Finite Group Databasebeta

← #27181300

Subgroups — #27181300

Order 12928.

386 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 117 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#20265708C211yesmaximal, characteristic
2C2×#15692220C211yesmaximal, characteristic, complemented
3#20265512C211yesmaximal, characteristic, complemented
4#20265487C212yesmaximal, complemented
5#20265398C212yesmaximal, complemented
6#20265335C211yesmaximal, characteristic, complemented
7#20265315C211yesmaximal, characteristic
8#20265283C212yesmaximal, complemented
9#20265274C212yesmaximal, complemented
10#20265231C211yesmaximal, characteristic, complemented
11#481×C101C211yesmaximal, characteristic
12C22×Dic202C2211yescharacteristic
13C2×Dic404C2212yes—
14C2×#13108062C2212yes—
15#15692248—21no—
16#15692246C2211yescharacteristic
17#15692242C2211yescharacteristic
18#15692236—21no—
19#15692234C2211yescharacteristic
20#15692220—21no—
21#15692220—21no—
22#15692220C2212yes—
23#15692220C2212yes—
24#15692220C2211yescharacteristic
25D4×Dic101—21no—
26D4×Dic101C2211yescharacteristic
27#15692211—21no—
28C2×#13108053—21no—
29C2×C4×Dic101—21no—
30C2×C4×Dic101C2211yescharacteristic
31#15692188C2212yes—
32#15692184C2212yes—
33#15692180C2212yes—
34#15692178C2212yes—
35#15692178C2212yes—
36C22⋊#11717579C2211yescharacteristic
37#15692165C2212yes—
38#121×C101C2211yescharacteristic
39C2×C4⋊C4×C101C2211yescharacteristic
40#106×C101C2212yes—
41#102×C101C2212yes—
42C22⋊C8×C101C2211yescharacteristic
43C2×Dic202—21no—
44C2×Dic202—24no—
45C2×Dic202C2311yescharacteristic
46C2×Dic202C2312yes—
47C22×Dic101—21no—
48C22×Dic101—21no—
49C22×Dic101D411yescharacteristic
50C22×Dic101D411yescharacteristic
51Dic404—24no—
52#13108062—24no—
53C2×#11717579C2312yes—
54C4⋊Dic101—21no—
55C4⋊Dic101C2311yescharacteristic
56#13108053—21no—
57#13108053—22no—
58#13108053—22no—
59#13108053—21no—
60#13108053—21no—
61C22⋊Dic101—21no—
62C22⋊Dic101—21no—
63C22⋊Dic101—21no—
64C22⋊Dic101—21no—
65C4×Dic101—22no—
66C4×Dic101—22no—
67C4×Dic101—21no—
68C4×Dic101D411yescharacteristic
69C4×Dic101D411yescharacteristic
70C4×Dic101C2311yescharacteristic
71C4×Dic101C2311yescharacteristic
72C2×D4×C101—21no—
73C2×D4×C101C2311yescharacteristic
74C22×C4×C101—21no—
75C22×C4×C101C2311yescharacteristic
76C2×C8×C101C2312yes—
77C4⋊C4×C101—21no—
78C4⋊C4×C101C2312yes—
79C4⋊C4×C101C2311yescharacteristic
80C22⋊C4×C101—21no—
81C23×C101—21no—
82C23×C101C2×D411yescharacteristic
83D4×C101—22no—
84D4×C101—42no—
85C2×C4×C101—22no—
86C2×C4×C101—21no—
87C2×C4×C101—22no—
88C2×C4×C101—42no—
89C2×C4×C101C2×D411yescharacteristic
90C2×C4×C101C2411yescharacteristic
91C2×Dic101—24no—
92C2×Dic101—22no—
93C2×Dic101—22no—
94C2×Dic101—21no—
95C2×Dic101—21no—
96C2×Dic101—41no—
97C2×Dic101—42no—
98C2×Dic101C2×D411yescharacteristic
99C2×Dic101C2×D411yescharacteristic
100C2×Dic101C2×D411yescharacteristic
101C2×Dic101C2×D411yescharacteristic
102Dic202—24no—
103Dic202—22no—
104Dic202—22no—
105Dic202#4114yes—
106C8×C101—22no—
107#11717579—22no—
108C22×C101—21no—
109C22×C101—21no—
110C22×C101—41no—
111C22×C101—41no—
112C22×C101—41no—
113C22×C101#13712yes—
114C22×C101C22×D411yescharacteristic
115C4×C101—21no—
116C4×C101—41no—
117C4×C101—42no—
118C4×C101C2×#4112yes—
119Dic101—24no—
120Dic101—21no—
121Dic101—21no—
122Dic101—21no—
123Dic101—21no—
124Dic101—41no—
125Dic101—41no—
126C2×C101—22no—
127C2×C101—81no—
128C2×C101#52511yescharacteristic
129C2×C101C2×#13711yescharacteristic
130C2×C101#57511yescharacteristic
131#3117—1011nomaximal, Sylow
132C101#311711yescharacteristic, complemented, Sylow
133#529—1011no—
134C2×C4×Q8—1011no—
135#484—1011no—
136#481D10111yescharacteristic
137#479—1012no—
138#464—1012no—
139#451—1011no—
140#448—1011no—
141#441—1012no—
142#440—1012no—
143#432—1011no—
144C22×Q8—1011no—
145C2×Q16—1012no—
146C2×SD16—1012no—
147#126—2021no—
148#125—1011no—
149#124—1011no—
150#123—2021no—
151C22⋊Q8—1011no—
152#121C2×D10111yescharacteristic
153C4×Q8—1012no—
154C4×Q8—1012no—
155C4×Q8—1011no—
156C4×Q8—2021no—
157C4×Q8—2021no—
158C4×D4—1011no—
159C4×D4—2021no—
160#117—2021no—
161C2×C4⋊C4—2021no—
162C2×C4⋊C4C2×D10111yescharacteristic
163C2×C42—1011no—
164C2×C42—2021no—
165#107—1012no—
166#106C2×D10112yes—
167C4⋊C8—1012no—
168#103—1012no—
169#103—1012no—
170#103—1012no—
171#102C2×D10112yes—
172C22⋊C8—1011no—
173C22⋊C8C2×D10111yescharacteristic
174#97—1012no—
175C2×Q8—1012no—
176C2×Q8—1011no—
177C2×Q8—2024no—
178C2×Q8—2021no—
179C2×D4—21no—
180C2×D4C22×D10111yescharacteristic
181C22×C4—21no—
182C22×C4—1011no—
183C22×C4—1011no—
184C22×C4—2021no—
185C22×C4—2021no—
186C22×C4C22×D10111yescharacteristic
187Q16—2024no—
188SD16—2024no—
189C2×C8—1012no—
190C2×C8C22×D10112yes—
191C4⋊C4—21no—
192C4⋊C4—1011no—
193C4⋊C4—2022no—
194C4⋊C4—2022no—
195C4⋊C4—2021no—
196C4⋊C4—2021no—
197C4⋊C4—2021no—
198C4⋊C4—2021no—
199C4⋊C4C22×D10112yes—
200C4⋊C4C22×D10111yescharacteristic
201C22⋊C4—21no—
202C22⋊C4—2021no—
203C22⋊C4—2021no—
204C22⋊C4—2021no—
205C22⋊C4—2021no—
206C42—1011no—
207C42—1011no—
208C42—1011no—
209C42—1011no—
210C42—2022no—
211C42—2022no—
212C42—2021no—
213C23—21no—
214C23D4×D10111yescharacteristic
215Q8—1014no—
216Q8—2024no—
217Q8—2022no—
218Q8—2022no—
219D4—22no—
220D4—42no—
221C2×C4—22no—
222C2×C4—22no—
223C2×C4—21no—
224C2×C4—42no—
225C2×C4—1011no—
226C2×C4—1011no—
227C2×C4—1011no—
228C2×C4—1011no—
229C2×C4—4042no—
230C2×C4—4041no—
231C2×C4—2021no—
232C2×C4—2021no—
233C2×C4—2024no—
234C2×C4—2022no—
235C2×C4—2022no—
236C2×C4D4×D10111yescharacteristic
237C2×C4C23×D10111yescharacteristic
238C8—22no—
239C8—2022no—
240C22—21no—
241C22—21no—
242C22—41no—
243C22—41no—
244C22—41no—
245C22#1569228412yes—
246C22C2×D4×D10111yescharacteristic
247C4—21no—
248C4—41no—
249C4—42no—
250C4—4041no—
251C4—4041no—
252C4—2024no—
253C4—2021no—
254C4—2021no—
255C4—2021no—
256C4—2021no—
257C4#41×D10112yes—
258C2—22no—
259C2—81no—
260C2#2026568511yescharacteristic
261C2C2×#1569228411yescharacteristic
262C2#2026597311yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#481×C101C2
socleC22×C101C22×D4
prime radical#121×C101C22
hypercenter#481D101
hypocenterC101#3117
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #481

Complements

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

NH ≅ G/Ntype
2. C2×#15692220C2semidirect
3. #20265512C2semidirect
4. #20265487C2semidirect
5. #20265398C2semidirect
6. #20265335C2semidirect
8. #20265283C2semidirect
9. #20265274C2semidirect
10. #20265231C2semidirect
132. C101#3117semidirect

Maximal subgroups

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

subgroupindex
#202657082
C2×#156922202
#202655122
#202654872
#202653982
#202653352
#202653152
#202652832
#202652742
#202652312
#481×C1012
#3117101