Finite Group Databasebeta

← #27174401

Subgroups — #27174401

Order 12928.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#15692220C211yesmaximal, characteristic, complemented
2C2×#15692211C211yesmaximal, characteristic
3#20265115C211yesmaximal, characteristic, complemented
4#20265108C212yesmaximal, complemented
5#20265100C212yesmaximal
6#20265098C211yesmaximal, characteristic, complemented
7#20265089C211yesmaximal, characteristic
8C4⋊C4×Dic101C212yesmaximal, complemented
9#20265028C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic101C212yesmaximal
11#399×C101C211yesmaximal, characteristic
12C22×Dic202C2211yescharacteristic
13C23×Dic101C2211yescharacteristic
14#15692220—24no—
15#15692211—24no—
16C2×C4⋊Dic101C2212yes—
17C2×#13108053C2212yes—
18C2×#13108053C2211yescharacteristic
19C2×#13108053C2212yes—
20C2×#13108053C2211yescharacteristic
21C2×C22⋊Dic101C2212yes—
22C2×C22⋊Dic101C2211yescharacteristic
23C2×C4×Dic101C2212yes—
24C2×C4×Dic101C2212yes—
25C2×C4×Dic101C2211yescharacteristic
26C2×C4×Dic101C2211yescharacteristic
27#15692162C2212yes—
28#15692162C2212yes—
29#15692162C2211yescharacteristic
30#15692162C2212yes—
31#15692162C2212yes—
32C2×C4⋊C4×C101C2212yes—
33C2×C4⋊C4×C101C2211yescharacteristic
34C2×C22⋊C4×C101C2212yes—
35C2×C22⋊C4×C101C2211yescharacteristic
36#95×C101C2211yescharacteristic
37C2×Dic202—22no—
38C2×Dic202—24no—
39C22×Dic101—22no—
40C22×Dic101—22no—
41C22×Dic101Q814yescomplemented
42C22×Dic101C2312yes—
43C22×Dic101C2311yescharacteristic
44C22×Dic101C2311yescharacteristic
45C22×Dic101C2311yescharacteristic
46C22×Dic101C2312yes—
47C22×Dic101C2311yescharacteristic
48C4⋊Dic101—24no—
49#13108053—22no—
50#13108053—24no—
51#13108053—22no—
52#13108053—24no—
53C22⋊Dic101—22no—
54C22⋊Dic101—24no—
55C4×Dic101—22no—
56C4×Dic101—24no—
57C4×Dic101—24no—
58C4×Dic101D414yes—
59C24×C101C2311yescharacteristic
60C22×C4×C101C2311yescharacteristic
61C22×C4×C101C2312yes—
62C22×C4×C101C2311yescharacteristic
63C22×C4×C101C2312yes—
64C4⋊C4×C101—22no—
65C4⋊C4×C101—24no—
66C22⋊C4×C101—22no—
67C22⋊C4×C101—24no—
68C23×C101—21no—
69C23×C101—22no—
70C23×C101—22no—
71C23×C101—21no—
72C23×C101C2×Q812yes—
73C23×C101C2411yescharacteristic
74C2×C4×C101—22no—
75C2×C4×C101—22no—
76C2×C4×C101—22no—
77C2×C4×C101—21no—
78C2×C4×C101—22no—
79C2×C4×C101—22no—
80C2×C4×C101—22no—
81C2×C4×C101—22no—
82C2×C4×C101C2×D412yes—
83C2×C4×C101#4114yes—
84C2×Dic101—22no—
85C2×Dic101—21no—
86C2×Dic101—22no—
87C2×Dic101—21no—
88C2×Dic101—22no—
89C2×Dic101—28no—
90C2×Dic101—21no—
91C2×Dic101—22no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—22no—
95C2×Dic101C2×D412yes—
96C2×Dic101C2×D412yes—
97C2×Dic101C2×Q812yes—
98C2×Dic101C2×Q812yes—
99C2×Dic101#4114yes—
100C2×Dic101#4112yes—
101Dic202—28no—
102C22×C101—22no—
103C22×C101—24no—
104C22×C101—41no—
105C22×C101—42no—
106C22×C101—41no—
107C22×C101C22×D411yescharacteristic
108C22×C101C22×Q811yescharacteristic
109C22×C101C2×#4111yescharacteristic
110C22×C101C2×#4112yes—
111C22×C101C2×#4112yes—
112C4×C101—22no—
113C4×C101—24no—
114C4×C101—41no—
115C4×C101—42no—
116Dic101—22no—
117Dic101—24no—
118Dic101—22no—
119Dic101—41no—
120Dic101—42no—
121Dic101C22⋊Q814yes—
122C2×C101—42no—
123C2×C101C2×C22⋊Q811yescharacteristic
124C2×C101#52511yescharacteristic
125C2×C101#52912yes—
126C2×C101#53211yescharacteristic
127C2×C101#53312yes—
128#2396—1011nomaximal, Sylow
129C101#239611yescharacteristic, complemented, Sylow
130C2×C4×Q8—1011no—
131C2×#117—1011no—
132#400—1011no—
133#399D10111yescharacteristic
134#398—1012no—
135#396—1012no—
136#395—1011no—
137#393—1011no—
138C4×C4⋊C4—1012no—
139C4×C4⋊C4—1011no—
140C4×C22⋊C4—1012no—
141C22×Q8—1011no—
142C23×C4—1011no—
143C4×Q8—2024no—
144#117—2024no—
145C2×C4⋊C4—1011no—
146C2×C4⋊C4—1011no—
147C2×C4⋊C4—1012no—
148C2×C4⋊C4—1012no—
149C2×C4⋊C4—1012no—
150C2×C4⋊C4C2×D10111yescharacteristic
151C2×C4⋊C4C2×D10112yes—
152C2×C22⋊C4—1011no—
153C2×C22⋊C4—1012no—
154C2×C22⋊C4C2×D10111yescharacteristic
155C2×C22⋊C4C2×D10112yes—
156C2×C42—1011no—
157C2×C42—1011no—
158C2×C42—1012no—
159C2×C42—1012no—
160#95—1012no—
161#95—1012no—
162#95—1012no—
163#95—1011no—
164#95—1012no—
165#95C2×D10111yescharacteristic
166C24C22×D10111yescharacteristic
167C2×Q8—2024no—
168C2×Q8—2022no—
169C22×C4—1011no—
170C22×C4—1014no—
171C22×C4—1011no—
172C22×C4—1012no—
173C22×C4—1011no—
174C22×C4—1012no—
175C22×C4—1011no—
176C22×C4—2022no—
177C22×C4—2022no—
178C22×C4C22×D10111yescharacteristic
179C22×C4C22×D10112yes—
180C22×C4C22×D10111yescharacteristic
181C22×C4C22×D10112yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—2022no—
185C4⋊C4—2024no—
186C4⋊C4—2024no—
187C4⋊C4—2022no—
188C4⋊C4—2024no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—2024no—
192C22⋊C4—2022no—
193C42—1014no—
194C42—2024no—
195C42—2022no—
196C42—2024no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D10112yes—
202C23C23×D10111yescharacteristic
203Q8—2028no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—1012no—
213C2×C4—1012no—
214C2×C4—1012no—
215C2×C4—1012no—
216C2×C4—1012no—
217C2×C4—1014no—
218C2×C4—2028no—
219C2×C4—2021no—
220C2×C4—2022no—
221C2×C4—2022no—
222C2×C4—2022no—
223C2×C4—2022no—
224C2×C4—2022no—
225C2×C4—2021no—
226C2×C4—2021no—
227C2×C4—2022no—
228C2×C4—2022no—
229C2×C4D4×D10112yes—
230C2×C4#1310807314yes—
231C22—24no—
232C22—22no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D10111yescharacteristic
237C22C2×Q8×D10111yescharacteristic
238C22#41×D10112yes—
239C22#41×D10111yescharacteristic
240C22C2×#1310807312yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—1014no—
246C4—4041no—
247C4—4042no—
248C4—2022no—
249C4—2024no—
250C4—2022no—
251C2—42no—
252C2C22⋊Q8×D10111yescharacteristic
253C2#2026568511yescharacteristic
254C2#2026570412yes—
255C2#2026572511yescharacteristic
256C2#2026572912yes—
subgroupG/H
center, FrattiniC23C23×D101
commutator, socle, cosocleC23×C101C24
Fitting#399×C101C2
prime radicalC24×C101C23
hypercenter#399D101
hypocenterC101#2396
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#15692220C2semidirect
3. #20265115C2semidirect
4. #20265108C2semidirect
6. #20265098C2semidirect
8. C4⋊C4×Dic101C2semidirect
9. #20265028C2semidirect
41. C22×Dic101Q8semidirect
129. C101#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#156922202
C2×#156922112
#202651152
#202651082
#202651002
#202650982
#202650892
C4⋊C4×Dic1012
#202650282
C22⋊C4×Dic1012
#399×C1012
#2396101