Finite Group Databasebeta

← #16858360

Subgroups — #16858360

Order 3968.

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

normal subgroups only

This group has 350 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#14521772C211yesmaximal, characteristic, complemented
2#14521406C212yesmaximal, complemented
3#14521401C211yesmaximal, characteristic, complemented
4#14521353C212yesmaximal, complemented
5#14521336C211yesmaximal, characteristic, complemented
6#14521264C212yesmaximal
7#14521254C211yesmaximal, characteristic, complemented
8Q8×C31⋊C8C211yesmaximal, characteristic, complemented
9C4×#92480C212yesmaximal, complemented
10C4×#92479C211yesmaximal, characteristic
11C31×#554C211yesmaximal, characteristic, complemented
12C31×C4⋊Q8C2211yescharacteristic
13C31×#125C2212yes—
14C31×C22⋊Q8—21no—
15C31×C22⋊Q8—22no—
16C4×Q8×C31C2211yescharacteristic
17C4×D4×C31C2211yescharacteristic
18C4×D4×C31C2212yes—
19C2×C4⋊C4×C31—21no—
20#11758523—21no—
21C2×#92480C2212yes—
22C2×#92479C2211yescharacteristic
23#11758475—22no—
24#11758469—21no—
25#11758452C2212yes—
26C4×D124C2211yescharacteristic
27#11758446C2211yescharacteristic
28C4×Dic62C2212yes—
29#11758420C2211yescharacteristic
30#11758398C2212yes—
31#11758398C2212yes—
32#11758398C2212yes—
33#11758397C2211yescharacteristic
34#11758397C2212yes—
35#11758396C2212yes—
36#11758395C2211yescharacteristic
37C4⋊(C31⋊C8)C2212yes—
38C4⋊(C31⋊C8)C2211yescharacteristic
39C4×C31⋊C8C2211yescharacteristic
40C4×C31⋊C8C2212yes—
41C2×Q8×C31—21no—
42C2×Q8×C31D411yescharacteristic
43C2×D4×C31C2311yescharacteristic
44C22×C4×C31—21no—
45C22×C4×C31—22no—
46C2×D124C2311yescharacteristic
47C2×C4×D31—21no—
48C2×Dic62C2312yes—
49C4⋊C4×C31—22no—
50C4⋊C4×C31—21no—
51C4⋊C4×C31—22no—
52C4⋊C4×C31D411yescharacteristic
53C4⋊C4×C31D412yes—
54C4⋊C4×C31C2311yescharacteristic
55C4⋊C4×C31C2312yes—
56C22⋊C4×C31—21no—
57C22⋊C4×C31—22no—
58C42×C31C2312yes—
59C42×C31C2311yescharacteristic
60#92480—24no—
61#92479—22no—
62(C2×D31)⋊C4—21no—
63(C2×D31)⋊C4—22no—
64(C2×D31)⋊C4—22no—
65C4⋊Dic31—21no—
66C4⋊Dic31—21no—
67C4⋊Dic31C2311yescharacteristic
68Dic31⋊C4—22no—
69C4×Dic31—22no—
70C2×C31⋊C8C2311yescharacteristic
71C2×C31⋊C8C2311yescharacteristic
72C2×C31⋊C8C2312yes—
73C23×C31—21no—
74C22×D31—21no—
75Q8×C31—22no—
76Q8×C31—81no—
77D4×C31—21no—
78D4×C31#4112yes—
79C2×C4×C31—21no—
80C2×C4×C31—22no—
81C2×C4×C31—42no—
82C2×C4×C31—82no—
83C2×C4×C31C2×D412yes—
84C2×C4×C31C2×D412yes—
85C2×C4×C31C2×D411yescharacteristic
86C2×C4×C31C2×D411yescharacteristic
87C2×C4×C31C2411yescharacteristic
88C2×Dic31—21no—
89C2×Dic31—22no—
90D124—21no—
91D124—21no—
92C4×D31—42no—
93Dic62—22no—
94Dic62—22no—
95C31⋊C8—21no—
96C31⋊C8—22no—
97C31⋊C8—21no—
98#3360—311nomaximal, Sylow
99C22×C31—22no—
100C22×C31—41no—
101C22×C31—41no—
102C22×C31C22×D411yescharacteristic
103C2×D31—41no—
104C2×D31—41no—
105C2×D31—41no—
106C4×C31—22no—
107C4×C31—22no—
108C4×C31—21no—
109C4×C31—41no—
110C4×C31—42no—
111C4×C31#13512yes—
112C4×C31C2×#4111yescharacteristic
113C4×C312−1+411yescharacteristic
114Dic31—42no—
115Dic31—41no—
116#556—311no—
117#554D3111yescharacteristic, complemented
118#488—312no—
119#487—311no—
120#479—312no—
121#476—311no—
122#463—312no—
123#461—311no—
124C8×Q8—311no—
125C4×SD16—312no—
126C4×D8—311no—
127C2×C31—42no—
128C2×C31#54811yescharacteristic
129C2×C31C2×#13511yescharacteristic
130C2×C31#57711yescharacteristic
131D31—81no—
132C2×SD16—312no—
133C2×D8—311no—
134C4⋊Q8—311no—
135C4⋊Q8C2×D3111yescharacteristic
136#126—622no—
137#125C2×D3112yes—
138#124—312no—
139C22⋊Q8—21no—
140C22⋊Q8—22no—
141C22⋊Q8—621no—
142C4×Q8—312no—
143C4×Q8C2×D3111yescharacteristic
144C4×D4—311no—
145C4×D4C2×D3111yescharacteristic
146C4×D4C2×D3112yes—
147#117—621no—
148C2×C4⋊C4—21no—
149#107—311no—
150#106—312no—
151C4⋊C8—311no—
152C4⋊C8—312no—
153#103—312no—
154#103—312no—
155#103—312no—
156#102—311no—
157#102—311no—
158#102—312no—
159C4×C8—311no—
160C4×C8—312no—
161C31#336011yescharacteristic, complemented, Sylow
162C2×Q8—21no—
163C2×Q8—312no—
164C2×Q8C22⋊D3111yescharacteristic
165C2×D4—311no—
166C2×D4C22×D3111yescharacteristic
167C22×C4—21no—
168C22×C4—22no—
169C22×C4—621no—
170SD16—624no—
171D8—622no—
172C2×C8—311no—
173C2×C8—312no—
174C2×C8—311no—
175C4⋊C4—22no—
176C4⋊C4—21no—
177C4⋊C4—22no—
178C4⋊C4—311no—
179C4⋊C4—622no—
180C4⋊C4—621no—
181C4⋊C4—621no—
182C4⋊C4C22⋊D3111yescharacteristic
183C4⋊C4C22⋊D3112yes—
184C4⋊C4C22×D3112yes—
185C4⋊C4C22×D3111yescharacteristic
186C22⋊C4—21no—
187C22⋊C4—22no—
188C22⋊C4—622no—
189C22⋊C4—621no—
190C22⋊C4—622no—
191C42—622no—
192C42C22×D3111yescharacteristic
193C42C22×D3112yes—
194C23—21no—
195C23—621no—
196Q8—22no—
197Q8—81no—
198Q8—622no—
199Q8—622no—
200D4—21no—
201D4—621no—
202D4—621no—
203D4#9249912yes—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—42no—
207C2×C4—82no—
208C2×C4—621no—
209C2×C4—622no—
210C2×C4—1242no—
211C2×C4C2×C22⋊D3112yes—
212C2×C4C2×C22⋊D3112yes—
213C2×C4C2×C22⋊D3111yescharacteristic
214C2×C4C2×C22⋊D3111yescharacteristic
215C2×C4C23×D3111yescharacteristic
216C8—621no—
217C8—622no—
218C8—621no—
219C22—22no—
220C22—41no—
221C22—41no—
222C22—1241no—
223C22—1241no—
224C22—1241no—
225C22C22×C22⋊D3111yescharacteristic
226C4—22no—
227C4—21no—
228C4—22no—
229C4—42no—
230C4—41no—
231C4—1242no—
232C4—1241no—
233C4#1175852712yes—
234C4C2×#9249911yescharacteristic
235C4#1175856411yescharacteristic
236C2—42no—
237C2—2481no—
238C2#1452171011yescharacteristic
239C2C2×#1175852711yescharacteristic
240C2#1452186011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#554C2
socleC22×C31C22×D4
prime radicalC2×#92479C22
hypercenter#554D31
hypocenterC31#3360
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #554

Complements

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

NH ≅ G/Ntype
1. #14521772C2semidirect
2. #14521406C2semidirect
3. #14521401C2semidirect
4. #14521353C2semidirect
5. #14521336C2semidirect
7. #14521254C2semidirect
8. Q8×C31⋊C8C2semidirect
9. C4×#92480C2semidirect
11. C31×#554C2semidirect
117. #554D31semidirect
161. C31#3360semidirect

Maximal subgroups

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

subgroupindex
#145217722
#145214062
#145214012
#145213532
#145213362
#145212642
#145212542
Q8×C31⋊C82
C4×#924802
C4×#924792
C31×#5542
#336031