Finite Group Databasebeta

← #27169559

Subgroups — #27169559

Order 12928.

248 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C4×Q8×D101C211yesmaximal, characteristic
2#20265228C211yesmaximal, characteristic
3#20265199C211yesmaximal, characteristic
4#20264909C211yesmaximal, characteristic, complemented
5Q8⋊#11717579C211yesmaximal, characteristic, complemented
6#20264907C211yesmaximal, characteristic, complemented
7Q8⋊C8×C101C211yesmaximal, characteristic, complemented
8C2×Q8×D101C411yescharacteristic
9C2×#13108057—21no—
10Q8×Dic101C411yescharacteristic
11C4×Dic202—21no—
12C4×Dic202C2211yescharacteristic
13#15692220—21no—
14#15692220C411yescharacteristic
15C4⋊C4×D101—21no—
16C4⋊C4×D101C411yescharacteristic
17C42×D101—21no—
18C42×D101C2211yescharacteristic
19C4⋊#11717579C2211yescharacteristic
20#15692184—21no—
21#15692184—21no—
22#15692167—21no—
23C4×#11717579C2211yescharacteristic
24C4×Q8×C101C2211yescharacteristic
25C4⋊C8×C101C2211yescharacteristic
26C4×C8×C101C2211yescharacteristic
27C2×Dic202—21no—
28C2×Dic202C2×C411yescharacteristic
29Q8×D101—22no—
30Q8×D101—22no—
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101C2×C411yescharacteristic
34C2×C4×D101D411yescharacteristic
35C2×C4×D101D411yescharacteristic
36#13108057—42no—
37C2×#11717579—21no—
38C2×#11717579—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101C2×C411yescharacteristic
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44C4×Dic101—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C4×Dic101C2×C411yescharacteristic
48C4×Dic101D411yescharacteristic
49C4×Dic101D411yescharacteristic
50C2×Q8×C101C2×C411yescharacteristic
51C2×C8×C101—21no—
52C2×C8×C101—21no—
53C4⋊C4×C101—21no—
54C4⋊C4×C101C2×C411yescharacteristic
55C42×C101—21no—
56C42×C101C2311yescharacteristic
57C22×D101C22⋊C411yescharacteristic
58Q8×C101—21no—
59Q8×C101M1612yescomplemented
60C2×C4×C101—21no—
61C2×C4×C101—21no—
62C2×C4×C101C22×C411yescharacteristic
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101C2×D411yescharacteristic
65C2×Dic101—21no—
66C2×Dic101—21no—
67C2×Dic101C22⋊C411yescharacteristic
68C2×Dic101C22⋊C411yescharacteristic
69C2×Dic101C22⋊C411yescharacteristic
70C4×D101—24no—
71C4×D101—21no—
72C4×D101—21no—
73C4×D101—41no—
74C4×D101—42no—
75C4×D101SD1612yes—
76C4×D101Q1612yes—
77Dic202—24no—
78Dic202—21no—
79Dic202M1612yescomplemented
80C8×C101—41no—
81C8×C101—41no—
82#11717579—41no—
83#11717579—41no—
84C22×C101C2×C22⋊C411yescharacteristic
85C2×D101—21no—
86C2×D101—21no—
87C2×D101#10312yes—
88C4×C101—22no—
89C4×C101—21no—
90C4×C101—41no—
91C4×C101C2×M1611yescharacteristic
92C4×C101C2×M1611yescharacteristic
93C4×C101C2×SD1611yescharacteristic
94C4×C101C2×Q1611yescharacteristic
95Dic101—21no—
96Dic101—22no—
97Dic101—21no—
98Dic101—41no—
99Dic101#10312yes—
100C2×C101#40711yescharacteristic
101C2×C101C2×#10311yescharacteristic
102C2×C101#42111yescharacteristic
103D101—22no—
104#1487—1011nomaximal, Sylow
105C101#148711yescharacteristic, complemented, Sylow
106C2×C4×Q8—1011no—
107#431—1011no—
108#423—1011no—
109Q8⋊C8—1011no—
110Q8⋊C8—1011no—
111Q8⋊C8—1011no—
112Q8⋊C8D10111yescharacteristic, complemented
113C22×Q8—1011no—
114C2×M16—2021no—
115C4×Q8—1011no—
116C4×Q8—1011no—
117C4×Q8—1011no—
118C4×Q8—2021no—
119C4×Q8—2021no—
120C4×Q8C2×D10111yescharacteristic
121C2×C4⋊C4—1011no—
122C2×C4⋊C4—2021no—
123C2×C42—1011no—
124C2×C42—2021no—
125C4⋊C8—1011no—
126C4⋊C8—2021no—
127C4⋊C8—2021no—
128C4⋊C8C2×D10111yescharacteristic
129C22⋊C8—2021no—
130C4×C8—1011no—
131C4×C8C2×D10111yescharacteristic
132C2×Q8—1011no—
133C2×Q8—2021no—
134C2×Q8—2022no—
135C2×Q8—2022no—
136C2×Q8C4×D10111yescharacteristic
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—1011no—
140C22×C4—2021no—
141C22×C4—2021no—
142M16—4042no—
143C2×C8—21no—
144C2×C8—21no—
145C2×C8—2021no—
146C2×C8—2021no—
147C4⋊C4—21no—
148C4⋊C4—1011no—
149C4⋊C4—2021no—
150C4⋊C4—2021no—
151C4⋊C4—2021no—
152C4⋊C4—2021no—
153C4⋊C4C4×D10111yescharacteristic
154C42—21no—
155C42—1011no—
156C42—1011no—
157C42—1011no—
158C42—2021no—
159C42—2021no—
160C42—2021no—
161C42C22×D10111yescharacteristic
162C23—1011no—
163Q8—21no—
164Q8—1012no—
165Q8—2024no—
166Q8—2021no—
167Q8#1310805712yescomplemented
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—1011no—
171C2×C4—1012no—
172C2×C4—1012no—
173C2×C4—1011no—
174C2×C4—1011no—
175C2×C4—4041no—
176C2×C4—4042no—
177C2×C4—2024no—
178C2×C4—2021no—
179C2×C4—2021no—
180C2×C4—2021no—
181C2×C4—2021no—
182C2×C4C2×C4×D10111yescharacteristic
183C2×C4D4×D10111yescharacteristic
184C2×C4D4×D10111yescharacteristic
185C8—41no—
186C8—41no—
187C8—4041no—
188C8—4041no—
189C22—1012no—
190C22—2021no—
191C22—2021no—
192C22C22⋊C4×D10111yescharacteristic
193C4—22no—
194C4—21no—
195C4—41no—
196C4—1012no—
197C4—4041no—
198C4—2021no—
199C4—2022no—
200C4—2021no—
201C4M16×D10111yescharacteristic
202C4C2×#1310805711yescharacteristic
203C4SD16×D10111yescharacteristic
204C4Q16×D10111yescharacteristic
205C2—2022no—
206C2#2026513911yescharacteristic
207C2#103×D10111yescharacteristic
208C2#2026519211yescharacteristic
subgroupG/H
centerC2×C4D4×D101
commutatorC2×C4×C101C22×C4
FrattiniC42C22×D101
FittingQ8⋊C8×C101C2
socleC22×C101C2×C22⋊C4
prime radicalC22×D101C22⋊C4
cosocleC42×C101C23
hypercenterQ8⋊C8D101
hypocenterC101#1487
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C42 ◁ Q8⋊C8

Complements

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

NH ≅ G/Ntype
4. #20264909C2semidirect
5. Q8⋊#11717579C2semidirect
6. #20264907C2semidirect
7. Q8⋊C8×C101C2semidirect
59. Q8×C101M16semidirect
79. Dic202M16semidirect
105. C101#1487semidirect
112. Q8⋊C8D101semidirect
167. Q8#13108057semidirect

Maximal subgroups

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

subgroupindex
C4×Q8×D1012
#202652282
#202651992
#202649092
Q8⋊#117175792
#202649072
Q8⋊C8×C1012
#1487101