Finite Group Databasebeta

← #16785298

Subgroups — #16785298

Order 3328.

1374 conjugacy classes of subgroups; only the 228 normal subgroups are shown.

normal subgroups only

This group has 1374 conjugacy classes of subgroups — too many to draw. Only its 228 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×#11719126C212yesmaximal
2C2×#11718291C214yesmaximal, complemented
3C22×#88126C211yesmaximal, characteristic
4C23×C4×D13C2211yescharacteristic
5C22×#6122C2214yes—
6C22×C42×C13C411yescharacteristic
7C22×#6103C412yes—
8C22×#6101C2212yes—
9C22×C4×Dic13C411yescharacteristic
10C2×#88126—24no—
11C2×#88126—28no—
12#11719126—24no—
13#11719126C418yescomplemented
14C2×#88111—24no—
15#11718291—28no—
16C24×D13C2×C411yescharacteristic
17C23×C4×C13C2×C412yes—
18C23×C4×C13C2×C411yescharacteristic
19C23×Dic13C2×C412yes—
20C23×Dic13C2311yescharacteristic
21C22×C4×D13—24no—
22C22×C4×D13—24no—
23C22×C4×D13C2×C414yes—
24C22×C4×D13D412yes—
25C22×C4×D13D414yes—
26C22×C4×D13Q812yes—
27C2×#6122—24no—
28C2×#6122—28no—
29C2×#6122—28no—
30C2×#6122C2×C418yes—
31C22×#1057—24no—
32C2×C42×C13—24no—
33C2×C42×C13—22no—
34C2×#6103—24no—
35C2×#6103—24no—
36C2×#6103—24no—
37C2×#6101—24no—
38C2×#6101—24no—
39C2×#6101D418yes—
40C2×C4×Dic13—22no—
41C2×C4×Dic13—24no—
42#88126—232no—
43#88111—216no—
44#10728—131nomaximal, Sylow
45C24×C13C22×C411yescharacteristic
46C23×D13—22no—
47C23×D13—22no—
48C23×D13—44no—
49C23×D13C4212yes—
50C23×D13C22⋊C412yes—
51C23×D13C4⋊C412yes—
52C22×C4×C13—22no—
53C22×C4×C13—22no—
54C22×C4×C13—22no—
55C22×C4×C13—22no—
56C22×C4×C13—44no—
57C22×C4×C13C4212yes—
58C22×C4×C13C22⋊C412yes—
59C22×C4×C13C22⋊C414yes—
60C22×C4×C13C4⋊C412yes—
61C22×Dic13—22no—
62C22×Dic13—22no—
63C22×Dic13—44no—
64C22×Dic13C22⋊C414yes—
65C22×Dic13C22×C412yes—
66C22×Dic13C2×D411yescharacteristic
67C22×Dic13C2×D412yes—
68C22×Dic13C2×D414yes—
69C22×Dic13C2×Q811yescharacteristic
70C22×Dic13#4114yes—
71C2×C4×D13—216no—
72C2×C4×D13—28no—
73C2×C4×D13—28no—
74C2×C4×D13—28no—
75C2×C4×D13—28no—
76C2×C4×D13C22⋊C418yes—
77C2×C4×D13C4⋊C418yes—
78#6122—216no—
79#6122—48no—
80C2×#1057—28no—
81C2×#1057—44no—
82C2×#1057—44no—
83C42×C13—28no—
84#6103—216no—
85#6101—216no—
86C4×Dic13—28no—
87C22×#117—131no—
88C2×#407—132no—
89C2×#343—134no—
90C23×C13—24no—
91C23×C13C2×C4211yescharacteristic
92C23×C13C2×C22⋊C412yes—
93C23×C13C2×C22⋊C411yescharacteristic
94C23×C13C2×C4⋊C411yescharacteristic
95C23×C13#11712yes—
96C22×D13—24no—
97C22×D13—22no—
98C22×D13—24no—
99C22×D13—22no—
100C22×D13—28no—
101C22×D13—44no—
102C22×D13—44no—
103C22×D13—48no—
104C22×D13—44no—
105C22×D13—44no—
106C22×D13#9514yes—
107C2×C4×C13—24no—
108C2×C4×C13—22no—
109C2×C4×C13—24no—
110C2×C4×C13—22no—
111C2×C4×C13—24no—
112C2×C4×C13—44no—
113C2×C4×C13—44no—
114C2×C4×C13—44no—
115C2×C4×C13#9514yes—
116C2×Dic13—24no—
117C2×Dic13—44no—
118C2×Dic13—44no—
119C2×Dic13—44no—
120C2×Dic13C2×C22⋊C412yes—
121C2×Dic13C2×C4⋊C412yes—
122C2×Dic13C4×D418yes—
123C2×Dic13#12014yes—
124C2×Dic13#12114yes—
125C2×Dic13C22⋊Q814yes—
126C2×Dic13#12314yes—
127C4×D13—216no—
128C4×D13—216no—
129#1057—48no—
130C24×C4—131no—
131C22×M16—134no—
132C2×#117—268no—
133C2×#117—264no—
134C22×C4⋊C4—132no—
135C22×C22⋊C4—132no—
136C22×C42—131no—
137C22×C42#26111yescharacteristic
138#407—138no—
139#407—264no—
140C2×C22⋊C8—264no—
141#343—268no—
142C22×C13—24no—
143C22×C13—22no—
144C22×C13—22no—
145C22×C13—22no—
146C22×C13—22no—
147C22×C13C2×#9511yescharacteristic
148C22×C13C4×C22⋊C412yes—
149C22×C13#37911yescharacteristic
150C22×C13#38012yes—
151C22×C13#38111yescharacteristic
152C22×C13#41314yes—
153C2×D13—24no—
154C2×D13—28no—
155C2×D13—48no—
156C2×D13—48no—
157C2×D13—44no—
158C2×D13—44no—
159C4×C13—24no—
160C4×C13—44no—
161Dic13—44no—
162Dic13#38514yes—
163Dic13#38614yes—
164C25—131no—
165C23×C4—131no—
166C23×C4—134no—
167C23×C4—132no—
168C23×C4—132no—
169C23×C4—134no—
170C23×C4—132no—
171C23×C4—264no—
172C23×C4—264no—
173C23×C4C2×#26112yes—
174C23×C4C2×#26111yescharacteristic
175C2×M16—138no—
176C2×M16—264no—
177C2×M16—268no—
178C2×M16—268no—
179C22×C8—264no—
180#117—2632no—
181C2×C4⋊C4—138no—
182C2×C4⋊C4—264no—
183C2×C4⋊C4—264no—
184C2×C22⋊C4—264no—
185C2×C22⋊C4—264no—
186C2×C22⋊C4—264no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—264no—
190C2×C42—262no—
191C22⋊C8—2616no—
192C2×C13—22no—
193C2×C13—22no—
194C2×C13C22⋊#9511yescharacteristic
195C2×C13#174212yes—
196C2×C13#178512yes—
197C2×C13#182212yes—
198D13—44no—
199C24—132no—
200C24—132no—
201C24—132no—
202C24—262no—
203C24—262no—
204C24—524no—
205C24C22×#26111yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—44no—
211C22×C4—132no—
212C22×C4—131no—
213C22×C4—131no—
214C22×C4—134no—
215C22×C4—134no—
216C22×C4—132no—
217C22×C4—138no—
218C22×C4—138no—
219C22×C4—134no—
220C22×C4—268no—
221C22×C4—2616no—
222C22×C4—268no—
223C22×C4—268no—
224C22×C4—268no—
225C22×C4—262no—
226C22×C4—262no—
227C22×C4—524no—
228C22×C4C4×#26112yes—
229C22×C4C4⋊#26112yes—
230C22×C4C22⋊#26112yes—
231C22×C4C22⋊#26114yes—
232M16—2616no—
233M16—528no—
234C2×C8—268no—
235C2×C8—524no—
236C2×C8—524no—
237C4⋊C4—2616no—
238C22⋊C4—2616no—
239C42—28no—
240C42—268no—
241C13#1072811yescharacteristic, complemented, Sylow
242C23—24no—
243C23—134no—
244C23—262no—
245C23—268no—
246C23—264no—
247C23—262no—
248C23—264no—
249C23—524no—
250C23—524no—
251C23—524no—
252C23—524no—
253C23—528no—
254C23C2×C4×#26111yescharacteristic
255C23C2×C4⋊#26111yescharacteristic
256C23#8823712yes—
257C23C2×C22⋊#26112yes—
258C23C2×C22⋊#26111yescharacteristic
259C2×C4—22no—
260C2×C4—24no—
261C2×C4—22no—
262C2×C4—24no—
263C2×C4—24no—
264C2×C4—44no—
265C2×C4—44no—
266C2×C4—44no—
267C2×C4—134no—
268C2×C4—132no—
269C2×C4—132no—
270C2×C4—138no—
271C2×C4—134no—
272C2×C4—134no—
273C2×C4—134no—
274C2×C4—2616no—
275C2×C4—2616no—
276C2×C4—264no—
277C2×C4—524no—
278C2×C4—524no—
279C2×C4—524no—
280C2×C4(C2×#261)⋊C414yes—
281C8—528no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—22no—
286C22—24no—
287C22—264no—
288C22—268no—
289C22—528no—
290C22—528no—
291C22—524no—
292C22—524no—
293C22#1171913214yes—
294C22C2×(C2×#261)⋊C411yescharacteristic
295C22C4×C22⋊#26112yes—
296C22#1171913511yescharacteristic
297C22#1171913612yes—
298C22#1171917111yescharacteristic
299C4—24no—
300C4—44no—
301C4—134no—
302C4—134no—
303C4—524no—
304C2—22no—
305C2—22no—
306C2—524no—
307C2#1313001011yescharacteristic
308C2#1313054112yes—
309C2#1313067212yes—
310C2#1313073712yes—
subgroupG/H
centerC23C2×C22⋊#261
commutatorC23×C13C2×C42
FrattiniC24C22×#261
FittingC22×C42×C13C4
socleC23×C13C2×C22⋊C4
prime radicalC24×D13C2×C4
cosocleC23×Dic13C23
hypercenterC22×C42#261
hypocenterC13#10728
LCSG ▷ C23×C13 ▷ C2×C13 ▷ C13
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#11718291C2semidirect
13. #11719126C4semidirect
241. C13#10728semidirect

Maximal subgroups

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

subgroupindex
C2×#117191262
C2×#117182912
C22×#881262
#1072813