Finite Group Databasebeta

← #16846212

Subgroups — #16846212

Order 3968.

332 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 89 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#14521847C211yesmaximal, characteristic
2#14521066C211yesmaximal, characteristic, complemented
3C2×#11758393C211yesmaximal, characteristic, complemented
4#14521022C211yesmaximal, characteristic, complemented
5#14520819C211yesmaximal, characteristic, complemented
6#14520809C211yesmaximal, characteristic, complemented
7C4×#41×C31C211yesmaximal, characteristic, complemented
8C2×#41×C31C2211yescharacteristic
9C2×#92495C2211yescharacteristic
10C4×Q8×C31—21no—
11C4×D4×C31—21no—
12C4×D4×C31—22no—
13C31×#117—21no—
14C31×#117C2211yescharacteristic
15C2×C42×C31—21no—
16C2×C42×C31C2211yescharacteristic
17#11758528C412yescomplemented
18#11758527—22no—
19#11758526C412yescomplemented
20#11758519—21no—
21#11758509—21no—
22#11758503C2211yescharacteristic
23C2×#92474C2211yescharacteristic
24C2×#92474C2211yescharacteristic
25#11758425—22no—
26#11758394—22no—
27#11758393—21no—
28#11758393—21no—
29#41×C31—21no—
30#41×C31—21no—
31#41×C31C2×C412yes—
32#41×C31D412yes—
33C2×Q8×C31D411yescharacteristic
34C2×D4×C31—21no—
35C2×D4×C31D411yescharacteristic
36C22×C4×C31—21no—
37C22×C4×C31—22no—
38C22×C4×C31—21no—
39C22×C4×C31C2311yescharacteristic
40C2×C22⋊D31—21no—
41#92495—21no—
42#92495—21no—
43#92495—21no—
44#92495—21no—
45C2×D124C2×C411yescharacteristic, complemented
46C2×C4×D31—21no—
47C2×Dic62C2×C411yescharacteristic, complemented
48C4⋊C4×C31—22no—
49C4⋊C4×C31—21no—
50C22⋊C4×C31—22no—
51C22⋊C4×C31D412yes—
52C42×C31—22no—
53C42×C31—21no—
54C42×C31—21no—
55C42×C31D411yescharacteristic
56C42×C31D411yescharacteristic
57C22⋊Dic31—21no—
58#92482—22no—
59#92481—22no—
60#92480—22no—
61#92479—22no—
62C4⋊Dic31—21no—
63C4×Dic31—21no—
64#92474—21no—
65#92474—21no—
66#92474—21no—
67#92474C2×C412yescomplemented
68C2×C31⋊C8—21no—
69C2×C31⋊C8—21no—
70C23×C31—21no—
71C23×C31C2×D411yescharacteristic
72C22×D31—21no—
73Q8×C31—21no—
74Q8×C31C22⋊C412yes—
75D4×C31—22no—
76D4×C31—22no—
77D4×C31—21no—
78D4×C31C22⋊C412yes—
79C2×C4×C31—21no—
80C2×C4×C31—21no—
81C2×C4×C31—22no—
82C2×C4×C31—22no—
83C2×C4×C31—22no—
84C2×C4×C31—21no—
85C2×C4×C31—42no—
86C2×C4×C31—44no—
87C2×C4×C31C22×C411yescharacteristic
88C2×C4×C31C2×D411yescharacteristic
89C2×C4×C31C2×D411yescharacteristic
90C2×C4×C31C2×D411yescharacteristic
91C2×C4×C31#4111yescharacteristic
92C2×C4×C31#4111yescharacteristic
93C22⋊D31—41no—
94C22⋊D31—41no—
95C2×Dic31—21no—
96C2×Dic31—41no—
97D124—21no—
98D124—21no—
99C4×D31—41no—
100C4×D31—41no—
101Dic62—21no—
102Dic62—21no—
103C31⋊C8—22no—
104C31⋊C8—41no—
105#1906—311nomaximal, Sylow
106C22×C31—22no—
107C22×C31—41no—
108C22×C31—42no—
109C22×C31C4×D411yescharacteristic
110C22×C31#12011yescharacteristic
111C22×C31#12111yescharacteristic
112C2×D31—41no—
113C2×D31—81no—
114C4×C31—22no—
115C4×C31—22no—
116C4×C31—42no—
117C4×C31—41no—
118C4×C31C2×C22⋊C411yescharacteristic
119C4×C31C4×D411yescharacteristic
120C4×C31#12111yescharacteristic
121C4×C31#12311yescharacteristic
122Dic31—41no—
123Dic31—81no—
124#575—311no—
125C4×#41D3111yescharacteristic, complemented
126#421—311no—
127C2×#104—311no—
128#410—311no—
129#344—311no—
130#339—311no—
131C2×C31—21no—
132C2×C31—21no—
133C2×C31—21no—
134C2×C31—42no—
135C2×C31#38611yescharacteristic
136D31—81no—
137C2×#41—311no—
138C2×#41C2×D3111yescharacteristic
139#137—312no—
140#137—621no—
141AGL(1,ℤ/8)—312no—
142AGL(1,ℤ/8)—621no—
143#135—622no—
144C2×M16—311no—
145C2×M16—311no—
146C4×Q8—21no—
147C4×D4—21no—
148C4×D4—22no—
149#117—21no—
150#117—311no—
151#117C2×D3111yescharacteristic
152C2×C42—21no—
153C2×C42C2×D3111yescharacteristic
154#104—622no—
155#104—621no—
156#104—621no—
157#99—622no—
158C31#190611yescharacteristic, complemented, Sylow
159#41—21no—
160#41—21no—
161#41—621no—
162#41—621no—
163#41—621no—
164#41—621no—
165#41C4×D3112yes—
166#41D12412yes—
167C2×Q8—311no—
168C2×Q8C22⋊D3111yescharacteristic
169C2×D4—21no—
170C2×D4—311no—
171C2×D4—621no—
172C2×D4C22⋊D3111yescharacteristic
173C22×C4—21no—
174C22×C4—22no—
175C22×C4—21no—
176C22×C4—621no—
177C22×C4C22×D3111yescharacteristic
178Q16—622no—
179SD16—622no—
180SD16—622no—
181D8—622no—
182M16—312no—
183M16—621no—
184M16—621no—
185M16—621no—
186C2×C8—621no—
187C2×C8—621no—
188C4⋊C4—22no—
189C4⋊C4—21no—
190C4⋊C4—621no—
191C22⋊C4—22no—
192C22⋊C4—621no—
193C22⋊C4C22⋊D3112yes—
194C42—22no—
195C42—21no—
196C42—21no—
197C42—621no—
198C42C22⋊D3111yescharacteristic
199C42C22⋊D3111yescharacteristic
200C23—21no—
201C23—621no—
202C23C2×C22⋊D3111yescharacteristic
203Q8—21no—
204Q8—621no—
205Q8—621no—
206Q8(C2×D31)⋊C412yes—
207D4—22no—
208D4—22no—
209D4—21no—
210D4—621no—
211D4—621no—
212D4—1241no—
213D4—1241no—
214D4(C2×D31)⋊C412yes—
215C2×C4—21no—
216C2×C4—21no—
217C2×C4—22no—
218C2×C4—22no—
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—42no—
222C2×C4—44no—
223C2×C4—621no—
224C2×C4—1241no—
225C2×C4—1241no—
226C2×C4—1241no—
227C2×C4C2×C4×D3111yescharacteristic
228C2×C4C2×D12411yescharacteristic
229C2×C4#9249511yescharacteristic
230C2×C4#9249511yescharacteristic
231C2×C4C2×C22⋊D3111yescharacteristic
232C2×C4C2×C22⋊D3111yescharacteristic
233C8—622no—
234C8—1241no—
235C22—22no—
236C22—41no—
237C22—42no—
238C22—1241no—
239C22—2481no—
240C22C4×C22⋊D3111yescharacteristic
241C22#1175850711yescharacteristic
242C22#1175853011yescharacteristic
243C4—22no—
244C4—22no—
245C4—42no—
246C4—41no—
247C4—1241no—
248C4—2481no—
249C4C2×(C2×D31)⋊C411yescharacteristic
250C4C4×C22⋊D3111yescharacteristic
251C4#1175850611yescharacteristic
252C4#1175850711yescharacteristic
253C2—21no—
254C2—21no—
255C2—21no—
256C2—42no—
257C2—2481no—
258C2#1452092811yescharacteristic
subgroupG/H
centerC4#11758507
commutatorC2×C4×C31C22×C4
FrattiniC22×C4C22×D31
FittingC4×#41×C31C2
socleC2×C31#386
prime radical#14521847C2
cosocleC22×C4×C31C23
hypercenterC4×#41D31
hypocenterC31#1906
LCSG ▷ C2×C4×C31 ▷ C2×C31 ▷ C31
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#41

Complements

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

NH ≅ G/Ntype
2. #14521066C2semidirect
3. C2×#11758393C2semidirect
4. #14521022C2semidirect
5. #14520819C2semidirect
6. #14520809C2semidirect
7. C4×#41×C31C2semidirect
17. #11758528C4semidirect
19. #11758526C4semidirect
45. C2×D124C2×C4semidirect
47. C2×Dic62C2×C4semidirect
67. #92474C2×C4semidirect
125. C4×#41D31semidirect
158. C31#1906semidirect

Maximal subgroups

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

subgroupindex
#145218472
#145210662
C2×#117583932
#145210222
#145208192
#145208092
C4×#41×C312
#190631