Finite Group Databasebeta

← #16855009

Subgroups — #16855009

Order 3968.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#11758567C211yesmaximal, characteristic
2C2×#11758528C211yesmaximal, characteristic, complemented
3C22×Dic124C211yesmaximal, characteristic, complemented
4#14521625C211yesmaximal, characteristic, complemented
5#14521212C214yesmaximal, complemented
6#14521206C214yesmaximal, complemented
7#14521057C211yesmaximal, characteristic, complemented
8#14521015C211yesmaximal, characteristic, complemented
9C31×#416C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11#11758567—24no—
12#11758567—24no—
13C2×Q8×D31C2212yes—
14C2×#92497—22no—
15C2×#92497C2212yes—
16C2×#92495—21no—
17C2×#92495C2211yescharacteristic
18C22×Dic62—21no—
19C22×Dic62C2211yescharacteristic
20C22×Dic62C2211yescharacteristic
21C22×C8×C31C2211yescharacteristic
22C31×#117C2211yescharacteristic
23#11758528—24no—
24C2×#92482C2212yes—
25C2×#92480C2212yes—
26C2×#92474C2211yescharacteristic
27C2×Dic124—24no—
28C2×Dic124C2214yes—
29#11758474C2214yes—
30#11758463—22no—
31#11758452—21no—
32#11758446—21no—
33C31×#103C2212yes—
34C31×#102C2212yes—
35(C2×D31)⋊C8C2214yes—
36#11758398C2214yes—
37#41×C31—22no—
38#41×C31D414yes—
39C2×Q8×C31C2311yescharacteristic
40C2×D4×C31—21no—
41C2×D4×C31C2311yescharacteristic
42C22×C4×C31—21no—
43C22×C4×C31C2311yescharacteristic
44C2×C22⋊D31—22no—
45C2×C22⋊D31D412yes—
46C22×Dic31—21no—
47C22×Dic31—22no—
48C22×Dic31—21no—
49Q8×D31—24no—
50Q8×D31—24no—
51#92497—216no—
52#92497—24no—
53#92497—24no—
54#92495—28no—
55#92495—22no—
56#92495—22no—
57C2×D124D411yescharacteristic
58C2×C4×D31—22no—
59C2×C4×D31C2312yes—
60C2×Dic62—28no—
61C2×Dic62—21no—
62C2×Dic62—22no—
63C2×Dic62—22no—
64C2×Dic62—22no—
65C2×Dic62—22no—
66C2×Dic62—21no—
67C2×Dic62D411yescharacteristic
68C2×Dic62D414yes—
69C2×Dic62C2312yes—
70C2×Dic62C2312yes—
71C2×C8×C31—22no—
72C2×C8×C31C2312yes—
73C4⋊C4×C31C2312yes—
74C22⋊C4×C31—21no—
75C42×C31—21no—
76#92482—24no—
77#92480—24no—
78(C2×D31)⋊C4—24no—
79C4⋊Dic31—24no—
80#92474—22no—
81C2×C31⋊C8C2312yes—
82Dic124—28no—
83C23×C31—21no—
84C23×C31C2×D411yescharacteristic
85C22×D31C2×D412yes—
86Q8×C31—21no—
87Q8×C31C2×D412yes—
88D4×C31—24no—
89D4×C31—21no—
90D4×C31C2×D412yes—
91C2×C4×C31—21no—
92C2×C4×C31—24no—
93C2×C4×C31—22no—
94C2×C4×C31C2×D412yes—
95C2×C4×C31C2×D411yescharacteristic
96C2×C4×C31C2×D412yes—
97C2×C4×C31C2411yescharacteristic
98C22⋊D31—28no—
99C22⋊D31—24no—
100C2×Dic31—28no—
101C2×Dic31—24no—
102C2×Dic31—22no—
103C2×Dic31—22no—
104C2×Dic31—42no—
105C2×Dic31C2×D412yes—
106D124—22no—
107C4×D31—28no—
108C4×D31—22no—
109C4×D31—22no—
110Dic62—28no—
111Dic62—22no—
112Dic62—24no—
113Dic62—22no—
114Dic62—22no—
115Dic62—22no—
116Dic62—42no—
117Dic62C2×D414yes—
118C8×C31—22no—
119C31⋊C8—22no—
120#3011—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31—22no—
123C22×C31—42no—
124C22×C31#12012yes—
125C22×C31C22×D411yescharacteristic
126C2×D31—22no—
127C2×D31—24no—
128C4×C31—22no—
129C4×C31—42no—
130C4×C31#12012yes—
131C4×C31C22×D411yescharacteristic
132C4×C31C22×D411yescharacteristic
133Dic31—22no—
134Dic31—24no—
135Dic31—42no—
136C2×2−1+4—311no—
137C2×#137—311no—
138C22×Q16—311no—
139#536—311no—
140#452—314no—
141#451—314no—
142#419—311no—
143#416D3111yescharacteristic, complemented
144#408—311no—
145C2×C31—22no—
146C2×C31—42no—
147C2×C31C2×#12011yescharacteristic
148C2×C31#57812yes—
149D31—42no—
1502−1+4—624no—
1512−1+4—624no—
152C2×#41—311no—
153C2×#41—312no—
154C2×#41—621no—
155C2×#41—622no—
156C2×#41C2×D3111yescharacteristic
157C22×Q8—312no—
158C22×Q8—311no—
159C22×Q8—311no—
160C22×Q8—621no—
161#137—624no—
162C2×Q16—312no—
163C2×Q16—314no—
164C2×Q16—624no—
165C2×SD16—312no—
166C2×M16—311no—
167C22×C8C2×D3111yescharacteristic
168C4⋊Q8—621no—
169#124—621no—
170#123—622no—
171C22⋊Q8—314no—
172#117C2×D3111yescharacteristic
173#103—314no—
174#103C2×D3112yes—
175#102C2×D3112yes—
176C22⋊C8—314no—
177C31#301111yescharacteristic, complemented, Sylow
178#41—22no—
179#41—628no—
180#41—624no—
181#41—6216no—
182#41—624no—
183#41—622no—
184#41—622no—
185#41D12414yes—
186C2×Q8—311no—
187C2×Q8—314no—
188C2×Q8—312no—
189C2×Q8—312no—
190C2×Q8—622no—
191C2×Q8—628no—
192C2×Q8—624no—
193C2×Q8—624no—
194C2×Q8—622no—
195C2×Q8—621no—
196C2×Q8—621no—
197C2×Q8—622no—
198C2×Q8—622no—
199C2×Q8C22×D3111yescharacteristic
200C2×D4—21no—
201C2×D4—312no—
202C2×D4—311no—
203C2×D4—622no—
204C2×D4C22×D3111yescharacteristic
205C22×C4—21no—
206C22×C4—312no—
207C22×C4—622no—
208C22×C4—622no—
209C22×C4—621no—
210C22×C4—621no—
211C22×C4C22×D3111yescharacteristic
212Q16—628no—
213Q16—624no—
214SD16—624no—
215M16—622no—
216C2×C8—22no—
217C2×C8—312no—
218C2×C8C22×D3112yes—
219C4⋊C4—624no—
220C4⋊C4C22×D3112yes—
221C22⋊C4—21no—
222C22⋊C4—624no—
223C42—21no—
224C23—21no—
225C23—312no—
226C23D4×D3111yescharacteristic
227Q8—21no—
228Q8—314no—
229Q8—628no—
230Q8—622no—
231Q8—624no—
232Q8—622no—
233Q8—622no—
234Q8—622no—
235Q8—1242no—
236Q8C2×D12412yes—
237D4—24no—
238D4—21no—
239D4—628no—
240D4—624no—
241D4—622no—
242D4C2×D12412yes—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—312no—
247C2×C4—628no—
248C2×C4—628no—
249C2×C4—624no—
250C2×C4—622no—
251C2×C4—622no—
252C2×C4—622no—
253C2×C4—622no—
254C2×C4—1242no—
255C2×C4C2×D12412yes—
256C2×C4D4×D3111yescharacteristic
257C2×C4D4×D3112yes—
258C2×C4C23×D3111yescharacteristic
259C8—22no—
260C8—622no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—624no—
265C22—622no—
266C22#1175845912yes—
267C22C2×D4×D3111yescharacteristic
268C4—22no—
269C4—42no—
270C4—622no—
271C4—624no—
272C4—1242no—
273C4#1175845912yes—
274C4C22×D12411yescharacteristic
275C4C2×D4×D3111yescharacteristic
276C2—22no—
277C2—42no—
278C2—1242no—
279C2C2×#1175845911yescharacteristic
280C2#1452186312yes—
subgroupG/H
centerC22C2×D4×D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#416C2
socleC22×C31C22×D4
prime radicalC2×#11758567C2
hypercenter#416D31
hypocenterC31#3011
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
2. C2×#11758528C2semidirect
3. C22×Dic124C2semidirect
4. #14521625C2semidirect
5. #14521212C2semidirect
6. #14521206C2semidirect
7. #14521057C2semidirect
8. #14521015C2semidirect
9. C31×#416C2semidirect
143. #416D31semidirect
177. C31#3011semidirect

Maximal subgroups

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

subgroupindex
C2×#117585672
C2×#117585282
C22×Dic1242
#145216252
#145212122
#145212062
#145210572
#145210152
C31×#4162
#301131