Finite Group Databasebeta

← #27180963

Subgroups — #27180963

Order 12928.

398 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 149 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#20265805C211yesmaximal, characteristic, complemented
2#20265477C212yesmaximal, complemented
3#20265477C212yesmaximal, complemented
4#20265467C212yesmaximal, complemented
5#20265457C212yesmaximal, complemented
6#20265220C211yesmaximal, characteristic, complemented
7#20265213C211yesmaximal, characteristic, complemented
8#20265204C211yesmaximal, characteristic, complemented
9#20265182C211yesmaximal, characteristic
10C2×#15692176C211yesmaximal, characteristic
11#518×C101C211yesmaximal, characteristic
12C2×#13108058C2211yescharacteristic
13C22×#11717579C2211yescharacteristic
14#15692255C2212yescomplemented
15#15692254—21no—
16#15692247C2212yescomplemented
17#15692246—21no—
18#15692232—22no—
19#15692214C2211yescharacteristic
20C2×C4⋊Dic101C2211yescharacteristic
21#15692188C2212yescomplemented
22#15692188C2211yescharacteristic, complemented
23#15692188C2211yescharacteristic, complemented
24#15692186C2212yescomplemented
25#15692186C2211yescharacteristic, complemented
26#15692186C2211yescharacteristic, complemented
27C4⋊#11717579C2212yescomplemented
28C4⋊#11717579C2212yescomplemented
29#15692176C2212yes—
30#15692176C2212yes—
31#15692176C2212yes—
32#15692176C2212yes—
33C22×D4×C101C2211yescharacteristic
34C4×D4×C101—21no—
35C4×D4×C101—21no—
36C4×D4×C101C2212yes—
37C4×D4×C101C2212yes—
38#117×C101C2211yescharacteristic
39#117×C101C2211yescharacteristic
40C2×C22⋊C4×C101—22no—
41C2×Dic202—22no—
42C22×Dic101—21no—
43#13108058—22no—
44C2×#11717579—22no—
45C2×#11717579C2311yescharacteristic
46C2×#11717579C2312yes—
47C2×#11717579C2311yescharacteristic
48C4⋊Dic101—21no—
49C4⋊Dic101—22no—
50C4⋊Dic101C2312yes—
51C4⋊Dic101C2311yescharacteristic
52C4⋊Dic101C2311yescharacteristic
53#13108053—22no—
54#13108053—22no—
55C22⋊Dic101—21no—
56C4×Dic101—21no—
57C24×C101—21no—
58C2×D4×C101—21no—
59C2×D4×C101—22no—
60C2×D4×C101Q814yescomplemented
61C2×D4×C101C2312yes—
62C22×C4×C101—22no—
63C22×C4×C101—22no—
64C22×C4×C101C2311yescharacteristic
65C4⋊C4×C101D411yescharacteristic
66C4⋊C4×C101D411yescharacteristic
67C4⋊C4×C101C2311yescharacteristic
68C4⋊C4×C101C2311yescharacteristic
69C22⋊C4×C101—21no—
70C22⋊C4×C101—22no—
71C22⋊C4×C101—22no—
72C22⋊C4×C101D412yes—
73C42×C101—21no—
74C42×C101C2312yes—
75C23×C101—22no—
76C23×C101—24no—
77C23×C101—81no—
78C23×C101—81no—
79C23×C101C2×D411yescharacteristic
80D4×C101—24no—
81D4×C101—22no—
82D4×C101C2×Q814yes—
83C2×C4×C101—21no—
84C2×C4×C101—21no—
85C2×C4×C101—82no—
86C2×C4×C101—82no—
87C2×C4×C101C2×D412yes—
88C2×C4×C101C2×D412yes—
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101C2×Q812yes—
91C2×C4×C101#4112yes—
92C2×C4×C101C2411yescharacteristic
93C2×Dic101—21no—
94C2×Dic101—21no—
95C2×Dic101—22no—
96C2×Dic101—81no—
97Dic202—82no—
98#11717579—21no—
99#11717579—21no—
100#11717579—22no—
101C22×C101—21no—
102C22×C101—21no—
103C22×C101—24no—
104C22×C101—42no—
105C22×C101—42no—
106C22×C101—84no—
107C22×C101C22⋊Q812yes—
108C22×C101C22×D411yescharacteristic
109C4×C101—22no—
110C4×C101—22no—
111C4×C101—41no—
112C4×C101—41no—
113C4×C101C22⋊Q812yes—
114C4×C101C22×Q811yescharacteristic
115C4×C101C2×#4111yescharacteristic
116Dic101—41no—
117Dic101—42no—
118Dic101—41no—
119C2×C101—22no—
120C2×C101—44no—
121C2×C101C2×C22⋊Q811yescharacteristic
122C2×C101#57611yescharacteristic
123C2×C101#57711yescharacteristic
124#3081—1011nomaximal, Sylow
125C101#308111yescharacteristic, complemented, Sylow
126#544—1011no—
127#518D10111yescharacteristic
128#478—1012no—
129#478—1012no—
130#476—1012no—
131#474—1012no—
132#428—1011no—
133#427—1011no—
134#424—1011no—
135#418—1011no—
136C2×#102—1011no—
137C22×D4C2×D10111yescharacteristic
138C2×M16—1011no—
139C22×C8—1011no—
140C4⋊Q8—1012no—
141C4⋊Q8—2021no—
142#125—1012no—
143#125—2021no—
144C22⋊Q8—2022no—
145C4×D4—21no—
146C4×D4—21no—
147C4×D4C2×D10112yes—
148C4×D4C2×D10112yes—
149#117—1011no—
150#117C2×D10111yescharacteristic
151#117C2×D10111yescharacteristic
152C2×C4⋊C4—1011no—
153C2×C22⋊C4—22no—
154#107—1011no—
155#107—1011no—
156#107—1012no—
157#106—1011no—
158#106—1011no—
159#106—1012no—
160C4⋊C8—1012no—
161C4⋊C8—1012no—
162#102—1012no—
163#102—1012no—
164#102—1012no—
165#102—1012no—
166C24—21no—
167C2×Q8—2022no—
168C2×D4—21no—
169C2×D4—22no—
170C2×D4Dic20214yescomplemented
171C2×D4C22×D10112yes—
172C22×C4—22no—
173C22×C4—22no—
174C22×C4—2021no—
175C22×C4C22×D10111yescharacteristic
176M16—2022no—
177C2×C8—1011no—
178C2×C8—1012no—
179C2×C8—1011no—
180C2×C8—2022no—
181C4⋊C4—1012no—
182C4⋊C4—1011no—
183C4⋊C4—1011no—
184C4⋊C4—2022no—
185C4⋊C4—2022no—
186C4⋊C4—2021no—
187C4⋊C4—2022no—
188C4⋊C4C22⋊D10111yescharacteristic
189C4⋊C4C22⋊D10111yescharacteristic
190C4⋊C4C22×D10111yescharacteristic
191C4⋊C4C22×D10111yescharacteristic
192C22⋊C4—21no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2021no—
196C22⋊C4C22⋊D10112yes—
197C42—21no—
198C42—2021no—
199C42C22×D10112yes—
200C23—22no—
201C23—24no—
202C23—81no—
203C23—81no—
204C23C2×C22⋊D10111yescharacteristic
205Q8—8082no—
206D4—22no—
207D4—24no—
208D4C2×Dic20214yes—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—82no—
212C2×C4—82no—
213C2×C4—8081no—
214C2×C4—2021no—
215C2×C4—2021no—
216C2×C4—2022no—
217C2×C4C2×C22⋊D10112yes—
218C2×C4C2×C22⋊D10112yes—
219C2×C4C2×C22⋊D10111yescharacteristic
220C2×C4C2×Dic20212yes—
221C2×C4#1310807412yes—
222C2×C4C23×D10111yescharacteristic
223C8—2021no—
224C8—2021no—
225C8—2022no—
226C22—21no—
227C22—21no—
228C22—24no—
229C22—42no—
230C22—42no—
231C22—84no—
232C22#1569223412yes—
233C22C22×C22⋊D10111yescharacteristic
234C4—22no—
235C4—22no—
236C4—41no—
237C4—41no—
238C4—4041no—
239C4—4042no—
240C4—4041no—
241C4#1569223412yes—
242C4C22×Dic20211yescharacteristic
243C4C2×#1310807411yescharacteristic
244C2—22no—
245C2—44no—
246C2C2×#1569223411yescharacteristic
247C2#2026598011yescharacteristic
248C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#518×C101C2
socleC22×C101C22×D4
prime radicalC22×D4×C101C22
hypercenter#518D101
hypocenterC101#3081
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #518

Complements

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

NH ≅ G/Ntype
1. #20265805C2semidirect
2. #20265477C2semidirect
3. #20265477C2semidirect
4. #20265467C2semidirect
5. #20265457C2semidirect
6. #20265220C2semidirect
7. #20265213C2semidirect
8. #20265204C2semidirect
14. #15692255C22semidirect
16. #15692247C22semidirect
21. #15692188C22semidirect
22. #15692188C22semidirect
23. #15692188C22semidirect
24. #15692186C22semidirect
25. #15692186C22semidirect
26. #15692186C22semidirect
27. C4⋊#11717579C22semidirect
28. C4⋊#11717579C22semidirect
60. C2×D4×C101Q8semidirect
125. C101#3081semidirect
170. C2×D4Dic202semidirect

Maximal subgroups

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

subgroupindex
#202658052
#202654772
#202654672
#202654572
#202652202
#202652132
#202652042
#202651822
C2×#156921762
#518×C1012
#3081101