Finite Group Databasebeta

← #30606568

Subgroups — #30606568

Order 15616.

512 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 94 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#21462309C211yesmaximal, characteristic
2D61⋊#355C211yesmaximal, characteristic
3#101×#6923C212yesmaximal, complemented
4#21460915C212yesmaximal, complemented
5#355×D61C211yesmaximal, characteristic
6C2×Q8×#6923C2211yescharacteristic
7C2×D4×#6923C411yescharacteristic
8D61⋊#101C2212yes—
9M16×#6923—22no—
10(C2×#6923)⋊Q8C2211yescharacteristic
11#16840714C411yescharacteristic
12#16840712—22no—
13#16840709—22no—
14#16840708C411yescharacteristic
15#16840698C411yescharacteristic
16#16840671C412yescomplemented
17#16840670C412yescomplemented
18#16840641—22no—
19#124×D61C2211yescharacteristic
20#101×D61C2212yes—
21#16839559C411yescharacteristic, complemented
22#16839558C412yescomplemented
23C61×#355C411yescharacteristic, complemented
24C23×#6923—21no—
25D61⋊M16—22no—
26Q8×#6923—42no—
27D4×#6923—42no—
28C2×C4⋊#6923—22no—
29C2×C4⋊#6923C2×C411yescharacteristic
30C2×C4⋊#6923C2×C411yescharacteristic
31C2×C4⋊#6923D412yes—
32C2×C22⋊#6923—21no—
33C2×C22⋊#6923—22no—
34C2×C4×#6923C2×C411yescharacteristic
35C2×C4×#6923C2×C411yescharacteristic
36C2×C4×#6923D412yes—
37#14519838—22no—
38#14519838C2×C414yescomplemented
39#14519831—42no—
40C8×#6923—42no—
41(C2×#6923)⋊C4—21no—
42(C2×#6923)⋊C4—21no—
43(C2×#6923)⋊C4—22no—
44C2×Q8×D61C2311yescharacteristic
45C2×D4×D61C2×C411yescharacteristic
46M16×D61—22no—
47#14519766C2×C411yescharacteristic
48#14519765C2×C411yescharacteristic
49#14519764C2×C411yescharacteristic
50#14519763—22no—
51C22⋊C4×D61—22no—
52C42×D61C2×C411yescharacteristic
53#14519696C2×C412yescomplemented
54#14519695—22no—
55#124×C61C2×C411yescharacteristic
56#101×C61C2×C412yescomplemented
57C23×D61—21no—
58C22×#6923—21no—
59C22×#6923—21no—
60C22×#6923—42no—
61C22×#6923—82no—
62C22×#6923C22⋊C411yescharacteristic
63C22×#6923C22⋊C412yes—
64C22×#6923C22⋊C411yescharacteristic
65C2×Q8×C61C22×C411yescharacteristic
66C2×D4×C61C4211yescharacteristic
67C2×C22⋊D61—21no—
68Q8×D61—41no—
69Q8×D61—41no—
70D4×D61—41no—
71D4×D61—41no—
72C2×D244C4211yescharacteristic
73C2×C4×D61—21no—
74C2×C4×D61C22×C411yescharacteristic
75C2×C4×D61C2×D412yes—
76C2×Dic122C22×C411yescharacteristic
77C2×Dic122#4112yes—
78C22⋊#6923—42no—
79C22⋊#6923—82no—
80#11758217—24no—
81C4⋊#6923—41no—
82C4⋊#6923—42no—
83C4⋊#6923—41no—
84C4⋊#6923—82no—
85C4×#6923—22no—
86C4×#6923—22no—
87C4×#6923—22no—
88C4×#6923—41no—
89#11758213—42no—
90D61⋊C8—42no—
91M16×C61—22no—
92C22⋊C4×C61—22no—
93C42×C61C4211yescharacteristic
94C22⋊Dic61—22no—
95#11758198—22no—
96#11758198—22no—
97C4×Dic61—21no—
98C4×Dic61C4211yescharacteristic
99#11758194—22no—
100#11758189—42no—
101C8×D61—42no—
102C23×C61—21no—
103C22×D61—21no—
104C22×D61—42no—
105C22×D61—42no—
106C22×D61—41no—
107C22×D61—41no—
108C22×D61C2×C22⋊C411yescharacteristic
109C2×#6923—21no—
110C2×#6923—22no—
111C2×#6923—41no—
112C2×#6923—42no—
113C2×#6923—41no—
114C2×#6923—42no—
115C2×#6923—41no—
116C2×#6923—41no—
117C2×#6923—41no—
118C2×#6923—41no—
119C2×#6923—84no—
120C2×#6923#9912yes—
121Q8×C61—41no—
122D4×C61—41no—
123C2×C4×C61—21no—
124C2×C4×C61C2×C4211yescharacteristic
125C2×C4×C61C4×D412yes—
126C22⋊D61—42no—
127C2×Dic61—21no—
128C2×Dic61#11711yescharacteristic
129C2×Dic61C4×D412yes—
130D244—41no—
131C4×D61—22no—
132C4×D61—21no—
133C4×D61—22no—
134C4×D61—21no—
135C4×D61—41no—
136C4×D61—41no—
137Dic122—41no—
138Dic122—42no—
139#92420—44no—
140C8×C61—42no—
141#92418—42no—
142#12632—611nomaximal, Sylow
143C22×C61—42no—
144C22×C61—41no—
145C22×C61C4×C22⋊C411yescharacteristic
146C2×D61—21no—
147C2×D61—21no—
148C2×D61—42no—
149C2×D61—41no—
150C2×D61—82no—
151C2×D61—81no—
152C2×D61—81no—
153C2×D61C2×#9911yescharacteristic
154C2×D61#41011yescharacteristic
155#6923—22no—
156#6923—42no—
157#6923—41no—
158#6923—81no—
159#6923—81no—
160C4×C61—22no—
161C4×C61—21no—
162C4×C61—41no—
163Dic61—22no—
164Dic61—21no—
165Dic61—41no—
166#2326—611no—
167C2×#355—611no—
168C2×#355—611no—
169C4×#101—612no—
170#1394—612no—
171C2×C61—21no—
172C2×C61—81no—
173C2×C61C4×#9911yescharacteristic
174D61—21no—
175D61—81no—
176D61#213311yescharacteristic
177D61#213411yescharacteristic
178C2×#124—611no—
179C2×C4×Q8—611no—
180C2×C4×D4—611no—
181C2×#101—612no—
182C2×#101—612no—
183C4×M16—1222no—
184#391—611no—
185#390—611no—
186#388—1222no—
187#385—1222no—
188#384—611no—
189#376—611no—
190#355—612no—
191#355—612no—
192#355—611no—
193#355—612no—
194#355#692311yescharacteristic, complemented
195#343—1222no—
196C61#1263211yescharacteristic, complemented, Sylow
197C22×Q8—611no—
198C22×D4—611no—
199C23×C4—1221no—
200C2×M16—1222no—
201C2×M16—1222no—
202#124—611no—
203#124—611no—
204#124—611no—
205#124—1222no—
206#124C2×#692311yescharacteristic
207C4×Q8—2442no—
208C4×D4—2442no—
209C2×C4⋊C4—611no—
210C2×C4⋊C4—612no—
211C2×C4⋊C4—611no—
212C2×C4⋊C4—1222no—
213C2×C22⋊C4—1222no—
214C2×C22⋊C4—1222no—
215C2×C22⋊C4—1221no—
216C2×C42—611no—
217C2×C42—611no—
218C2×C42—612no—
219C2×C42—611no—
220#101—614no—
221#101—612no—
222#101—1222no—
223#101—1222no—
224#101C2×#692312yescomplemented
225#97—2442no—
226C4×C8—2442no—
227#95—1221no—
228#95—1221no—
229#95—1222no—
230C24—1221no—
231C2×Q8—611no—
232C2×Q8—612no—
233C2×Q8—2441no—
234C2×Q8—2441no—
235C2×Q8C22×#692311yescharacteristic
236C2×D4—611no—
237C2×D4—1221no—
238C2×D4—2441no—
239C2×D4—2441no—
240C2×D4C4×#692311yescharacteristic
241C22×C4—611no—
242C22×C4—611no—
243C22×C4—612no—
244C22×C4—611no—
245C22×C4—612no—
246C22×C4—1221no—
247C22×C4—1221no—
248C22×C4—1221no—
249C22×C4—4882no—
250C22×C4—2442no—
251M16—22no—
252M16—1224no—
253M16—1222no—
254M16—2442no—
255M16—2442no—
256C2×C8—2442no—
257C2×C8—2442no—
258C4⋊C4—4882no—
259C4⋊C4—2441no—
260C4⋊C4—2442no—
261C4⋊C4—2441no—
262C22⋊C4—22no—
263C22⋊C4—1222no—
264C22⋊C4—1222no—
265C22⋊C4—1222no—
266C22⋊C4—4882no—
267C22⋊C4—2442no—
268C42—611no—
269C42—1221no—
270C42—1222no—
271C42—1222no—
272C42—1222no—
273C42—2441no—
274C42C4×#692311yescharacteristic
275C23—21no—
276C23—611no—
277C23—1221no—
278C23—2441no—
279C23—2441no—
280C23—2442no—
281C23—2442no—
282Q8—41no—
283Q8—2442no—
284Q8—2441no—
285D4—41no—
286D4—2441no—
287D4—2442no—
288C2×C4—21no—
289C2×C4—612no—
290C2×C4—611no—
291C2×C4—612no—
292C2×C4—1221no—
293C2×C4—1222no—
294C2×C4—1221no—
295C2×C4—1222no—
296C2×C4—1222no—
297C2×C4—1221no—
298C2×C4—1221no—
299C2×C4—4884no—
300C2×C4—2441no—
301C2×C4—2441no—
302C2×C4—2441no—
303C2×C4—2441no—
304C2×C4—2441no—
305C2×C4—2441no—
306C2×C4—2442no—
307C2×C4—2441no—
308C2×C4—2442no—
309C2×C4—2441no—
310C2×C4C2×C4×#692311yescharacteristic
311C2×C4D4×#692312yes—
312C8—42no—
313C8—2442no—
314C8—2444no—
315C22—42no—
316C22—41no—
317C22—611no—
318C22—611no—
319C22—1221no—
320C22—1221no—
321C22—4881no—
322C22—4881no—
323C22—4882no—
324C22—2442no—
325C22—2441no—
326C22C22⋊C4×#692311yescharacteristic
327C4—22no—
328C4—21no—
329C4—41no—
330C4—1222no—
331C4—1222no—
332C4—1221no—
333C4—4881no—
334C4—4881no—
335C4—2441no—
336C4—2441no—
337C4—2442no—
338C2—21no—
339C2—81no—
340C2—611no—
341C2—611no—
342C2—1221no—
343C2—4881no—
344C2#99×#692311yescharacteristic
subgroupG/H
centerC2#99×#6923
commutatorC2×C4×C61C2×C42
FrattiniC2×Q8C22×#6923
FittingC61×#355C4
socleC2×C61C4×#99
prime radicalC2×D4×D61C2×C4
cosocleC2×Q8×D61C23
hypercenter#355#6923
hypocenterC61#12632
LCSG ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×Q8 ◁ #355

Complements

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

NH ≅ G/Ntype
3. #101×#6923C2semidirect
4. #21460915C2semidirect
16. #16840671C4semidirect
17. #16840670C4semidirect
21. #16839559C4semidirect
22. #16839558C4semidirect
23. C61×#355C4semidirect
38. #14519838C2×C4semidirect
53. #14519696C2×C4semidirect
56. #101×C61C2×C4semidirect
194. #355#6923semidirect
196. C61#12632semidirect
224. #101C2×#6923semidirect

Maximal subgroups

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

subgroupindex
#214623092
D61⋊#3552
#101×#69232
#214609152
#355×D612
#1263261