Finite Group Databasebeta

← #356×#6923 #30606558

Subgroups — #356×#6923 #30606558

Order 15616.

500 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 96 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⋊Q8×#6923C211yesmaximal, characteristic
2D61⋊#356C211yesmaximal, characteristic
3#101×#6923C212yesmaximal
4#21460915C212yesmaximal
5#356×D61C211yesmaximal, characteristic
6C2×Q8×#6923C411yescharacteristic
7C2×Q8×#6923C2211yescharacteristic
8D61⋊#101C2212yes—
9M16×#6923—22no—
10(C2×#6923)⋊Q8C411yescharacteristic
11(C2×#6923)⋊Q8C2211yescharacteristic
12#16840713—22no—
13#16840708C411yescharacteristic
14C4⋊C4×#6923—22no—
15C42×#6923C411yescharacteristic
16#16840673C412yescomplemented
17#16840672C412yescomplemented
18#16840641—22no—
19C4⋊Q8×D61C2211yescharacteristic
20#101×D61C2212yes—
21#16839561C411yescharacteristic, complemented
22#16839560C412yescomplemented
23C61×#356C411yescharacteristic, complemented
24D61⋊M16—22no—
25Q8×#6923—41no—
26Q8×#6923—41no—
27Q8×#6923—41no—
28Q8×#6923—41no—
29C2×C4⋊#6923—22no—
30C2×C4⋊#6923—21no—
31C2×C4⋊#6923C2×C411yescharacteristic
32C2×C4⋊#6923C2×C411yescharacteristic
33C2×C4⋊#6923D412yes—
34C2×C4×#6923—21no—
35C2×C4×#6923—21no—
36C2×C4×#6923—21no—
37C2×C4×#6923C2×C411yescharacteristic
38C2×C4×#6923C2×C411yescharacteristic
39C2×C4×#6923D412yes—
40#14519838—22no—
41#14519838C2×C414yes—
42#14519831—42no—
43C8×#6923—42no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4—22no—
46C2×Q8×D61C2×C411yescharacteristic
47C2×Q8×D61C2311yescharacteristic
48M16×D61—22no—
49#14519777C2×C411yescharacteristic
50#14519776—22no—
51#14519775C2×C411yescharacteristic
52#14519775C2×C411yescharacteristic
53C4⋊C4×D61—22no—
54C42×D61C2×C411yescharacteristic
55#14519696C2×C412yes—
56#14519695—22no—
57C4⋊Q8×C61C2×C411yescharacteristic
58#101×C61C2×C412yes—
59C22×#6923—21no—
60C22×#6923—21no—
61C22×#6923C22⋊C411yescharacteristic
62C22×#6923C22⋊C412yes—
63C22×#6923C22⋊C411yescharacteristic
64C2×Q8×C61C4211yescharacteristic
65C2×Q8×C61C22×C411yescharacteristic
66Q8×D61—41no—
67Q8×D61—41no—
68Q8×D61—41no—
69Q8×D61—41no—
70C2×C4×D61—21no—
71C2×C4×D61—21no—
72C2×C4×D61C22×C411yescharacteristic
73C2×C4×D61C2×D412yes—
74C2×Dic122—21no—
75C2×Dic122C4211yescharacteristic
76C2×Dic122C22×C411yescharacteristic
77C2×Dic122#4112yes—
78#11758217—24no—
79C4⋊#6923—41no—
80C4⋊#6923—42no—
81C4⋊#6923—44no—
82C4⋊#6923—41no—
83C4⋊#6923—42no—
84C4×#6923—24no—
85C4×#6923—22no—
86C4×#6923—22no—
87C4×#6923—21no—
88C4×#6923—21no—
89C4×#6923—42no—
90C4×#6923—41no—
91C4×#6923—41no—
92C4×#6923—41no—
93#11758213—42no—
94D61⋊C8—42no—
95M16×C61—22no—
96C4⋊C4×C61—22no—
97C42×C61C4211yescharacteristic
98C4⋊Dic61—22no—
99#11758196—22no—
100#11758196—22no—
101C4×Dic61—21no—
102C4×Dic61C4211yescharacteristic
103#11758194—22no—
104#11758189—42no—
105C8×D61—42no—
106C22×D61C2×C22⋊C411yescharacteristic
107C2×#6923—22no—
108C2×#6923—22no—
109C2×#6923—21no—
110C2×#6923—22no—
111C2×#6923—41no—
112C2×#6923—41no—
113C2×#6923—41no—
114C2×#6923—42no—
115C2×#6923—42no—
116C2×#6923—41no—
117C2×#6923#9911yescharacteristic
118C2×#6923#9911yescharacteristic
119Q8×C61—41no—
120Q8×C61—41no—
121C2×C4×C61—21no—
122C2×C4×C61—21no—
123C2×C4×C61C2×C4211yescharacteristic
124C2×C4×C61C4×D412yes—
125C2×Dic61—21no—
126C2×Dic61—21no—
127C2×Dic61#11711yescharacteristic
128C2×Dic61C4×D412yes—
129C4×D61—22no—
130C4×D61—22no—
131C4×D61—22no—
132C4×D61—21no—
133C4×D61—22no—
134C4×D61—21no—
135C4×D61—41no—
136C4×D61—41no—
137Dic122—41no—
138Dic122—42no—
139Dic122—42no—
140Dic122—41no—
141#92420—44no—
142C8×C61—42no—
143#92418—42no—
144C4×#356—611nomaximal, Sylow
145C22×C61C4×C22⋊C411yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61C2×#9911yescharacteristic
149C2×D61#41011yescharacteristic
150#6923—21no—
151#6923—42no—
152#6923—41no—
153#6923—42no—
154#6923—81no—
155#6923#35612yesdirect product factor
156C4×C61—22no—
157C4×C61—22no—
158C4×C61—21no—
159C4×C61—41no—
160Dic61—22no—
161Dic61—22no—
162Dic61—21no—
163Dic61—41no—
164C4×C4⋊Q8—611no—
165C2×#356—611no—
166C2×#356—611no—
167C4×#101—612no—
168#1394—612no—
169C2×C61—21no—
170C2×C61C4×#9911yescharacteristic
171D61—21no—
172D61C2×#35611yescharacteristic
173D61#213211yescharacteristic
174C2×C4⋊Q8—611no—
175C2×C4×Q8—611no—
176C2×C4×Q8—611no—
177C2×#101—612no—
178C2×#101—612no—
179C4×M16—1222no—
180#391—611no—
181#391—611no—
182#389—1222no—
183#384—611no—
184C4×C4⋊C4—1222no—
185C43—611no—
186#356—612no—
187#356—612no—
188#356—611no—
189#356—612no—
190#356#692311yescharacteristic, direct product factor
191#343—1222no—
192C61C4×#35611yescharacteristic, complemented, Sylow
193C22×Q8—611no—
194C22×Q8—611no—
195C2×M16—1222no—
196C2×M16—1222no—
197C4⋊Q8—611no—
198C4⋊Q8—611no—
199C4⋊Q8—611no—
200C4⋊Q8—1222no—
201C4⋊Q8C2×#692311yescharacteristic
202C4×Q8—2441no—
203C4×Q8—2441no—
204C4×Q8—2441no—
205C4×Q8—2441no—
206C2×C4⋊C4—611no—
207C2×C4⋊C4—612no—
208C2×C4⋊C4—611no—
209C2×C4⋊C4—1222no—
210C2×C4⋊C4—1221no—
211C2×C4⋊C4—1222no—
212C2×C42—611no—
213C2×C42—611no—
214C2×C42—612no—
215C2×C42—611no—
216C2×C42—1221no—
217C2×C42—1221no—
218C2×C42—1221no—
219#101—614no—
220#101—612no—
221#101—1222no—
222#101—1222no—
223#101C2×#692312yes—
224#97—2442no—
225C4×C8—2442no—
226#95—1222no—
227#95—1222no—
228C2×Q8—611no—
229C2×Q8—612no—
230C2×Q8—611no—
231C2×Q8—1221no—
232C2×Q8—2441no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C2×Q8—2441no—
236C2×Q8C4×#692311yescharacteristic
237C2×Q8C22×#692311yescharacteristic
238C22×C4—611no—
239C22×C4—611no—
240C22×C4—612no—
241C22×C4—612no—
242C22×C4—611no—
243C22×C4—1221no—
244C22×C4—1221no—
245C22×C4—1221no—
246C22×C4—1221no—
247M16—22no—
248M16—1224no—
249M16—1222no—
250M16—2442no—
251M16—2442no—
252C2×C8—2442no—
253C2×C8—2442no—
254C4⋊C4—22no—
255C4⋊C4—1222no—
256C4⋊C4—1222no—
257C4⋊C4—1222no—
258C4⋊C4—2441no—
259C4⋊C4—2442no—
260C4⋊C4—2441no—
261C4⋊C4—2442no—
262C4⋊C4—2444no—
263C42—611no—
264C42—1221no—
265C42—1222no—
266C42—1222no—
267C42—1221no—
268C42—1221no—
269C42—1224no—
270C42—2441no—
271C42—2441no—
272C42—2441no—
273C42—2442no—
274C42C4×#692311yescharacteristic
275C23—611no—
276Q8—41no—
277Q8—41no—
278Q8—2442no—
279Q8—2441no—
280Q8—2441no—
281Q8—2442no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—611no—
285C2×C4—611no—
286C2×C4—611no—
287C2×C4—612no—
288C2×C4—1221no—
289C2×C4—1222no—
290C2×C4—1221no—
291C2×C4—1222no—
292C2×C4—1222no—
293C2×C4—1222no—
294C2×C4—1222no—
295C2×C4—1221no—
296C2×C4—1221no—
297C2×C4—1222no—
298C2×C4—1222no—
299C2×C4—1221no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2442no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2441no—
307C2×C4—2442no—
308C2×C4C2×C4×#692311yescharacteristic
309C2×C4D4×#692312yes—
310C8—42no—
311C8—2442no—
312C8—2444no—
313C22—611no—
314C22—611no—
315C22—1221no—
316C22—1221no—
317C22C22⋊C4×#692311yescharacteristic
318C4—22no—
319C4—22no—
320C4—21no—
321C4—41no—
322C4—612no—
323C4—1222no—
324C4—1221no—
325C4—1222no—
326C4—1221no—
327C4—4881no—
328C4—2441no—
329C4—2442no—
330C4—2442no—
331C4—2441no—
332C2—21no—
333C2—611no—
334C2—611no—
335C2—1221no—
336C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#356C4
socleC2×C61C4×#99
prime radicalC22×D61C2×C22⋊C4
cosocleC2×Q8×D61C23
hypercenter#356#6923
hypocenterC61C4×#356
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #356

Complements

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

NH ≅ G/Ntype
16. #16840673C4semidirect
17. #16840672C4semidirect
21. #16839561C4semidirect
22. #16839560C4semidirect
23. C61×#356C4semidirect
155. #6923#356direct (N × H)
190. #356#6923direct (N × H)
192. C61C4×#356semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×#69232
D61⋊#3562
#101×#69232
#214609152
#356×D612
C4×#35661