Finite Group Databasebeta

← #27170312

Subgroups — #27170312

Order 12928.

308 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 57 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#20265812C211yesmaximal, characteristic
2#20265523C211yesmaximal, characteristic, complemented
3#20265458C211yesmaximal, characteristic, complemented
4#20265383C211yesmaximal, characteristic, complemented
5#20264904C211yesmaximal, characteristic, complemented
6#20264904C211yesmaximal, characteristic, complemented
7#331×C101C211yesmaximal, characteristic, complemented
8C2×D4×D101—21no—
9C2×D4×D101—21no—
10C2×#13108063—21no—
11#15692255C2211yescharacteristic, complemented
12#15692226—21no—
13#15692226—21no—
14#15692223—22no—
15C4×D404C2211yescharacteristic
16C4×D404C2211yescharacteristic
17#15692189—21no—
18#15692180—21no—
19#15692177—21no—
20#15692176—22no—
21C4×#11717579C2211yescharacteristic, complemented
22#127×C101C2211yescharacteristic
23C4⋊C8×C101C2211yescharacteristic, complemented
24C4⋊C8×C101C2211yescharacteristic, complemented
25C23×D101—22no—
26C2×Dic202—21no—
27C2×D404D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—22no—
30C2×C22⋊D101—22no—
31D4×D101—42no—
32D4×D101—42no—
33D4×D101—42no—
34D4×D101—42no—
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37#13108063—42no—
38C2×#11717579—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101—21no—
41C4⋊Dic101D411yescharacteristic
42C4⋊Dic101D411yescharacteristic
43C22⋊Dic101—22no—
44#13108051—21no—
45#13108051—21no—
46C2×D4×C101—21no—
47C2×D4×C101—21no—
48C2×D4×C101D412yes—
49C2×C8×C101—21no—
50C2×C8×C101—21no—
51C42×C101C2311yescharacteristic
52C23×C101—22no—
53C22×D101—21no—
54C22×D101—21no—
55C22×D101—44no—
56C22×D101—44no—
57C22×D101—82no—
58D4×C101—42no—
59D4×C101—41no—
60D4×C101—41no—
61D4×C101—41no—
62D4×C101—41no—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×C4×C101C2×D411yescharacteristic
66C22⋊D101—44no—
67C22⋊D101—44no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—41no—
71D404—21no—
72D404—21no—
73D404D812yes—
74D404SD1612yescomplemented
75C4×D101—41no—
76C4×D101—41no—
77C4×D101—41no—
78C4×D101—41no—
79Dic202—41no—
80Dic202—41no—
81C8×C101—41no—
82C8×C101—41no—
83#11717579—41no—
84C22×C101—42no—
85C22×C101—42no—
86C22×C101—42no—
87C22×C101#12011yescharacteristic
88C2×D101—22no—
89C2×D101—22no—
90C2×D101—41no—
91C2×D101—41no—
92C2×D101—41no—
93C2×D101—41no—
94C2×D101—88no—
95C4×C101—21no—
96C4×C101C2×D811yescharacteristic
97C4×C101C2×SD1611yescharacteristic
98C4×C101AGL(1,ℤ/8)11yescharacteristic
99C4×C101AGL(1,ℤ/8)11yescharacteristic
100Dic101—41no—
101Dic101—41no—
102Dic101—81no—
103C2×C101—82no—
104C2×C101#44711yescharacteristic
105C2×C101#45011yescharacteristic
106C2×C101#45511yescharacteristic
107D101—42no—
108D101—42no—
109#1639—1011nomaximal, Sylow
110C101#163911yescharacteristic, complemented, Sylow
111D42—1011no—
112#486—1011no—
113#474—1011no—
114#461—1011no—
115#331D10111yescharacteristic, complemented
116D4⋊C8—1011no—
117D4⋊C8—1011no—
118C22×D4—2021no—
119C22×D4—2021no—
120C2×SD16—2021no—
121C4⋊Q8—1011no—
122#127C2×D10111yescharacteristic
123#121—2021no—
124#121—2021no—
125#120—2022no—
126C4×D4—1011no—
127C4×D4—1011no—
128#107—2021no—
129C4⋊C8C2×D10111yescharacteristic, complemented
130C4⋊C8C2×D10111yescharacteristic, complemented
131#103—2021no—
132#102—2021no—
133#102—2022no—
134C4×C8—1011no—
135C24—2022no—
136C2×Q8—2021no—
137C2×D4—21no—
138C2×D4—21no—
139C2×D4—1011no—
140C2×D4—1011no—
141C2×D4—4042no—
142C2×D4—4042no—
143C2×D4—4042no—
144C2×D4—4042no—
145C2×D4—2022no—
146C2×D4—2022no—
147C2×D4D40412yes—
148C22×C4—2021no—
149C22×C4—2021no—
150SD16—4042no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—2021no—
154C4⋊C4—1011no—
155C4⋊C4—1011no—
156C4⋊C4—2021no—
157C4⋊C4—2021no—
158C22⋊C4—2022no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C42C22×D10111yescharacteristic
162C23—22no—
163C23—8082no—
164C23—4044no—
165C23—4044no—
166C23—2021no—
167C23—2021no—
168Q8—4041no—
169Q8—4041no—
170D4—42no—
171D4—41no—
172D4—41no—
173D4—41no—
174D4—41no—
175D4—1012no—
176D4—1012no—
177D4—4044no—
178D4—4044no—
179D4—2021no—
180D4—2021no—
181C2×C4—4041no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—4041no—
185C2×C4—4041no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4D4×D10111yescharacteristic
189C2×C4D4×D10111yescharacteristic
190C2×C4C2×D40411yescharacteristic
191C8—41no—
192C8—41no—
193C8—4041no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22—8088no—
198C22—4041no—
199C22—4041no—
200C22—4041no—
201C22—4041no—
202C22—2022no—
203C22—2022no—
204C22#1569222511yescharacteristic
205C4—21no—
206C4—8081no—
207C4—4041no—
208C4—4041no—
209C4D8×D10111yescharacteristic
210C4SD16×D10111yescharacteristic
211C4#1569227911yescharacteristic
212C4#1569228211yescharacteristic
213C2—82no—
214C2—4042no—
215C2—4042no—
216C2#2026531211yescharacteristic
217C2#2026532911yescharacteristic
218C2#2026535711yescharacteristic
subgroupG/H
centerC22#15692225
commutator, cosocleC42×C101C23
FrattiniC42C22×D101
Fitting#331×C101C2
socleC22×C101#120
prime radical#20265812C2
hypercenter#331D101
hypocenterC101#1639
LCSG ▷ C42×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #331

Complements

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

NH ≅ G/Ntype
2. #20265523C2semidirect
3. #20265458C2semidirect
4. #20265383C2semidirect
5. #20264904C2semidirect
6. #20264904C2semidirect
7. #331×C101C2semidirect
11. #15692255C22semidirect
21. C4×#11717579C22semidirect
23. C4⋊C8×C101C22semidirect
24. C4⋊C8×C101C22semidirect
74. D404SD16semidirect
110. C101#1639semidirect
115. #331D101semidirect
129. C4⋊C8C2×D101semidirect
130. C4⋊C8C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202658122
#202655232
#202654582
#202653832
#202649042
#331×C1012
#1639101