Finite Group Databasebeta

← C31×#8327 #21465086

Subgroups — C31×#8327 #21465086

Order 7936.

684 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 684 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C2×C31×#541C211yesmaximal, characteristic
2C31×#1479C214yesmaximal, complemented
3C31×#1458C212yesmaximal
4C31×#541—28no—
5C2×C31×C4⋊Q8C411yescharacteristic
6C2×C31×#126—22no—
7C2×C31×#125—22no—
8C2×C31×#125C411yescharacteristic
9C2×C31×C22⋊Q8C412yes—
10C2×C4×Q8×C31C2212yes—
11C2×C31×#117C2211yescharacteristic
12C31×#431C2214yes—
13C31×Q8⋊C8—28no—
14C31×#324—22no—
15C31×#323—22no—
16C22×Q8×C31C2×C412yes—
17C23×C4×C31C2×C411yescharacteristic
18C31×C4⋊Q8—24no—
19C31×#126—216no—
20C31×#125—216no—
21C31×#125—24no—
22C31×C22⋊Q8—24no—
23C31×C22⋊Q8—24no—
24C4×Q8×C31—24no—
25C4×Q8×C31D418yes—
26C31×#117—24no—
27C2×C4⋊C4×C31—22no—
28C2×C4⋊C4×C31—22no—
29C2×C4⋊C4×C31—22no—
30C2×C4⋊C4×C31—22no—
31C2×C4⋊C4×C31C2×C412yes—
32C2×C4⋊C4×C31D412yes—
33C2×C22⋊C4×C31—22no—
34C2×C22⋊C4×C31D412yes—
35C2×C42×C31—22no—
36C2×C42×C31C2×C411yescharacteristic
37C2×C42×C31C2311yescharacteristic
38C31×C4⋊C8—24no—
39C31×C22⋊C8—24no—
40C4×C8×C31—24no—
41C24×C31C22⋊C411yescharacteristic
42C2×Q8×C31—22no—
43C2×Q8×C31—28no—
44C2×Q8×C31C22⋊C414yes—
45C22×C4×C31—22no—
46C22×C4×C31—22no—
47C22×C4×C31—22no—
48C22×C4×C31—24no—
49C22×C4×C31C22⋊C411yescharacteristic
50C22×C4×C31C22⋊C412yes—
51C22×C4×C31C22×C411yescharacteristic
52C22×C4×C31C2×D412yes—
53C2×C8×C31—44no—
54C4⋊C4×C31—28no—
55C4⋊C4×C31—28no—
56C4⋊C4×C31—28no—
57C4⋊C4×C31—22no—
58C4⋊C4×C31—28no—
59C4⋊C4×C31—24no—
60C4⋊C4×C31C22⋊C414yes—
61C22⋊C4×C31—28no—
62C22⋊C4×C31—24no—
63C42×C31—22no—
64C42×C31—28no—
65C42×C31C2×D414yes—
66#8327C3111yesmaximal, characteristic, direct product factor, Sylow
67C23×C31—21no—
68C23×C31—22no—
69C23×C31—42no—
70C23×C31C2×C22⋊C411yescharacteristic
71Q8×C31—28no—
72Q8×C31—44no—
73C2×C4×C31—22no—
74C2×C4×C31—24no—
75C2×C4×C31—24no—
76C2×C4×C31—21no—
77C2×C4×C31—22no—
78C2×C4×C31—42no—
79C2×C4×C31—44no—
80C2×C4×C31—44no—
81C2×C4×C31—44no—
82C2×C4×C31—42no—
83C2×C4×C31C2×C22⋊C412yes—
84C2×C4×C31#12014yes—
85C2×C4×C31#13114yes—
86C8×C31—84no—
87C2×#541C2×C3111yescharacteristic
88#1479C2×C3114yescomplemented
89#1458C2×C3112yes—
90C22×C31—22no—
91C22×C31—42no—
92C22×C31—41no—
93C22×C31—82no—
94C22×C31#37911yescharacteristic
95C22×C31#40812yes—
96C4×C31—24no—
97C4×C31—42no—
98C4×C31—44no—
99C4×C31—41no—
100C4×C31—44no—
101C4×C31#40814yes—
102#541—28no—
103C2×C4⋊Q8C4×C3111yescharacteristic
104C2×#126—22no—
105C2×#125—22no—
106C2×#125C4×C3111yescharacteristic
107C2×C22⋊Q8C4×C3112yes—
108C2×C4×Q8C22×C3112yes—
109C2×#117C22×C3111yescharacteristic
110#431C22×C3114yes—
111Q8⋊C8—28no—
112#324—22no—
113#323—22no—
114C2×C31—22no—
115C2×C31—81no—
116C2×C31#178011yescharacteristic
117C2×C31#179612yes—
118C22×Q8C2×C4×C3112yes—
119C23×C4C2×C4×C3111yescharacteristic
120C4⋊Q8—24no—
121#126—216no—
122#125—216no—
123#125—24no—
124C22⋊Q8—24no—
125C22⋊Q8—24no—
126C4×Q8—24no—
127C4×Q8D4×C3118yes—
128#117—24no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4C2×C4×C3112yes—
134C2×C4⋊C4D4×C3112yes—
135C2×C22⋊C4—22no—
136C2×C22⋊C4D4×C3112yes—
137C2×C42—22no—
138C2×C42C2×C4×C3111yescharacteristic
139C2×C42C23×C3111yescharacteristic
140C4⋊C8—24no—
141C22⋊C8—24no—
142C4×C8—24no—
143C31#832711yescharacteristic, direct product factor, Sylow
144C24C22⋊C4×C3111yescharacteristic
145C2×Q8—22no—
146C2×Q8—28no—
147C2×Q8C22⋊C4×C3114yes—
148C22×C4—22no—
149C22×C4—22no—
150C22×C4—22no—
151C22×C4—24no—
152C22×C4C22⋊C4×C3111yescharacteristic
153C22×C4C22⋊C4×C3112yes—
154C22×C4C22×C4×C3111yescharacteristic
155C22×C4C2×D4×C3112yes—
156C2×C8—44no—
157C4⋊C4—28no—
158C4⋊C4—28no—
159C4⋊C4—28no—
160C4⋊C4—22no—
161C4⋊C4—28no—
162C4⋊C4—24no—
163C4⋊C4C22⋊C4×C3114yes—
164C22⋊C4—28no—
165C22⋊C4—24no—
166C42—22no—
167C42—28no—
168C42C2×D4×C3114yes—
169C23—21no—
170C23—22no—
171C23—42no—
172C23C2×C22⋊C4×C3111yescharacteristic
173Q8—28no—
174Q8—44no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—44no—
182C2×C4—44no—
183C2×C4—44no—
184C2×C4—42no—
185C2×C4C2×C22⋊C4×C3112yes—
186C2×C4C31×#12014yes—
187C2×C4C31×#13114yes—
188C8—84no—
189C22—22no—
190C22—42no—
191C22—41no—
192C22—82no—
193C22C31×#37911yescharacteristic
194C22C31×#40812yes—
195C4—24no—
196C4—44no—
197C4—42no—
198C4—41no—
199C4—44no—
200C4C31×#40814yes—
201C2—22no—
202C2—81no—
203C2C31×#178011yescharacteristic
204C2C31×#179612yes—
subgroupG/H
center, socleC22×C31#379
commutatorC22×C4C22×C4×C31
Frattini, cosocleC2×C42C23×C31
prime radicalC24×C31C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C31 ◁ C2×C42×C31 ◁ G

Complements

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

NH ≅ G/Ntype
2. C31×#1479C2semidirect
66. #8327C31direct (N × H)
88. #1479C2×C31semidirect
143. C31#8327direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C31×#5412
C31×#14792
C31×#14582
#832731