21no—
331C2—2441no—
332C2—2441no—
333C2—2442no—
334C2#2146125911yescharacteristic
335C2#2146208111yescharacteristic
336C2C2×#1684066611yescharacteristic
subgroupG/H
centerC22C2×#14519835
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC2×C4⋊Q8×C61C4
socleC22×C61C2×#99
prime radicalC22×D244D4
cosocleC2×#11758198C23
hypercenterC2×C4⋊Q8#6923
hypocenterC61#11901
LCSG ▷ C22×C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
3. #21460816C2semidirect
14. #16839697C4semidirect
18. C4×C4⋊Dic61C4semidirect
19. C2×C4⋊Q8×C61C4semidirect
173. C2×C4⋊Q8#6923semidirect
188. C61#11901semidirect

Maximal subgroups

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

subgroupindex
#214616802
#214610562
#214608162
#214507732
#1190161