Finite Group Databasebeta

← #29571759

Subgroups — #29571759

Order 15616.

706 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16840491C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic
3#21444499C212yesmaximal
4#21444249C212yesmaximal
5C61×#1461C211yesmaximal, characteristic
6#16840491—24no—
7#16840491—24no—
8C2×#14519777C412yes—
9C2×#14519777C2211yescharacteristic
10C2×#14519768—21no—
11C2×#14519768—22no—
12C2×Q8×Dic61—21no—
13C2×C4×Dic122C412yes—
14C2×C4×Dic122C2211yescharacteristic
15#16839786C2212yes—
16C2×#14519696—21no—
17#16839531—21no—
18#16839510—24no—
19#16839494—24no—
20C2×C4×Q8×C61C2211yescharacteristic
21C61×#431C2212yes—
22C61×#324—22no—
23C22×Dic122C2×C412yes—
24C22×Dic122C2×C411yescharacteristic
25C2×#11758194—22no—
26#14519777—24no—
27#14519777—24no—
28#14519777—22no—
29#14519777D414yes—
30#14519768—216no—
31#14519768—28no—
32Q8×Dic61—28no—
33C4×Dic122—24no—
34C4×Dic122—22no—
35C4×Dic122—24no—
36C4×Dic122D414yes—
37C2×C4⋊Dic61—22no—
38C2×C4⋊Dic61—22no—
39C2×C4⋊Dic61—21no—
40C2×C4⋊Dic61—21no—
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—22no—
69C22×Dic61—21no—
70C22×Dic61—21no—
71C2×Dic122—21no—
72C2×Dic122—22no—
73C2×Dic122—42no—
74C2×Dic122—44no—
75C2×Dic122C22⋊C412yes—
76C2×Dic122C22⋊C414yes—
77C2×C8×C61—42no—
78C4⋊C4×C61—24no—
79C4⋊C4×C61—22no—
80C42×C61—22no—
81C42×C61—22no—
82C42×C61C2×D412yes—
83C42×C61C2×D412yes—
84C4⋊Dic61—28no—
85C4⋊Dic61—28no—
86C4⋊Dic61—21no—
87C4⋊Dic61—24no—
88C4⋊Dic61—24no—
89C4⋊Dic61C22⋊C412yes—
90#11758196—28no—
91#11758196—24no—
92C4×Dic61—28no—
93C4×Dic61—24no—
94#11758194—42no—
95#11758194—42no—
96C2×#92418—42no—
97C23×C61C2×C22⋊C411yescharacteristic
98Q8×C61—24no—
99Q8×C61—24no—
100C2×C4×C61—22no—
101C2×C4×C61—22no—
102C2×C4×C61—22no—
103C2×C4×C61—22no—
104C2×C4×C61—41no—
105C2×C4×C61—42no—
106C2×C4×C61#10314yes—
107C2×C4×C61C2×C22⋊C411yescharacteristic
108C2×C4×C61C2×C22⋊C411yescharacteristic
109C2×C4×C61#12012yes—
110C2×C4×C61#12012yes—
111C2×C4×C61C2×SD1611yescharacteristic
112C2×C4×C61C2×Q1611yescharacteristic
113C2×C4×C61#13712yes—
114C2×Dic61—22no—
115C2×Dic61—22no—
116C2×Dic61—24no—
117C2×Dic61—42no—
118C2×Dic61—44no—
119C2×Dic61—42no—
120Dic122—24no—
121Dic122—42no—
122Dic122—44no—
123Dic122—44no—
124C8×C61—82no—
125#92418—82no—
126#8392—611nomaximal, Sylow
127C22×C61—21no—
128C22×C61—21no—
129C22×C61#37911yescharacteristic
130C22×C61C2×#10311yescharacteristic
131C22×C61#41911yescharacteristic
132C4×C61—22no—
133C4×C61—21no—
134C4×C61—24no—
135C4×C61—42no—
136C4×C61#41312yes—
137C4×C61#44812yes—
138C4×C61#45112yes—
139Dic61—42no—
140Dic61—42no—
141Dic61—44no—
142C2×#557—611no—
143#1801—611no—
144#1548—612no—
145#1487—612no—
146#1461D6111yescharacteristic
147C2×C61—21no—
148C2×C61—21no—
149C2×C61#178511yescharacteristic
150C2×C61#178711yescharacteristic
151C2×C61#179611yescharacteristic
152#557—1224no—
153#557—1224no—
154C2×C4⋊Q8—612no—
155C2×C4⋊Q8—611no—
156C2×#125—1221no—
157C2×#125—1222no—
158C2×C4×Q8—612no—
159C2×C4×Q8—611no—
160C2×C4×Q8—1221no—
161C2×C4×Q8C2×D6111yescharacteristic
162#431C2×D6112yes—
163#423—612no—
164C2×#101—1221no—
165#343—1221no—
166#333—1224no—
167Q8⋊C8—1224no—
168#324—22no—
169C61#839211yescharacteristic, complemented, Sylow
170C22×Q8—611no—
171C22×Q8—612no—
172C22×Q8D24411yescharacteristic
173C2×M16—1222no—
174C4⋊Q8—614no—
175C4⋊Q8—1224no—
176C4⋊Q8—1224no—
177C4⋊Q8—1222no—
178#125—12216no—
179#125—1228no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—614no—
183C4×Q8—1224no—
184C4×Q8—1224no—
185C4×Q8—1228no—
186C4×Q8—1222no—
187C2×C4⋊C4—611no—
188C2×C4⋊C4—1222no—
189C2×C4⋊C4—1222no—
190C2×C4⋊C4—1222no—
191C2×C4⋊C4—1221no—
192C2×C4⋊C4—1221no—
193C2×C4⋊C4—1221no—
194C2×C4⋊C4D24411yescharacteristic
195C2×C4⋊C4C22⋊D6112yes—
196C2×C42—1222no—
197C2×C42—1221no—
198C2×C42C4×D6112yes—
199C2×C42C22×D6111yescharacteristic
200C4⋊C8—22no—
201C4⋊C8—1222no—
202C4⋊C8—1222no—
203#101—2442no—
204C22⋊C8—22no—
205C4×C8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—612no—
209C2×Q8—614no—
210C2×Q8—1221no—
211C2×Q8—1222no—
212C2×Q8—2442no—
213C2×Q8—2444no—
214C2×Q8#1175819012yes—
215C2×Q8Dic24412yes—
216C22×C4—1221no—
217C22×C4—1222no—
218C22×C4—1221no—
219C22×C4#1175819812yes—
220C22×C4#1175819812yes—
221C22×C4C2×C4×D6111yescharacteristic
222C22×C4C2×D24411yescharacteristic
223C22×C4C2×C22⋊D6111yescharacteristic
224M16—2442no—
225M16—2442no—
226C2×C8—42no—
227C2×C8—2442no—
228C4⋊C4—24no—
229C4⋊C4—22no—
230C4⋊C4—612no—
231C4⋊C4—1228no—
232C4⋊C4—1228no—
233C4⋊C4—1228no—
234C4⋊C4—1221no—
235C4⋊C4—1224no—
236C4⋊C4—1224no—
237C4⋊C4—1224no—
238C42—22no—
239C42—22no—
240C42—1228no—
241C42—1224no—
242C42D4×D6112yes—
243C42D4×D6112yes—
244C23C2×#1175819811yescharacteristic
245Q8—24no—
246Q8—24no—
247Q8—1224no—
248Q8—2444no—
249Q8—2442no—
250Q8—2444no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—42no—
257C2×C4—1222no—
258C2×C4—1222no—
259C2×C4—1224no—
260C2×C4—2442no—
261C2×C4—2442no—
262C2×C4—2444no—
263C2×C4#1451970214yes—
264C2×C4C22⋊C4×D6111yescharacteristic
265C2×C4C22⋊C4×D6111yescharacteristic
266C2×C4#1451974512yes—
267C2×C4#1451974712yes—
268C2×C4C2×#1175819011yescharacteristic
269C2×C4C2×Dic24411yescharacteristic
270C2×C4#1451981012yes—
271C8—82no—
272C8—4882no—
273C22—21no—
274C22—21no—
275C22#1683961711yescharacteristic
276C22C2×#1451970211yescharacteristic
277C22#1683977211yescharacteristic
278C4—21no—
279C4—24no—
280C4—22no—
281C4—42no—
282C4—2442no—
283C4—2444no—
284C4—2442no—
285C4#1683974512yes—
286C4#1683990312yes—
287C4#1683992012yes—
288C2—21no—
289C2—21no—
290C2#2144574611yescharacteristic
291C2#2144575511yescharacteristic
292C2#2144579211yescharacteristic
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#8392
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
169. C61#8392semidirect

Maximal subgroups

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

subgroupindex
C2×#168404912
#214458152
#214444992
#214442492
C61×#14612
#839261