Finite Group Databasebeta

← #27169728

Subgroups — #27169728

Order 12928.

264 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 61 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#124×D101C211yesmaximal, characteristic
2#20265232C211yesmaximal, characteristic, complemented
3#20265129C211yesmaximal, characteristic, complemented
4#20264917C211yesmaximal, characteristic, complemented
5#20264916C211yesmaximal, characteristic, complemented
6#20264916C211yesmaximal, characteristic, complemented
7#329×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101C411yescharacteristic
9C2×D4×D101C411yescharacteristic
10C2×#13108057—21no—
11#15692244C411yescharacteristic
12#15692243C2211yescharacteristic
13#15692242C411yescharacteristic
14#15692241—21no—
15#15692241—21no—
16C22⋊C4×D101—21no—
17C22⋊C4×D101—21no—
18C42×D101C2211yescharacteristic
19#15692184—21no—
20#15692167—21no—
21#15692166C2211yescharacteristic, complemented
22#15692166C2211yescharacteristic, complemented
23C8×Dic101—21no—
24#124×C101C2211yescharacteristic
25#97×C101C2211yescharacteristic, complemented
26#97×C101C2211yescharacteristic, complemented
27C23×D101—21no—
28C2×Dic202—21no—
29C2×Dic202C2×C411yescharacteristic
30Q8×D101—42no—
31C2×D404C2×C411yescharacteristic
32C2×C22⋊D101—21no—
33D4×D101—42no—
34C2×C4×D101—21no—
35C2×C4×D101C2×C411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38#13108057—24no—
39C2×#11717579—21no—
40C2×#11717579—21no—
41C22⋊Dic101—21no—
42C22⋊Dic101—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46#13108051—21no—
47C4×Dic101C2×C411yescharacteristic
48C4×Dic101D411yescharacteristic
49C4×Dic101D411yescharacteristic
50C2×Q8×C101C2×C411yescharacteristic
51C2×D4×C101C2×C411yescharacteristic
52C2×C8×C101—21no—
53C2×C8×C101—21no—
54C22⋊C4×C101—21no—
55C22⋊C4×C101—21no—
56C42×C101C2311yescharacteristic
57C23×C101—21no—
58C22×D101—21no—
59C22×D101—42no—
60C22×D101—42no—
61C22×D101—42no—
62C22×D101C22⋊C411yescharacteristic
63Q8×C101—41no—
64D4×C101—41no—
65C2×C4×C101—21no—
66C2×C4×C101C22×C411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C22⋊D101—42no—
70C2×Dic101—21no—
71C2×Dic101C22⋊C411yescharacteristic
72C2×Dic101C22⋊C411yescharacteristic
73C2×Dic101C22⋊C411yescharacteristic
74D404—41no—
75C4×D101—22no—
76C4×D101—22no—
77C4×D101—41no—
78C4×D101—42no—
79Dic202—42no—
80Dic202—41no—
81C8×C101—22no—
82C8×C101—41no—
83#11717579—22no—
84#11717579—41no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101C2×C22⋊C411yescharacteristic
89C2×D101—21no—
90C2×D101—21no—
91C2×D101—41no—
92C2×D101—41no—
93C2×D101—41no—
94C2×D101—84no—
95C2×D101#10412yes—
96C4×C101—21no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—41no—
100Dic101—21no—
101Dic101—21no—
102Dic101—41no—
103Dic101#10012yes—
104C2×C101—81no—
105C2×C101C2×#10011yescharacteristic
106C2×C101C2×#10411yescharacteristic
107C2×C101#42111yescharacteristic
108D101—22no—
109D101—81no—
110#1527—1011nomaximal, Sylow
111C101#152711yescharacteristic, complemented, Sylow
112C2×#124—1011no—
113#432—1011no—
114C4×M16—1011no—
115#329—1011no—
116#329—1011no—
117#329—1011no—
118#329D10111yescharacteristic, complemented
119C22×Q8—1011no—
120C22×D4—1011no—
121C2×M16—2021no—
122#124—1011no—
123#124—1011no—
124#124—1011no—
125#124—2021no—
126#124—2021no—
127#124C2×D10111yescharacteristic
128C2×C22⋊C4—2021no—
129C2×C22⋊C4—2021no—
130C2×C42—1011no—
131C4⋊C8—2021no—
132C22⋊C8—2021no—
133#97—1011no—
134#97—1011no—
135#97C2×D10111yescharacteristic, complemented
136#97C2×D10111yescharacteristic, complemented
137C4×C8—2021no—
138C24—2021no—
139C2×Q8—1011no—
140C2×Q8—4042no—
141C2×Q8—2021no—
142C2×Q8C4×D10111yescharacteristic
143C2×D4—1011no—
144C2×D4—4042no—
145C2×D4—2021no—
146C2×D4C4×D10111yescharacteristic
147C22×C4—1011no—
148C22×C4—1011no—
149C22×C4—1011no—
150C22×C4—2021no—
151M16—2024no—
152C2×C8—21no—
153C2×C8—21no—
154C2×C8—2021no—
155C2×C8—2021no—
156C22⋊C4—21no—
157C22⋊C4—21no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C22⋊C4—2021no—
164C42—1011no—
165C42—1011no—
166C42—1011no—
167C42C22×D10111yescharacteristic
168C23—21no—
169C23—1011no—
170C23—4042no—
171C23—4042no—
172C23—4042no—
173C23—2021no—
174Q8—41no—
175Q8—4041no—
176Q8—4042no—
177D4—41no—
178D4—4041no—
179D4—4042no—
180C2×C4—21no—
181C2×C4—1011no—
182C2×C4—1011no—
183C2×C4—1011no—
184C2×C4—4041no—
185C2×C4—4042no—
186C2×C4—2022no—
187C2×C4—2022no—
188C2×C4—2021no—
189C2×C4C2×C4×D10111yescharacteristic
190C2×C4D4×D10111yescharacteristic
191C2×C4D4×D10111yescharacteristic
192C8—22no—
193C8—41no—
194C8—4041no—
195C8—2022no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22—8084no—
200C22—1012no—
201C22—4041no—
202C22—4041no—
203C22—4041no—
204C22—2021no—
205C22—2021no—
206C22C22⋊C4×D10111yescharacteristic
207C4—21no—
208C4—21no—
209C4—21no—
210C4—41no—
211C4—1012no—
212C4—4041no—
213C4—2021no—
214C4—2021no—
215C2—81no—
216C2—8081no—
217C2—2022no—
218C2#100×D10111yescharacteristic
219C2#104×D10111yescharacteristic
220C2#2026519211yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
Fitting#329×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC42×C101C23
hypercenter#329D101
hypocenterC101#1527
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #329

Complements

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

NH ≅ G/Ntype
2. #20265232C2semidirect
3. #20265129C2semidirect
4. #20264917C2semidirect
5. #20264916C2semidirect
6. #20264916C2semidirect
7. #329×C101C2semidirect
21. #15692166C22semidirect
22. #15692166C22semidirect
25. #97×C101C22semidirect
26. #97×C101C22semidirect
111. C101#1527semidirect
118. #329D101semidirect
135. #97C2×D101semidirect
136. #97C2×D101semidirect

Maximal subgroups

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

subgroupindex
#124×D1012
#202652322
#202651292
#202649172
#202649162
#329×C1012
#1527101