Finite Group Databasebeta

← #27187147

Subgroups — #27187147

Order 12928.

360 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 78 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×D101⋊M16C211yesmaximal, characteristic
2C2×#15692318C212yesmaximal, complemented
3#20266119C211yesmaximal, characteristic
4#20266050C212yesmaximal, complemented
5C22⋊Q8×D101C211yesmaximal, characteristic
6C2×#13108084—21no—
7D101⋊M16—24no—
8C2×#13108083—21no—
9C2×D101⋊C8C2212yes—
10C2×C4⋊#87938C2212yes—
11C2×C22⋊#87938—21no—
12#15692318—24no—
13(C2×#87938)⋊C4—21no—
14C2×Q8×D101C2211yescharacteristic
15C22×C4×D101C2211yescharacteristic
16#15692235C411yescharacteristic, complemented
17#15692234C411yescharacteristic, complemented
18#15692233—21no—
19#15692232—21no—
20#15692231C411yescharacteristic, complemented
21C4⋊C4×D101—21no—
22C4⋊C4×D101C2211yescharacteristic
23C22⋊C4×D101—21no—
24C22⋊Q8×C101C411yescharacteristic, complemented
25C22×#87938—22no—
26#13108084—24no—
27#13108083—24no—
28D101⋊C8—22no—
29D101⋊C8D414yes—
30C2×#11717581—22no—
31C4⋊#87938—22no—
32C4⋊#87938—22no—
33C22⋊#87938—42no—
34C23×D101D411yescharacteristic
35C2×Dic202—21no—
36C2×Dic202C2×C411yescharacteristic
37Q8×D101—22no—
38Q8×D101—22no—
39C22×Dic101D411yescharacteristic
40C2×C4×D101—22no—
41C2×C4×D101—22no—
42C2×C4×D101—21no—
43C2×C4×D101—21no—
44C2×C4×D101C2×C411yescharacteristic
45C2×C4×D101D411yescharacteristic
46C2×C4×D101D411yescharacteristic
47C2×C4×D101C2311yescharacteristic
48C4⋊Dic101—21no—
49C4⋊Dic101C2×C411yescharacteristic
50#13108053—21no—
51#13108053—21no—
52#13108053—21no—
53C22⋊Dic101—21no—
54#13108051—21no—
55#13108051—21no—
56C2×Q8×C101C2×C411yescharacteristic
57C22×C4×C101C2×C411yescharacteristic
58C4⋊C4×C101—21no—
59C4⋊C4×C101C2×C411yescharacteristic
60C22⋊C4×C101—21no—
61C23×C101C22⋊C411yescharacteristic
62C22×D101—22no—
63C22×D101—21no—
64C22×D101—21no—
65C22×D101—22no—
66C22×D101C22⋊C411yescharacteristic
67C22×D101C2×D411yescharacteristic
68C2×#87938—42no—
69C2×#87938—42no—
70C2×#87938—42no—
71Q8×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101C22⋊C411yescharacteristic
76C2×C4×C101C22×C411yescharacteristic
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C2×D411yescharacteristic
83C4×D101—22no—
84C4×D101—22no—
85C4×D101—21no—
86C4×D101—21no—
87C4×D101—42no—
88C4×D101—41no—
89C4×D101—41no—
90C4×D101C2×D412yes—
91C4×D101#4112yes—
92Dic202—22no—
93Dic202—41no—
94Dic202—41no—
95#11717581—24no—
96C22×C101—21no—
97C22×C101—21no—
98C22×C101—21no—
99C22×C101C2×C22⋊C411yescharacteristic
100C2×D101—21no—
101C2×D101—21no—
102C2×D101—21no—
103C2×D101—42no—
104C2×D101—42no—
105C2×D101#12112yes—
106C2×D101AGL(1,ℤ/8)11yescharacteristic
107C2×D101#13712yes—
108C2×D101#13711yescharacteristic
109#87938—82no—
110C4×C101—21no—
111C4×C101—41no—
112C4×C101—41no—
113C4×C101C4×D412yes—
114Dic101—21no—
115Dic101—41no—
116Dic101—41no—
117Dic101#12011yescharacteristic
118Dic101#12311yescharacteristic
119C2×C101—41no—
120C2×C101#38611yescharacteristic
121C2×C101#41711yescharacteristic
122C2×C101#41911yescharacteristic
123D101—41no—
124D101#46812yes—
125D101#47012yes—
126#1894—1011nomaximal, Sylow
127C101#189411yescharacteristic, complemented, Sylow
128C22×M16—1011no—
129C2×C22⋊Q8—1011no—
130C2×#103—1012no—
131#380—1011no—
132#340—1012no—
133C22×Q8—1011no—
134C23×C4—1011no—
135C2×M16—2024no—
136C2×M16—2021no—
137C2×M16—2021no—
138C22×C8—1012no—
139C22⋊Q8—1011no—
140C22⋊Q8—1011no—
141C22⋊Q8—1011no—
142C22⋊Q8—2021no—
143C22⋊Q8—2021no—
144C22⋊Q8#8793811yescharacteristic, complemented
145C2×C4⋊C4—1011no—
146C2×C4⋊C4—1012no—
147C2×C4⋊C4—2021no—
148C2×C22⋊C4—2021no—
149C2×C22⋊C4—2021no—
150#103—2024no—
151#95—2021no—
152C24—1011no—
153C2×Q8—1011no—
154C2×Q8—2021no—
155C2×Q8—2022no—
156C2×Q8—2022no—
157C2×Q8C2×#8793811yescharacteristic
158C22×C4—1011no—
159C22×C4—1011no—
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163C22×C4—2022no—
164C22×C4—2022no—
165C22×C4—2021no—
166C22×C4—2021no—
167C22×C4—2022no—
168C22×C4C2×#8793811yescharacteristic
169M16—2024no—
170M16—2024no—
171C2×C8—1014no—
172C2×C8—2022no—
173C2×C8—2022no—
174C4⋊C4—21no—
175C4⋊C4—1011no—
176C4⋊C4—2021no—
177C4⋊C4—2021no—
178C4⋊C4—2021no—
179C4⋊C4—2022no—
180C4⋊C4—2022no—
181C4⋊C4—2021no—
182C4⋊C4C2×#8793811yescharacteristic
183C22⋊C4—21no—
184C22⋊C4—4042no—
185C22⋊C4—2021no—
186C22⋊C4—2021no—
187C22⋊C4—2021no—
188C23—1011no—
189C23—1011no—
190C23—2021no—
191C23—2022no—
192C23—2021no—
193C23—2022no—
194C23C22⋊#8793811yescharacteristic
195Q8—22no—
196Q8—4041no—
197Q8—4041no—
198Q8—2022no—
199C2×C4—22no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—1011no—
203C2×C4—1012no—
204C2×C4—1012no—
205C2×C4—1011no—
206C2×C4—4042no—
207C2×C4—4041no—
208C2×C4—4041no—
209C2×C4—4042no—
210C2×C4—4042no—
211C2×C4—4042no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4—2022no—
215C2×C4—2021no—
216C2×C4—2021no—
217C2×C4—2021no—
218C2×C4—2021no—
219C2×C4—2022no—
220C2×C4C22⋊#8793811yescharacteristic
221C2×C4C22×#8793811yescharacteristic
222C8—2024no—
223C22—21no—
224C22—21no—
225C22—21no—
226C22—1012no—
227C22—1011no—
228C22—1011no—
229C22—1012no—
230C22—4042no—
231C22—4042no—
232C22—2021no—
233C22—2021no—
234C22—2021no—
235C22C2×C22⋊#8793811yescharacteristic
236C4—21no—
237C4—41no—
238C4—41no—
239C4—8082no—
240C4—1011no—
241C4—1011no—
242C4—4041no—
243C4—4041no—
244C4—2021no—
245C4D4×#8793812yes—
246C2—41no—
247C2—1012no—
248C2—1012no—
249C2—4041no—
250C2#2026612511yescharacteristic
251C2#2026616811yescharacteristic
252C2#2026617011yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
FittingC22⋊Q8×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenterC22⋊Q8#87938
hypocenterC101#1894
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
2. C2×#15692318C2semidirect
4. #20266050C2semidirect
16. #15692235C4semidirect
17. #15692234C4semidirect
20. #15692231C4semidirect
24. C22⋊Q8×C101C4semidirect
127. C101#1894semidirect
144. C22⋊Q8#87938semidirect

Maximal subgroups

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

subgroupindex
C2×D101⋊M162
C2×#156923182
#202661192
#202660502
C22⋊Q8×D1012
#1894101