Finite Group Databasebeta

← #16857570

Subgroups — #16857570

Order 3968.

420 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 139 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
1D4×Dic62C211yesmaximal, characteristic, complemented
2#14521401C211yesmaximal, characteristic, complemented
3#14521379C212yesmaximal, complemented
4#14521331C211yesmaximal, characteristic, complemented
5#14521276C212yesmaximal, complemented
6#14521254C211yesmaximal, characteristic, complemented
7#14521187C212yesmaximal, complemented
8C4×#92480C211yesmaximal, characteristic, complemented
9D4×C31⋊C8C211yesmaximal, characteristic, complemented
10C2×#11758398C212yesmaximal, complemented
11C31×#547C211yesmaximal, characteristic
12C2×#41×C31—21no—
13C22×Dic62C2212yes—
14C31×C4⋊Q8C2211yescharacteristic
15C31×#123—22no—
16C31×C22⋊Q8—21no—
17C31×#121C2212yes—
18C4×D4×C31C2211yescharacteristic
19C4×D4×C31C2211yescharacteristic
20C2×C4⋊C4×C31C2212yes—
21D4×Dic31—21no—
22C2×#92480C2211yescharacteristic
23C2×#92480C2212yes—
24#11758501C2212yes—
25C22×C31⋊C8C2212yes—
26#11758465—21no—
27#11758455—22no—
28#11758446C2211yescharacteristic
29C4×Dic62C2211yescharacteristic
30#11758420C2211yescharacteristic
31C22⋊(C31⋊C8)C2212yes—
32#11758398C2212yes—
33#11758398C2212yes—
34#11758398C2211yescharacteristic
35#11758398C2212yes—
36#11758398C2212yes—
37#11758396C2211yescharacteristic
38#11758396C2212yes—
39C4⋊(C31⋊C8)C2211yescharacteristic
40C4×C31⋊C8C2211yescharacteristic
41#41×C31—44no—
42C2×Q8×C31—21no—
43C2×D4×C31—22no—
44C2×D4×C31D411yescharacteristic
45C2×D4×C31C2311yescharacteristic
46C22×C4×C31—22no—
47C22×C4×C31—21no—
48C22×C4×C31C2312yes—
49C22×Dic31—22no—
50C2×Dic62—24no—
51C2×Dic62—22no—
52C2×Dic62—42no—
53C2×Dic62C2312yes—
54C2×Dic62C2311yescharacteristic
55C4⋊C4×C31—22no—
56C4⋊C4×C31—21no—
57C4⋊C4×C31D411yescharacteristic
58C4⋊C4×C31C2311yescharacteristic
59C4⋊C4×C31C2312yes—
60C22⋊C4×C31—21no—
61C22⋊C4×C31—22no—
62C22⋊C4×C31D412yes—
63C42×C31C2311yescharacteristic
64C22⋊Dic31—22no—
65#92480—22no—
66#92480—42no—
67C4⋊Dic31—21no—
68C4⋊Dic31—21no—
69C4⋊Dic31C2311yescharacteristic
70Dic31⋊C4—21no—
71Dic31⋊C4—22no—
72C4×Dic31—21no—
73C2×C31⋊C8—42no—
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2311yescharacteristic
76C2×C31⋊C8C2312yes—
77C23×C31—21no—
78C23×C31C2×D412yes—
79Q8×C31—42no—
80D4×C31—21no—
81D4×C31—21no—
82D4×C31—44no—
83D4×C31SD1614yescomplemented
84C2×C4×C31—22no—
85C2×C4×C31—22no—
86C2×C4×C31—21no—
87C2×C4×C31—22no—
88C2×C4×C31—44no—
89C2×C4×C31—42no—
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2411yescharacteristic
94C2×Dic31—21no—
95C2×Dic31—21no—
96C2×Dic31—22no—
97C2×Dic31—44no—
98Dic62—21no—
99Dic62—22no—
100Dic62—22no—
101Dic62—44no—
102Dic62#4112yescomplemented
103C31⋊C8—21no—
104C31⋊C8—21no—
105C31⋊C8—22no—
106#3298—311nomaximal, Sylow
107C22×C31—22no—
108C22×C31—22no—
109C22×C31—41no—
110C22×C31—41no—
111C22×C31—41no—
112C22×C31C2×SD1614yes—
113C22×C31C22×D411yescharacteristic
114C4×C31—22no—
115C4×C31—21no—
116C4×C31—41no—
117C4×C31—42no—
118C4×C31C2×SD1612yes—
119C4×C31C2×#4111yescharacteristic
120C4×C31H(5,𝔽2)11yescharacteristic
121Dic31—22no—
122Dic31—42no—
123Dic31—41no—
124D4×Q8—311no—
125#547D3111yescharacteristic
126#487—311no—
127#483—312no—
128#475—311no—
129#465—312no—
130#461—311no—
131#448—312no—
132C4×SD16—311no—
133C8×D4—311no—
134C2×#103—312no—
135C2×C31—24no—
136C2×C31—81no—
137C2×C31#54611yescharacteristic
138C2×C31C22×SD1611yescharacteristic
139C2×C31#57811yescharacteristic
140C2×#41—21no—
141C22×Q8—312no—
142C2×SD16—311no—
143C2×SD16—312no—
144C22×C8—312no—
145C4⋊Q8—311no—
146C4⋊Q8—621no—
147C4⋊Q8C2×D3111yescharacteristic
148#123—22no—
149C22⋊Q8—21no—
150C22⋊Q8—312no—
151C22⋊Q8—622no—
152#121C2×D3112yes—
153C4×Q8—311no—
154C4×D4—621no—
155C4×D4C2×D3111yescharacteristic
156C4×D4C2×D3111yescharacteristic
157C2×C4⋊C4C2×D3112yes—
158#106—311no—
159#106—312no—
160C4⋊C8—311no—
161#103—311no—
162#103—312no—
163#103—312no—
164#103—312no—
165#103—312no—
166#102—311no—
167C22⋊C8—312no—
168C4×C8—311no—
169C31#329811yescharacteristic, complemented, Sylow
170#41—44no—
171C2×Q8—21no—
172C2×Q8—311no—
173C2×Q8—312no—
174C2×Q8—624no—
175C2×Q8—622no—
176C2×Q8—1242no—
177C2×D4—22no—
178C2×D4C22⋊D3111yescharacteristic
179C2×D4C22×D3111yescharacteristic
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—622no—
183C22×C4C22×D3112yes—
184SD16—622no—
185SD16—1242no—
186C2×C8—311no—
187C2×C8—311no—
188C2×C8—312no—
189C2×C8—1242no—
190C4⋊C4—22no—
191C4⋊C4—21no—
192C4⋊C4—311no—
193C4⋊C4—621no—
194C4⋊C4—622no—
195C4⋊C4—621no—
196C4⋊C4—621no—
197C4⋊C4C22⋊D3111yescharacteristic
198C4⋊C4C22×D3111yescharacteristic
199C4⋊C4C22×D3112yes—
200C22⋊C4—21no—
201C22⋊C4—22no—
202C22⋊C4—622no—
203C22⋊C4C22⋊D3112yes—
204C42—621no—
205C42C22×D3111yescharacteristic
206C23—21no—
207C23C2×C22⋊D3112yes—
208Q8—42no—
209Q8—312no—
210Q8—621no—
211Q8—622no—
212Q8—622no—
213Q8—1244no—
214D4—21no—
215D4—21no—
216D4—44no—
217D4#9248014yescomplemented
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—44no—
223C2×C4—42no—
224C2×C4—621no—
225C2×C4—621no—
226C2×C4—622no—
227C2×C4—1244no—
228C2×C4C2×C22⋊D3111yescharacteristic
229C2×C4C2×C22⋊D3112yes—
230C2×C4C2×C22⋊D3111yescharacteristic
231C2×C4C23×D3111yescharacteristic
232C8—621no—
233C8—621no—
234C8—622no—
235C22—22no—
236C22—22no—
237C22—41no—
238C22—41no—
239C22—41no—
240C22C2×#9248014yes—
241C22C22×C22⋊D3111yescharacteristic
242C4—21no—
243C4—22no—
244C4—41no—
245C4—42no—
246C4—622no—
247C4—1241no—
248C4—1242no—
249C4C2×#9248012yes—
250C4#41×D3111yescharacteristic
251C4#1175856611yescharacteristic
252C2—24no—
253C2—81no—
254C2#1452168711yescharacteristic
255C2C22×#9248011yescharacteristic
256C2#1452186411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
Fitting, prime radicalC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3298
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. D4×Dic62C2semidirect
2. #14521401C2semidirect
3. #14521379C2semidirect
4. #14521331C2semidirect
5. #14521276C2semidirect
6. #14521254C2semidirect
7. #14521187C2semidirect
8. C4×#92480C2semidirect
9. D4×C31⋊C8C2semidirect
10. C2×#11758398C2semidirect
83. D4×C31SD16semidirect
102. Dic62#41semidirect
169. C31#3298semidirect
217. D4#92480semidirect

Maximal subgroups

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

subgroupindex
D4×Dic622
#145214012
#145213792
#145213312
#145212762
#145212542
#145211872
C4×#924802
D4×C31⋊C82
C2×#117583982
C31×#5472
#329831