Finite Group Databasebeta

← #354×#6923 #30606538

Subgroups — #354×#6923 #30606538

Order 15616.

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

normal subgroups only

This group has 584 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
1#124×#6923C211yesmaximal, characteristic
2D61⋊#354C211yesmaximal, characteristic
3#99×#6923C212yesmaximal
4#21460912C212yesmaximal
5#354×D61C211yesmaximal, characteristic
6C2×Q8×#6923C411yescharacteristic, complemented
7C2×D4×#6923C2211yescharacteristic
8D61⋊#99C2212yes—
9(C2×#6923)⋊Q8C411yescharacteristic, complemented
10#16840714C2211yescharacteristic
11#16840712—22no—
12#16840706C411yescharacteristic, complemented
13C22⋊C4×#6923—22no—
14C22⋊C4×#6923—22no—
15C42×#6923C411yescharacteristic, complemented
16#16840669C412yescomplemented
17#16840668C412yescomplemented
18#16840639—22no—
19#124×D61C2211yescharacteristic
20#99×D61C2212yes—
21#16839557C411yescharacteristic, complemented
22#16839556C412yescomplemented
23C61×#354C411yescharacteristic, complemented
24C23×#6923D412yes—
25Q8×#6923—41no—
26Q8×#6923—41no—
27D4×#6923—41no—
28D4×#6923—41no—
29C2×C4⋊#6923—21no—
30C2×C4⋊#6923C2×C411yescharacteristic
31C2×C4⋊#6923C2×C411yescharacteristic
32C2×C22⋊#6923—22no—
33C2×C22⋊#6923—22no—
34C2×C22⋊#6923D412yes—
35C2×C4×#6923—21no—
36C2×C4×#6923—21no—
37C2×C4×#6923—21no—
38C2×C4×#6923—42no—
39C2×C4×#6923C2×C411yescharacteristic
40C2×C4×#6923C2×C411yescharacteristic
41#14519836C2×C414yes—
42#14519835—22no—
43(C2×#6923)⋊C4—22no—
44(C2×#6923)⋊C4—22no—
45(C2×#6923)⋊C4—42no—
46C2×Q8×D61C2×C411yescharacteristic
47C2×D4×D61C2311yescharacteristic
48#14519766C2×C411yescharacteristic
49#14519765C2×C411yescharacteristic
50#14519764C2×C411yescharacteristic
51#14519763—22no—
52C22⋊C4×D61—22no—
53C22⋊C4×D61—22no—
54C42×D61C2×C411yescharacteristic
55#14519692C2×C412yes—
56#14519691—22no—
57#124×C61C2×C411yescharacteristic
58#99×C61C2×C412yes—
59C23×D61C2×D412yes—
60C22×#6923—21no—
61C22×#6923—22no—
62C22×#6923—22no—
63C22×#6923—21no—
64C22×#6923—44no—
65C22×#6923—42no—
66C22×#6923C22⋊C411yescharacteristic
67C22×#6923C22⋊C412yes—
68C22×#6923C22⋊C411yescharacteristic
69C2×Q8×C61C4211yescharacteristic, complemented
70C2×D4×C61C22×C411yescharacteristic
71C2×C22⋊D61#4112yes—
72Q8×D61—41no—
73Q8×D61—41no—
74D4×D61—41no—
75D4×D61—41no—
76C2×D244C22×C411yescharacteristic
77C2×C4×D61—21no—
78C2×C4×D61—21no—
79C2×C4×D61—42no—
80C2×C4×D61C22×C411yescharacteristic
81C2×Dic122—21no—
82C2×Dic122C4211yescharacteristic, complemented
83C22⋊#6923—24no—
84C22⋊#6923—42no—
85C22⋊#6923—44no—
86C22⋊#6923—42no—
87C4⋊#6923—41no—
88C4⋊#6923—41no—
89C4⋊#6923—42no—
90C4×#6923—21no—
91C4×#6923—21no—
92C4×#6923—41no—
93C4×#6923—42no—
94C4×#6923—42no—
95C4×#6923—41no—
96C4×#6923—41no—
97C4×#6923—84no—
98C22⋊C4×C61—22no—
99C22⋊C4×C61—22no—
100C42×C61C4211yescharacteristic, complemented
101C22⋊Dic61—22no—
102C22⋊Dic61—22no—
103#11758198—22no—
104#11758198—22no—
105#11758198—42no—
106C4×Dic61—21no—
107C4×Dic61C4211yescharacteristic, complemented
108C23×C61C4×D412yes—
109C22×D61—22no—
110C22×D61—22no—
111C22×D61—42no—
112C22×D61—42no—
113C22×D61C2×C22⋊C411yescharacteristic
114C22×D61C4×D412yes—
115C2×#6923—22no—
116C2×#6923—22no—
117C2×#6923—21no—
118C2×#6923—22no—
119C2×#6923—41no—
120C2×#6923—41no—
121C2×#6923—44no—
122C2×#6923—41no—
123C2×#6923—42no—
124C2×#6923—44no—
125C2×#6923—42no—
126C2×#6923—41no—
127C2×#6923—82no—
128C2×#6923—82no—
129C2×#6923—82no—
130C2×#6923#9911yescharacteristic
131C2×#6923#9911yescharacteristic
132Q8×C61—41no—
133D4×C61—41no—
134C2×C4×C61—21no—
135C2×C4×C61—21no—
136C2×C4×C61—42no—
137C2×C4×C61C2×C4211yescharacteristic
138C22⋊D61—42no—
139C2×Dic61—21no—
140C2×Dic61—21no—
141C2×Dic61—42no—
142C2×Dic61#11711yescharacteristic
143D244—41no—
144C4×D61—21no—
145C4×D61—21no—
146C4×D61—41no—
147C4×D61—41no—
148C4×D61—41no—
149C4×D61—41no—
150C4×D61—82no—
151C4×D61—82no—
152Dic122—42no—
153Dic122—41no—
154C4×#354—611nomaximal, Sylow
155C22×C61—22no—
156C22×C61—42no—
157C22×C61C4×C22⋊C411yescharacteristic
158C2×D61—21no—
159C2×D61—21no—
160C2×D61—22no—
161C2×D61—42no—
162C2×D61—42no—
163C2×D61—42no—
164C2×D61—82no—
165C2×D61C2×#9911yescharacteristic
166C2×D61#41011yescharacteristic
167#6923—21no—
168#6923—42no—
169#6923—42no—
170#6923—41no—
171#6923—84no—
172#6923—81no—
173#6923#35412yesdirect product factor
174C4×C61—21no—
175C4×C61—41no—
176C4×C61—41no—
177C4×C61—82no—
178Dic61—21no—
179Dic61—41no—
180Dic61—41no—
181Dic61—82no—
182C4×#124—611no—
183C2×#354—611no—
184C2×#354—611no—
185C4×#99—612no—
186#1393—612no—
187C2×C61—21no—
188C2×C61—42no—
189C2×C61C4×#9911yescharacteristic
190D61—21no—
191D61—42no—
192D61C2×#35411yescharacteristic
193D61#212611yescharacteristic
194C2×#124—611no—
195C2×C4×Q8—611no—
196C2×C4×D4—611no—
197C2×#99—612no—
198C2×#99—612no—
199#391—611no—
200#390—611no—
201#388—1222no—
202#382—611no—
203C4×C22⋊C4—1222no—
204C4×C22⋊C4—1222no—
205C43—611no—
206#354—612no—
207#354—612no—
208#354—611no—
209#354—612no—
210#354#692311yescharacteristic, direct product factor
211#342—1222no—
212C61C4×#35411yescharacteristic, complemented, Sylow
213C22×Q8—611no—
214C22×D4—611no—
215C23×C4—612no—
216#124—611no—
217#124—611no—
218#124—611no—
219#124—1222no—
220#124C2×#692311yescharacteristic
221C4×Q8—2441no—
222C4×Q8—2441no—
223C4×D4—2441no—
224C4×D4—2441no—
225C2×C4⋊C4—611no—
226C2×C4⋊C4—611no—
227C2×C4⋊C4—1221no—
228C2×C22⋊C4—612no—
229C2×C22⋊C4—1222no—
230C2×C22⋊C4—1222no—
231C2×C22⋊C4—1222no—
232C2×C22⋊C4—1222no—
233C2×C42—611no—
234C2×C42—611no—
235C2×C42—611no—
236C2×C42—1221no—
237C2×C42—1221no—
238C2×C42—1221no—
239C2×C42—2442no—
240#99—614no—
241#99—612no—
242#99—1222no—
243#99—1222no—
244#99C2×#692312yes—
245#95—1222no—
246#95—1222no—
247#95—2442no—
248C24—612no—
249C2×Q8—611no—
250C2×Q8—1221no—
251C2×Q8—2441no—
252C2×Q8—2441no—
253C2×Q8C4×#692311yescharacteristic, complemented
254C2×D4—612no—
255C2×D4—611no—
256C2×D4—2441no—
257C2×D4—2441no—
258C2×D4C22×#692311yescharacteristic
259C22×C4—611no—
260C22×C4—611no—
261C22×C4—612no—
262C22×C4—611no—
263C22×C4—1221no—
264C22×C4—1221no—
265C22×C4—1222no—
266C22×C4—1222no—
267C22×C4—1221no—
268C22×C4—1221no—
269C22×C4—2442no—
270C22×C4—2444no—
271C22×C4—2442no—
272C4⋊C4—2441no—
273C4⋊C4—2441no—
274C4⋊C4—2442no—
275C22⋊C4—22no—
276C22⋊C4—22no—
277C22⋊C4—1222no—
278C22⋊C4—1222no—
279C22⋊C4—1222no—
280C22⋊C4—1224no—
281C22⋊C4—1222no—
282C22⋊C4—2442no—
283C22⋊C4—2442no—
284C22⋊C4—2444no—
285C22⋊C4—2442no—
286C42—611no—
287C42—1221no—
288C42—1221no—
289C42—1221no—
290C42—4884no—
291C42—2441no—
292C42—2441no—
293C42—2441no—
294C42—2442no—
295C42—2442no—
296C42C4×#692311yescharacteristic, complemented
297C23—611no—
298C23—612no—
299C23—1222no—
300C23—1222no—
301C23—2442no—
302C23—2442no—
303C23D4×#692312yes—
304Q8—41no—
305Q8—2441no—
306Q8—2442no—
307D4—41no—
308D4—2441no—
309D4—2442no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—42no—
313C2×C4—611no—
314C2×C4—611no—
315C2×C4—611no—
316C2×C4—1221no—
317C2×C4—1221no—
318C2×C4—1222no—
319C2×C4—1221no—
320C2×C4—1222no—
321C2×C4—1222no—
322C2×C4—1221no—
323C2×C4—1221no—
324C2×C4—4882no—
325C2×C4—4882no—
326C2×C4—4882no—
327C2×C4—4882no—
328C2×C4—4882no—
329C2×C4—2441no—
330C2×C4—2441no—
331C2×C4—2441no—
332C2×C4—2441no—
333C2×C4—2444no—
334C2×C4—2442no—
335C2×C4—2441no—
336C2×C4—2444no—
337C2×C4—2442no—
338C2×C4—2441no—
339C2×C4—2441no—
340C2×C4—2441no—
341C2×C4—2442no—
342C2×C4C2×C4×#692311yescharacteristic
343C22—22no—
344C22—42no—
345C22—611no—
346C22—611no—
347C22—1221no—
348C22—1221no—
349C22—1222no—
350C22—4882no—
351C22—2442no—
352C22—2442no—
353C22—2442no—
354C22C22⋊C4×#692311yescharacteristic
355C4—21no—
356C4—41no—
357C4—41no—
358C4—82no—
359C4—612no—
360C4—1221no—
361C4—1221no—
362C4—4882no—
363C4—4881no—
364C4—4884no—
365C4—2441no—
366C4—2442no—
367C4—2441no—
368C4—2442no—
369C4—2441no—
370C2—21no—
371C2—42no—
372C2—611no—
373C2—611no—
374C2—1221no—
375C2—2442no—
376C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×D4C22×#6923
FittingC61×#354C4
socleC2×C61C4×#99
prime radical, cosocleC2×D4×D61C23
hypercenter#354#6923
hypocenterC61C4×#354
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #354

Complements

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

NH ≅ G/Ntype
6. C2×Q8×#6923C4semidirect
9. (C2×#6923)⋊Q8C4semidirect
12. #16840706C4semidirect
15. C42×#6923C4semidirect
16. #16840669C4semidirect
17. #16840668C4semidirect
21. #16839557C4semidirect
22. #16839556C4semidirect
23. C61×#354C4semidirect
69. C2×Q8×C61C42semidirect
82. C2×Dic122C42semidirect
100. C42×C61C42semidirect
107. C4×Dic61C42semidirect
173. #6923#354direct (N × H)
210. #354#6923direct (N × H)
212. C61C4×#354semidirect
253. C2×Q8C4×#6923semidirect
296. C42C4×#6923semidirect

Maximal subgroups

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

subgroupindex
#124×#69232
D61⋊#3542
#99×#69232
#214609122
#354×D612
C4×#35461