Finite Group Databasebeta

← #27179594

Subgroups — #27179594

Order 12928.

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

normal subgroups only

This group has 716 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#20265654C212yesmaximal, complemented
3#20265151C211yesmaximal, characteristic, complemented
4#20265150C212yesmaximal, complemented
5#20265149C218yesmaximal, complemented
6#410×C101C211yesmaximal, characteristic, complemented
7#15692302—24no—
8#15692302—24no—
9C2×#13108074C412yescomplemented
10C2×#13108074C2211yescharacteristic
11C2×#13108073C412yescomplemented
12C2×#13108073C2214yes—
13C22×Dic202C412yescomplemented
14C22×Dic202C2211yescharacteristic
15C2×Q8×D101C412yescomplemented
16C4×Dic202—24no—
17#15692220—24no—
18#15692214C2212yes—
19#15692211C2218yes—
20#15692170C2214yes—
21#15692169C2218yes—
22C2×#41×C101C2211yescharacteristic
23#117×C101C2212yes—
24#99×C101C2214yes—
25#13108074—22no—
26#13108074—22no—
27#13108074—24no—
28#13108074—24no—
29#13108073—216no—
30#13108073—28no—
31C2×Dic202—24no—
32C2×Dic202—24no—
33C2×Dic202C2×C414yes—
34C2×Dic202C2×C411yescharacteristic
35C2×Dic202C2×C412yes—
36C2×Dic202C2×C412yes—
37C2×Dic202D414yes—
38C2×Dic202D414yes—
39Q8×D101—28no—
40C2×D404C2×C411yescharacteristic
41C2×C22⋊D101C2×C414yes—
42C2×C22⋊D101C2312yes—
43C22×Dic101C2×C414yes—
44C22×Dic101C2312yes—
45C2×C4×D101C2×C412yes—
46C2×C4×D101C2×C414yes—
47C4⋊Dic101—22no—
48#13108053—28no—
49C22⋊Dic101C2314yes—
50C4×Dic101—22no—
51C4×Dic101—28no—
52#41×C101—22no—
53#41×C101—22no—
54C2×Q8×C101C2×C411yescharacteristic
55C2×D4×C101C2×C411yescharacteristic
56C2×D4×C101C2312yes—
57C22×C4×C101C2×C412yes—
58C22×C4×C101C2311yescharacteristic
59C4⋊C4×C101—22no—
60C22⋊C4×C101C2314yes—
61C42×C101—22no—
62C23×C101C22×C412yes—
63C23×C101C2411yescharacteristic
64C22×D101C22×C412yes—
65Q8×C101—22no—
66D4×C101—24no—
67D4×C101—22no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101—24no—
71C2×C4×C101C22×C412yes—
72C2×C4×C101C22×C411yescharacteristic
73C2×C4×C101C22×C411yescharacteristic
74C2×C4×C101C2×D412yes—
75C2×C4×C101C2×D412yes—
76C22⋊D101—24no—
77C22⋊D101—28no—
78C2×Dic101—28no—
79C2×Dic101—24no—
80C2×Dic101C22×C412yes—
81C2×Dic101C22×C414yes—
82C2×Dic101C2×D418yes—
83D404—21no—
84D404—21no—
85C4×D101—24no—
86C4×D101—28no—
87Dic202—21no—
88Dic202—21no—
89Dic202—28no—
90Dic202C22⋊C418yes—
91Dic202C22⋊C414yes—
92Dic202C22⋊C414yes—
93C22×C101—22no—
94C22×C101—41no—
95C22×C101—42no—
96C22×C101C23×C411yescharacteristic
97C22×C101C22×D412yes—
98C2×D101—22no—
99C2×D101—42no—
100C4×C101—22no—
101C4×C101—44no—
102C4×C101C2×C22⋊C411yescharacteristic
103C4×C101C2×C22⋊C411yescharacteristic
104C4×C101C2×C22⋊C412yes—
105Dic101—22no—
106Dic101—44no—
107Dic101C2×C22⋊C418yes—
108C2×C101—22no—
109C2×C101—21no—
110C2×C101—42no—
111C2×C101C22×C22⋊C411yescharacteristic
112D101—42no—
113#2884—1011nomaximal, Sylow
114C101#288411yescharacteristic, complemented, Sylow
115C2×2−1+4—1011no—
116#519—1012no—
117#410—1011no—
118#410—1012no—
119#410—1018no—
120#410D10111yescharacteristic, complemented
1212−1+4—2024no—
1222−1+4—2024no—
123C2×#41—1012no—
124C2×#41—1014no—
125C2×#41—1012no—
126C2×#41—1011no—
127C2×#41C2×D10111yescharacteristic
128C22×Q8—1012no—
129C22×Q8—1012no—
130C22×Q8—1011no—
131C4×Q8—2024no—
132C4×Q8—2024no—
133#117—1012no—
134#117—1018no—
135#117C2×D10112yes—
136#99—1014no—
137#99—1018no—
138#99C2×D10114yes—
139#41—22no—
140#41—22no—
141#41—2024no—
142#41—2024no—
143#41—20216no—
144#41—2028no—
145#41—2022no—
146#41—2022no—
147C2×Q8—1012no—
148C2×Q8—1012no—
149C2×Q8—1014no—
150C2×Q8—1011no—
151C2×Q8—1014no—
152C2×Q8—1014no—
153C2×Q8—2024no—
154C2×Q8—2024no—
155C2×Q8—2028no—
156C2×Q8C4×D10111yescharacteristic
157C2×D4—1014no—
158C2×D4—1012no—
159C2×D4—1011no—
160C2×D4C4×D10111yescharacteristic
161C2×D4C22×D10112yes—
162C22×C4—1012no—
163C22×C4—1014no—
164C22×C4—1014no—
165C22×C4—1012no—
166C22×C4C4×D10112yes—
167C22×C4C22×D10111yescharacteristic
168C4⋊C4—22no—
169C4⋊C4—2028no—
170C4⋊C4—2022no—
171C22⋊C4—1014no—
172C22⋊C4C22×D10114yes—
173C42—22no—
174C42—2028no—
175C42—2022no—
176C23—1012no—
177C23C2×C4×D10112yes—
178C23C23×D10111yescharacteristic
179Q8—22no—
180Q8—1014no—
181Q8—1014no—
182Q8—1018no—
183Q8—2028no—
184Q8—2021no—
185Q8—2021no—
186D4—24no—
187D4—22no—
188D4—2028no—
189D4—2024no—
190D4—2021no—
191D4—2021no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—24no—
195C2×C4—1014no—
196C2×C4—1018no—
197C2×C4—1012no—
198C2×C4—2028no—
199C2×C4—2028no—
200C2×C4—2024no—
201C2×C4—2024no—
202C2×C4C2×C4×D10112yes—
203C2×C4C2×C4×D10111yescharacteristic
204C2×C4C2×C4×D10111yescharacteristic
205C2×C4D4×D10112yes—
206C2×C4D4×D10112yes—
207C22—22no—
208C22—41no—
209C22—42no—
210C22—4042no—
211C22—2022no—
212C22C22×C4×D10111yescharacteristic
213C22C2×D4×D10112yes—
214C4—22no—
215C4—44no—
216C4—1018no—
217C4—4044no—
218C4—2022no—
219C4C22⋊C4×D10111yescharacteristic
220C4C22⋊C4×D10111yescharacteristic
221C4C22⋊C4×D10112yes—
222C2—22no—
223C2—21no—
224C2—42no—
225C2—4042no—
226C2C2×C22⋊C4×D10111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D101
commutatorC22×C101C23×C4
FrattiniC23C23×D101
Fitting#410×C101C2
socleC2×C101C22×C22⋊C4
prime radicalC2×#15692302C2
cosocleC23×C101C24
hypercenter#410D101
hypocenterC101#2884
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #20265654C2semidirect
3. #20265151C2semidirect
4. #20265150C2semidirect
5. #20265149C2semidirect
6. #410×C101C2semidirect
9. C2×#13108074C4semidirect
11. C2×#13108073C4semidirect
13. C22×Dic202C4semidirect
15. C2×Q8×D101C4semidirect
114. C101#2884semidirect
120. #410D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923022
#202656542
#202651512
#202651502
#202651492
#410×C1012
#2884101