Finite Group Databasebeta

← #27172260

Subgroups — #27172260

Order 12928.

430 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 430 conjugacy classes of subgroups — too many to draw. Only its 75 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#20265736C211yesmaximal, characteristic
2C2×#15692238C211yesmaximal, characteristic, complemented
3C2×#15692170C211yesmaximal, characteristic, complemented
4C2×#15692169C211yesmaximal, characteristic, complemented
5#20265040C211yesmaximal, characteristic, complemented
6#20264943C211yesmaximal, characteristic, complemented
7C2×#123×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101C2211yescharacteristic
9#15692238—22no—
10#15692238—21no—
11#15692238—22no—
12#15692236—21no—
13#15692227—22no—
14#15692224—21no—
15#15692223—21no—
16C2×#13108053—21no—
17C2×C22⋊Dic101C2211yescharacteristic
18C2×C22⋊Dic101C2211yescharacteristic
19C2×C22⋊Dic101C2211yescharacteristic
20C2×#13108051—21no—
21#15692170—22no—
22#15692169—22no—
23#15692162—22no—
24C22×D4×C101C2211yescharacteristic
25C23×C4×C101C2211yescharacteristic
26#123×C101—22no—
27#123×C101—22no—
28C2×C4⋊C4×C101—21no—
29C2×C22⋊C4×C101—21no—
30C2×C22⋊C4×C101C2211yescharacteristic
31C23×D101D411yescharacteristic
32C2×C22⋊D101—21no—
33C2×C22⋊D101—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101—21no—
36C2×C22⋊D101—21no—
37C2×C22⋊D101—21no—
38C2×C22⋊D101—22no—
39C22×Dic101—21no—
40C22×Dic101—21no—
41C22×Dic101D411yescharacteristic
42C22×Dic101D412yes—
43#13108053—24no—
44#13108053—22no—
45C22⋊Dic101—22no—
46C22⋊Dic101—21no—
47C22⋊Dic101—21no—
48C22⋊Dic101—21no—
49C22⋊Dic101—21no—
50C22⋊Dic101—21no—
51C22⋊Dic101D412yescomplemented
52#13108051—24no—
53#13108051—22no—
54C24×C101D411yescharacteristic
55C24×C101C2311yescharacteristic
56C2×D4×C101—21no—
57C2×D4×C101—21no—
58C2×D4×C101—21no—
59C2×D4×C101—21no—
60C2×D4×C101—21no—
61C2×D4×C101—21no—
62C22×C4×C101—21no—
63C22×C4×C101—21no—
64C22×C4×C101—21no—
65C22×C4×C101—21no—
66C22×C4×C101—21no—
67C22×C4×C101—21no—
68C22×C4×C101—21no—
69C22×C4×C101D411yescharacteristic
70C22×C4×C101D412yescomplemented
71C4⋊C4×C101—24no—
72C22⋊C4×C101—24no—
73C22⋊C4×C101—21no—
74C22⋊C4×C101D412yescomplemented
75C23×C101—21no—
76C23×C101—21no—
77C23×C101—21no—
78C23×C101—42no—
79C23×C101—42no—
80C23×C101C2×D411yescharacteristic
81C23×C101C2×D411yescharacteristic
82C23×C101C2×D411yescharacteristic
83C23×C101C2×D411yescharacteristic
84C23×C101C2×D411yescharacteristic
85C23×C101C2×D411yescharacteristic
86C23×C101#4111yescharacteristic
87C22×D101—21no—
88C22×D101—21no—
89C22×D101—21no—
90C22×D101—42no—
91D4×C101—41no—
92D4×C101—41no—
93D4×C101—41no—
94D4×C101—41no—
95C2×C4×C101—22no—
96C2×C4×C101—22no—
97C2×C4×C101—21no—
98C2×C4×C101—22no—
99C2×C4×C101—21no—
100C2×C4×C101—22no—
101C2×C4×C101—21no—
102C2×C4×C101—42no—
103C2×C4×C101—41no—
104C2×C4×C101—41no—
105C2×C4×C101—41no—
106C2×C4×C101—41no—
107C22⋊D101—41no—
108C22⋊D101—41no—
109C22⋊D101—41no—
110C22⋊D101—41no—
111C22⋊D101—82no—
112C2×Dic101—22no—
113C2×Dic101—22no—
114C2×Dic101—21no—
115C2×Dic101—21no—
116C2×Dic101—21no—
117C2×Dic101—42no—
118C2×Dic101—42no—
119C2×Dic101—42no—
120C2×Dic101—41no—
121C22×C101—22no—
122C22×C101—22no—
123C22×C101—22no—
124C22×C101—41no—
125C22×C101—41no—
126C22×C101—41no—
127C22×C101—41no—
128C22×C101—42no—
129C22×C101—41no—
130C22×C101—42no—
131C22×C101—81no—
132C22×C101#12011yescharacteristic
133C22×C101#12011yescharacteristic
134C22×C101#12011yescharacteristic
135C22×C101#12111yescharacteristic
136C22×C101#12111yescharacteristic
137C22×C101#12111yescharacteristic
138C22×C101#12711yescharacteristic
139C2×D101—42no—
140C2×D101—41no—
141C2×D101—42no—
142C2×D101—81no—
143C4×C101—42no—
144C4×C101—42no—
145C4×C101—41no—
146Dic101—42no—
147Dic101—41no—
148Dic101—42no—
149Dic101—81no—
150C2×C101—21no—
151C2×C101—21no—
152C2×C101—21no—
153C2×C101—21no—
154C2×C101—21no—
155C2×C101—21no—
156C2×C101—81no—
157C2×C101#39211yescharacteristic
158C2×C101#45812yes—
159D101—81no—
160#2022—1011nomaximal, Sylow
161C101#202211yescharacteristic, complemented, Sylow
162#534—1011no—
163C2×#123—1011no—
164C2×#123D10111yescharacteristic, complemented
165C2×#99—1011no—
166C2×#99—1011no—
167#381—1011no—
168#342—1011no—
169C22×D4—1011no—
170C22×D4C2×D10111yescharacteristic
171C23×C4C2×D10111yescharacteristic
172#123—22no—
173#123—22no—
174#123—2022no—
175#123—2021no—
176#123—2021no—
177#123—2022no—
178#121—2022no—
179#120—2021no—
180#120—2021no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—2021no—
183C2×C22⋊C4—21no—
184C2×C22⋊C4—1011no—
185C2×C22⋊C4—1011no—
186C2×C22⋊C4—1011no—
187C2×C22⋊C4—2021no—
188C2×C22⋊C4C2×D10111yescharacteristic
189#99—2022no—
190#99—2022no—
191#95—2022no—
192C24—1011no—
193C24C22⋊D10111yescharacteristic
194C24C22×D10111yescharacteristic
195C2×D4—21no—
196C2×D4—21no—
197C2×D4—21no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—21no—
201C2×D4—2022no—
202C2×D4—2021no—
203C2×D4—2021no—
204C2×D4—2021no—
205C2×D4—2021no—
206C2×D4—2021no—
207C2×D4—2021no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—1012no—
216C22×C4—1011no—
217C22×C4—2021no—
218C22×C4—2021no—
219C22×C4C22⋊D10111yescharacteristic
220C22×C4C22⋊D10112yescomplemented
221C4⋊C4—24no—
222C4⋊C4—2024no—
223C4⋊C4—2022no—
224C22⋊C4—24no—
225C22⋊C4—21no—
226C22⋊C4—1012no—
227C22⋊C4—2024no—
228C22⋊C4—2022no—
229C22⋊C4—2021no—
230C22⋊C4—2021no—
231C22⋊C4—2022no—
232C22⋊C4—2021no—
233C22⋊C4—2021no—
234C22⋊C4—2021no—
235C22⋊C4C22⋊D10112yescomplemented
236C23—21no—
237C23—21no—
238C23—21no—
239C23—42no—
240C23—42no—
241C23—4042no—
242C23—2021no—
243C23—2021no—
244C23—2021no—
245C23D4×D10111yescharacteristic
246C23D4×D10111yescharacteristic
247C23D4×D10111yescharacteristic
248C23C2×C22⋊D10111yescharacteristic
249C23C2×C22⋊D10111yescharacteristic
250C23C2×C22⋊D10111yescharacteristic
251C23#1310807311yescharacteristic
252D4—41no—
253D4—41no—
254D4—41no—
255D4—41no—
256D4—8082no—
257D4—4041no—
258D4—4041no—
259D4—4041no—
260D4—4041no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—4042no—
274C2×C4—4041no—
275C2×C4—4042no—
276C2×C4—4042no—
277C2×C4—2022no—
278C2×C4—2021no—
279C2×C4—2022no—
280C2×C4—2021no—
281C2×C4—2021no—
282C22—22no—
283C22—22no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—41no—
291C22—42no—
292C22—81no—
293C22—8081no—
294C22—4042no—
295C22—4041no—
296C22—4042no—
297C22#1569222311yescharacteristic
298C22#1569222311yescharacteristic
299C22#1569222411yescharacteristic
300C22#1569222611yescharacteristic
301C22#1569222711yescharacteristic
302C22#1569222711yescharacteristic
303C22#1569225111yescharacteristic
304C4—42no—
305C4—42no—
306C4—41no—
307C4—8081no—
308C4—4042no—
309C4—4041no—
310C4—4042no—
311C2—21no—
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2—21no—
317C2—81no—
318C2—8081no—
319C2#2026508511yescharacteristic
320C2#2026536612yes—
subgroupG/H
centerC22#15692223
commutator, cosocleC24×C101C23
FrattiniC24C22×D101
FittingC2×#123×C101C2
socleC22×C101#120
prime radical#20265736C2
hypercenterC2×#123D101
hypocenterC101#2022
LCSG ▷ C24×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C24 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×#15692238C2semidirect
3. C2×#15692170C2semidirect
4. C2×#15692169C2semidirect
5. #20265040C2semidirect
6. #20264943C2semidirect
7. C2×#123×C101C2semidirect
51. C22⋊Dic101D4semidirect
70. C22×C4×C101D4semidirect
74. C22⋊C4×C101D4semidirect
161. C101#2022semidirect
164. C2×#123D101semidirect
220. C22×C4C22⋊D101semidirect
235. C22⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202657362
C2×#156922382
C2×#156921702
C2×#156921692
#202650402
#202649432
C2×#123×C1012
#2022101