Finite Group Databasebeta

← #27169654

Subgroups — #27169654

Order 12928.

242 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 63 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#20265708C211yesmaximal, characteristic
2#20265231C211yesmaximal, characteristic, complemented
3#20265028C211yesmaximal, characteristic, complemented
4#20264915C211yesmaximal, characteristic, complemented
5#20264913C211yesmaximal, characteristic, complemented
6#20264911C211yesmaximal, characteristic
7#327×C101C211yesmaximal, characteristic
8#15692248—21no—
9#15692246C411yescharacteristic, complemented
10#15692242C411yescharacteristic, complemented
11#15692236—21no—
12#15692234C2211yescharacteristic
13#15692220C411yescharacteristic, complemented
14D4×Dic101—21no—
15D4×Dic101C411yescharacteristic, complemented
16#15692211—21no—
17C2×#13108053—21no—
18C2×C4×Dic101—21no—
19C2×C4×Dic101C2211yescharacteristic
20#15692184—21no—
21C22⋊#11717579C2211yescharacteristic
22#15692165—21no—
23#15692162C2211yescharacteristic
24#121×C101C2211yescharacteristic
25C22⋊C8×C101C2211yescharacteristic
26#95×C101C2211yescharacteristic
27C2×Dic202C2×C411yescharacteristic
28C22×Dic101—21no—
29C22×Dic101—21no—
30C22×Dic101D411yescharacteristic
31C22×Dic101D411yescharacteristic
32C2×#11717579—21no—
33C4⋊Dic101—21no—
34C4⋊Dic101C2×C411yescharacteristic
35#13108053—24no—
36#13108053—21no—
37#13108053—21no—
38#13108053—21no—
39C22⋊Dic101—21no—
40C22⋊Dic101—21no—
41C22⋊Dic101—21no—
42C22⋊Dic101—21no—
43C4×Dic101—21no—
44C4×Dic101—42no—
45C4×Dic101C2×C411yescharacteristic
46C4×Dic101C2×C411yescharacteristic
47C4×Dic101D411yescharacteristic
48C4×Dic101D411yescharacteristic
49C2×D4×C101—21no—
50C2×D4×C101C2×C411yescharacteristic
51C22×C4×C101—21no—
52C22×C4×C101C2311yescharacteristic
53C2×C8×C101—21no—
54C4⋊C4×C101C2×C411yescharacteristic
55C22⋊C4×C101—21no—
56C23×C101—21no—
57C23×C101C2×D411yescharacteristic
58D4×C101—41no—
59D4×C101—41no—
60D4×C101—41no—
61C2×C4×C101—22no—
62C2×C4×C101—21no—
63C2×C4×C101—41no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C22×C411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101—22no—
71C2×Dic101—22no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—41no—
75C2×Dic101—41no—
76C2×Dic101—42no—
77C2×Dic101C22⋊C411yescharacteristic
78C2×Dic101C22⋊C411yescharacteristic
79C2×Dic101C22⋊C411yescharacteristic
80C2×Dic101C22⋊C411yescharacteristic
81Dic202—41no—
82C8×C101—41no—
83#11717579—41no—
84C22×C101—21no—
85C22×C101—21no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101—41no—
89C22×C101C2×C22⋊C411yescharacteristic
90C22×C101AGL(1,ℤ/8)11yescharacteristic
91C22×C101#13711yescharacteristic
92C4×C101—21no—
93C4×C101—21no—
94C4×C101—42no—
95C4×C101—41no—
96Dic101—21no—
97Dic101—21no—
98Dic101—21no—
99Dic101—42no—
100Dic101—41no—
101Dic101—41no—
102Dic101#10412yes—
103C2×C101—21no—
104C2×C101—21no—
105C2×C101—81no—
106C2×C101#41011yescharacteristic
107C2×C101#41711yescharacteristic
108C2×C101C2×#10411yescharacteristic
109#1509—1011nomaximal, Sylow
110C101#150911yescharacteristic, complemented, Sylow
111#529—1011no—
112#432—1011no—
113C4×C4⋊C4—1011no—
114#328—1011no—
115#328—1011no—
116#327—1011no—
117#327D10111yescharacteristic
118#126—2021no—
119#125—1011no—
120#124—1011no—
121#123—2021no—
122C22⋊Q8—1011no—
123#121C2×D10111yescharacteristic
124C4×Q8—1011no—
125C4×D4—1011no—
126C4×D4—2021no—
127#117—2021no—
128C2×C4⋊C4—2021no—
129C2×C42—1011no—
130C2×C42—2021no—
131C4⋊C8—2021no—
132C22⋊C8—1011no—
133C22⋊C8C2×D10111yescharacteristic
134#97—2021no—
135#95—1011no—
136#95C2×D10111yescharacteristic
137C2×Q8—1011no—
138C2×D4—21no—
139C2×D4C4×D10111yescharacteristic
140C22×C4—21no—
141C22×C4—1011no—
142C22×C4—1011no—
143C22×C4—2021no—
144C22×C4—2021no—
145C22×C4C22×D10111yescharacteristic
146C2×C8—21no—
147C2×C8—2021no—
148C4⋊C4—1011no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2024no—
153C4⋊C4—2021no—
154C4⋊C4C4×D10111yescharacteristic
155C22⋊C4—21no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C22⋊C4—2021no—
159C22⋊C4—2021no—
160C42—1011no—
161C42—1011no—
162C42—1011no—
163C42—1011no—
164C42—4042no—
165C42—2021no—
166C23—21no—
167C23D4×D10111yescharacteristic
168Q8—4041no—
169D4—41no—
170D4—41no—
171D4—41no—
172C2×C4—22no—
173C2×C4—21no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—1011no—
178C2×C4—1011no—
179C2×C4—1011no—
180C2×C4—1011no—
181C2×C4—4042no—
182C2×C4—4041no—
183C2×C4—4041no—
184C2×C4—2022no—
185C2×C4—2022no—
186C2×C4—2021no—
187C2×C4—2021no—
188C2×C4—2021no—
189C2×C4—2021no—
190C2×C4C2×C4×D10111yescharacteristic
191C2×C4D4×D10111yescharacteristic
192C8—41no—
193C8—4041no—
194C22—21no—
195C22—21no—
196C22—41no—
197C22—41no—
198C22—41no—
199C22C22⋊C4×D10111yescharacteristic
200C22#1569227911yescharacteristic
201C22#1569228411yescharacteristic
202C4—21no—
203C4—21no—
204C4—42no—
205C4—41no—
206C4—1012no—
207C4—4042no—
208C4—4041no—
209C4—4041no—
210C4—2021no—
211C4—2021no—
212C4—2021no—
213C2—21no—
214C2—21no—
215C2—81no—
216C2#2026514911yescharacteristic
217C2#2026517511yescharacteristic
218C2#104×D10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#327×C101C2
socleC22×C101C2×C22⋊C4
prime radical#121×C101C22
cosocleC22×C4×C101C23
hypercenter#327D101
hypocenterC101#1509
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #327

Complements

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

NH ≅ G/Ntype
2. #20265231C2semidirect
3. #20265028C2semidirect
4. #20264915C2semidirect
5. #20264913C2semidirect
9. #15692246C4semidirect
10. #15692242C4semidirect
13. #15692220C4semidirect
15. D4×Dic101C4semidirect
110. C101#1509semidirect

Maximal subgroups

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

subgroupindex
#202657082
#202652312
#202650282
#202649152
#202649132
#202649112
#327×C1012
#1509101