Finite Group Databasebeta

← #16857639

Subgroups — #16857639

Order 3968.

398 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 125 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#14521764C211yesmaximal, characteristic, complemented
2#14521399C211yesmaximal, characteristic, complemented
3#14521387C212yesmaximal, complemented
4#14521339C211yesmaximal, characteristic, complemented
5#14521280C212yesmaximal, complemented
6#14521251C211yesmaximal, characteristic, complemented
7#14521194C212yesmaximal, complemented
8C4×#92479C211yesmaximal, characteristic, complemented
9D4×C31⋊C8C211yesmaximal, characteristic, complemented
10#14521045C212yesmaximal, complemented
11C31×#547C211yesmaximal, characteristic, complemented
12C2×#41×C31C2212yes—
13C31×C4⋊Q8C2211yescharacteristic
14C31×#123—22no—
15C31×C22⋊Q8C2212yes—
16C31×#121—21no—
17C4×D4×C31C2211yescharacteristic
18C4×D4×C31C2211yescharacteristic
19C2×C4⋊C4×C31—21no—
20C2×#92481C2212yes—
21#11758515—21no—
22C2×#92479C2211yescharacteristic
23#11758507C2212yes—
24#11758503C2212yes—
25C22×C31⋊C8C2212yes—
26#11758469—21no—
27#11758462—22no—
28C4×D124C2211yescharacteristic
29#11758446C2211yescharacteristic
30#11758423C2212yes—
31#11758423C2212yes—
32#11758423C2212yes—
33#11758420C2211yescharacteristic
34#11758420C2212yes—
35C22⋊(C31⋊C8)C2212yes—
36#11758397C2211yescharacteristic
37#11758396C2212yes—
38#11758395C2211yescharacteristic
39C4⋊(C31⋊C8)C2211yescharacteristic
40C4×C31⋊C8C2211yescharacteristic
41#41×C31—24no—
42#41×C31—42no—
43C2×Q8×C31C2312yes—
44C2×D4×C31—22no—
45C2×D4×C31D411yescharacteristic
46C2×D4×C31C2311yescharacteristic
47C22×C4×C31—21no—
48C22×C4×C31—22no—
49C22×C4×C31C2312yes—
50C2×C22⋊D31—22no—
51C2×D124C2311yescharacteristic
52C2×C4×D31—21no—
53C2×Dic62—21no—
54C4⋊C4×C31—21no—
55C4⋊C4×C31—22no—
56C4⋊C4×C31—21no—
57C4⋊C4×C31D411yescharacteristic
58C4⋊C4×C31C2311yescharacteristic
59C22⋊C4×C31—22no—
60C22⋊C4×C31—21no—
61C22⋊C4×C31D412yes—
62C42×C31C2311yescharacteristic
63C22⋊Dic31—22no—
64#92481—42no—
65#92479—22no—
66(C2×D31)⋊C4—21no—
67(C2×D31)⋊C4—22no—
68C4⋊Dic31—21no—
69C4⋊Dic31C2311yescharacteristic
70C4⋊Dic31C2312yes—
71C4×Dic31—22no—
72C2×C31⋊C8—42no—
73C2×C31⋊C8C2311yescharacteristic
74C2×C31⋊C8C2311yescharacteristic
75C2×C31⋊C8C2312yes—
76C23×C31—21no—
77C23×C31C2×D412yes—
78C22×D31—21no—
79Q8×C31—22no—
80Q8×C31—22no—
81D4×C31—22no—
82D4×C31—21no—
83D4×C31—44no—
84D4×C31#4112yes—
85C2×C4×C31—22no—
86C2×C4×C31—22no—
87C2×C4×C31—21no—
88C2×C4×C31—22no—
89C2×C4×C31—44no—
90C2×C4×C31—42no—
91C2×C4×C31C2×D412yes—
92C2×C4×C31C2×D411yescharacteristic
93C2×C4×C31C2×D411yescharacteristic
94C2×C4×C31C2411yescharacteristic
95C22⋊D31—82no—
96C2×Dic31—21no—
97C2×Dic31—22no—
98D124—21no—
99D124—21no—
100C4×D31—42no—
101Dic62—81no—
102C31⋊C8—21no—
103C31⋊C8—21no—
104C31⋊C8—22no—
105#3303—311nomaximal, Sylow
106C22×C31—22no—
107C22×C31—22no—
108C22×C31—22no—
109C22×C31—22no—
110C22×C31—41no—
111C22×C31—41no—
112C22×C31C22×D411yescharacteristic
113C2×D31—41no—
114C2×D31—41no—
115C2×D31—41no—
116C4×C31—22no—
117C4×C31—21no—
118C4×C31—42no—
119C4×C31—41no—
120C4×C31#13512yes—
121C4×C31C2×#4111yescharacteristic
122C4×C31H(5,𝔽2)11yescharacteristic
123Dic31—41no—
124Dic31—42no—
125#555—311no—
126#547D3111yescharacteristic, complemented
127#487—311no—
128#485—312no—
129#476—311no—
130#465—312no—
131#461—311no—
132#449—312no—
133C4×D8—311no—
134C8×D4—311no—
135#416—312no—
136C2×C31—42no—
137C2×C31—42no—
138C2×C31#54611yescharacteristic
139C2×C31C2×#13511yescharacteristic
140C2×C31#57711yescharacteristic
141D31—81no—
142C2×#41C2×D3112yes—
143C2×SD16—312no—
144C2×D8—311no—
145C22×C8—312no—
146C4⋊Q8—311no—
147C4⋊Q8C2×D3111yescharacteristic
148#124—622no—
149#123—22no—
150C22⋊Q8C2×D3112yes—
151#121—21no—
152#121—312no—
153#121—621no—
154C4×D4—311no—
155C4×D4C2×D3111yescharacteristic
156C4×D4C2×D3111yescharacteristic
157#117—312no—
158#117—621no—
159C2×C4⋊C4—21no—
160#107—311no—
161#106—312no—
162C4⋊C8—311no—
163#103—312no—
164#103—312no—
165#103—312no—
166#102—311no—
167#102—312no—
168#102—311no—
169C22⋊C8—312no—
170C4×C8—311no—
171C31#330311yescharacteristic, complemented, Sylow
172#41—24no—
173#41—42no—
174C2×Q8—621no—
175C2×Q8C22×D3112yes—
176C2×D4—22no—
177C2×D4—311no—
178C2×D4—622no—
179C2×D4C22⋊D3111yescharacteristic
180C2×D4C22×D3111yescharacteristic
181C22×C4—21no—
182C22×C4—22no—
183C22×C4—621no—
184C22×C4C22×D3112yes—
185SD16—1242no—
186D8—622no—
187C2×C8—311no—
188C2×C8—311no—
189C2×C8—312no—
190C2×C8—1242no—
191C4⋊C4—21no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—312no—
195C4⋊C4—311no—
196C4⋊C4—621no—
197C4⋊C4C22⋊D3111yescharacteristic
198C4⋊C4C22×D3111yescharacteristic
199C22⋊C4—22no—
200C22⋊C4—21no—
201C22⋊C4—622no—
202C22⋊C4—622no—
203C22⋊C4—621no—
204C22⋊C4C22⋊D3112yes—
205C42—622no—
206C42C22×D3111yescharacteristic
207C23—21no—
208C23—621no—
209C23C2×C22⋊D3112yes—
210Q8—22no—
211Q8—22no—
212Q8—2481no—
213D4—22no—
214D4—21no—
215D4—44no—
216D4—621no—
217D4—621no—
218D4—2482no—
219D4#9249712yes—
220C2×C4—22no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—44no—
225C2×C4—42no—
226C2×C4—621no—
227C2×C4—622no—
228C2×C4—1242no—
229C2×C4C2×C22⋊D3112yes—
230C2×C4C2×C22⋊D3111yescharacteristic
231C2×C4C2×C22⋊D3111yescharacteristic
232C2×C4C23×D3111yescharacteristic
233C8—621no—
234C8—621no—
235C8—622no—
236C22—22no—
237C22—22no—
238C22—22no—
239C22—22no—
240C22—41no—
241C22—41no—
242C22—1241no—
243C22—1241no—
244C22—1241no—
245C22C22×C22⋊D3111yescharacteristic
246C4—21no—
247C4—22no—
248C4—42no—
249C4—41no—
250C4—1242no—
251C4—1241no—
252C4#1175852712yes—
253C4C2×#9249711yescharacteristic
254C4#1175856111yescharacteristic
255C2—42no—
256C2—42no—
257C2—2481no—
258C2#1452169311yescharacteristic
259C2C2×#1175852711yescharacteristic
260C2#1452185711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#547C2
socleC22×C31C22×D4
hypercenter#547D31
hypocenterC31#3303
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. #14521764C2semidirect
2. #14521399C2semidirect
3. #14521387C2semidirect
4. #14521339C2semidirect
5. #14521280C2semidirect
6. #14521251C2semidirect
7. #14521194C2semidirect
8. C4×#92479C2semidirect
9. D4×C31⋊C8C2semidirect
10. #14521045C2semidirect
11. C31×#547C2semidirect
126. #547D31semidirect
171. C31#3303semidirect

Maximal subgroups

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

subgroupindex
#145217642
#145213992
#145213872
#145213392
#145212802
#145212512
#145211942
C4×#924792
D4×C31⋊C82
#145210452
C31×#5472
#330331