Finite Group Databasebeta

← #27174422

Subgroups — #27174422

Order 12928.

504 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 167 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×Q8×Dic101C211yesmaximal, characteristic
2C2×#15692220C211yesmaximal, characteristic
3#20265110C212yesmaximal
4#20265109C211yesmaximal, characteristic
5#20265107C211yesmaximal, characteristic
6#20265098C212yesmaximal
7#20265096C211yesmaximal, characteristic
8C4⋊C4×Dic101C212yesmaximal
9#20265028C212yesmaximal
10#20265022C211yesmaximal, characteristic
11#395×C101C211yesmaximal, characteristic
12C22×Dic202C2211yescharacteristic
13Q8×Dic101—24no—
14#15692220—24no—
15C2×C4⋊Dic101C2212yes—
16C2×C4⋊Dic101C2212yes—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108053C2212yes—
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×C4×Dic101C2211yescharacteristic
23C2×C4×Dic101C2212yes—
24C2×C4×Dic101C2211yescharacteristic
25C2×C4×Dic101C2211yescharacteristic
26C2×C4×Dic101C2212yes—
27#15692162C2211yescharacteristic
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C22×Q8×C101C2211yescharacteristic
33C2×C4⋊C4×C101C2212yes—
34C2×C4⋊C4×C101C2211yescharacteristic
35#95×C101C2211yescharacteristic
36#95×C101C2212yes—
37C2×Dic202—22no—
38C2×Dic202—22no—
39C2×Dic202—22no—
40C22×Dic101C2311yescharacteristic
41C22×Dic101C2312yes—
42C22×Dic101C2311yescharacteristic
43C22×Dic101C2312yes—
44C22×Dic101C2311yescharacteristic
45C22×Dic101C2311yescharacteristic
46C4⋊Dic101—22no—
47C4⋊Dic101—24no—
48C4⋊Dic101—24no—
49#13108053—22no—
50#13108053—24no—
51#13108053—24no—
52#13108053—22no—
53C4×Dic101—22no—
54C4×Dic101—22no—
55C4×Dic101—24no—
56C4×Dic101D414yes—
57C4×Dic101Q818yescomplemented
58C2×Q8×C101—22no—
59C2×Q8×C101—24no—
60C22×C4×C101C2311yescharacteristic
61C22×C4×C101C2311yescharacteristic
62C22×C4×C101C2312yes—
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2312yes—
65C4⋊C4×C101—22no—
66C4⋊C4×C101—24no—
67C23×C101C2411yescharacteristic
68Q8×C101—28no—
69C2×C4×C101—21no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—22no—
74C2×C4×C101—22no—
75C2×C4×C101—21no—
76C2×C4×C101—21no—
77C2×C4×C101—22no—
78C2×C4×C101C2×D412yes—
79C2×C4×C101C2×Q814yes—
80C2×C4×C101#4114yes—
81C2×C4×C101#4112yes—
82C2×Dic101—21no—
83C2×Dic101—22no—
84C2×Dic101—22no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—22no—
90C2×Dic101—22no—
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101C2×D412yes—
94C2×Dic101C2×D412yes—
95C2×Dic101C2×Q814yes—
96C2×Dic101C2×Q814yes—
97C2×Dic101#4112yes—
98Dic202—28no—
99C22×C101C22×D411yescharacteristic
100C22×C101C22×Q812yes—
101C22×C101C2×#4111yescharacteristic
102C22×C101C2×#4112yes—
103C22×C101C2×#4111yescharacteristic
104C4×C101—22no—
105C4×C101—22no—
106C4×C101—24no—
107C4×C101—24no—
108C4×C101—41no—
109Dic101—22no—
110Dic101—22no—
111Dic101—24no—
112Dic101—41no—
113Dic101—42no—
114Dic101C4⋊Q814yes—
115C2×C101#52511yescharacteristic
116C2×C101#52511yescharacteristic
117C2×C101C2×C4⋊Q811yescharacteristic
118C2×C101#53312yes—
119C2×C101#53312yes—
120#2399—1011nomaximal, Sylow
121C101#239911yescharacteristic, complemented, Sylow
122C2×C4×Q8—1011no—
123C2×C4×Q8—1011no—
124#398—1011no—
125#398—1012no—
126#398—1011no—
127#395—1011no—
128#395—1012no—
129#395D10111yescharacteristic
130C4×C4⋊C4—1012no—
131C4×C4⋊C4—1012no—
132#376—1011no—
133C22×Q8—1011no—
134C22×Q8C2×D10111yescharacteristic
135C4×Q8—2024no—
136C4×Q8—2024no—
137C2×C4⋊C4—1011no—
138C2×C4⋊C4—1012no—
139C2×C4⋊C4—1011no—
140C2×C4⋊C4—1011no—
141C2×C4⋊C4—1012no—
142C2×C4⋊C4—1012no—
143C2×C4⋊C4—1012no—
144C2×C4⋊C4C2×D10111yescharacteristic
145C2×C4⋊C4C2×D10112yes—
146C2×C42—1011no—
147C2×C42—1012no—
148C2×C42—1011no—
149C2×C42—1012no—
150C2×C42—1011no—
151#95—1011no—
152#95—1012no—
153#95—1012no—
154#95—1012no—
155#95—1012no—
156#95C2×D10111yescharacteristic
157#95C2×D10112yes—
158C2×Q8—22no—
159C2×Q8—24no—
160C2×Q8—2022no—
161C2×Q8—2022no—
162C2×Q8—2022no—
163C22×C4—1011no—
164C22×C4—1012no—
165C22×C4—1011no—
166C22×C4—1011no—
167C22×C4—1012no—
168C22×C4—1011no—
169C22×C4C22×D10111yescharacteristic
170C22×C4C22×D10111yescharacteristic
171C22×C4C22×D10112yes—
172C22×C4C22×D10112yes—
173C22×C4C22×D10111yescharacteristic
174C4⋊C4—22no—
175C4⋊C4—24no—
176C4⋊C4—2022no—
177C4⋊C4—2024no—
178C4⋊C4—2024no—
179C4⋊C4—2022no—
180C4⋊C4—2022no—
181C4⋊C4—2024no—
182C4⋊C4—2024no—
183C42—1014no—
184C42—1018no—
185C42—2022no—
186C42—2024no—
187C42—2022no—
188C23C23×D10111yescharacteristic
189Q8—28no—
190Q8—2028no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—21no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—22no—
200C2×C4—1012no—
201C2×C4—1014no—
202C2×C4—1012no—
203C2×C4—1014no—
204C2×C4—1012no—
205C2×C4—2021no—
206C2×C4—2022no—
207C2×C4—2022no—
208C2×C4—2022no—
209C2×C4—2022no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2022no—
213C2×C4—2022no—
214C2×C4—2021no—
215C2×C4—2021no—
216C2×C4D4×D10112yes—
217C2×C4Q8×D10114yes—
218C2×C4#1310807314yes—
219C2×C4#1310807512yes—
220C22C2×D4×D10111yescharacteristic
221C22C2×Q8×D10112yes—
222C22#41×D10111yescharacteristic
223C22C2×#1310807312yes—
224C22C2×#1310807511yescharacteristic
225C4—24no—
226C4—22no—
227C4—24no—
228C4—22no—
229C4—41no—
230C4—1014no—
231C4—4041no—
232C4—4042no—
233C4—2024no—
234C4—2022no—
235C4—2022no—
236C2#2026568311yescharacteristic
237C2#2026569011yescharacteristic
238C2C4⋊Q8×D10111yescharacteristic
239C2#2026572912yes—
240C2#2026573212yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, prime radical, cosocleC23×C101C24
Fitting#395×C101C2
hypercenter#395D101
hypocenterC101#2399
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
57. C4×Dic101Q8semidirect
121. C101#2399semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic1012
C2×#156922202
#202651102
#202651092
#202651072
#202650982
#202650962
C4⋊C4×Dic1012
#202650282
#202650222
#395×C1012
#2399101