Finite Group Databasebeta

← #16781157

Subgroups — #16781157

Order 3328.

1030 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1030 conjugacy classes of subgroups — too many to draw. Only its 160 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#13130692C212yesmaximal
2C2×#11718303C214yesmaximal, complemented
3C22×#88198C211yesmaximal, characteristic
4C23×Q8×C13C411yescharacteristic
5C23×Dic26C411yescharacteristic
6C2×#88198—28no—
7C2×#88198—22no—
8C2×#88198C412yes—
9C2×#88198C2212yes—
10C22×#6101—22no—
11C22×C4×Dic13C2211yescharacteristic
12C2×C22⋊#1057—22no—
13C2×#88112C2214yes—
14#11718303—28no—
15#11718290—22no—
16C22×Q8×C13—24no—
17C22×Q8×C13—21no—
18C22×Q8×C13—42no—
19C22×Q8×C13—42no—
20C22×Q8×C13C2×C412yes—
21C23×C4×C13—21no—
22C23×C4×C13C2×C411yescharacteristic
23C23×Dic13—21no—
24C23×Dic13D412yes—
25C22×Dic26—21no—
26C22×Dic26—24no—
27C22×Dic26—42no—
28C22×Dic26—42no—
29C22×Dic26C2×C412yes—
30C22×#1057—24no—
31#88198—216no—
32#88198—24no—
33#88198—24no—
34#88198—24no—
35#88198D418yes—
36C2×#6101—216no—
37C2×#6101—24no—
38C2×#6101—24no—
39C2×C4×Dic13—24no—
40C2×C4×Dic13C2×C411yescharacteristic
41C2×C4×Dic13D412yes—
42C2×C4×Dic13C2311yescharacteristic
43C22⋊#1057—44no—
44#88112—28no—
45#8342—131nomaximal, Sylow
46C24×C13C22⋊C411yescharacteristic
47C2×Q8×C13—24no—
48C2×Q8×C13—22no—
49C2×Q8×C13—416no—
50C2×Q8×C13—42no—
51C2×Q8×C13—42no—
52C2×Q8×C13—42no—
53C2×Q8×C13—42no—
54C2×Q8×C13C22⋊C414yes—
55C22×C4×C13—22no—
56C22×C4×C13—22no—
57C22×C4×C13—22no—
58C22×C4×C13—24no—
59C22×C4×C13—22no—
60C22×C4×C13—42no—
61C22×C4×C13—42no—
62C22×C4×C13C22⋊C411yescharacteristic
63C22×C4×C13C22×C411yescharacteristic
64C22×Dic13—24no—
65C22×Dic13—24no—
66C22×Dic13—24no—
67C22×Dic13—22no—
68C22×Dic13—42no—
69C22×Dic13—42no—
70C22×Dic13C22⋊C412yes—
71C22×Dic13SD1614yes—
72C22×Dic13Q1614yes—
73C22×Dic13C2×D412yes—
74C2×Dic26—24no—
75C2×Dic26—22no—
76C2×Dic26—42no—
77C2×Dic26—416no—
78C2×Dic26—42no—
79C2×Dic26—42no—
80C2×Dic26—42no—
81C2×Dic26C22⋊C414yes—
82C2×#1057—44no—
83C2×#1057—44no—
84C2×#1057—44no—
85#6101—216no—
86#6101—216no—
87C4×Dic13—22no—
88C4×Dic13—24no—
89C4×Dic13C2×D414yes—
90C22×C4⋊Q8—131no—
91#1797—132no—
92C2×#333—134no—
93C23×C13—21no—
94C23×C13—21no—
95C23×C13—22no—
96C23×C13#10012yes—
97C23×C13#10314yes—
98C23×C13C2×C22⋊C411yescharacteristic
99Q8×C13—48no—
100Q8×C13—44no—
101Q8×C13—44no—
102C2×C4×C13—28no—
103C2×C4×C13—21no—
104C2×C4×C13—21no—
105C2×C4×C13—21no—
106C2×C4×C13—21no—
107C2×C4×C13—24no—
108C2×C4×C13—48no—
109C2×C4×C13—42no—
110C2×C4×C13—42no—
111C2×C4×C13#10012yes—
112C2×C4×C13C2×C22⋊C412yes—
113C2×Dic13—22no—
114C2×Dic13—22no—
115C2×Dic13—28no—
116C2×Dic13—28no—
117C2×Dic13—24no—
118C2×Dic13—48no—
119C2×Dic13—42no—
120C2×Dic13—42no—
121C2×Dic13#10314yes—
122C2×Dic13#12014yes—
123C2×Dic13C2×SD1612yes—
124C2×Dic13C2×Q1612yes—
125C2×Dic13#13714yes—
126Dic26—48no—
127Dic26—44no—
128Dic26—44no—
129#1057—84no—
130C23×Q8—131no—
131C23×Q8#26111yescharacteristic
132C2×C4⋊Q8—132no—
133C2×C4⋊Q8—132no—
134C2×C4⋊Q8—262no—
135C2×C4⋊Q8—268no—
136C22×C4⋊C4—262no—
137C22×C42—131no—
138C2×C4⋊C8—134no—
139C2×C22⋊C8—262no—
140#336—262no—
141#333—268no—
142C22×C13—24no—
143C22×C13—22no—
144C22×C13—22no—
145C22×C13—24no—
146C22×C13#33314yescomplemented
147C22×C13#37911yescharacteristic
148C22×C13C2×#10011yescharacteristic
149C22×C13C2×#10011yescharacteristic
150C22×C13C2×#10312yes—
151C22×C13#41912yes—
152C4×C13—22no—
153C4×C13—22no—
154C4×C13—44no—
155Dic13—24no—
156Dic13—44no—
157Dic13#44814yes—
158Dic13#45114yes—
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—42no—
162C22×Q8—42no—
163C22×Q8—132no—
164C22×Q8—264no—
165C22×Q8—261no—
166C22×Q8—522no—
167C22×Q8—522no—
168C22×Q8C2×#26112yes—
169C23×C4—21no—
170C23×C4—132no—
171C23×C4—261no—
172C23×C4C2×#26111yescharacteristic
173C22×C8—264no—
174C4⋊Q8—138no—
175C4⋊Q8—264no—
176C4⋊Q8—264no—
177C4⋊Q8—2616no—
178C4⋊Q8—264no—
179C2×C4⋊C4—264no—
180C2×C4⋊C4—264no—
181C2×C4⋊C4—2616no—
182C2×C42—131no—
183C2×C42—132no—
184C2×C42—131no—
185C2×C42—264no—
186C4⋊C8—268no—
187C22⋊C8—524no—
188C2×C13—24no—
189C2×C13C2×#33311yescharacteristic
190C2×C13#154812yes—
191C2×C13#155811yescharacteristic
192C2×C13#178311yescharacteristic
193C2×C13#178712yes—
194C24C22⋊#26111yescharacteristic
195C2×Q8—24no—
196C2×Q8—22no—
197C2×Q8—416no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8—134no—
203C2×Q8—264no—
204C2×Q8—262no—
205C2×Q8—5216no—
206C2×Q8—522no—
207C2×Q8—522no—
208C2×Q8—522no—
209C2×Q8—522no—
210C2×Q8C22⋊#26114yes—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—132no—
219C22×C4—134no—
220C22×C4—134no—
221C22×C4—132no—
222C22×C4—264no—
223C22×C4—264no—
224C22×C4—264no—
225C22×C4—262no—
226C22×C4—522no—
227C22×C4—522no—
228C22×C4C22⋊#26111yescharacteristic
229C22×C4C22×#26111yescharacteristic
230C2×C8—524no—
231C2×C8—524no—
232C2×C8—524no—
233C4⋊C4—2616no—
234C4⋊C4—2616no—
235C42—134no—
236C42—264no—
237C42—262no—
238C13#834211yescharacteristic, complemented, Sylow
239C23—21no—
240C23—21no—
241C23—22no—
242C23#8811714yes—
243C23#8812112yes—
244C23C2×C22⋊#26111yescharacteristic
245Q8—44no—
246Q8—48no—
247Q8—44no—
248Q8—524no—
249Q8—528no—
250Q8—524no—
251C2×C4—28no—
252C2×C4—21no—
253C2×C4—21no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—24no—
257C2×C4—48no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—134no—
261C2×C4—132no—
262C2×C4—132no—
263C2×C4—134no—
264C2×C4—134no—
265C2×C4—268no—
266C2×C4—268no—
267C2×C4—262no—
268C2×C4—262no—
269C2×C4—264no—
270C2×C4—528no—
271C2×C4—522no—
272C2×C4—522no—
273C2×C4#8812112yes—
274C2×C4C2×C22⋊#26112yes—
275C8—1044no—
276C22—24no—
277C22—22no—
278C22—22no—
279C22—24no—
280C22#1171830314yescomplemented
281C22C2×#8811712yes—
282C22#1171915312yes—
283C22C2×#8812111yescharacteristic
284C22C2×#8812111yescharacteristic
285C22#1171917111yescharacteristic
286C4—22no—
287C4—22no—
288C4—44no—
289C4—134no—
290C4—134no—
291C4—264no—
292C4—524no—
293C2—24no—
294C2C2×#1171830311yescharacteristic
295C2#1313034912yes—
296C2#1313036911yescharacteristic
297C2#1313066811yescharacteristic
298C2#1313067512yes—
subgroupG/H
centerC23C2×C22⋊#261
commutatorC22×C4×C13C22×C4
FrattiniC22×C4C22×#261
FittingC23×Q8×C13C4
socleC23×C13C2×C22⋊C4
prime radicalC24×C13C22⋊C4
cosocleC2×C4×Dic13C23
hypercenterC23×Q8#261
hypocenterC13#8342
LCSG ▷ C22×C4×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
2. C2×#11718303C2semidirect
146. C22×C13#333semidirect
238. C13#8342semidirect
280. C22#11718303semidirect

Maximal subgroups

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

subgroupindex
#131306922
C2×#117183032
C22×#881982
#834213