Finite Group Databasebeta

← #27179469

Subgroups — #27179469

Order 12928.

564 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 169 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×#15692232C211yesmaximal, characteristic
2C22×C4×Dic101C211yesmaximal, characteristic
3#20265081C212yesmaximal, complemented
4#20265074C212yesmaximal, complemented
5#20265064C212yesmaximal, complemented
6#20265060C212yesmaximal, complemented
7#20265052C211yesmaximal, characteristic
8#20265049C211yesmaximal, characteristic
9#20265040C211yesmaximal, characteristic
10#20265035C211yesmaximal, characteristic
11#381×C101C211yesmaximal, characteristic
12C22×Dic202C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14C23×Dic101C2211yescharacteristic
15#15692232—24no—
16C2×C4⋊Dic101C2212yes—
17C2×C4⋊Dic101C2211yescharacteristic
18C2×#13108053C2212yes—
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C22⋊Dic101C2211yescharacteristic
24C2×C4×Dic101—24no—
25C2×C4×Dic101C2212yes—
26C2×C4×Dic101C2212yes—
27#15692162C2212yes—
28#15692162C2212yes—
29#15692162C2212yes—
30#15692162C2212yes—
31#15692162C2212yes—
32C23×C4×C101C2211yescharacteristic
33C2×C22⋊C4×C101C2211yescharacteristic
34C2×C22⋊C4×C101C2211yescharacteristic
35#95×C101C2212yes—
36#95×C101C2212yes—
37C2×Dic202—22no—
38C2×Dic202—24no—
39C22×Dic101—22no—
40C22×Dic101—24no—
41C22×Dic101—24no—
42C22×Dic101D414yes—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2312yes—
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2311yescharacteristic
47C22×Dic101C2312yes—
48C22×Dic101C2311yescharacteristic
49C4⋊Dic101—22no—
50C4⋊Dic101—24no—
51#13108053—22no—
52#13108053—22no—
53#13108053—24no—
54#13108053—24no—
55C22⋊Dic101—22no—
56C22⋊Dic101—22no—
57C4×Dic101—24no—
58C4×Dic101—24no—
59C24×C101C2311yescharacteristic
60C22×C4×C101—22no—
61C22×C4×C101—22no—
62C22×C4×C101Q814yescomplemented
63C22×C4×C101C2311yescharacteristic
64C22×C4×C101C2312yes—
65C22×C4×C101C2311yescharacteristic
66C22×C4×C101C2312yes—
67C22⋊C4×C101—22no—
68C22⋊C4×C101—22no—
69C23×C101—22no—
70C23×C101—22no—
71C23×C101C2×D412yes—
72C23×C101C2×Q812yes—
73C23×C101#4112yes—
74C23×C101C2411yescharacteristic
75C2×C4×C101—28no—
76C2×C4×C101—22no—
77C2×C4×C101—22no—
78C2×C4×C101—21no—
79C2×C4×C101—21no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101—22no—
83C2×C4×C101—22no—
84C2×C4×C101—22no—
85C2×C4×C101C2×Q814yes—
86C2×Dic101—28no—
87C2×Dic101—22no—
88C2×Dic101—22no—
89C2×Dic101—22no—
90C2×Dic101—28no—
91C2×Dic101—21no—
92C2×Dic101—21no—
93C2×Dic101—22no—
94C2×Dic101—22no—
95C2×Dic101—22no—
96C2×Dic101—22no—
97C2×Dic101—22no—
98C2×Dic101C2×D412yes—
99C2×Dic101C2×D412yes—
100C2×Dic101#4114yes—
101C2×Dic101#4114yes—
102Dic202—44no—
103C22×C101—22no—
104C22×C101—22no—
105C22×C101—24no—
106C22×C101—24no—
107C22×C101C22⋊Q814yes—
108C22×C101C22×D411yescharacteristic
109C22×C101C22×Q811yescharacteristic
110C22×C101C2×#4112yes—
111C22×C101C2×#4112yes—
112C22×C101C2×#4111yescharacteristic
113C4×C101—24no—
114C4×C101—41no—
115C4×C101—41no—
116C4×C101—42no—
117Dic101—22no—
118Dic101—24no—
119Dic101—41no—
120Dic101—41no—
121Dic101—42no—
122Dic101#12314yes—
123C2×C101—24no—
124C2×C101C2×C22⋊Q811yescharacteristic
125C2×C101C2×#12311yescharacteristic
126C2×C101#52511yescharacteristic
127C2×C101#52912yes—
128C2×C101#53312yes—
129#2855—1011nomaximal, Sylow
130C101#285511yescharacteristic, complemented, Sylow
131C2×C22⋊Q8—1011no—
132C22×C42—1011no—
133#391—1012no—
134#389—1012no—
135#387—1012no—
136#387—1012no—
137#385—1011no—
138#385—1011no—
139#381—1011no—
140#381D10111yescharacteristic
141#380—1011no—
142C22×Q8—1011no—
143C23×C4—1011no—
144C23×C4—1011no—
145C23×C4C2×D10111yescharacteristic
146C22⋊Q8—2024no—
147C2×C4⋊C4—1011no—
148C2×C4⋊C4—1011no—
149C2×C4⋊C4—1011no—
150C2×C4⋊C4—1012no—
151C2×C4⋊C4—1012no—
152C2×C4⋊C4—1012no—
153C2×C22⋊C4—1011no—
154C2×C22⋊C4—1011no—
155C2×C22⋊C4C2×D10111yescharacteristic
156C2×C22⋊C4C2×D10111yescharacteristic
157C2×C42—1012no—
158C2×C42—1012no—
159C2×C42—2024no—
160#95—1012no—
161#95—1012no—
162#95—1012no—
163#95—1012no—
164#95—1012no—
165#95C2×D10112yes—
166#95C2×D10112yes—
167C24C22×D10111yescharacteristic
168C2×Q8—2022no—
169C2×Q8—2024no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—1011no—
173C22×C4—1014no—
174C22×C4—1011no—
175C22×C4—1012no—
176C22×C4—1012no—
177C22×C4—1011no—
178C22×C4—1011no—
179C22×C4—2024no—
180C22×C4—2024no—
181C22×C4—2022no—
182C22×C4Dic20214yescomplemented
183C22×C4C22×D10112yes—
184C22×C4C22×D10111yescharacteristic
185C22×C4C22×D10111yescharacteristic
186C22×C4C22×D10112yes—
187C4⋊C4—2024no—
188C4⋊C4—2024no—
189C4⋊C4—2022no—
190C4⋊C4—2022no—
191C4⋊C4—2024no—
192C4⋊C4—2022no—
193C22⋊C4—22no—
194C22⋊C4—22no—
195C22⋊C4—2022no—
196C22⋊C4—2022no—
197C42—2024no—
198C42—2024no—
199C23—22no—
200C23—22no—
201C23D4×D10112yes—
202C23C2×Dic20212yes—
203C23#1310807312yes—
204C23C23×D10111yescharacteristic
205Q8—4044no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—28no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—22no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—1012no—
217C2×C4—1014no—
218C2×C4—1012no—
219C2×C4—1014no—
220C2×C4—2028no—
221C2×C4—2021no—
222C2×C4—2021no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4—2022no—
226C2×C4—2022no—
227C2×C4—2022no—
228C2×C4—2028no—
229C2×C4—2022no—
230C2×C4—2022no—
231C2×C4—2022no—
232C2×C4C2×Dic20214yes—
233C22—24no—
234C22—22no—
235C22—22no—
236C22—24no—
237C22#1569223214yes—
238C22C2×D4×D10111yescharacteristic
239C22C22×Dic20211yescharacteristic
240C22#41×D10112yes—
241C22#41×D10112yes—
242C22C2×#1310807311yescharacteristic
243C4—24no—
244C4—41no—
245C4—41no—
246C4—42no—
247C4—1014no—
248C4—4041no—
249C4—4041no—
250C4—4042no—
251C4—2022no—
252C4—2024no—
253C2—24no—
254C2C2×#1569223211yescharacteristic
255C2#123×D10111yescharacteristic
256C2#2026568511yescharacteristic
257C2#2026570412yes—
258C2#2026573012yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#381×C101C2
prime radicalC24×C101C23
hypercenter#381D101
hypocenterC101#2855
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #381

Complements

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

NH ≅ G/Ntype
3. #20265081C2semidirect
4. #20265074C2semidirect
5. #20265064C2semidirect
6. #20265060C2semidirect
62. C22×C4×C101Q8semidirect
130. C101#2855semidirect
182. C22×C4Dic202semidirect

Maximal subgroups

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

subgroupindex
C2×#156922322
C22×C4×Dic1012
#202650812
#202650742
#202650642
#202650602
#202650522
#202650492
#202650402
#202650352
#381×C1012
#2855101