Finite Group Databasebeta

← #27179612

Subgroups — #27179612

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×#15692303C211yesmaximal, characteristic
2#20265943C212yesmaximal, complemented
3#20265160C214yesmaximal, complemented
4#20265159C212yesmaximal, complemented
5C2×#15692174C211yesmaximal, characteristic, complemented
6#101×D101C214yesmaximal, complemented
7C2×#101×C101C211yesmaximal, characteristic, complemented
8#15692303—24no—
9#15692303—24no—
10C2×#13108075C412yes—
11C2×#13108075C2212yes—
12C2×#13108074C414yes—
13C2×#13108074C2211yescharacteristic
14C2×#13108074C2211yescharacteristic
15C2×Q8×D101C412yes—
16C2×Q8×D101C2212yes—
17#15692263—24no—
18#15692261—24no—
19C2×#13108058C2212yes—
20M16×D101C2218yes—
21#15692174C2214yes—
22#15692173C2214yes—
23#15692171C2214yes—
24C22×Q8×C101C2211yescharacteristic
25C2×M16×C101C2212yes—
26#101×C101C2214yes—
27#13108075—28no—
28#13108075—28no—
29#13108074—22no—
30#13108074—22no—
31#13108074—28no—
32#13108074—28no—
33#13108074D414yes—
34#13108074D414yes—
35C2×Dic202C2×C414yes—
36C2×Dic202C2×C411yescharacteristic
37C2×Dic202C2311yescharacteristic
38Q8×D101—28no—
39Q8×D101—28no—
40C2×D404C2×C414yes—
41C2×D404C2×C411yescharacteristic
42C2×D404C2311yescharacteristic
43C2×C22⋊D101C2×C412yes—
44C2×C22⋊D101C2×C412yes—
45C2×C4×D101C2×C414yes—
46C2×C4×D101C2×C412yes—
47C2×C4×D101C2×C414yes—
48C2×C4×D101C2312yes—
49#13108058—22no—
50#13108058C2314yes—
51#13108057—28no—
52C2×#11717579—22no—
53C8×D101—28no—
54C2×Q8×C101—22no—
55C2×Q8×C101—22no—
56C2×Q8×C101C2×C412yes—
57C2×Q8×C101C2312yes—
58C22×C4×C101C2×C412yes—
59C22×C4×C101C2311yescharacteristic
60M16×C101—22no—
61M16×C101C2314yes—
62C2×C8×C101—22no—
63C23×C101C22×C411yescharacteristic
64C22×D101C22×C412yes—
65C22×D101C22×C412yes—
66Q8×C101—24no—
67Q8×C101—24no—
68C2×C4×C101—22no—
69C2×C4×C101—22no—
70C2×C4×C101C22×C414yes—
71C2×C4×C101C22×C411yescharacteristic
72C2×C4×C101C2×D412yes—
73C2×C4×C101C2×D412yes—
74C2×C4×C101C2411yescharacteristic
75C22⋊D101—22no—
76C22⋊D101—22no—
77C22⋊D101C22⋊C414yes—
78C22⋊D101C22⋊C414yes—
79C2×Dic101C22×C412yes—
80C2×Dic101C22×C412yes—
81D404—28no—
82D404—22no—
83D404C22⋊C414yes—
84C4×D101—28no—
85C4×D101—24no—
86C4×D101—28no—
87C4×D101C2×D418yes—
88Dic202—22no—
89Dic202—28no—
90Dic202C22⋊C414yes—
91C8×C101—24no—
92#11717579—24no—
93C22×C101—41no—
94C22×C101C2×C22⋊C411yescharacteristic
95C22×C101C2×C22⋊C411yescharacteristic
96C22×C101C23×C411yescharacteristic
97C2×D101—22no—
98C2×D101—42no—
99C2×D101—42no—
100C2×D101C2×C22⋊C414yes—
101C4×C101—24no—
102C4×C101C2×C22⋊C412yes—
103C4×C101C22×D412yes—
104Dic101—22no—
105Dic101C2×C22⋊C414yes—
106C2×C101—21no—
107C2×C101—21no—
108C2×C101—21no—
109C2×C101C22×C22⋊C411yescharacteristic
110D101—24no—
111D101—42no—
112#2889—1011nomaximal, Sylow
113C101#288911yescharacteristic, complemented, Sylow
114C2×2−1+4—1011no—
115#568—1012no—
116#413—1012no—
117#413—1014no—
118C2×#101—1011no—
119C2×#101—1014no—
120C2×#101D10111yescharacteristic, complemented
1212−1+4—2024no—
1222−1+4—2024no—
123C2×#41—1012no—
124C2×#41—1012no—
125C2×#41—1014no—
126C2×#41—1011no—
127C2×#41—1011no—
128C22×Q8—1012no—
129C22×Q8—1012no—
130C22×Q8C2×D10111yescharacteristic
131#131—2024no—
132#131—2024no—
133C2×M16—1012no—
134C2×M16—1018no—
135C2×M16C2×D10112yes—
136#101—1014no—
137#101—1014no—
138#101C2×D10114yes—
139#100—1014no—
140#41—1014no—
141#41—1014no—
142#41—2028no—
143#41—2028no—
144#41—2028no—
145#41—2028no—
146#41—2022no—
147#41—2022no—
148C2×Q8—22no—
149C2×Q8—22no—
150C2×Q8—1011no—
151C2×Q8—1014no—
152C2×Q8—1011no—
153C2×Q8—2028no—
154C2×Q8—2028no—
155C2×Q8C4×D10112yes—
156C2×Q8C22×D10112yes—
157C2×D4—1012no—
158C2×D4—1012no—
159C2×D4—1011no—
160C2×D4—1014no—
161C2×D4—1011no—
162C22×C4—1012no—
163C22×C4—1014no—
164C22×C4—1014no—
165C22×C4—1012no—
166C22×C4C4×D10112yes—
167C22×C4C22×D10111yescharacteristic
168M16—22no—
169M16—1014no—
170M16—2028no—
171M16—2022no—
172M16C22×D10114yes—
173C2×C8—22no—
174C2×C8—2028no—
175C2×C8—2022no—
176C23—1012no—
177C23—1012no—
178C23C2×C4×D10111yescharacteristic
179Q8—24no—
180Q8—24no—
181Q8—1014no—
182Q8—2028no—
183Q8—2022no—
184D4—1014no—
185D4—1014no—
186D4—1014no—
187D4—2028no—
188D4—2022no—
189D4—2022no—
190D4—2022no—
191C2×C4—22no—
192C2×C4—22no—
193C2×C4—1012no—
194C2×C4—1018no—
195C2×C4—1012no—
196C2×C4—2028no—
197C2×C4—2028no—
198C2×C4—2024no—
199C2×C4C2×C4×D10114yes—
200C2×C4C2×C4×D10111yescharacteristic
201C2×C4D4×D10112yes—
202C2×C4D4×D10112yes—
203C2×C4C23×D10111yescharacteristic
204C8—24no—
205C8—2024no—
206C22—41no—
207C22—1014no—
208C22—4042no—
209C22—4042no—
210C22—2022no—
211C22C22⋊C4×D10111yescharacteristic
212C22C22⋊C4×D10111yescharacteristic
213C22C22×C4×D10111yescharacteristic
214C4—24no—
215C4—1014no—
216C4—2022no—
217C4C22⋊C4×D10112yes—
218C4C2×D4×D10112yes—
219C2—21no—
220C2—21no—
221C2—21no—
222C2—4042no—
223C2—2024no—
224C2C2×C22⋊C4×D10111yescharacteristic
subgroupG/H
centerC2C2×C22⋊C4×D101
commutatorC22×C101C23×C4
FrattiniC2×C4C23×D101
FittingC2×#101×C101C2
socleC2×C101C22×C22⋊C4
prime radicalC2×#15692303C2
cosocleC2×C4×C101C24
hypercenterC2×#101D101
hypocenterC101#2889
LCSG ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#101

Complements

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

NH ≅ G/Ntype
2. #20265943C2semidirect
3. #20265160C2semidirect
4. #20265159C2semidirect
5. C2×#15692174C2semidirect
6. #101×D101C2semidirect
7. C2×#101×C101C2semidirect
113. C101#2889semidirect
120. C2×#101D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156923032
#202659432
#202651602
#202651592
C2×#156921742
#101×D1012
C2×#101×C1012
#2889101