Finite Group Databasebeta

← #27187079

Subgroups — #27187079

Order 12928.

324 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C22×#13108083C211yesmaximal, characteristic
2C2×#15692322C211yesmaximal, characteristic, complemented
3C2×C4⋊#11717581C211yesmaximal, characteristic, complemented
4C2×C22⋊#11717581C211yesmaximal, characteristic
5#20266043C211yesmaximal, characteristic, complemented
6#20266043C211yesmaximal, characteristic, complemented
7C22⋊C4×Dic101C211yesmaximal, characteristic
8C2×#13108083—22no—
9C2×#13108083—22no—
10C2×#13108083—22no—
11C22×#11717581C2211yescharacteristic
12C22×#11717581C2211yescharacteristic
13C22×#11717581C2211yescharacteristic
14C22×#11717581C2211yescharacteristic
15#15692322—22no—
16C4⋊#11717581—22no—
17C22⋊#11717581—22no—
18C23×Dic101C2211yescharacteristic
19C2×C22⋊Dic101C411yescharacteristic
20C2×C4×Dic101C2211yescharacteristic
21C2×C4×Dic101C2211yescharacteristic
22#15692162C411yescharacteristic
23#15692162C411yescharacteristic
24C2×C22⋊C4×C101C411yescharacteristic
25#13108083—28no—
26C2×#11717581—21no—
27C2×#11717581—21no—
28C2×#11717581—21no—
29C2×#11717581—21no—
30C2×#11717581—21no—
31C2×#11717581—21no—
32C2×#11717581—21no—
33C2×#11717581—21no—
34C2×#11717581—21no—
35C2×#11717581—21no—
36C2×#11717581D412yes—
37C2×#11717581D412yes—
38C22×Dic101—21no—
39C22×Dic101—21no—
40C22×Dic101—21no—
41C22×Dic101—21no—
42C22×Dic101C2×C411yescharacteristic
43C22×Dic101C2×C411yescharacteristic
44C22×Dic101C2×C411yescharacteristic
45C22×Dic101D412yes—
46C22×Dic101Q812yes—
47C22×Dic101C2311yescharacteristic
48C22⋊Dic101—22no—
49C4×Dic101—21no—
50C4×Dic101—21no—
51C4×Dic101—21no—
52C4×Dic101—21no—
53C24×C101C2×C411yescharacteristic
54C22×C4×C101C2×C411yescharacteristic
55C22×C4×C101C2×C411yescharacteristic
56C22⋊C4×C101—22no—
57C23×C101—21no—
58C23×C101—21no—
59C23×C101—21no—
60C23×C101—21no—
61C23×C101—21no—
62C23×C101—21no—
63C23×C101C4⋊C412yes—
64C23×C101C22×C411yescharacteristic
65C2×C4×C101—21no—
66C2×C4×C101—21no—
67C2×C4×C101—21no—
68C2×C4×C101—21no—
69C2×C4×C101—21no—
70C2×C4×C101M1612yescomplemented
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101—22no—
78C2×Dic101—22no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101C4⋊C412yes—
83C2×Dic101M1612yescomplemented
84C2×Dic101C2×D411yescharacteristic
85C2×Dic101C2×D411yescharacteristic
86C2×Dic101C2×D411yescharacteristic
87C2×Dic101C2×Q811yescharacteristic
88C2×Dic101#4111yescharacteristic
89C2×Dic101#4111yescharacteristic
90#11717581—22no—
91#11717581—22no—
92#11717581—41no—
93#11717581—41no—
94C22×C101—22no—
95C22×C101—22no—
96C22×C101—22no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101—41no—
101C22×C101C2×C4⋊C411yescharacteristic
102C22×C101C4×D411yescharacteristic
103C22×C101C4×D411yescharacteristic
104C22×C101C2×M1611yescharacteristic
105C22×C101C2×M1611yescharacteristic
106C22×C101#13111yescharacteristic
107C22×C101#13111yescharacteristic
108C4×C101—22no—
109C4×C101—41no—
110Dic101—22no—
111Dic101—22no—
112Dic101—41no—
113Dic101#12011yescharacteristic
114Dic101C22⋊Q811yescharacteristic
115Dic101C22⋊Q811yescharacteristic
116Dic101#12311yescharacteristic
117C2×C101—42no—
118C2×C101#38511yescharacteristic
119C2×C101#42311yescharacteristic
120C2×C101#42411yescharacteristic
121C2×C101#43511yescharacteristic
122C2×C101#43511yescharacteristic
123C2×C101#43611yescharacteristic
124C2×C101#43611yescharacteristic
125#1851—1011nomaximal, Sylow
126C101#185111yescharacteristic, complemented, Sylow
127C22×M16—1011no—
128C2×C4⋊C8—1011no—
129C2×C4⋊C8—1011no—
130C2×C22⋊C8—1011no—
131C4×C22⋊C4—1011no—
132#336—1011no—
133#336—1011no—
134C23×C4—1011no—
135C2×M16—2022no—
136C2×M16—2022no—
137C2×M16—2022no—
138C22×C8—1011no—
139C22×C8—1011no—
140C22×C8—1011no—
141C22×C8—1011no—
142C2×C22⋊C4—1011no—
143C2×C22⋊C4#8793811yescharacteristic
144C2×C42—1011no—
145C2×C42—1011no—
146C4⋊C8—2022no—
147C4⋊C8—2022no—
148C22⋊C8—2022no—
149#95—1011no—
150#95—1011no—
151C24C2×#8793811yescharacteristic
152C22×C4—1011no—
153C22×C4—1012no—
154C22×C4—1012no—
155C22×C4—1011no—
156C22×C4—1011no—
157C22×C4—1011no—
158C22×C4—2021no—
159C22×C4—2021no—
160C22×C4—2021no—
161C22×C4—2021no—
162C22×C4C2×#8793811yescharacteristic
163C22×C4C2×#8793811yescharacteristic
164M16—2028no—
165C2×C8—1012no—
166C2×C8—1012no—
167C2×C8—2021no—
168C2×C8—2021no—
169C2×C8—2021no—
170C2×C8—2021no—
171C2×C8—2021no—
172C2×C8—2021no—
173C2×C8—2021no—
174C2×C8—2021no—
175C2×C8—2021no—
176C2×C8—2021no—
177C22⋊C4—22no—
178C22⋊C4—2022no—
179C42—2021no—
180C42—2021no—
181C42—2021no—
182C42—2021no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—21no—
188C23—21no—
189C23C4⋊#8793812yes—
190C23C22×#8793811yescharacteristic
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—21no—
195C2×C4—21no—
196C2×C4—1011no—
197C2×C4—1011no—
198C2×C4—1011no—
199C2×C4—1011no—
200C2×C4—1011no—
201C2×C4—1011no—
202C2×C4—1012no—
203C2×C4—1012no—
204C2×C4—2022no—
205C2×C4—2022no—
206C2×C4—2022no—
207C2×C4—2022no—
208C2×C4—2021no—
209C2×C4—2021no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4—2021no—
215C2×C4#1310808312yescomplemented
216C8—4041no—
217C8—4041no—
218C8—2022no—
219C8—2022no—
220C22—22no—
221C22—22no—
222C22—22no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22C2×C4⋊#8793811yescharacteristic
228C22D4×#8793811yescharacteristic
229C22D4×#8793811yescharacteristic
230C22C2×#1310808311yescharacteristic
231C22D101⋊M1611yescharacteristic
232C22#1569234211yescharacteristic
233C22#1569234211yescharacteristic
234C4—22no—
235C4—41no—
236C4—1011no—
237C4—1011no—
238C4—1011no—
239C4—1011no—
240C4—4041no—
241C4—2022no—
242C4—2022no—
243C2—42no—
244C2#2026612411yescharacteristic
245C2#2026618011yescharacteristic
246C2#2026618311yescharacteristic
247C2#2026620811yescharacteristic
248C2#2026620911yescharacteristic
249C2#2026621011yescharacteristic
250C2#2026621211yescharacteristic
subgroupG/H
center, FrattiniC23C22×#87938
commutator, socleC23×C101C22×C4
FittingC2×C22⋊C4×C101C4
prime radicalC24×C101C2×C4
cosocleC22×Dic101C23
hypercenterC2×C22⋊C4#87938
hypocenterC101#1851
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×#15692322C2semidirect
3. C2×C4⋊#11717581C2semidirect
5. #20266043C2semidirect
6. #20266043C2semidirect
70. C2×C4×C101M16semidirect
83. C2×Dic101M16semidirect
126. C101#1851semidirect
215. C2×C4#13108083semidirect

Maximal subgroups

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

subgroupindex
C22×#131080832
C2×#156923222
C2×C4⋊#117175812
C2×C22⋊#117175812
#202660432
C22⋊C4×Dic1012
#1851101