Finite Group Databasebeta

← #27171414

Subgroups — #27171414

Order 12928.

374 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 374 conjugacy classes of subgroups — too many to draw. Only its 83 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#20265650C211yesmaximal, characteristic
2C2×#15692169C211yesmaximal, characteristic
3#20265054C211yesmaximal, characteristic
4#20265052C211yesmaximal, characteristic, complemented
5#20265049C211yesmaximal, characteristic, complemented
6#20264943C211yesmaximal, characteristic, complemented
7#342×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C2211yescharacteristic
9C23×Dic101C2211yescharacteristic
10C4×C22⋊D101—21no—
11C4×C22⋊D101—21no—
12#15692215—21no—
13#15692215—21no—
14#15692211—21no—
15C2×C4⋊Dic101—21no—
16C2×#13108053—21no—
17C2×C22⋊Dic101C2211yescharacteristic
18C2×C22⋊Dic101C2211yescharacteristic
19C2×#13108051—21no—
20C22⋊C4×D101—21no—
21#15692169C414yescomplemented
22#15692162—21no—
23#15692162—21no—
24#15692162—21no—
25C2×C22⋊C4×C101C2211yescharacteristic
26C2×C22⋊C4×C101C2211yescharacteristic
27C2×C22⋊C4×C101C2211yescharacteristic
28C23×D101D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C2×C22⋊D101—21no—
33C2×C22⋊D101C2×C412yes—
34C2×C22⋊D101Q812yes—
35C22×Dic101—21no—
36C22×Dic101—21no—
37C22×Dic101—21no—
38C22×Dic101—21no—
39C22×Dic101—42no—
40C22×Dic101D411yescharacteristic
41C22×Dic101D411yescharacteristic
42C22×Dic101D411yescharacteristic
43C2×C4×D101—21no—
44C2×C4×D101—21no—
45C4⋊Dic101—42no—
46#13108053—21no—
47#13108053—21no—
48#13108053—42no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101—21no—
52C22⋊Dic101C2×C412yescomplemented
53#13108051—21no—
54#13108051—21no—
55#13108051—42no—
56C4×Dic101—21no—
57C4×Dic101—21no—
58C24×C101C2311yescharacteristic
59C22×C4×C101—21no—
60C22×C4×C101—21no—
61C22×C4×C101D411yescharacteristic
62C22×C4×C101D411yescharacteristic
63C22⋊C4×C101—21no—
64C22⋊C4×C101—21no—
65C22⋊C4×C101—21no—
66C22⋊C4×C101—21no—
67C22⋊C4×C101—21no—
68C22⋊C4×C101C2×C412yescomplemented
69C23×C101—41no—
70C23×C101—41no—
71C23×C101C22×C411yescharacteristic
72C23×C101C2×D411yescharacteristic
73C23×C101C2×D411yescharacteristic
74C23×C101C2×D411yescharacteristic
75C23×C101C2×Q811yescharacteristic
76C23×C101#4111yescharacteristic
77C23×C101#4111yescharacteristic
78C22×D101—21no—
79C22×D101—21no—
80C22×D101—41no—
81C22×D101—41no—
82C22×D101C4⋊C412yescomplemented
83C2×C4×C101—22no—
84C2×C4×C101—21no—
85C2×C4×C101—21no—
86C2×C4×C101—41no—
87C2×C4×C101—41no—
88C2×C4×C101—41no—
89C2×C4×C101—41no—
90C2×C4×C101—41no—
91C2×C4×C101—41no—
92C2×C4×C101—41no—
93C22⋊D101—22no—
94C22⋊D101—22no—
95C22⋊D101—22no—
96C22⋊D101—22no—
97C2×Dic101—21no—
98C2×Dic101—21no—
99C2×Dic101—22no—
100C2×Dic101—22no—
101C2×Dic101—22no—
102C2×Dic101—22no—
103C2×Dic101—21no—
104C2×Dic101—21no—
105C2×Dic101—41no—
106C2×Dic101—41no—
107C2×Dic101—41no—
108C2×Dic101—41no—
109C2×Dic101—41no—
110C2×Dic101—41no—
111C2×Dic101—41no—
112C2×Dic101—41no—
113C2×Dic101C4⋊C412yescomplemented
114C4×D101—81no—
115C4×D101—81no—
116C22×C101—21no—
117C22×C101—21no—
118C22×C101—21no—
119C22×C101—21no—
120C22×C101—21no—
121C22×C101—21no—
122C22×C101—41no—
123C22×C101—41no—
124C22×C101—41no—
125C22×C101—41no—
126C22×C101C2×C4⋊C411yescharacteristic
127C22×C101C4×D411yescharacteristic
128C22×C101C4×D411yescharacteristic
129C22×C101#12011yescharacteristic
130C22×C101C22⋊Q811yescharacteristic
131C22×C101C22⋊Q811yescharacteristic
132C22×C101#12311yescharacteristic
133C2×D101—22no—
134C2×D101—41no—
135C2×D101—41no—
136C2×D101—42no—
137C2×D101—81no—
138C4×C101—42no—
139C4×C101—41no—
140C4×C101—41no—
141C4×C101—81no—
142Dic101—22no—
143Dic101—42no—
144Dic101—41no—
145Dic101—42no—
146Dic101—41no—
147C2×C101—21no—
148C2×C101—21no—
149C2×C101—21no—
150C2×C101—21no—
151C2×C101—21no—
152C2×C101—21no—
153C2×C101#38511yescharacteristic
154C2×C101B(4,𝔽2)11yescharacteristic
155C2×C101#45811yescharacteristic
156D101—42no—
157#1867—1011nomaximal, Sylow
158C101#186711yescharacteristic, complemented, Sylow
159#518—1011no—
160C2×#99—1011no—
161#386—1011no—
162#385—1011no—
163#385—1011no—
164#342—1011no—
165#342D10111yescharacteristic, complemented
166C22×D4—1011no—
167C23×C4—1011no—
168C4×D4—2021no—
169C4×D4—2021no—
170C4×D4—2021no—
171C4×D4—2021no—
172#117—2021no—
173C2×C4⋊C4—2021no—
174C2×C4⋊C4—2021no—
175C2×C22⋊C4—1011no—
176C2×C22⋊C4—1011no—
177C2×C22⋊C4—2021no—
178C2×C22⋊C4—2021no—
179C2×C22⋊C4C2×D10111yescharacteristic
180C2×C22⋊C4C2×D10111yescharacteristic
181C2×C22⋊C4C2×D10111yescharacteristic
182#99—1014no—
183#95—2021no—
184#95—2021no—
185#95—2021no—
186C24—1011no—
187C24C22×D10111yescharacteristic
188C2×D4—1012no—
189C2×D4—1012no—
190C2×D4—2021no—
191C2×D4—2021no—
192C2×D4—2021no—
193C2×D4—2021no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—1011no—
197C22×C4—1011no—
198C22×C4—1011no—
199C22×C4—4042no—
200C22×C4—2021no—
201C22×C4—2021no—
202C22×C4—2021no—
203C22×C4—2021no—
204C22×C4—2021no—
205C22×C4—2021no—
206C22×C4D40411yescharacteristic
207C22×C4D40411yescharacteristic
208C4⋊C4—4042no—
209C4⋊C4—4042no—
210C4⋊C4—2021no—
211C4⋊C4—2021no—
212C22⋊C4—21no—
213C22⋊C4—21no—
214C22⋊C4—21no—
215C22⋊C4—21no—
216C22⋊C4—21no—
217C22⋊C4—1012no—
218C22⋊C4—4042no—
219C22⋊C4—2021no—
220C22⋊C4—2021no—
221C22⋊C4—2021no—
222C22⋊C4—2021no—
223C22⋊C4—2021no—
224C22⋊C4C4×D10112yescomplemented
225C42—2021no—
226C42—2021no—
227C23—41no—
228C23—41no—
229C23—1012no—
230C23—4041no—
231C23—4041no—
232C23—2021no—
233C23—2021no—
234C23C2×C4×D10111yescharacteristic
235C23D4×D10111yescharacteristic
236C23D4×D10111yescharacteristic
237C23C2×D40411yescharacteristic
238C23Q8×D10111yescharacteristic
239C23#1310807311yescharacteristic
240C23#1310807411yescharacteristic
241D4—2022no—
242D4—2022no—
243D4—2022no—
244D4—2022no—
245C2×C4—22no—
246C2×C4—21no—
247C2×C4—21no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—8081no—
256C2×C4—8081no—
257C2×C4—1012no—
258C2×C4—4041no—
259C2×C4—4041no—
260C2×C4—4041no—
261C2×C4—4041no—
262C2×C4—4041no—
263C2×C4—4041no—
264C2×C4—4041no—
265C2×C4—4041no—
266C2×C4—2022no—
267C2×C4—2022no—
268C2×C4—2022no—
269C2×C4—2022no—
270C2×C4—2021no—
271C2×C4—2021no—
272C2×C4—2021no—
273C2×C4—2021no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—8081no—
285C22—4042no—
286C22—4041no—
287C22—4041no—
288C22—2022no—
289C22C4⋊C4×D10111yescharacteristic
290C22#1569221511yescharacteristic
291C22C4×D40411yescharacteristic
292C22#1569222511yescharacteristic
293C22#1569223311yescharacteristic
294C22#1569223511yescharacteristic
295C22#1569223711yescharacteristic
296C4—42no—
297C4—41no—
298C4—41no—
299C4—81no—
300C4—4042no—
301C4—4041no—
302C4—4042no—
303C4—4041no—
304C4—2022no—
305C2—21no—
306C2—21no—
307C2—21no—
308C2—21no—
309C2—21no—
310C2—21no—
311C2—4042no—
312C2#2026505111yescharacteristic
313C2#2026536511yescharacteristic
314C2#2026536811yescharacteristic
subgroupG/H
centerC22#15692225
commutatorC23×C101C22×C4
FrattiniC24C22×D101
Fitting#342×C101C2
socleC22×C101#120
prime radicalC22×C22⋊D101C22
cosocleC24×C101C23
hypercenter#342D101
hypocenterC101#1867
LCSG ▷ C23×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ #342

Complements

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

NH ≅ G/Ntype
4. #20265052C2semidirect
5. #20265049C2semidirect
6. #20264943C2semidirect
7. #342×C101C2semidirect
21. #15692169C4semidirect
52. C22⋊Dic101C2×C4semidirect
68. C22⋊C4×C101C2×C4semidirect
82. C22×D101C4⋊C4semidirect
113. C2×Dic101C4⋊C4semidirect
158. C101#1867semidirect
165. #342D101semidirect
224. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
#202656502
C2×#156921692
#202650542
#202650522
#202650492
#202649432
#342×C1012
#1867101