Finite Group Databasebeta

← #15760874

Subgroups — #15760874

Order 3328.

682 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 111 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#13123877C211yesmaximal, characteristic
2#13114760C212yesmaximal, complemented
3#13113174C211yesmaximal, characteristic
4#13112742C212yesmaximal, complemented
5C13×#2409C211yesmaximal, characteristic
6C23×Q8×C13C2211yescharacteristic
7C2×C13×C22⋊Q8—21no—
8C2×C4×D4×C13C2211yescharacteristic
9C13×#395—21no—
10C13×#393—21no—
11C13×#391—21no—
12C13×#386C412yes—
13C13×#385C412yes—
14C13×#380C2211yescharacteristic
15C2×#88076—24no—
16#11718868—21no—
17#11718868C2212yes—
18#11718868C2212yes—
19#11718673—24no—
20#11718586—21no—
21#11718120—21no—
22#11718117—21no—
23C22×Q8×C13—21no—
24C22×Q8×C13—22no—
25C22×Q8×C13—21no—
26C22×Q8×C13—22no—
27C22×Q8×C13—44no—
28C22×D4×C13C2×C411yescharacteristic
29C23×C4×C13C2×C412yes—
30C23×C4×C13D411yescharacteristic
31C23×C4×C13C2311yescharacteristic
32C13×C22⋊Q8—44no—
33C4×D4×C13—22no—
34C4×D4×C13—22no—
35C2×C13×C4⋊C4—22no—
36C2×C13×C4⋊C4—21no—
37C2×C13×C4⋊C4D411yescharacteristic
38C2×C13×C22⋊C4—22no—
39C2×C13×C22⋊C4C2×C411yescharacteristic
40C2×C13×C22⋊C4C2×C412yes—
41C2×C13×C22⋊C4D411yescharacteristic
42C2×C42×C13D411yescharacteristic
43C2×#6099—22no—
44C2×#6099—22no—
45C2×#6099D414yes—
46C13×#95—22no—
47C13×#95—22no—
48C13×#95—21no—
49#88076—48no—
50C22⋊#1055—22no—
51C22⋊#1055—22no—
52C22⋊#1055—22no—
53C22⋊#1055—22no—
54C22⋊#1055—22no—
55C4⋊#1055—22no—
56C4×#1055—22no—
57#9506—131nomaximal, Sylow
58C24×C13C22⋊C411yescharacteristic
59C24×C13C22×C411yescharacteristic
60C2×Q8×C13—28no—
61C2×Q8×C13—24no—
62C2×Q8×C13—44no—
63C2×Q8×C13—42no—
64C2×Q8×C13—44no—
65C2×Q8×C13—48no—
66C2×Q8×C13—41no—
67C2×Q8×C13—42no—
68C2×Q8×C13—41no—
69C2×D4×C13—22no—
70C2×D4×C13—22no—
71C2×D4×C13—41no—
72C22×C4×C13—22no—
73C22×C4×C13—24no—
74C22×C4×C13—21no—
75C22×C4×C13—21no—
76C22×C4×C13—22no—
77C22×C4×C13—44no—
78C22×C4×C13—42no—
79C22×C4×C13—42no—
80C22×C4×C13C22⋊C411yescharacteristic
81C22×C4×C13C22⋊C411yescharacteristic
82C22×C4×C13C22⋊C411yescharacteristic
83C22×C4×C13C2×D412yes—
84C22×C4×C13C2×D411yescharacteristic
85C22×C4×C13C2×D411yescharacteristic
86C22×C4×C13#4112yes—
87C13×C4⋊C4—22no—
88C13×C4⋊C4—42no—
89C13×C4⋊C4—44no—
90C13×C22⋊C4—22no—
91C13×C22⋊C4—22no—
92C13×C22⋊C4—44no—
93C13×C22⋊C4—42no—
94C42×C13—21no—
95C42×C13—21no—
96#6099—44no—
97#6099—44no—
98#6099—82no—
99C2×#1055—24no—
100C2×#1055—42no—
101#2975—131no—
102#2409D1311yescharacteristic
103#1784—132no—
104#1465—131no—
105#1305—132no—
106C23×C13—21no—
107C23×C13—21no—
108C23×C13—21no—
109C23×C13—21no—
110C23×C13—21no—
111C23×C13—41no—
112C23×C13—41no—
113C23×C13—41no—
114C23×C13—41no—
115C23×C13C2×C22⋊C411yescharacteristic
116C23×C13C4×D412yes—
117Q8×C13—48no—
118Q8×C13—44no—
119Q8×C13—44no—
120D4×C13—44no—
121C2×C4×C13—21no—
122C2×C4×C13—21no—
123C2×C4×C13—24no—
124C2×C4×C13—22no—
125C2×C4×C13—24no—
126C2×C4×C13—28no—
127C2×C4×C13—21no—
128C2×C4×C13—21no—
129C2×C4×C13—41no—
130C2×C4×C13—42no—
131C2×C4×C13—42no—
132C2×C4×C13—44no—
133C2×C4×C13—42no—
134C2×C4×C13—42no—
135C2×C4×C13—41no—
136C2×C4×C13—41no—
137C2×C4×C13—42no—
138C2×C4×C13—42no—
139C2×C4×C13—42no—
140C2×C4×C13—81no—
141C2×C4×C13#10112yescomplemented
142C2×C4×C13#12011yescharacteristic
143C2×C4×C13#12112yes—
144C2×C4×C13#12311yescharacteristic
145#1055—44no—
146#1055—82no—
147C23×Q8C2×D1311yescharacteristic
148C2×C22⋊Q8—21no—
149C2×C4×D4C2×D1311yescharacteristic
150#436—264no—
151#431—261no—
152C2×#101—264no—
153#407—132no—
154#407—132no—
155#407—261no—
156#395—21no—
157#393—21no—
158#391—21no—
159#386Dic1312yes—
160#385Dic1312yes—
161#380C2×D1311yescharacteristic
162#324—261no—
163#323—261no—
164C22×C13—21no—
165C22×C13—21no—
166C22×C13—22no—
167C22×C13—22no—
168C22×C13—22no—
169C22×C13—42no—
170C22×C13—42no—
171C22×C13—42no—
172C22×C13—41no—
173C22×C13—41no—
174C22×C13—41no—
175C22×C13—81no—
176C22×C13#38611yescharacteristic
177C22×C13C2×#10111yescharacteristic
178C22×C13#41311yescharacteristic
179C4×C13—22no—
180C4×C13—21no—
181C4×C13—22no—
182C4×C13—44no—
183C4×C13—41no—
184C4×C13—41no—
185C4×C13—82no—
186C4×C13B(4,𝔽2)12yes—
187C22×Q8—22no—
188C22×Q8—22no—
189C22×Q8—21no—
190C22×Q8—21no—
191C22×Q8—44no—
192C22×D4C2×Dic1311yescharacteristic
193C23×C4C2×Dic1312yes—
194C23×C4C22⋊D1311yescharacteristic
195C23×C4C22×D1311yescharacteristic
196C2×M16—134no—
197C2×M16—262no—
198C2×M16—262no—
199C22⋊Q8—44no—
200C4×D4—22no—
201C4×D4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4C22⋊D1311yescharacteristic
205C2×C22⋊C4—22no—
206C2×C22⋊C4C2×Dic1311yescharacteristic
207C2×C22⋊C4C2×Dic1312yes—
208C2×C22⋊C4C22⋊D1311yescharacteristic
209C2×C42C22⋊D1311yescharacteristic
210C4⋊C8—262no—
211#101—528no—
212C22⋊C8—262no—
213C22⋊C8—262no—
214C22⋊C8—262no—
215C22⋊C8—262no—
216C22⋊C8—262no—
217C4×C8—262no—
218#95—22no—
219#95—22no—
220#95—21no—
221C2×C13—21no—
222C2×C13—21no—
223C2×C13—42no—
224C2×C13—81no—
225C2×C13#188911yescharacteristic
226C2×C13#190111yescharacteristic
227C2×C13#191511yescharacteristic
228C24C22⋊Dic1311yescharacteristic
229C24C22×Dic1311yescharacteristic
230C2×Q8—28no—
231C2×Q8—24no—
232C2×Q8—42no—
233C2×Q8—44no—
234C2×Q8—44no—
235C2×Q8—48no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—42no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—41no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—44no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4C22⋊Dic1311yescharacteristic
251C22×C4C22⋊Dic1311yescharacteristic
252C22×C4C22⋊Dic1311yescharacteristic
253C22×C4D4×D1312yes—
254C22×C4#612912yes—
255C22×C4C2×C22⋊D1311yescharacteristic
256C22×C4C2×C22⋊D1311yescharacteristic
257M16—524no—
258M16—524no—
259M16—1042no—
260C2×C8—264no—
261C2×C8—522no—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C4⋊C4—44no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—44no—
268C22⋊C4—42no—
269C42—21no—
270C42—21no—
271C13#950611yescharacteristic, complemented, Sylow
272C23—21no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—41no—
281C23D4×Dic1312yes—
282C23C2×C22⋊Dic1311yescharacteristic
283Q8—48no—
284Q8—44no—
285Q8—44no—
286D4—44no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—24no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—81no—
307C2×C4#8807612yescomplemented
308C2×C4#8818911yescharacteristic
309C2×C4#8819111yescharacteristic
310C2×C4#8819312yes—
311C8—524no—
312C8—1042no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—42no—
319C22—42no—
320C22—42no—
321C22—41no—
322C22—41no—
323C22—41no—
324C22—81no—
325C22#1171888111yescharacteristic
326C22C2×#8807611yescharacteristic
327C22#1171889811yescharacteristic
328C4—22no—
329C4—21no—
330C4—22no—
331C4—44no—
332C4—41no—
333C4—41no—
334C4—82no—
335C4#1171869312yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—81no—
340C2#1311524511yescharacteristic
341C2#1311530511yescharacteristic
342C2#1311538911yescharacteristic
subgroupG/H
centerC22#11718881
commutatorC24×C13C22×C4
FrattiniC23×C4C22×D13
FittingC13×#2409C2
socleC22×C13#386
prime radicalC22×D4×C13C2×C4
cosocleC23×C4×C13C23
hypercenter#2409D13
hypocenterC13#9506
LCSG ▷ C24×C13 ▷ C22×C13 ▷ C13
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2409

Complements

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

NH ≅ G/Ntype
2. #13114760C2semidirect
4. #13112742C2semidirect
141. C2×C4×C13#101semidirect
271. C13#9506semidirect
307. C2×C4#88076semidirect

Maximal subgroups

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

subgroupindex
#131238772
#131147602
#131131742
#131127422
C13×#24092
#950613