Finite Group Databasebeta

← #29571779

Subgroups — #29571779

Order 15616.

738 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 147 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
1C2×Q8×Dic122C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic
3#21444503C212yesmaximal
4#21444249C212yesmaximal
5C61×#1461C211yesmaximal, characteristic
6Q8×Dic122—24no—
7Q8×Dic122—24no—
8C2×#14519777C412yes—
9C2×#14519777C2211yescharacteristic
10C2×#14519776—21no—
11C2×#14519776—22no—
12C2×Q8×Dic61—21no—
13C2×C4×Dic122C412yes—
14C2×C4×Dic122C2211yescharacteristic
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839509—24no—
19#16839494—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122—22no—
24C22×Dic122—21no—
25C22×Dic122C2×C411yescharacteristic
26C2×#11758194—22no—
27#14519777—22no—
28#14519777—24no—
29#14519777—24no—
30#14519777D414yes—
31#14519776—216no—
32#14519776—28no—
33Q8×Dic61—28no—
34C4×Dic122—24no—
35C4×Dic122—22no—
36C4×Dic122—24no—
37C4×Dic122D414yes—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61C2×C412yes—
41C2×C4⋊Dic61C2×C411yescharacteristic
42C2×#11758196—22no—
43C2×#11758196—21no—
44C2×C4×Dic61—22no—
45C2×C4×Dic61—21no—
46C4⋊#92418—22no—
47C4⋊#92418—22no—
48#14519696—42no—
49C22×Q8×C61D411yescharacteristic
50C4×Q8×C61—22no—
51C4×Q8×C61—22no—
52C2×C4⋊C4×C61D411yescharacteristic
53C2×C4⋊C4×C61D412yes—
54C2×C42×C61C2×C412yes—
55C2×C42×C61C2311yescharacteristic
56C4⋊C8×C61—22no—
57C22⋊C8×C61—22no—
58C4×C8×C61—22no—
59C2×Q8×C61—22no—
60C2×Q8×C61—22no—
61C2×Q8×C61SD1612yes—
62C2×Q8×C61Q1612yes—
63C22×C4×C61C22⋊C412yes—
64C22×C4×C61C22⋊C412yes—
65C22×C4×C61C22×C411yescharacteristic
66C22×C4×C61C2×D411yescharacteristic
67C22×C4×C61C2×D411yescharacteristic
68C22×Dic61—21no—
69C22×Dic61—21no—
70C22×Dic61—22no—
71C2×Dic122—21no—
72C2×Dic122—24no—
73C2×Dic122—28no—
74C2×Dic122—24no—
75C2×Dic122—82no—
76C2×Dic122—84no—
77C2×Dic122C22⋊C412yes—
78C2×C8×C61—42no—
79C4⋊C4×C61—24no—
80C4⋊C4×C61—22no—
81C42×C61—22no—
82C42×C61—22no—
83C42×C61C2×D412yes—
84C42×C61C2×D412yes—
85C4⋊Dic61—28no—
86C4⋊Dic61—21no—
87C4⋊Dic61—22no—
88C4⋊Dic61—24no—
89C4⋊Dic61C22⋊C414yes—
90C4⋊Dic61C22⋊C412yes—
91#11758196—28no—
92#11758196—24no—
93C4×Dic61—28no—
94C4×Dic61—24no—
95#11758194—42no—
96#11758194—42no—
97C2×#92418—42no—
98C23×C61C2×C22⋊C411yescharacteristic
99Q8×C61—24no—
100Q8×C61—24no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—22no—
105C2×C4×C61—41no—
106C2×C4×C61—42no—
107C2×C4×C61#10314yes—
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61C2×C22⋊C411yescharacteristic
110C2×C4×C61#12012yes—
111C2×C4×C61#12012yes—
112C2×C4×C61C2×SD1611yescharacteristic
113C2×C4×C61C2×Q1611yescharacteristic
114C2×C4×C61#13712yes—
115C2×Dic61—24no—
116C2×Dic61—22no—
117C2×Dic61—24no—
118C2×Dic61—22no—
119C2×Dic61—44no—
120C2×Dic61—42no—
121Dic122—24no—
122Dic122—22no—
123Dic122—84no—
124Dic122—88no—
125Dic122#10114yescomplemented
126C8×C61—82no—
127#92418—82no—
128#8396—611nomaximal, Sylow
129C22×C61—21no—
130C22×C61—21no—
131C22×C61#37911yescharacteristic
132C22×C61C2×#10311yescharacteristic
133C22×C61#41911yescharacteristic
134C4×C61—22no—
135C4×C61—21no—
136C4×C61—24no—
137C4×C61—42no—
138C4×C61C2×#10112yes—
139C4×C61#44812yes—
140C4×C61#45112yes—
141Dic61—24no—
142Dic61—44no—
143Dic61—42no—
144C2×Q82—611no—
145#1801—611no—
146#1549—612no—
147#1487—612no—
148#1461D6111yescharacteristic
149C2×C61—21no—
150C2×C61—21no—
151C2×C61#178411yescharacteristic
152C2×C61#178711yescharacteristic
153C2×C61#179511yescharacteristic
154Q82—1224no—
155Q82—1224no—
156C2×C4⋊Q8—612no—
157C2×C4⋊Q8—611no—
158C2×C4⋊Q8—1221no—
159C2×C4⋊Q8—1222no—
160C2×C4×Q8—612no—
161C2×C4×Q8—611no—
162C2×C4×Q8—1221no—
163C2×C4×Q8C2×D6111yescharacteristic
164#431C2×D6112yes—
165#423—612no—
166C2×#101—1221no—
167#343—1221no—
168#332—1224no—
169Q8⋊C8—1224no—
170#324—22no—
171C61#839611yescharacteristic, complemented, Sylow
172C22×Q8—611no—
173C22×Q8—1222no—
174C22×Q8—1221no—
175C22×Q8D24411yescharacteristic
176C2×M16—1222no—
177C4⋊Q8—614no—
178C4⋊Q8—1224no—
179C4⋊Q8—1224no—
180C4⋊Q8—12216no—
181C4⋊Q8—1228no—
182C4⋊Q8—1222no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—614no—
186C4×Q8—1224no—
187C4×Q8—1224no—
188C4×Q8—1228no—
189C4×Q8—1222no—
190C2×C4⋊C4—611no—
191C2×C4⋊C4—612no—
192C2×C4⋊C4—1222no—
193C2×C4⋊C4—1222no—
194C2×C4⋊C4—1221no—
195C2×C4⋊C4—1221no—
196C2×C4⋊C4D24411yescharacteristic
197C2×C4⋊C4C22⋊D6112yes—
198C2×C42—1222no—
199C2×C42—1221no—
200C2×C42C4×D6112yes—
201C2×C42C22×D6111yescharacteristic
202C4⋊C8—22no—
203C4⋊C8—1222no—
204C4⋊C8—1222no—
205#101—2442no—
206C22⋊C8—22no—
207C4×C8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—612no—
211C2×Q8—1224no—
212C2×Q8—1228no—
213C2×Q8—1221no—
214C2×Q8—1224no—
215C2×Q8—4882no—
216C2×Q8—4884no—
217C2×Q8#1175819012yes—
218C2×Q8Dic24412yes—
219C22×C4—1221no—
220C22×C4—1222no—
221C22×C4—1221no—
222C22×C4#1175819812yes—
223C22×C4#1175819812yes—
224C22×C4C2×C4×D6111yescharacteristic
225C22×C4C2×D24411yescharacteristic
226C22×C4C2×C22⋊D6111yescharacteristic
227M16—2442no—
228M16—2442no—
229C2×C8—42no—
230C2×C8—2442no—
231C4⋊C4—24no—
232C4⋊C4—22no—
233C4⋊C4—612no—
234C4⋊C4—614no—
235C4⋊C4—1228no—
236C4⋊C4—1228no—
237C4⋊C4—1221no—
238C4⋊C4—1222no—
239C4⋊C4—1224no—
240C4⋊C4—1224no—
241C42—22no—
242C42—22no—
243C42—1228no—
244C42—1224no—
245C42D4×D6112yes—
246C42D4×D6112yes—
247C23C2×#1175819811yescharacteristic
248Q8—24no—
249Q8—24no—
250Q8—614no—
251Q8—1224no—
252Q8—1222no—
253Q8—4884no—
254Q8—4888no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—41no—
260C2×C4—42no—
261C2×C4—1224no—
262C2×C4—1222no—
263C2×C4—1224no—
264C2×C4—1222no—
265C2×C4—2444no—
266C2×C4—2442no—
267C2×C4#1451970214yes—
268C2×C4C22⋊C4×D6111yescharacteristic
269C2×C4C22⋊C4×D6111yescharacteristic
270C2×C4#1451974512yes—
271C2×C4#1451974712yes—
272C2×C4C2×#1175819011yescharacteristic
273C2×C4C2×Dic24411yescharacteristic
274C2×C4#1451981012yes—
275C8—82no—
276C8—4882no—
277C22—21no—
278C22—21no—
279C22#1683961711yescharacteristic
280C22C2×#1451970211yescharacteristic
281C22#1683977211yescharacteristic
282C4—21no—
283C4—24no—
284C4—22no—
285C4—42no—
286C4—1224no—
287C4—2444no—
288C4—2442no—
289C4#101×D6112yes—
290C4#1683990312yes—
291C4#1683992012yes—
292C2—21no—
293C2—21no—
294C2#2144574111yescharacteristic
295C2#2144575511yescharacteristic
296C2#2144578711yescharacteristic
subgroupG/H
centerC22#16839617
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#1461C2
socleC22×C61#379
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C42×C61C23
hypercenter#1461D61
hypocenterC61#8396
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
125. Dic122#101semidirect
171. C61#8396semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1222
#214458152
#214445032
#214442492
C61×#14612
#839661