Finite Group Databasebeta

← #38741827

Subgroups — #38741827

Order 25216.

314 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 59 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
1D4×D788C211yesmaximal, characteristic
2#27161453C211yesmaximal, characteristic, complemented
3#27161366C211yesmaximal, characteristic, complemented
4#27160822C211yesmaximal, characteristic, complemented
5D4⋊#13096802C211yesmaximal, characteristic, complemented
6#27160806C211yesmaximal, characteristic, complemented
7#476×C197C211yesmaximal, characteristic, complemented
8C22×D788—21no—
9C2×D4×D197—21no—
10C2×#15691345—22no—
11#20263323C2211yescharacteristic
12#20263301—21no—
13#20263300—21no—
14#20263296—22no—
15C4×D788C2211yescharacteristic
16#20263257—21no—
17C4⋊#13096802C2211yescharacteristic, complemented
18#20263246—21no—
19C4×#13096802C2211yescharacteristic, complemented
20C4⋊Q8×C197C2211yescharacteristic, complemented
21C4×D4×C197C2211yescharacteristic
22#106×C197—21no—
23C4⋊C8×C197C2211yescharacteristic, complemented
24#102×C197—21no—
25C23×D197—22no—
26C2×D788—22no—
27C2×D788—21no—
28C2×D788—21no—
29C2×D788—42no—
30C2×D788—42no—
31C2×D788D411yescharacteristic
32C2×D788D412yes—
33C2×C22⋊D197—22no—
34D4×D197—42no—
35D4×D197—42no—
36C2×C4×D197—21no—
37#15691345—44no—
38C2×#13096802—21no—
39C2×#13096802—21no—
40C4⋊Dic197D411yescharacteristic
41#15691335—22no—
42#15691335—21no—
43C2×Q8×C197—21no—
44C2×D4×C197D411yescharacteristic
45C22×C4×C197—21no—
46C2×C8×C197—21no—
47C4⋊C4×C197—21no—
48C4⋊C4×C197—21no—
49C4⋊C4×C197D411yescharacteristic
50C22⋊C4×C197—21no—
51C42×C197C2311yescharacteristic
52C22×D197—21no—
53C22×D197—22no—
54C22×D197—44no—
55C22×D197—44no—
56C22×D197—82no—
57D788—21no—
58D788—42no—
59D788—44no—
60D788—41no—
61D788—41no—
62D788—41no—
63D788—41no—
64D788SD1612yescomplemented
65C22⋊D197—44no—
66C2×Dic197—21no—
67C4×D197—41no—
68C4×D197—41no—
69#13096802—22no—
70#13096802—41no—
71C23×C197—21no—
72Q8×C197—42no—
73D4×C197—21no—
74D4×C197SD1612yescomplemented
75C2×C4×C197—21no—
76C2×C4×C197—41no—
77C2×C4×C197—41no—
78C2×C4×C197—41no—
79C2×C4×C197C2×D411yescharacteristic
80C2×C4×C197C2×D411yescharacteristic
81C2×C4×C197C2×D411yescharacteristic
82C8×C197—41no—
83C22×C197—22no—
84C22×C197—41no—
85C22×C197—41no—
86C22×C197#12011yescharacteristic
87C2×D197—22no—
88C2×D197—41no—
89C2×D197—41no—
90C2×D197—42no—
91C2×D197—42no—
92C2×D197—42no—
93C2×D197—88no—
94C4×C197—21no—
95C4×C197—41no—
96C4×C197—81no—
97C4×C197C2×SD1611yescharacteristic
98C4×C197C2×SD1611yescharacteristic
99C4×C197AGL(1,ℤ/8)11yescharacteristic
100C4×C197AGL(1,ℤ/8)11yescharacteristic
101Dic197—41no—
102C2×C197—42no—
103C2×C197#45011yescharacteristic
104C2×C197#45011yescharacteristic
105C2×C197#45311yescharacteristic
106D197—42no—
107D197—82no—
108C197#162911yescharacteristic, complemented, Sylow
109#1629—1971nomaximal, Sylow
110D42—1971no—
111#492—1971no—
112#476—1971no—
113#476D19711yescharacteristic, complemented
114#331—1971no—
115D4⋊C8—1971no—
116D4⋊C8—1971no—
117C22×D4—3941no—
118C22×D4—3941no—
119C2×SD16—3942no—
120C4⋊Q8C2×D19711yescharacteristic, complemented
121#127—1971no—
122#121—3941no—
123#121—3941no—
124#120—3942no—
125C4×D4—1971no—
126C4×D4C2×D19711yescharacteristic
127#106—21no—
128#106—3941no—
129C4⋊C8—1971no—
130C4⋊C8C2×D19711yescharacteristic, complemented
131#102—21no—
132#102—3941no—
133C4×C8—1971no—
134C24—3942no—
135C2×Q8—21no—
136C2×D4—7882no—
137C2×D4—7882no—
138C2×D4—7882no—
139C2×D4—7882no—
140C2×D4—3942no—
141C2×D4—3942no—
142C2×D4—3941no—
143C2×D4—3941no—
144C2×D4—1972no—
145C2×D4—1971no—
146C2×D4C22⋊D19711yescharacteristic
147C22×C4—21no—
148C22×C4—3941no—
149SD16—7884no—
150C2×C8—21no—
151C2×C8—3941no—
152C2×C8—3941no—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—1971no—
156C4⋊C4C22⋊D19711yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—3942no—
159C22⋊C4—3941no—
160C42C22×D19711yescharacteristic
161C23—21no—
162C23—7884no—
163C23—7884no—
164C23—15762no—
165C23—3942no—
166C23—3941no—
167Q8—42no—
168D4—21no—
169D4—7882no—
170D4—7884no—
171D4—7884no—
172D4—7881no—
173D4—7881no—
174D4—7881no—
175D4—7881no—
176D4—3941no—
177D4—1972no—
178D4#1569134512yescomplemented
179C2×C4—21no—
180C2×C4—41no—
181C2×C4—41no—
182C2×C4—41no—
183C2×C4—7881no—
184C2×C4—7881no—
185C2×C4—3941no—
186C2×C4D4×D19711yescharacteristic
187C2×C4D4×D19711yescharacteristic
188C2×C4C2×C22⋊D19711yescharacteristic
189C8—41no—
190C8—7881no—
191C8—3942no—
192C22—22no—
193C22—41no—
194C22—41no—
195C22—7881no—
196C22—7881no—
197C22—7882no—
198C22—7882no—
199C22—7882no—
200C22—15768no—
201C22—3942no—
202C22#2026329411yescharacteristic
203C4—21no—
204C4—41no—
205C4—81no—
206C4—7881no—
207C4SD16×D19711yescharacteristic
208C4C2×#1569134511yescharacteristic
209C4#2026335111yescharacteristic
210C4#2026335311yescharacteristic
211C2—42no—
212C2—7882no—
213C2—15762no—
214C2#2716122811yescharacteristic
215C2#2716122911yescharacteristic
216C2#2716124511yescharacteristic
subgroupG/H
centerC22#20263294
commutator, cosocleC42×C197C23
FrattiniC42C22×D197
Fitting#476×C197C2
socleC22×C197#120
prime radicalD4×D788C2
hypercenter#476D197
hypocenterC197#1629
LCSG ▷ C42×C197 ▷ C22×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #476

Complements

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

NH ≅ G/Ntype
2. #27161453C2semidirect
3. #27161366C2semidirect
4. #27160822C2semidirect
5. D4⋊#13096802C2semidirect
6. #27160806C2semidirect
7. #476×C197C2semidirect
17. C4⋊#13096802C22semidirect
19. C4×#13096802C22semidirect
20. C4⋊Q8×C197C22semidirect
23. C4⋊C8×C197C22semidirect
64. D788SD16semidirect
74. D4×C197SD16semidirect
108. C197#1629semidirect
113. #476D197semidirect
120. C4⋊Q8C2×D197semidirect
130. C4⋊C8C2×D197semidirect
178. D4#15691345semidirect

Maximal subgroups

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

subgroupindex
D4×D7882
#271614532
#271613662
#271608222
D4⋊#130968022
#271608062
#476×C1972
#1629197