Finite Group Databasebeta

← #16855516

Subgroups — #16855516

Order 3968.

376 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 153 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#14521606C211yesmaximal, characteristic
2#14521356C214yesmaximal, complemented
3#14521344C212yesmaximal, complemented
4#14521334C212yesmaximal, complemented
5#14521089C212yesmaximal
6#14521073C211yesmaximal, characteristic
7#14521058C212yesmaximal
8C2×C4×Q8×C31C211yesmaximal, characteristic
9C22×Q8×C31C2211yescharacteristic
10C4×Q8×C31—21no—
11C4×Q8×C31—21no—
12C4×Q8×C31C2212yes—
13C4×Q8×C31C2212yes—
14C2×C4⋊C4×C31—21no—
15C2×C4⋊C4×C31C2211yescharacteristic
16C2×C42×C31—21no—
17C2×C42×C31C2211yescharacteristic
18#11758503C2212yes—
19#11758501—21no—
20#11758501—21no—
21C2×#92474C2212yes—
22#11758447C2212yes—
23#11758446C2211yescharacteristic
24#11758446C2211yescharacteristic
25C4×Dic62—21no—
26C4×Dic62—21no—
27#11758423C2214yes—
28#11758423C2214yes—
29#11758396C2214yes—
30#11758395C2214yes—
31C4⋊(C31⋊C8)C2212yes—
32C4⋊(C31⋊C8)C2212yes—
33C2×Q8×C31—21no—
34C2×Q8×C31—22no—
35C2×Q8×C31Q814yescomplemented
36C2×Q8×C31C2312yes—
37C22×C4×C31—21no—
38C22×C4×C31—21no—
39C22×C4×C31D411yescharacteristic
40C22×C4×C31D411yescharacteristic
41C22×C4×C31C2311yescharacteristic
42C2×Dic62—21no—
43C2×Dic62—21no—
44C4⋊C4×C31—22no—
45C4⋊C4×C31—22no—
46C4⋊C4×C31—21no—
47C4⋊C4×C31C2312yes—
48C42×C31—22no—
49C42×C31—22no—
50C42×C31D411yescharacteristic
51C42×C31D411yescharacteristic
52C42×C31C2311yescharacteristic
53C42×C31C2311yescharacteristic
54C22⋊Dic31—22no—
55C4⋊Dic31—21no—
56C4⋊Dic31—21no—
57C4⋊Dic31C2312yes—
58C4⋊Dic31C2312yes—
59Dic31⋊C4—22no—
60Dic31⋊C4—22no—
61C4×Dic31—22no—
62#92474—24no—
63C2×C31⋊C8C2314yes—
64C23×C31C2×D411yescharacteristic
65Q8×C31—24no—
66Q8×C31—22no—
67Q8×C31C2×Q814yes—
68C2×C4×C31—24no—
69C2×C4×C31—22no—
70C2×C4×C31—22no—
71C2×C4×C31—22no—
72C2×C4×C31—41no—
73C2×C4×C31—42no—
74C2×C4×C31C2×D411yescharacteristic
75C2×C4×C31C2×D411yescharacteristic
76C2×C4×C31C2×D411yescharacteristic
77C2×C4×C31C2×D411yescharacteristic
78C2×C4×C31C2×D411yescharacteristic
79C2×C4×C31C2×Q812yes—
80C2×C4×C31#4112yes—
81C2×C4×C31C2411yescharacteristic
82C2×Dic31—22no—
83C2×Dic31—22no—
84Dic62—42no—
85Dic62—42no—
86C31⋊C8—24no—
87#3078—311nomaximal, Sylow
88C22×C31—21no—
89C22×C31—21no—
90C22×C31C22⋊Q812yes—
91C22×C31C22×D411yescharacteristic
92C4×C31—24no—
93C4×C31—21no—
94C4×C31—21no—
95C4×C31—21no—
96C4×C31—42no—
97C4×C31C22⋊Q812yes—
98C4×C31#13712yes—
99C4×C31C22×Q811yescharacteristic
100C4×C31C2×#4111yescharacteristic
101Dic31—42no—
102Dic31—42no—
103#533—311no—
104C2×C4×Q8D3111yescharacteristic
105#479—314no—
106#477—312no—
107#475—312no—
108#428—312no—
109#423—311no—
110#419—312no—
111C2×C31—22no—
112C2×C31C2×C22⋊Q811yescharacteristic
113C2×C31C2×#13711yescharacteristic
114C2×C31#57511yescharacteristic
115C22×Q8C2×D3111yescharacteristic
116C2×M16—312no—
117C4⋊Q8—311no—
118C4⋊Q8—311no—
119#125—312no—
120C22⋊Q8—621no—
121C22⋊Q8—621no—
122C4×Q8—21no—
123C4×Q8—21no—
124C4×Q8—621no—
125C4×Q8—621no—
126C4×Q8C2×D3112yes—
127C4×Q8C2×D3112yes—
128#117—312no—
129C2×C4⋊C4—21no—
130C2×C4⋊C4C2×D3111yescharacteristic
131C2×C42—21no—
132C2×C42C2×D3111yescharacteristic
133#107—314no—
134#106—314no—
135C4⋊C8—312no—
136C4⋊C8—312no—
137#103—314no—
138#103—314no—
139C31#307811yescharacteristic, complemented, Sylow
140C2×Q8—22no—
141C2×Q8—21no—
142C2×Q8—621no—
143C2×Q8—621no—
144C2×Q8Dic6214yescomplemented
145C2×Q8C22×D3112yes—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4C22⋊D3111yescharacteristic
149C22×C4C22⋊D3111yescharacteristic
150C22×C4C22×D3111yescharacteristic
151M16—624no—
152C2×C8—314no—
153C4⋊C4—22no—
154C4⋊C4—22no—
155C4⋊C4—21no—
156C4⋊C4—312no—
157C4⋊C4—312no—
158C4⋊C4—622no—
159C4⋊C4—622no—
160C4⋊C4—621no—
161C4⋊C4—621no—
162C4⋊C4C22×D3112yes—
163C22⋊C4—622no—
164C42—22no—
165C42—22no—
166C42—622no—
167C42C22⋊D3111yescharacteristic
168C42C22⋊D3111yescharacteristic
169C42C22×D3111yescharacteristic
170C42C22×D3111yescharacteristic
171C23C2×C22⋊D3111yescharacteristic
172Q8—24no—
173Q8—22no—
174Q8—1242no—
175Q8—1242no—
176Q8C2×Dic6214yes—
177C2×C4—24no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—41no—
182C2×C4—42no—
183C2×C4—622no—
184C2×C4—622no—
185C2×C4C2×Dic6212yes—
186C2×C4#9249512yes—
187C2×C4C2×C22⋊D3111yescharacteristic
188C2×C4C2×C22⋊D3111yescharacteristic
189C2×C4C2×C22⋊D3111yescharacteristic
190C2×C4C2×C22⋊D3111yescharacteristic
191C2×C4C2×C22⋊D3111yescharacteristic
192C2×C4C23×D3111yescharacteristic
193C8—624no—
194C22—21no—
195C22—21no—
196C22#1175850112yes—
197C22C22×C22⋊D3111yescharacteristic
198C4—24no—
199C4—21no—
200C4—21no—
201C4—21no—
202C4—42no—
203C4—1242no—
204C4—1242no—
205C4#1175850112yes—
206C4#1175851912yes—
207C4C22×Dic6211yescharacteristic
208C4C2×#9249511yescharacteristic
209C2—22no—
210C2C2×#1175850111yescharacteristic
211C2C2×#1175851911yescharacteristic
212C2#1452184711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC2×C4×Q8×C31C2
socleC22×C31C22×D4
prime radicalC23×C31C2×D4
hypercenterC2×C4×Q8D31
hypocenterC31#3078
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #14521356C2semidirect
3. #14521344C2semidirect
4. #14521334C2semidirect
35. C2×Q8×C31Q8semidirect
139. C31#3078semidirect
144. C2×Q8Dic62semidirect

Maximal subgroups

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

subgroupindex
#145216062
#145213562
#145213442
#145213342
#145210892
#145210732
#145210582
C2×C4×Q8×C312
#307831