Finite Group Databasebeta

← #27169621

Subgroups — #27169621

Order 12928.

304 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 77 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
1#20265688C211yesmaximal, characteristic
2C2×C22⋊#11717579C211yesmaximal, characteristic, complemented
3#20264913C212yesmaximal, complemented
4#20264910C212yesmaximal, complemented
5C4×C22⋊C4×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C411yescharacteristic, complemented
7C22×#11717579—21no—
8#15692238—21no—
9#15692234C411yescharacteristic, complemented
10#15692234C2211yescharacteristic
11#15692229C411yescharacteristic, complemented
12#15692229C2211yescharacteristic
13#15692224—21no—
14C4×C22⋊D101—21no—
15C4×C22⋊D101—21no—
16#15692214C411yescharacteristic, complemented
17C22⋊#11717579—21no—
18C22⋊#11717579C2212yes—
19C23×C4×C101C2211yescharacteristic
20C2×C22⋊C4×C101—21no—
21C2×C42×C101—21no—
22#95×C101C2212yes—
23#13108074—42no—
24C2×Dic202C2×C411yescharacteristic
25C2×D404C2×C411yescharacteristic
26C2×C22⋊D101—21no—
27C2×C22⋊D101—21no—
28C2×C22⋊D101—21no—
29C2×C4×D101—21no—
30C2×#11717579—22no—
31C2×#11717579—42no—
32C4⋊Dic101C2×C411yescharacteristic
33C4⋊Dic101C2×C411yescharacteristic
34#13108053—21no—
35#13108053—21no—
36C22⋊Dic101—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41C4×Dic101—21no—
42C24×C101D411yescharacteristic
43C22×C4×C101—22no—
44C22×C4×C101—42no—
45C22×C4×C101C2×C411yescharacteristic
46C22×C4×C101C2×C411yescharacteristic
47C22×C4×C101D411yescharacteristic
48C22×C4×C101D411yescharacteristic
49C22×C4×C101D411yescharacteristic
50C22×C4×C101C2311yescharacteristic
51C22⋊C4×C101—24no—
52C42×C101—42no—
53C23×C101—22no—
54C23×C101—22no—
55C23×C101—41no—
56C23×C101C22⋊C411yescharacteristic
57C23×C101C22⋊C411yescharacteristic
58C23×C101C2×D411yescharacteristic
59C22×D101—21no—
60C2×C4×C101—22no—
61C2×C4×C101—22no—
62C2×C4×C101—22no—
63C2×C4×C101—24no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101—41no—
67C2×C4×C101—42no—
68C2×C4×C101—42no—
69C2×C4×C101C22⋊C411yescharacteristic
70C2×C4×C101C22⋊C411yescharacteristic
71C2×C4×C101C22×C411yescharacteristic
72C2×C4×C101C2×D411yescharacteristic
73C22⋊D101—42no—
74C22⋊D101—42no—
75C22⋊D101—42no—
76C2×Dic101—21no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79D404—41no—
80C4×D101—42no—
81Dic202—41no—
82#11717579—42no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101—21no—
87C22×C101—21no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—44no—
91C22×C101#10412yes—
92C22×C101C2×C22⋊C411yescharacteristic
93C22×C101#13512yes—
94C2×D101—41no—
95C2×D101—41no—
96C2×D101—41no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—21no—
100C4×C101—44no—
101C4×C101#9912yes—
102Dic101—41no—
103Dic101—41no—
104Dic101—41no—
105C2×C101—22no—
106C2×C101—22no—
107C2×C101—41no—
108C2×C101C2×#9911yescharacteristic
109C2×C101#41611yescharacteristic
110C2×C101C2×#10411yescharacteristic
111D101—81no—
112#1501—1011nomaximal, Sylow
113C101#150111yescharacteristic, complemented, Sylow
114#525—1011no—
115C2×C22⋊C8—1011no—
116C4×C22⋊C4D10111yescharacteristic, complemented
117#328—1012no—
118#327—1012no—
119C2×#41—1011no—
120C23×C4C2×D10111yescharacteristic
121C22×C8—2021no—
122#123—2021no—
123C22⋊Q8—1011no—
124C22⋊Q8—1011no—
125#121—1011no—
126#121—1011no—
127#120—2021no—
128C4×D4—2021no—
129C4×D4—2021no—
130#117—1011no—
131C2×C22⋊C4—21no—
132C2×C42—21no—
133C22⋊C8—1012no—
134C22⋊C8—2021no—
135#95C2×D10112yes—
136C24C22⋊D10111yescharacteristic
137#41—4042no—
138C2×Q8—1011no—
139C2×D4—1011no—
140C2×D4—2021no—
141C2×D4—2021no—
142C2×D4—2021no—
143C22×C4—22no—
144C22×C4—42no—
145C22×C4—2021no—
146C22×C4C4×D10111yescharacteristic
147C22×C4C4×D10111yescharacteristic
148C22×C4D40411yescharacteristic
149C22×C4D40411yescharacteristic
150C22×C4C22⋊D10111yescharacteristic
151C22×C4C22×D10111yescharacteristic
152C2×C8—4042no—
153C2×C8—2022no—
154C4⋊C4—1011no—
155C4⋊C4—1011no—
156C4⋊C4—2021no—
157C4⋊C4—2021no—
158C22⋊C4—24no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C42—42no—
165C42—2021no—
166C23—22no—
167C23—22no—
168C23—41no—
169C23—2021no—
170C23#1310805111yescharacteristic
171C23#1310805111yescharacteristic
172C23C2×C22⋊D10111yescharacteristic
173Q8—4041no—
174D4—4041no—
175D4—4042no—
176D4—4042no—
177D4—4042no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—24no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—4042no—
188C2×C4—2021no—
189C2×C4—2021no—
190C2×C4—2021no—
191C2×C4#1310805111yescharacteristic
192C2×C4#1310805111yescharacteristic
193C2×C4C2×C4×D10111yescharacteristic
194C2×C4C2×D40411yescharacteristic
195C8—4042no—
196C22—21no—
197C22—21no—
198C22—21no—
199C22—21no—
200C22—21no—
201C22—21no—
202C22—21no—
203C22—44no—
204C22—4041no—
205C22—4041no—
206C22—4041no—
207C22#1569218312yes—
208C22C2×#1310805111yescharacteristic
209C22#1569227512yes—
210C4—21no—
211C4—21no—
212C4—21no—
213C4—44no—
214C4—4041no—
215C4—4041no—
216C4—4041no—
217C4#1569216912yes—
218C2—22no—
219C2—22no—
220C2—41no—
221C2—8081no—
222C2C2×#1569216911yescharacteristic
223C2#2026517111yescharacteristic
224C2C2×#1569218311yescharacteristic
subgroupG/H
centerC2×C4#13108051
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
FittingC4×C22⋊C4×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265688C2
cosocleC22×C4×C101C23
hypercenterC4×C22⋊C4D101
hypocenterC101#1501
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C22⋊#11717579C2semidirect
3. #20264913C2semidirect
4. #20264910C2semidirect
5. C4×C22⋊C4×C101C2semidirect
6. C2×#13108074C4semidirect
9. #15692234C4semidirect
11. #15692229C4semidirect
16. #15692214C4semidirect
113. C101#1501semidirect
116. C4×C22⋊C4D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
C2×C22⋊#117175792
#202649132
#202649102
C4×C22⋊C4×C1012
#1501101