Finite Group Databasebeta

← #16855004

Subgroups — #16855004

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×#11758519C211yesmaximal, characteristic, complemented
3C22×#92482C211yesmaximal, characteristic, complemented
4#14521209C214yesmaximal, complemented
5#14521204C214yesmaximal, complemented
6#14521057C211yesmaximal, characteristic, complemented
7#14521043C211yesmaximal, characteristic, complemented
8#14521016C211yesmaximal, characteristic, complemented
9C31×#536C211yesmaximal, characteristic, complemented
10C2×#41×C31C2211yescharacteristic
11C22×Q8×C31C2211yescharacteristic
12#11758567—24no—
13#11758567—24no—
14C2×Q8×D31—21no—
15C2×#92497—22no—
16C2×#92497—21no—
17C2×#92495C2212yes—
18C2×#92495C2211yescharacteristic
19C22×Dic62C2212yes—
20C22×Dic62C2211yescharacteristic
21C31×C4⋊Q8—21no—
22C31×#124—21no—
23C31×#123—22no—
24C31×C22⋊Q8C2214yes—
25C31×#117C2211yescharacteristic
26C2×#92482—24no—
27C2×#92482C2212yes—
28C2×#92482C2214yes—
29#11758519—24no—
30C2×#92481C2212yes—
31C2×#92474C2211yescharacteristic
32C22×C31⋊C8C2211yescharacteristic
33C22⋊(C31⋊C8)C2214yes—
34#11758398C2212yes—
35#11758398C2214yes—
36#11758397C2212yes—
37#41×C31—22no—
38#41×C31—22no—
39C2×Q8×C31—21no—
40C2×Q8×C31—22no—
41C2×Q8×C31—22no—
42C2×Q8×C31D411yescharacteristic
43C2×Q8×C31C2312yes—
44C2×D4×C31D412yes—
45C2×D4×C31D411yescharacteristic
46C22×C4×C31—21no—
47C22×C4×C31C2312yes—
48C22×C4×C31C2311yescharacteristic
49C2×C22⋊D31—22no—
50C2×C22⋊D31—21no—
51C22×Dic31—22no—
52C22×Dic31—21no—
53Q8×D31—28no—
54#92497—216no—
55#92497—28no—
56#92495—22no—
57#92495—24no—
58#92495—24no—
59#92495D414yes—
60C2×D124C2311yescharacteristic
61C2×C4×D31—22no—
62C2×C4×D31—21no—
63C2×Dic62—22no—
64C2×Dic62—21no—
65C2×Dic62—22no—
66C2×Dic62—24no—
67C2×Dic62—24no—
68C2×Dic62D414yes—
69C2×Dic62C2311yescharacteristic
70C2×Dic62C2312yes—
71C4⋊C4×C31—24no—
72C4⋊C4×C31C2312yes—
73C22⋊C4×C31—21no—
74C22⋊C4×C31—24no—
75C42×C31—21no—
76#92482—24no—
77#92482—28no—
78#92481—24no—
79#92474—22no—
80C2×C31⋊C8—22no—
81C2×C31⋊C8C2312yes—
82C2×C31⋊C8C2312yes—
83C23×C31C2×D412yes—
84C23×C31C2×D411yescharacteristic
85C22×D31—21no—
86Q8×C31—22no—
87Q8×C31—22no—
88Q8×C31—22no—
89Q8×C31—42no—
90D4×C31—24no—
91D4×C31—22no—
92C2×C4×C31—22no—
93C2×C4×C31—22no—
94C2×C4×C31—22no—
95C2×C4×C31—22no—
96C2×C4×C31—42no—
97C2×C4×C31C2×D412yes—
98C2×C4×C31C2×D412yes—
99C2×C4×C31C2×D412yes—
100C2×C4×C31C2×D411yescharacteristic
101C2×C4×C31C2411yescharacteristic
102C22⋊D31—28no—
103C22⋊D31—24no—
104C2×Dic31—28no—
105C2×Dic31—24no—
106C2×Dic31—21no—
107C2×Dic31—22no—
108D124—21no—
109D124C2×D412yes—
110C4×D31—28no—
111C4×D31—24no—
112Dic62—28no—
113Dic62—24no—
114Dic62—22no—
115Dic62—21no—
116Dic62C2×D412yes—
117Dic62C2×D414yes—
118C31⋊C8—22no—
119C31⋊C8—22no—
120#3011—311nomaximal, Sylow
121C22×C31—22no—
122C22×C31—22no—
123C22×C31—24no—
124C22×C31#12012yes—
125C22×C31C22×D411yescharacteristic
126C2×D31—22no—
127C2×D31—42no—
128C4×C31—22no—
129C4×C31—42no—
130C4×C31—42no—
131C4×C31#12012yes—
132C4×C31C22×D411yescharacteristic
133C4×C31C22×D411yescharacteristic
134Dic31—22no—
135Dic31—24no—
136C2×2−1+4—311no—
137C2×#137—311no—
138C22×Q16—311no—
139#536D3111yescharacteristic, complemented
140#452—314no—
141#451—314no—
142#419—311no—
143#416—311no—
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—622no—
155C2×#41—621no—
156C2×#41C2×D3111yescharacteristic
157C22×Q8—311no—
158C22×Q8—312no—
159C22×Q8—621no—
160C22×Q8C2×D3111yescharacteristic
161#137—624no—
162C2×Q16—314no—
163C2×Q16—312no—
164C2×Q16—624no—
165C2×SD16—312no—
166C2×M16—311no—
167C22×C8—311no—
168C4⋊Q8—21no—
169#124—21no—
170#123—22no—
171C22⋊Q8C2×D3114yes—
172#117C2×D3111yescharacteristic
173#103—312no—
174#103—314no—
175#102—312no—
176C22⋊C8—314no—
177C31#301111yescharacteristic, complemented, Sylow
178#41—22no—
179#41—22no—
180#41—314no—
181#41—624no—
182#41—624no—
183#41—6216no—
184#41—628no—
185#41—622no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—22no—
189C2×Q8—314no—
190C2×Q8—312no—
191C2×Q8—311no—
192C2×Q8—624no—
193C2×Q8—624no—
194C2×Q8—628no—
195C2×Q8—622no—
196C2×Q8—622no—
197C2×Q8—621no—
198C2×Q8C22⋊D3111yescharacteristic
199C2×Q8C22×D3112yes—
200C2×D4—311no—
201C2×D4—622no—
202C2×D4—621no—
203C2×D4C22⋊D3112yes—
204C2×D4C22⋊D3111yescharacteristic
205C22×C4—21no—
206C22×C4—622no—
207C22×C4—621no—
208C22×C4—622no—
209C22×C4—621no—
210C22×C4C22×D3112yes—
211C22×C4C22×D3111yescharacteristic
212Q16—628no—
213Q16—624no—
214SD16—624no—
215M16—622no—
216C2×C8—312no—
217C2×C8—312no—
218C2×C8—622no—
219C4⋊C4—24no—
220C4⋊C4C22×D3112yes—
221C22⋊C4—21no—
222C22⋊C4—24no—
223C42—21no—
224C23—621no—
225C23C2×C22⋊D3112yes—
226C23C2×C22⋊D3111yescharacteristic
227Q8—22no—
228Q8—22no—
229Q8—22no—
230Q8—42no—
231Q8—314no—
232Q8—312no—
233Q8—628no—
234Q8—624no—
235Q8—622no—
236Q8—621no—
237D4—24no—
238D4—22no—
239D4—312no—
240D4—628no—
241D4—624no—
242D4—621no—
243C2×C4—22no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—22no—
247C2×C4—42no—
248C2×C4—628no—
249C2×C4—628no—
250C2×C4—624no—
251C2×C4—622no—
252C2×C4—621no—
253C2×C4—624no—
254C2×C4D4×D3112yes—
255C2×C4D4×D3112yes—
256C2×C4C2×C22⋊D3112yes—
257C2×C4C2×C22⋊D3111yescharacteristic
258C2×C4C23×D3111yescharacteristic
259C8—622no—
260C8—622no—
261C22—22no—
262C22—24no—
263C22—22no—
264C22—622no—
265C22—1242no—
266C22#1175851412yes—
267C22C22×C22⋊D3111yescharacteristic
268C4—22no—
269C4—42no—
270C4—42no—
271C4—624no—
272C4—622no—
273C4#1175851412yes—
274C4C2×D4×D3111yescharacteristic
275C4C2×D4×D3111yescharacteristic
276C2—22no—
277C2—42no—
278C2—1242no—
279C2C2×#1175851411yescharacteristic
280C2#1452186412yes—
subgroupG/H
centerC22C22×C22⋊D31
commutator, cosocleC2×C4×C31C24
FrattiniC2×C4C23×D31
FittingC31×#536C2
socleC22×C31C22×D4
prime radicalC2×#11758567C2
hypercenter#536D31
hypocenterC31#3011
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C22 ◁ C2×#41 ◁ #536

Complements

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

NH ≅ G/Ntype
2. C2×#11758519C2semidirect
3. C22×#92482C2semidirect
4. #14521209C2semidirect
5. #14521204C2semidirect
6. #14521057C2semidirect
7. #14521043C2semidirect
8. #14521016C2semidirect
9. C31×#536C2semidirect
139. #536D31semidirect
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×#117585192
C22×#924822
#145212092
#145212042
#145210572
#145210432
#145210162
C31×#5362
#301131