Finite Group Databasebeta

← #16846747

Subgroups — #16846747

Order 3968.

362 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 75 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×#11758519C211yesmaximal, characteristic
2#14521623C211yesmaximal, characteristic
3C2×#11758393C211yesmaximal, characteristic
4#14521058C211yesmaximal, characteristic, complemented
5C2×#11758424C211yesmaximal, characteristic, complemented
6#14520809C211yesmaximal, characteristic, complemented
7C2×C31×C4⋊Q8C211yesmaximal, characteristic, complemented
8C22×Q8×C31C2211yescharacteristic
9C22×Q8×C31C2211yescharacteristic
10C2×#92495C2211yescharacteristic
11C31×C4⋊Q8—22no—
12C31×C4⋊Q8—22no—
13C2×C4⋊C4×C31—22no—
14C2×C42×C31C2211yescharacteristic
15#11758524—21no—
16#11758523—22no—
17#11758521—21no—
18C2×#92482—21no—
19#11758519—22no—
20#11758519—22no—
21C2×#92481—21no—
22#11758506—22no—
23#11758503C2211yescharacteristic
24C2×#92474C2211yescharacteristic
25C2×#92474C2211yescharacteristic
26#11758424—22no—
27#11758423—22no—
28#11758393—22no—
29C2×Q8×C31—21no—
30C2×Q8×C31—21no—
31C2×Q8×C31—21no—
32C2×Q8×C31—21no—
33C2×Q8×C31—21no—
34C2×Q8×C31—21no—
35C2×Q8×C31—21no—
36C2×Q8×C31—21no—
37C2×Q8×C31—21no—
38C2×Q8×C31—21no—
39C2×Q8×C31—21no—
40C2×Q8×C31D412yes—
41C22×C4×C31—21no—
42C22×C4×C31—21no—
43C22×C4×C31D412yes—
44C22×C4×C31C2311yescharacteristic
45C2×C22⋊D31—21no—
46#92495—21no—
47#92495—21no—
48#92495—21no—
49#92495—21no—
50C2×D124D411yescharacteristic
51C2×C4×D31—21no—
52C2×Dic62D411yescharacteristic
53C4⋊C4×C31—28no—
54C42×C31—21no—
55C42×C31D412yes—
56C22⋊Dic31—21no—
57#92482—24no—
58#92481—24no—
59(C2×D31)⋊C4—24no—
60C4⋊Dic31D412yes—
61Dic31⋊C4—24no—
62C4×Dic31—21no—
63#92474—21no—
64#92474—21no—
65#92474—21no—
66#92474D412yes—
67C2×C31⋊C8—21no—
68C2×C31⋊C8—21no—
69C23×C31C2×D411yescharacteristic
70C22×D31—21no—
71Q8×C31—22no—
72Q8×C31—22no—
73Q8×C31—41no—
74Q8×C31—41no—
75Q8×C31—41no—
76Q8×C31—41no—
77Q8×C31—41no—
78Q8×C31—41no—
79C2×C4×C31—24no—
80C2×C4×C31—22no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—22no—
85C2×C4×C31—41no—
86C2×C4×C31—41no—
87C2×C4×C31C2×D411yescharacteristic
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31C2×D411yescharacteristic
92C2×C4×C31#4111yescharacteristic
93C22⋊D31—41no—
94C22⋊D31—41no—
95C2×Dic31—22no—
96C2×Dic31—21no—
97D124—21no—
98D124—21no—
99C4×D31—41no—
100C4×D31—41no—
101Dic62—21no—
102Dic62—21no—
103C31⋊C8—22no—
104C31⋊C8—41no—
105#2005—311nomaximal, Sylow
106C22×C31—22no—
107C22×C31#12011yescharacteristic
108C22×C31#12011yescharacteristic
109C22×C31#12111yescharacteristic
110C2×D31—41no—
111C2×D31—42no—
112C4×C31—24no—
113C4×C31—42no—
114C4×C31—42no—
115C4×C31#12011yescharacteristic
116C4×C31#12111yescharacteristic
117C4×C31#12111yescharacteristic
118C4×C31#12711yescharacteristic
119Dic31—41no—
120Dic31—42no—
121C2×#137—311no—
122#536—311no—
123C2×C4⋊Q8D3111yescharacteristic, complemented
124C2×#104—311no—
125#419—311no—
126C2×#101—311no—
127#339—311no—
128C2×C31—21no—
129C2×C31—21no—
130C2×C31#39211yescharacteristic
131C2×C31#45612yes—
132D31—81no—
133C2×#41—311no—
134C22×Q8C2×D3111yescharacteristic
135C22×Q8C2×D3111yescharacteristic
136#137—622no—
137#137—622no—
138C2×Q16—621no—
139C2×SD16—621no—
140C2×M16—311no—
141C2×M16—311no—
142C4⋊Q8—22no—
143C4⋊Q8—22no—
144C4⋊Q8—621no—
145#124—621no—
146#123—622no—
147C22⋊Q8—622no—
148#117—311no—
149C2×C4⋊C4—22no—
150C2×C42C2×D3111yescharacteristic
151#104—622no—
152#103—622no—
153#101—622no—
154C31#200511yescharacteristic, complemented, Sylow
155#41—621no—
156#41—621no—
157#41—621no—
158#41—621no—
159C2×Q8—21no—
160C2×Q8—21no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8—21no—
168C2×Q8—21no—
169C2×Q8—21no—
170C2×Q8—311no—
171C2×Q8C22⋊D3112yes—
172C2×D4—311no—
173C2×D4—621no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—621no—
177C22×C4C22⋊D3112yes—
178C22×C4C22×D3111yescharacteristic
179Q16—624no—
180SD16—624no—
181M16—312no—
182M16—621no—
183M16—621no—
184M16—621no—
185C2×C8—621no—
186C2×C8—621no—
187C4⋊C4—28no—
188C4⋊C4—312no—
189C4⋊C4—624no—
190C22⋊C4—624no—
191C22⋊C4—621no—
192C42—21no—
193C42—621no—
194C42C22⋊D3112yes—
195C23—621no—
196C23C2×C22⋊D3111yescharacteristic
197Q8—22no—
198Q8—22no—
199Q8—41no—
200Q8—41no—
201Q8—41no—
202Q8—41no—
203Q8—41no—
204Q8—41no—
205Q8—621no—
206Q8—621no—
207D4—621no—
208D4—621no—
209D4—1241no—
210D4—1241no—
211C2×C4—24no—
212C2×C4—22no—
213C2×C4—22no—
214C2×C4—22no—
215C2×C4—22no—
216C2×C4—22no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—622no—
220C2×C4—621no—
221C2×C4—1241no—
222C2×C4—1241no—
223C2×C4D4×D3111yescharacteristic
224C2×C4D4×D3111yescharacteristic
225C2×C4D4×D3111yescharacteristic
226C2×C4#9249711yescharacteristic
227C2×C4C2×C22⋊D3111yescharacteristic
228C2×C4C2×C22⋊D3111yescharacteristic
229C8—622no—
230C8—1241no—
231C22—22no—
232C22—1241no—
233C22—1242no—
234C22#1175851411yescharacteristic
235C22#1175851611yescharacteristic
236C22#1175853011yescharacteristic
237C4—24no—
238C4—42no—
239C4—42no—
240C4—1241no—
241C4—1242no—
242C4#1175851411yescharacteristic
243C4#1175851511yescharacteristic
244C4#1175851611yescharacteristic
245C4#1175851711yescharacteristic
246C2—21no—
247C2—21no—
248C2—2481no—
249C2#1452095711yescharacteristic
250C2#1452123112yes—
subgroupG/H
centerC22#11758514
commutator, cosocleC22×C4×C31C23
FrattiniC22×C4C22×D31
FittingC2×C31×C4⋊Q8C2
socleC22×C31#120
prime radicalC2×#92495C22
hypercenterC2×C4⋊Q8D31
hypocenterC31#2005
LCSG ▷ C22×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
4. #14521058C2semidirect
5. C2×#11758424C2semidirect
6. #14520809C2semidirect
7. C2×C31×C4⋊Q8C2semidirect
123. C2×C4⋊Q8D31semidirect
154. C31#2005semidirect

Maximal subgroups

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

subgroupindex
C2×#117585192
#145216232
C2×#117583932
#145210582
C2×#117584242
#145208092
C2×C31×C4⋊Q82
#200531