Finite Group Databasebeta

← #16845977

Subgroups — #16845977

Order 3968.

298 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 107 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×Q8×Dic31C211yesmaximal, characteristic
2C2×#11758396C211yesmaximal, characteristic
3C2×#11758423C211yesmaximal, characteristic
4#14520945C211yesmaximal, characteristic
5#14520812C211yesmaximal, characteristic
6#14520803C211yesmaximal, characteristic
7C2×C31×#103C211yesmaximal, characteristic
8C22×Q8×C31C2211yescharacteristic
9C22×C8×C31C2211yescharacteristic
10C2×C4⋊C4×C31C2211yescharacteristic
11Q8×Dic31—22no—
12Q8×Dic31—22no—
13C2×C4⋊Dic31—21no—
14C2×C4⋊Dic31C2211yescharacteristic
15C2×C4⋊Dic31C2211yescharacteristic
16C2×Dic31⋊C4—21no—
17C2×C4×Dic31—21no—
18C2×C4×Dic31C2211yescharacteristic
19C22×C31⋊C8C2211yescharacteristic
20C31×#103—22no—
21#11758423C414yescomplemented
22C31⋊#95—21no—
23#11758396—22no—
24C2×Q8×C31—21no—
25C2×Q8×C31—21no—
26C2×Q8×C31—21no—
27C2×Q8×C31C2×C412yes—
28C2×Q8×C31D412yes—
29C2×Q8×C31Q812yes—
30C22×C4×C31—21no—
31C22×C4×C31—21no—
32C22×C4×C31C2311yescharacteristic
33C22×Dic31—21no—
34C22×Dic31—21no—
35C22×Dic31D411yescharacteristic
36C22×Dic31D411yescharacteristic
37C2×C8×C31—21no—
38C2×C8×C31—21no—
39C2×C8×C31—21no—
40C4⋊C4×C31—21no—
41C4⋊C4×C31—21no—
42C4⋊Dic31—24no—
43C4⋊Dic31—21no—
44C4⋊Dic31—21no—
45C4⋊Dic31C2×C412yes—
46C4⋊Dic31D412yes—
47Dic31⋊C4—42no—
48C4×Dic31—24no—
49C4×Dic31—21no—
50C4×Dic31—21no—
51C2×C31⋊C8—21no—
52C2×C31⋊C8—21no—
53C2×C31⋊C8C2×C412yes—
54C23×C31C2×D411yescharacteristic
55Q8×C31—22no—
56Q8×C31—22no—
57Q8×C31—22no—
58Q8×C31C4⋊C414yescomplemented
59C2×C4×C31—22no—
60C2×C4×C31—22no—
61C2×C4×C31—22no—
62C2×C4×C31—41no—
63C2×C4×C31—41no—
64C2×C4×C31—41no—
65C2×C4×C31C22×C411yescharacteristic
66C2×C4×C31C2×D411yescharacteristic
67C2×C4×C31C2×D411yescharacteristic
68C2×C4×C31C2×Q811yescharacteristic
69C2×C4×C31#4111yescharacteristic
70C2×C4×C31#4111yescharacteristic
71C2×Dic31—21no—
72C2×Dic31—21no—
73C2×Dic31—22no—
74C2×Dic31—21no—
75C2×Dic31—22no—
76C2×Dic31—21no—
77C2×Dic31—21no—
78C2×Dic31—41no—
79C2×Dic31—41no—
80C2×Dic31—41no—
81C2×Dic31—41no—
82C2×Dic31SD1612yes—
83C8×C31—41no—
84C31⋊C8—22no—
85#1865—311nomaximal, Sylow
86C22×C31C4×D411yescharacteristic
87C22×C31#12011yescharacteristic
88C22×C31C22⋊Q811yescharacteristic
89C22×C31C2×SD1611yescharacteristic
90C22×C31#13511yescharacteristic
91C22×C31#13711yescharacteristic
92C22×C31#13711yescharacteristic
93C4×C31—24no—
94C4×C31—81no—
95C4×C31C2×C4⋊C411yescharacteristic
96C4×C31C4×D411yescharacteristic
97C4×C31C22⋊Q811yescharacteristic
98C4×C31#12311yescharacteristic
99Dic31—22no—
100Dic31—42no—
101Dic31—41no—
102Dic31—42no—
103C2×C4×Q8—311no—
104C2×#106—311no—
105C2×#103—311no—
106C2×#103D3111yescharacteristic
107#389—311no—
108#340—311no—
109#336—311no—
110C2×C31#38511yescharacteristic
111C2×C31C4×SD1611yescharacteristic
112C2×C31#44111yescharacteristic
113C2×C31#44811yescharacteristic
114C2×C31#45211yescharacteristic
115C2×C31#47511yescharacteristic
116C2×C31#47911yescharacteristic
117C22×Q8C2×D3111yescharacteristic
118C22×C8—311no—
119C22×C8C2×D3111yescharacteristic
120C4×Q8—622no—
121C4×Q8—622no—
122C2×C4⋊C4—311no—
123C2×C4⋊C4—311no—
124C2×C4⋊C4—621no—
125C2×C4⋊C4—621no—
126C2×C4⋊C4C2×D3111yescharacteristic
127C2×C42—311no—
128C2×C42—621no—
129#106—622no—
130#103—22no—
131#103—314no—
132#95—621no—
133C31#186511yescharacteristic, complemented, Sylow
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8Dic6212yes—
138C2×Q8C4×D3112yes—
139C2×Q8C22⋊D3112yes—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—311no—
143C22×C4—311no—
144C22×C4—621no—
145C22×C4—621no—
146C22×C4C22×D3111yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—21no—
150C2×C8—312no—
151C2×C8—621no—
152C2×C8—621no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—312no—
156C4⋊C4—312no—
157C4⋊C4—624no—
158C4⋊C4—621no—
159C4⋊C4—621no—
160C4⋊C4—1242no—
161C42—624no—
162C42—621no—
163C42—621no—
164C23D4×D3111yescharacteristic
165Q8—22no—
166Q8—22no—
167Q8—22no—
168Q8Dic31⋊C414yescomplemented
169C2×C4—22no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—312no—
176C2×C4—621no—
177C2×C4—621no—
178C2×C4—622no—
179C2×C4—621no—
180C2×C4—621no—
181C2×C4—622no—
182C2×C4—621no—
183C2×C4—1241no—
184C2×C4—1241no—
185C2×C4—1241no—
186C2×C4—1241no—
187C2×C4C2×Dic6211yescharacteristic
188C2×C4C2×C4×D3111yescharacteristic
189C2×C4D4×D3111yescharacteristic
190C2×C4#9249711yescharacteristic
191C2×C4#9249711yescharacteristic
192C2×C4C2×C22⋊D3111yescharacteristic
193C8—41no—
194C8—622no—
195C22#1175845511yescharacteristic
196C22(C22⋊D31)⋊C411yescharacteristic
197C22SD16×D3111yescharacteristic
198C22#1175849211yescharacteristic
199C22#1175849511yescharacteristic
200C22#1175849611yescharacteristic
201C22#1175851411yescharacteristic
202C4—24no—
203C4—81no—
204C4—622no—
205C4—1242no—
206C4—1241no—
207C4—1242no—
208C4#1175845511yescharacteristic
209C4(C22⋊D31)⋊C411yescharacteristic
210C4C2×Dic31⋊C411yescharacteristic
211C4#1175851211yescharacteristic
212C2#1452092111yescharacteristic
213C2#1452113411yescharacteristic
214C2#1452115311yescharacteristic
215C2#1452118611yescharacteristic
216C2#1452121011yescharacteristic
217C2#1452133211yescharacteristic
218C2#1452135811yescharacteristic
subgroupG/H
centerC23D4×D31
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC2×C31×#103C2
socle, prime radicalC23×C31C2×D4
cosocleC22×C4×C31C23
hypercenterC2×#103D31
hypocenterC31#1865
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
21. #11758423C4semidirect
58. Q8×C31C4⋊C4semidirect
133. C31#1865semidirect
168. Q8Dic31⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×Dic312
C2×#117583962
C2×#117584232
#145209452
#145208122
#145208032
C2×C31×#1032
#186531