Finite Group Databasebeta

← #27171431

Subgroups — #27171431

Order 12928.

376 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 85 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×M16×D101C211yesmaximal, characteristic, complemented
2#20265684C211yesmaximal, characteristic
3C2×#15692182C211yesmaximal, characteristic, complemented
4C4×#13108051C211yesmaximal, characteristic, complemented
5#20264938C211yesmaximal, characteristic, complemented
6#20264937C211yesmaximal, characteristic, complemented
7C2×#104×C101C211yesmaximal, characteristic, complemented
8C22×C4×D101C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10C2×#13108057—21no—
11M16×D101—22no—
12M16×D101—22no—
13C2×C8×D101—21no—
14#15692236—21no—
15#15692231C411yescharacteristic, complemented
16#15692231C411yescharacteristic, complemented
17#15692226C411yescharacteristic, complemented
18#15692226C411yescharacteristic, complemented
19#15692223—21no—
20C4×C22⋊D101—21no—
21C4×C22⋊D101—21no—
22#15692214C2211yescharacteristic
23C2×#13108051—21no—
24C2×C4×Dic101C2211yescharacteristic
25#15692182—22no—
26#15692162—21no—
27C2×#41×C101C2211yescharacteristic
28C2×M16×C101C2211yescharacteristic
29C2×C42×C101C2211yescharacteristic
30#104×C101—22no—
31C23×D101D411yescharacteristic
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C22×Dic101—21no—
35C22×Dic101D411yescharacteristic
36C2×C4×D101—21no—
37C2×C4×D101—21no—
38C2×C4×D101—21no—
39C2×C4×D101—21no—
40C2×C4×D101C2×C411yescharacteristic
41C2×C4×D101C2×C411yescharacteristic
42C2×C4×D101D411yescharacteristic
43C2×C4×D101D411yescharacteristic
44#13108058—21no—
45#13108058D412yescomplemented
46#13108057—24no—
47C2×#11717579—21no—
48C8×D101—24no—
49C4⋊Dic101C2×C411yescharacteristic
50C4⋊Dic101C2×C411yescharacteristic
51#13108053—21no—
52#13108053—21no—
53C22⋊Dic101—21no—
54C22⋊Dic101—21no—
55C22⋊Dic101—21no—
56#13108051—24no—
57#13108051—21no—
58#13108051—21no—
59C4×Dic101—21no—
60C4×Dic101—21no—
61C4×Dic101—21no—
62#41×C101—21no—
63#41×C101—21no—
64#41×C101—21no—
65#41×C101—21no—
66C2×Q8×C101C2×C411yescharacteristic
67C2×D4×C101—21no—
68C2×D4×C101C2×C411yescharacteristic
69C22×C4×C101—21no—
70C22×C4×C101—21no—
71C22×C4×C101C2311yescharacteristic
72M16×C101—21no—
73M16×C101D412yescomplemented
74C2×C8×C101—21no—
75C42×C101—21no—
76C42×C101—21no—
77C23×C101—21no—
78C23×C101C2×D411yescharacteristic
79C22×D101—21no—
80C22×D101—21no—
81C22×D101—22no—
82C22×D101—41no—
83C22×D101C22⋊C411yescharacteristic
84C22×D101C22⋊C411yescharacteristic
85Q8×C101—21no—
86Q8×C101—21no—
87D4×C101—21no—
88D4×C101—21no—
89D4×C101—41no—
90D4×C101—41no—
91C2×C4×C101—22no—
92C2×C4×C101—21no—
93C2×C4×C101—41no—
94C2×C4×C101—41no—
95C2×C4×C101—41no—
96C2×C4×C101—41no—
97C2×C4×C101C22×C411yescharacteristic
98C2×C4×C101C2×D411yescharacteristic
99C2×C4×C101C2×D411yescharacteristic
100C2×C4×C101C2×D411yescharacteristic
101C2×C4×C101#4111yescharacteristic
102C2×C4×C101#4111yescharacteristic
103C22⋊D101—42no—
104C22⋊D101—42no—
105C2×Dic101—21no—
106C2×Dic101—21no—
107C2×Dic101—22no—
108C2×Dic101—21no—
109C2×Dic101—21no—
110C2×Dic101—41no—
111C2×Dic101—41no—
112C2×Dic101C22⋊C411yescharacteristic
113C2×Dic101C22⋊C411yescharacteristic
114C4×D101—21no—
115C4×D101—21no—
116C4×D101—21no—
117C4×D101—21no—
118C4×D101—21no—
119C4×D101—21no—
120C4×D101—21no—
121C4×D101—21no—
122C8×C101—22no—
123#11717579—22no—
124C22×C101—21no—
125C22×C101—21no—
126C22×C101—41no—
127C22×C101—41no—
128C22×C101—41no—
129C22×C101C2×C22⋊C411yescharacteristic
130C22×C101#12111yescharacteristic
131C22×C101#12111yescharacteristic
132C2×D101—21no—
133C2×D101—21no—
134C2×D101—21no—
135C2×D101—21no—
136C2×D101—21no—
137C2×D101—42no—
138C2×D101—42no—
139C2×D101#10412yescomplemented
140C4×C101—42no—
141C4×C101—41no—
142C4×C101C4×D411yescharacteristic
143C4×C101C4×D411yescharacteristic
144C4×C101#12011yescharacteristic
145C4×C101#12311yescharacteristic
146Dic101—21no—
147Dic101—21no—
148Dic101—42no—
149Dic101—41no—
150Dic101—41no—
151C2×C101—21no—
152C2×C101—21no—
153C2×C101—81no—
154C2×C101#38611yescharacteristic
155C2×C101C2×#10411yescharacteristic
156C2×C101#42111yescharacteristic
157D101—22no—
158D101—41no—
159#1871—1011nomaximal, Sylow
160C101#187111yescharacteristic, complemented, Sylow
161C22×M16—1011no—
162#525—1011no—
163C2×#104—1011no—
164C2×#104D10111yescharacteristic, complemented
165C4×C22⋊C4—1011no—
166#339—1011no—
167#339—1011no—
168C2×#41C2×D10111yescharacteristic
169C23×C4—1011no—
170C2×M16—1011no—
171C2×M16—2021no—
172C2×M16—2022no—
173C2×M16—2022no—
174C2×M16C2×D10111yescharacteristic
175C22×C8—2021no—
176#123—2021no—
177C22⋊Q8—1011no—
178C22⋊Q8—1011no—
179#121—1011no—
180#121—1011no—
181#120—2021no—
182C4×D4—2021no—
183C4×D4—2021no—
184#117—1011no—
185C2×C22⋊C4—2021no—
186C2×C42—1011no—
187C2×C42C2×D10111yescharacteristic
188#104—22no—
189#104—2022no—
190#95—2021no—
191C24—1011no—
192#41—21no—
193#41—21no—
194#41—21no—
195#41—21no—
196C2×Q8C4×D10111yescharacteristic
197C2×D4—21no—
198C2×D4—2021no—
199C2×D4—2021no—
200C2×D4C4×D10111yescharacteristic
201C22×C4—21no—
202C22×C4—21no—
203C22×C4—1011no—
204C22×C4—1011no—
205C22×C4—1011no—
206C22×C4—1011no—
207C22×C4—1011no—
208C22×C4—2021no—
209C22×C4—2021no—
210C22×C4—2021no—
211C22×C4—2021no—
212C22×C4—2021no—
213C22×C4C22×D10111yescharacteristic
214M16—21no—
215M16—1012no—
216M16—2024no—
217M16—2021no—
218M16C22⋊D10112yescomplemented
219C2×C8—21no—
220C2×C8—2024no—
221C2×C8—2021no—
222C4⋊C4—1011no—
223C4⋊C4—1011no—
224C4⋊C4—2021no—
225C4⋊C4—2021no—
226C22⋊C4—2021no—
227C22⋊C4—2021no—
228C22⋊C4—2024no—
229C22⋊C4—2021no—
230C22⋊C4—2021no—
231C22⋊C4—2021no—
232C42—21no—
233C42—21no—
234C42—2021no—
235C42—2021no—
236C42—2021no—
237C23—21no—
238C23—1011no—
239C23—1011no—
240C23—4041no—
241C23—2022no—
242C23—2021no—
243C23—2021no—
244C23D4×D10111yescharacteristic
245Q8—21no—
246Q8—21no—
247D4—21no—
248D4—21no—
249D4—41no—
250D4—41no—
251D4—4042no—
252D4—4042no—
253C2×C4—22no—
254C2×C4—21no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—1011no—
260C2×C4—1011no—
261C2×C4—4041no—
262C2×C4—4041no—
263C2×C4—2022no—
264C2×C4—2021no—
265C2×C4—2021no—
266C2×C4—2021no—
267C2×C4—2021no—
268C2×C4—2021no—
269C2×C4—2021no—
270C2×C4—2021no—
271C2×C4—2021no—
272C2×C4—2021no—
273C2×C4—2021no—
274C2×C4—2021no—
275C2×C4—2021no—
276C2×C4C2×C4×D10111yescharacteristic
277C2×C4D4×D10111yescharacteristic
278C2×C4D4×D10111yescharacteristic
279C2×C4C2×C22⋊D10111yescharacteristic
280C2×C4#1310807311yescharacteristic
281C2×C4#1310807411yescharacteristic
282C8—22no—
283C8—2022no—
284C22—21no—
285C22—21no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—1012no—
290C22—4042no—
291C22—4042no—
292C22—2021no—
293C22—2021no—
294C22—2021no—
295C22—2021no—
296C22—2021no—
297C22C22⋊C4×D10111yescharacteristic
298C22#1569222611yescharacteristic
299C22#1569222811yescharacteristic
300C4—42no—
301C4—41no—
302C4—4042no—
303C4—4041no—
304C4—4041no—
305C4—2021no—
306C4—2021no—
307C4#1569221511yescharacteristic
308C4C4×C22⋊D10111yescharacteristic
309C4#1569222311yescharacteristic
310C4#1569223711yescharacteristic
311C2—21no—
312C2—21no—
313C2—81no—
314C2—4041no—
315C2—2022no—
316C2#2026505311yescharacteristic
317C2#104×D10111yescharacteristic
318C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC2×#104×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265684C2
cosocleC22×C4×C101C23
hypercenterC2×#104D101
hypocenterC101#1871
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
1. C2×M16×D101C2semidirect
3. C2×#15692182C2semidirect
4. C4×#13108051C2semidirect
5. #20264938C2semidirect
6. #20264937C2semidirect
7. C2×#104×C101C2semidirect
15. #15692231C4semidirect
16. #15692231C4semidirect
17. #15692226C4semidirect
18. #15692226C4semidirect
45. #13108058D4semidirect
73. M16×C101D4semidirect
139. C2×D101#104semidirect
160. C101#1871semidirect
164. C2×#104D101semidirect
218. M16C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C2×M16×D1012
#202656842
C2×#156921822
C4×#131080512
#202649382
#202649372
C2×#104×C1012
#1871101