Finite Group Databasebeta

← #27179613

Subgroups — #27179613

Order 12928.

704 conjugacy classes of subgroups; only the 209 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 209 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×#15692302C211yesmaximal, characteristic
2#20265943C212yesmaximal, complemented
3#20265161C211yesmaximal, characteristic, complemented
4#20265158C214yesmaximal, complemented
5C2×#15692173C212yesmaximal, complemented
6#101×D101C214yesmaximal, complemented
7#413×C101C211yesmaximal, characteristic, complemented
8#15692302—24no—
9#15692302—24no—
10C2×#13108074C412yes—
11C2×#13108074C2211yescharacteristic
12C2×#13108073C414yes—
13C2×#13108073C2212yes—
14C22×Dic202C412yes—
15C22×Dic202C2211yescharacteristic
16C2×Q8×D101C2212yes—
17#15692263—24no—
18#15692261—24no—
19C2×#13108058C2212yes—
20M16×D101C2218yes—
21#15692174C2212yes—
22#15692173C2218yes—
23#15692172C2212yes—
24C2×#41×C101C2211yescharacteristic
25C2×M16×C101C2212yes—
26#101×C101C2212yes—
27#100×C101C2212yes—
28#13108074—24no—
29#13108074—24no—
30#13108074D414yes—
31#13108074D414yes—
32#13108073—216no—
33#13108073—28no—
34C2×Dic202—22no—
35C2×Dic202—22no—
36C2×Dic202—24no—
37C2×Dic202—24no—
38C2×Dic202C2×C414yes—
39C2×Dic202C2×C412yes—
40C2×Dic202C2×C411yescharacteristic
41C2×Dic202C2312yes—
42Q8×D101—28no—
43C2×D404C2×C411yescharacteristic
44C2×C22⋊D101C2×C414yes—
45C2×C22⋊D101C2×C412yes—
46C22×Dic101C2×C414yes—
47C22×Dic101C2×C412yes—
48C2×C4×D101C2×C414yes—
49C2×C4×D101C2312yes—
50#13108058—22no—
51#13108058C2314yes—
52#13108057—28no—
53C2×#11717579—22no—
54C8×D101—28no—
55#41×C101—22no—
56#41×C101—22no—
57C2×Q8×C101C2311yescharacteristic
58C2×D4×C101C2×C412yes—
59C2×D4×C101C2311yescharacteristic
60C22×C4×C101C2×C412yes—
61C22×C4×C101C2311yescharacteristic
62M16×C101—22no—
63M16×C101C2314yes—
64C2×C8×C101—22no—
65C23×C101C22×C412yes—
66C23×C101C22×C411yescharacteristic
67C22×D101C22×C412yes—
68Q8×C101—22no—
69D4×C101—24no—
70D4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101—22no—
73C2×C4×C101C22×C412yes—
74C2×C4×C101C22×C411yescharacteristic
75C2×C4×C101C2×D412yes—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2411yescharacteristic
78C22⋊D101—28no—
79C22⋊D101C22⋊C418yes—
80C2×Dic101—24no—
81C2×Dic101—28no—
82C2×Dic101C22×C414yes—
83C2×Dic101C22×C412yes—
84D404C22⋊C412yes—
85D404C22⋊C412yes—
86C4×D101—28no—
87C4×D101C2×D418yes—
88Dic202—22no—
89Dic202—22no—
90Dic202—24no—
91Dic202—28no—
92Dic202C22⋊C412yes—
93Dic202C22⋊C412yes—
94C8×C101—24no—
95#11717579—24no—
96C22×C101—22no—
97C22×C101—41no—
98C22×C101—42no—
99C22×C101C2×C22⋊C412yes—
100C22×C101C23×C411yescharacteristic
101C2×D101—42no—
102C2×D101C2×C22⋊C414yes—
103C4×C101—22no—
104C4×C101C2×C22⋊C411yescharacteristic
105C4×C101C2×C22⋊C411yescharacteristic
106C4×C101C22×D412yes—
107Dic101—24no—
108Dic101C2×C22⋊C414yes—
109C2×C101—21no—
110C2×C101—22no—
111C2×C101—42no—
112C2×C101C22×C22⋊C411yescharacteristic
113D101—24no—
114#2889—1011nomaximal, Sylow
115C101#288911yescharacteristic, complemented, Sylow
116C2×2−1+4—1011no—
117#568—1012no—
118#413—1011no—
119#413—1014no—
120#413D10111yescharacteristic, complemented
121C2×#101—1012no—
122C2×#101—1014no—
1232−1+4—2024no—
1242−1+4—2024no—
125C2×#41—1014no—
126C2×#41—1012no—
127C2×#41—1012no—
128C2×#41—1011no—
129C2×#41C2×D10111yescharacteristic
130C22×Q8—1012no—
131C22×Q8—1012no—
132C22×Q8—1011no—
133#131—2024no—
134#131—2024no—
135C2×M16—1012no—
136C2×M16—1018no—
137C2×M16C2×D10112yes—
138#101—1012no—
139#101—1018no—
140#101C2×D10112yes—
141#100—1012no—
142#100C2×D10112yes—
143#41—22no—
144#41—22no—
145#41—1014no—
146#41—1014no—
147#41—2024no—
148#41—2024no—
149#41—2028no—
150#41—20216no—
151C2×Q8—1011no—
152C2×Q8—1014no—
153C2×Q8—1012no—
154C2×Q8—1012no—
155C2×Q8—2024no—
156C2×Q8—2024no—
157C2×Q8—2028no—
158C2×Q8—2022no—
159C2×Q8—2022no—
160C2×Q8C22×D10111yescharacteristic
161C2×D4—1012no—
162C2×D4—1014no—
163C2×D4—1011no—
164C2×D4C4×D10112yes—
165C2×D4C22×D10111yescharacteristic
166C22×C4—1012no—
167C22×C4—1014no—
168C22×C4—1012no—
169C22×C4—1014no—
170C22×C4C4×D10112yes—
171C22×C4C22×D10111yescharacteristic
172M16—22no—
173M16—1014no—
174M16—2028no—
175M16—2022no—
176M16C22×D10114yes—
177C2×C8—22no—
178C2×C8—2028no—
179C2×C8—2022no—
180C23—1012no—
181C23C2×C4×D10112yes—
182C23C2×C4×D10111yescharacteristic
183Q8—22no—
184Q8—1012no—
185Q8—1012no—
186Q8—2028no—
187Q8—2024no—
188Q8—2022no—
189Q8—2022no—
190D4—24no—
191D4—22no—
192D4—1018no—
193D4—1012no—
194D4—1012no—
195D4—2028no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—1012no—
199C2×C4—1018no—
200C2×C4—1014no—
201C2×C4—2028no—
202C2×C4—2028no—
203C2×C4—2024no—
204C2×C4C2×C4×D10112yes—
205C2×C4C2×C4×D10111yescharacteristic
206C2×C4D4×D10112yes—
207C2×C4D4×D10112yes—
208C2×C4C23×D10111yescharacteristic
209C8—24no—
210C8—2024no—
211C22—22no—
212C22—41no—
213C22—42no—
214C22—1014no—
215C22—4042no—
216C22C22⋊C4×D10112yes—
217C22C22×C4×D10111yescharacteristic
218C4—22no—
219C4—1014no—
220C4—2024no—
221C4C22⋊C4×D10111yescharacteristic
222C4C22⋊C4×D10111yescharacteristic
223C4C2×D4×D10112yes—
224C2—21no—
225C2—22no—
226C2—42no—
227C2—2024no—
228C2C2×C22⋊C4×D10111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D101
commutatorC22×C101C23×C4
FrattiniC2×C4C23×D101
Fitting#413×C101C2
socleC2×C101C22×C22⋊C4
prime radicalC2×#15692302C2
cosocleC2×C4×C101C24
hypercenter#413D101
hypocenterC101#2889
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
2. #20265943C2semidirect
3. #20265161C2semidirect
4. #20265158C2semidirect
5. C2×#15692173C2semidirect
6. #101×D101C2semidirect
7. #413×C101C2semidirect
115. C101#2889semidirect
120. #413D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923022
#202659432
#202651612
#202651582
C2×#156921732
#101×D1012
#413×C1012
#2889101