Finite Group Databasebeta

← #27182873

Subgroups — #27182873

Order 12928.

482 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 482 conjugacy classes of subgroups — too many to draw. Only its 133 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×#15692278C211yesmaximal, characteristic, complemented
2#20265838C212yesmaximal, complemented
3#20265562C211yesmaximal, characteristic, complemented
4#20265415C211yesmaximal, characteristic, complemented
5#20265413C211yesmaximal, characteristic, complemented
6#20265401C212yesmaximal, complemented
7#20265338C212yesmaximal, complemented
8#20265316C212yesmaximal, complemented
9Q16×Dic101C211yesmaximal, characteristic, complemented
10C8×C22⋊D101C211yesmaximal, characteristic, complemented
11C22×Q16×C101C211yesmaximal, characteristic, complemented
12C2×#13108075C2212yes—
13#15692278—24no—
14C2×#13108061C2212yes—
15C2×#13108061C2212yes—
16C2×D808C2211yescharacteristic
17C2×C8×D101C2211yescharacteristic
18#15692244—22no—
19#15692244C2212yes—
20#15692231—22no—
21#15692231C2212yes—
22#15692229—22no—
23#15692229C2212yes—
24Q8×Dic101C2212yes—
25C4×C22⋊D101—22no—
26C4×C22⋊D101C2211yescharacteristic
27#15692189C2211yescharacteristic
28#15692184C2211yescharacteristic
29#15692179C2212yes—
30#15692179C2212yes—
31#15692177C2212yes—
32C22⋊#11717579C2211yescharacteristic
33#15692167C2211yescharacteristic
34C8×Dic101C2211yescharacteristic
35C22×Q8×C101C2212yes—
36C2×Q16×C101—24no—
37C2×Q16×C101C2211yescharacteristic
38C2×Q16×C101C2212yes—
39C2×Q16×C101C2211yescharacteristic
40C22×C8×C101C2211yescharacteristic
41#13108075—24no—
42#13108075—42no—
43C2×D404—22no—
44C2×D404C2312yes—
45C2×C22⋊D101—22no—
46C2×C22⋊D101D411yescharacteristic
47C2×C4×D101—22no—
48C2×C4×D101C2311yescharacteristic
49#13108061—24no—
50#13108061—42no—
51D808—22no—
52C2×#11717579C2311yescharacteristic
53C2×#11717579C2311yescharacteristic
54C8×D101—22no—
55C4⋊Dic101—22no—
56C4⋊Dic101C2312yes—
57#13108053—22no—
58#13108053D411yescharacteristic
59C22⋊Dic101D411yescharacteristic
60#13108051—22no—
61#13108051—22no—
62#13108051—22no—
63#13108051—22no—
64#13108051D411yescharacteristic
65C4×Dic101—22no—
66C4×Dic101C2311yescharacteristic
67C2×Q8×C101—22no—
68C2×Q8×C101—24no—
69C2×Q8×C101—42no—
70C2×Q8×C101C2312yes—
71C2×Q8×C101C2312yes—
72C22×C4×C101—22no—
73C22×C4×C101C2311yescharacteristic
74Q16×C101—24no—
75Q16×C101—22no—
76Q16×C101D414yescomplemented
77C2×C8×C101—22no—
78C2×C8×C101C2311yescharacteristic
79C2×C8×C101C2311yescharacteristic
80C23×C101C2×D411yescharacteristic
81C22×D101—22no—
82C22×D101C2×D411yescharacteristic
83Q8×C101—22no—
84Q8×C101—22no—
85Q8×C101—22no—
86Q8×C101—44no—
87Q8×C101C2×D414yes—
88C2×C4×C101—22no—
89C2×C4×C101—21no—
90C2×C4×C101—21no—
91C2×C4×C101—22no—
92C2×C4×C101—44no—
93C2×C4×C101C2×D411yescharacteristic
94C2×C4×C101C2411yescharacteristic
95C22⋊D101—22no—
96C22⋊D101—44no—
97C2×Dic101—22no—
98C2×Dic101C2×D411yescharacteristic
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101D404—22no—
102D404—22no—
103D404—44no—
104C4×D101—21no—
105C4×D101—21no—
106C4×D101—44no—
107C8×C101—21no—
108C8×C101C2×D412yes—
109#11717579—21no—
110#11717579—21no—
111C22×C101—21no—
112C22×C101—21no—
113C22×C101#13512yescomplemented
114C22×C101C22×D411yescharacteristic
115C2×D101—21no—
116C2×D101—21no—
117C2×D101—21no—
118C2×D101—42no—
119C2×D101—42no—
120C2×D101—42no—
121C4×C101—24no—
122C4×C101—21no—
123C4×C101—42no—
124C4×C101C22×D411yescharacteristic
125C4×C101H(5,𝔽2)11yescharacteristic
126Dic101—21no—
127Dic101—21no—
128Dic101—21no—
129Dic101—42no—
130C2×C101—22no—
131C2×C101D4211yescharacteristic
132C2×C101C2×#13511yescharacteristic
133C2×C101#57811yescharacteristic
134D101—41no—
135D101—82no—
136#3285—1011nomaximal, Sylow
137C101#328511yescharacteristic, complemented, Sylow
138C2×#135—1011no—
139C22×Q16D10111yescharacteristic, complemented
140#548—1012no—
141#495—1011no—
142#467—1011no—
143#466—1011no—
144#464—1012no—
145#452—1012no—
146#448—1012no—
147C4×Q16—1011no—
148C8×D4—1011no—
149C2×#41—1012no—
150C22×Q8C2×D10112yes—
151#135—2024no—
152C2×Q16—24no—
153C2×Q16C2×D10112yes—
154C2×Q16C2×D10111yescharacteristic
155C2×Q16C2×D10111yescharacteristic
156C2×SD16—1012no—
157C2×SD16—1012no—
158C2×D8—1011no—
159C22×C8—1011no—
160C22×C8C2×D10111yescharacteristic
161#124—1012no—
162#124—2022no—
163C22⋊Q8—1012no—
164C22⋊Q8—2022no—
165#121—1012no—
166#121—2022no—
167C4×Q8—1012no—
168C4×D4—1011no—
169C4×D4—2022no—
170#107—1011no—
171C4⋊C8—1011no—
172#103—1012no—
173#103—1012no—
174#102—1012no—
175C22⋊C8—1011no—
176C22⋊C8—1011no—
177C4×C8—1011no—
178#41—4042no—
179#41—2024no—
180C2×Q8—22no—
181C2×Q8—24no—
182C2×Q8—42no—
183C2×Q8C22×D10112yes—
184C2×Q8C22×D10112yes—
185C2×D4—1011no—
186C2×D4—1012no—
187C2×D4—2022no—
188C2×D4—2022no—
189C22×C4—22no—
190C22×C4—1011no—
191C22×C4—2022no—
192C22×C4C22×D10111yescharacteristic
193Q16—24no—
194Q16—22no—
195Q16C22⋊D10114yescomplemented
196SD16—4042no—
197SD16—2024no—
198D8—2022no—
199C2×C8—22no—
200C2×C8—1011no—
201C2×C8—1011no—
202C2×C8—2022no—
203C2×C8C22×D10111yescharacteristic
204C2×C8C22×D10111yescharacteristic
205C4⋊C4—1011no—
206C4⋊C4—1012no—
207C4⋊C4—2022no—
208C4⋊C4—2022no—
209C22⋊C4—1011no—
210C22⋊C4—1011no—
211C22⋊C4—2022no—
212C22⋊C4—2022no—
213C22⋊C4—2022no—
214C22⋊C4—2022no—
215C42—1011no—
216C42—2022no—
217C23—1011no—
218C23—2022no—
219C23D4×D10111yescharacteristic
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—44no—
224Q8C2×C22⋊D10114yes—
225D4—4044no—
226D4—4044no—
227D4—2022no—
228D4—2022no—
229D4—2022no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—22no—
234C2×C4—44no—
235C2×C4—1011no—
236C2×C4—1011no—
237C2×C4—1011no—
238C2×C4—4044no—
239C2×C4—2022no—
240C2×C4—2021no—
241C2×C4—2021no—
242C2×C4D4×D10111yescharacteristic
243C2×C4C23×D10111yescharacteristic
244C8—21no—
245C8—2021no—
246C8—2021no—
247C8C2×C22⋊D10112yes—
248C22—21no—
249C22—21no—
250C22—4042no—
251C22—4042no—
252C22—4042no—
253C22—2021no—
254C22—2021no—
255C22—2021no—
256C22#1569227812yescomplemented
257C22C2×D4×D10111yescharacteristic
258C4—21no—
259C4—24no—
260C4—42no—
261C4—4042no—
262C4—2021no—
263C4—2021no—
264C4—2021no—
265C4C22×C22⋊D10111yescharacteristic
266C4#1569230011yescharacteristic
267C2—22no—
268C2—8082no—
269C2—4041no—
270C2D4×C22⋊D10111yescharacteristic
271C2C2×#1569227811yescharacteristic
272C2#2026599011yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC22×Q16×C101C2
socleC22×C101C22×D4
hypercenterC22×Q16D101
hypocenterC101#3285
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
1. C2×#15692278C2semidirect
2. #20265838C2semidirect
3. #20265562C2semidirect
4. #20265415C2semidirect
5. #20265413C2semidirect
6. #20265401C2semidirect
7. #20265338C2semidirect
8. #20265316C2semidirect
9. Q16×Dic101C2semidirect
10. C8×C22⋊D101C2semidirect
11. C22×Q16×C101C2semidirect
76. Q16×C101D4semidirect
113. C22×C101#135semidirect
137. C101#3285semidirect
139. C22×Q16D101semidirect
195. Q16C22⋊D101semidirect
256. C22#15692278semidirect

Maximal subgroups

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

subgroupindex
C2×#156922782
#202658382
#202655622
#202654152
#202654132
#202654012
#202653382
#202653162
Q16×Dic1012
C8×C22⋊D1012
C22×Q16×C1012
#3285101