Finite Group Databasebeta

← #16781701

Subgroups — #16781701

Order 3328.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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#13130670C212yesmaximal
2#13130086C212yesmaximal
3#13130084C212yesmaximal
4C22×#88142C211yesmaximal, characteristic
5C23×Dic26C411yescharacteristic
6C23×Dic26C2211yescharacteristic
7C22×C13×C4⋊C4C411yescharacteristic
8C22×C4⋊Dic13C2211yescharacteristic
9C22×#6101C411yescharacteristic
10C22×#6101C2211yescharacteristic
11C22×C4×Dic13C411yescharacteristic
12C2×#88142—24no—
13C2×#88142—24no—
14C2×#88142—24no—
15#11719169C2214yes—
16C2×#88113—24no—
17#11718286—24no—
18#11718285—24no—
19C23×C4×C13C2×C411yescharacteristic
20C23×C4×C13C2×C411yescharacteristic
21C23×C4×C13C2×C411yescharacteristic
22C23×Dic13C2×C411yescharacteristic
23C23×Dic13C2×C411yescharacteristic
24C23×Dic13C2×C411yescharacteristic
25C23×Dic13C2311yescharacteristic
26C22×Dic26—22no—
27C22×Dic26—22no—
28C22×Dic26—22no—
29C22×Dic26—22no—
30C22×Dic26—22no—
31C22×Dic26—44no—
32C22×Dic26—44no—
33C22×Dic26D414yes—
34C2×#6122—24no—
35C2×C13×C4⋊C4—22no—
36C2×C13×C4⋊C4—22no—
37C2×C13×C4⋊C4—22no—
38C2×C4⋊Dic13—22no—
39C2×C4⋊Dic13—22no—
40C2×C4⋊Dic13D414yes—
41C2×#6101—22no—
42C2×#6101—22no—
43C2×#6101—22no—
44C2×#6101—22no—
45C2×#6101—22no—
46C2×#6101D414yes—
47C2×C4×Dic13—22no—
48C2×C4×Dic13—22no—
49C2×C4×Dic13—22no—
50#88142—232no—
51C22⋊#1057—24no—
52C22⋊#1057—24no—
53#88113—48no—
54#8551—131nomaximal, Sylow
55C24×C13C22×C411yescharacteristic
56C22×C4×C13—21no—
57C22×C4×C13—21no—
58C22×C4×C13—24no—
59C22×C4×C13—21no—
60C22×C4×C13—21no—
61C22×C4×C13—21no—
62C22×C4×C13—21no—
63C22×C4×C13—42no—
64C22×C4×C13—42no—
65C22×C4×C13C22⋊C412yes—
66C22×C4×C13C22⋊C412yes—
67C22×C4×C13C22⋊C412yes—
68C22×Dic13—21no—
69C22×Dic13—21no—
70C22×Dic13—21no—
71C22×Dic13—21no—
72C22×Dic13—24no—
73C22×Dic13—24no—
74C22×Dic13—21no—
75C22×Dic13—21no—
76C22×Dic13—42no—
77C22×Dic13—42no—
78C22×Dic13C22⋊C412yes—
79C22×Dic13C22⋊C412yes—
80C22×Dic13C22⋊C412yes—
81C22×Dic13C2×D412yes—
82C22×Dic13C2×D412yes—
83C22×Dic13C2×D412yes—
84C2×Dic26—28no—
85C2×Dic26—28no—
86C2×Dic26—28no—
87C2×Dic26—44no—
88C2×Dic26—48no—
89C2×Dic26—48no—
90C2×Dic26—48no—
91C2×Dic26—48no—
92C2×Dic26—44no—
93C2×Dic26—44no—
94#6122—48no—
95C2×#1057—44no—
96C13×C4⋊C4—28no—
97C4⋊Dic13—28no—
98#6101—28no—
99#6101—28no—
100C4×Dic13—28no—
101C22×C4⋊Q8—131no—
102#1784—132no—
103#1463—132no—
104#1463—132no—
105C23×C13—22no—
106C23×C13—22no—
107C23×C13—22no—
108C23×C13C2×C22⋊C411yescharacteristic
109C23×C13C2×C22⋊C411yescharacteristic
110C23×C13C2×C22⋊C411yescharacteristic
111C2×C4×C13—24no—
112C2×C4×C13—24no—
113C2×C4×C13—22no—
114C2×C4×C13—22no—
115C2×C4×C13—22no—
116C2×C4×C13—22no—
117C2×C4×C13—24no—
118C2×C4×C13—42no—
119C2×C4×C13—42no—
120C2×C4×C13—42no—
121C2×C4×C13—42no—
122C2×C4×C13—42no—
123C2×C4×C13#10114yescomplemented
124C2×Dic13—24no—
125C2×Dic13—24no—
126C2×Dic13—24no—
127C2×Dic13—24no—
128C2×Dic13—22no—
129C2×Dic13—22no—
130C2×Dic13—24no—
131C2×Dic13—22no—
132C2×Dic13—22no—
133C2×Dic13—22no—
134C2×Dic13—42no—
135C2×Dic13—42no—
136C2×Dic13—42no—
137C2×Dic13—42no—
138C2×Dic13—42no—
139C2×Dic13#9914yes—
140C2×Dic13#10114yescomplemented
141C2×Dic13#12014yes—
142Dic26—416no—
143Dic26—416no—
144#1057—84no—
145C23×Q8—131no—
146C23×Q8—131no—
147C2×C4⋊Q8—264no—
148C2×C4⋊Q8—264no—
149C2×C4⋊Q8—264no—
150C22×C4⋊C4—131no—
151C22×C4⋊C4—131no—
152C22×C4⋊C4—131no—
153C22×C4⋊C4#26111yescharacteristic
154C22×C42—131no—
155C2×#101—264no—
156#407—134no—
157#324—264no—
158#324—264no—
159C22×C13—28no—
160C22×C13—21no—
161C22×C13—21no—
162C22×C13—21no—
163C22×C13—21no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13#37911yescharacteristic
167C22×C13C2×#9912yes—
168C22×C13C2×#10112yes—
169C22×C13C2×#10112yes—
170C4×C13—24no—
171C4×C13—42no—
172C4×C13—42no—
173Dic13—24no—
174Dic13—24no—
175Dic13—24no—
176Dic13—42no—
177C22×Q8—134no—
178C22×Q8—262no—
179C22×Q8—262no—
180C22×Q8—262no—
181C22×Q8—262no—
182C22×Q8—262no—
183C22×Q8—524no—
184C22×Q8—524no—
185C23×C4—131no—
186C23×C4—131no—
187C23×C4—131no—
188C23×C4—131no—
189C23×C4C2×#26111yescharacteristic
190C23×C4C2×#26111yescharacteristic
191C23×C4C2×#26111yescharacteristic
192C2×M16—264no—
193C4⋊Q8—2632no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—134no—
198C2×C4⋊C4—134no—
199C2×C4⋊C4—262no—
200C2×C4⋊C4—262no—
201C2×C4⋊C4—262no—
202C2×C4⋊C4—262no—
203C2×C4⋊C4—262no—
204C2×C4⋊C4—262no—
205C2×C4⋊C4—262no—
206C2×C42—262no—
207C2×C42—262no—
208C2×C42—262no—
209#101—528no—
210C22⋊C8—264no—
211C22⋊C8—264no—
212C2×C13—22no—
213C2×C13—22no—
214C2×C13—22no—
215C2×C13#178111yescharacteristic
216C2×C13#178411yescharacteristic
217C2×C13#178411yescharacteristic
218C24C22×#26111yescharacteristic
219C2×Q8—268no—
220C2×Q8—268no—
221C2×Q8—268no—
222C2×Q8—524no—
223C2×Q8—524no—
224C2×Q8—528no—
225C2×Q8—528no—
226C2×Q8—528no—
227C2×Q8—524no—
228C2×Q8—528no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—24no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—132no—
239C22×C4—132no—
240C22×C4—132no—
241C22×C4—132no—
242C22×C4—132no—
243C22×C4—132no—
244C22×C4—264no—
245C22×C4—264no—
246C22×C4—261no—
247C22×C4—261no—
248C22×C4—261no—
249C22×C4—261no—
250C22×C4—261no—
251C22×C4—261no—
252C22×C4—522no—
253C22×C4—522no—
254C22×C4C22⋊#26112yes—
255C22×C4C22⋊#26112yes—
256C22×C4C22⋊#26112yes—
257M16—528no—
258C2×C8—524no—
259C4⋊C4—28no—
260C4⋊C4—268no—
261C4⋊C4—268no—
262C4⋊C4—268no—
263C42—268no—
264C13#855111yescharacteristic, complemented, Sylow
265C23—22no—
266C23—22no—
267C23—22no—
268C23C2×C22⋊#26111yescharacteristic
269C23C2×C22⋊#26111yescharacteristic
270C23C2×C22⋊#26111yescharacteristic
271Q8—5216no—
272Q8—5216no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—134no—
286C2×C4—134no—
287C2×C4—134no—
288C2×C4—264no—
289C2×C4—264no—
290C2×C4—264no—
291C2×C4—264no—
292C2×C4—262no—
293C2×C4—262no—
294C2×C4—264no—
295C2×C4—262no—
296C2×C4—262no—
297C2×C4—262no—
298C2×C4—522no—
299C2×C4—522no—
300C2×C4—522no—
301C2×C4—522no—
302C2×C4—522no—
303C2×C4#8811314yescomplemented
304C8—1044no—
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22C2×#8811312yes—
313C22D13⋊#9912yes—
314C22D13⋊#10112yes—
315C22#1171917111yescharacteristic
316C4—24no—
317C4—42no—
318C4—42no—
319C4—264no—
320C4—264no—
321C4—264no—
322C4—522no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2#1313066311yescharacteristic
327C2#1313067011yescharacteristic
328C2#1313067111yescharacteristic
subgroupG/H
centerC22#11719171
commutator, prime radicalC24×C13C22×C4
FrattiniC24C22×#261
FittingC22×C13×C4⋊C4C4
socleC22×C13#379
cosocleC23×Dic13C23
hypercenterC22×C4⋊C4#261
hypocenterC13#8551
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
123. C2×C4×C13#101semidirect
140. C2×Dic13#101semidirect
264. C13#8551semidirect
303. C2×C4#88113semidirect

Maximal subgroups

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

subgroupindex
#131306702
#131300862
#131300842
C22×#881422
#855113