Finite Group Databasebeta

← #38755385

Subgroups — #38755385

Order 25216.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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
1Q8×Dic394C211yesmaximal, characteristic
2#27161804C211yesmaximal, characteristic
3#27161429C211yesmaximal, characteristic
4#27161428C212yesmaximal
5#27161375C211yesmaximal, characteristic
6#27161372C212yesmaximal
7#27161287C212yesmaximal
8#27161203C211yesmaximal, characteristic
9Q16×Dic197C212yesmaximal
10C4×#15691348C211yesmaximal, characteristic
11#462×C197C211yesmaximal, characteristic
12C2×#15691348C2211yescharacteristic
13#20263326—21no—
14#20263326C2211yescharacteristic
15#20263325—22no—
16#20263325C2212yes—
17#20263324C2212yes—
18#20263317—21no—
19#20263316—22no—
20Q8×Dic197C2212yes—
21Q8×Dic197C2212yes—
22C4×Dic394—21no—
23C4×Dic394C2211yescharacteristic
24C4×Dic394C2211yescharacteristic
25#20263291—21no—
26#20263260C2212yes—
27#20263259C2211yescharacteristic
28#20263255C2212yes—
29#20263251C2212yes—
30#20263250C2212yes—
31#20263250C2211yescharacteristic
32#20263250C2212yes—
33#20263249C2211yescharacteristic
34C4×#13096802C2211yescharacteristic
35C8×Dic197C2212yes—
36C2×Q16×C197C2212yes—
37C4⋊Q8×C197C2211yescharacteristic
38C4×Q8×C197C2211yescharacteristic
39C4⋊C8×C197C2211yescharacteristic
40#103×C197C2212yes—
41C2×Dic394—22no—
42C2×Dic394—21no—
43C2×Dic394D411yescharacteristic
44C2×Dic394C2311yescharacteristic
45#15691348—22no—
46C2×#13096802C2311yescharacteristic
47C2×#13096802C2311yescharacteristic
48C4⋊Dic197—22no—
49C4⋊Dic197—22no—
50C4⋊Dic197—21no—
51C4⋊Dic197D411yescharacteristic
52C4⋊Dic197C2312yes—
53C4⋊Dic197C2311yescharacteristic
54#15691337—22no—
55#15691337—22no—
56#15691337—22no—
57#15691337—21no—
58#15691337D412yes—
59C4×Dic197—22no—
60C4×Dic197—22no—
61C4×Dic197—21no—
62C4×Dic197C2312yes—
63C2×Q8×C197C2312yes—
64C2×Q8×C197C2311yescharacteristic
65Q16×C197—24no—
66C2×C8×C197C2312yes—
67C4⋊C4×C197—21no—
68C4⋊C4×C197—21no—
69C4⋊C4×C197—21no—
70C4⋊C4×C197C2311yescharacteristic
71C42×C197—21no—
72C42×C197C2311yescharacteristic
73Dic394—21no—
74Dic394—21no—
75Dic394—82no—
76Dic394—81no—
77Dic394Q1614yescomplemented
78C2×Dic197—21no—
79C2×Dic197—21no—
80C2×Dic197—22no—
81C2×Dic197C2×D412yes—
82C2×Dic197C2×D412yes—
83#13096802—21no—
84#13096802—21no—
85Q8×C197—21no—
86Q8×C197—22no—
87Q8×C197—22no—
88Q8×C197#4112yes—
89C2×C4×C197—21no—
90C2×C4×C197—21no—
91C2×C4×C197—22no—
92C2×C4×C197C2×D411yescharacteristic
93C2×C4×C197C2×D411yescharacteristic
94C2×C4×C197C2411yescharacteristic
95C8×C197—22no—
96C22×C197C22×D411yescharacteristic
97C4×C197—22no—
98C4×C197—21no—
99C4×C197—42no—
100C4×C197—41no—
101C4×C197C2×Q1612yes—
102C4×C197C2×#4111yescharacteristic
103C4×C1972−1+411yescharacteristic
104Dic197—22no—
105Dic197—41no—
106Dic197—41no—
107Dic197—42no—
108Dic197C2×Q1614yes—
109C2×C197#54811yescharacteristic
110C2×C197C22×Q1611yescharacteristic
111C2×C197#57711yescharacteristic
112C197#336311yescharacteristic, complemented, Sylow
113#3363—1971nomaximal, Sylow
114Q82—1971no—
115#557—1971no—
116#487—1971no—
117#487—1972no—
118#477—1971no—
119#477—1972no—
120#462—1972no—
121#462D19711yescharacteristic
122C8×Q8—1971no—
123C4×Q16—1972no—
124C4×Q16—1971no—
125C2×Q16—1971no—
126C2×Q16C2×D19712yes—
127C4⋊Q8—3941no—
128C4⋊Q8—3942no—
129C4⋊Q8—1972no—
130C4⋊Q8—1972no—
131C4⋊Q8—1971no—
132C4⋊Q8C2×D19711yescharacteristic
133#125—3942no—
134#125—3941no—
135C4×Q8—3941no—
136C4×Q8—3941no—
137C4×Q8—1972no—
138C4×Q8—1972no—
139C4×Q8—1971no—
140C4×Q8—1971no—
141C4×Q8C2×D19711yescharacteristic
142#107—1972no—
143#107—1971no—
144C4⋊C8—1972no—
145C4⋊C8C2×D19711yescharacteristic
146#103—1971no—
147#103—1972no—
148#103—1972no—
149#103—1972no—
150#103—1971no—
151#103C2×D19712yes—
152C4×C8—1972no—
153C4×C8—1971no—
154C2×Q8—3942no—
155C2×Q8—3941no—
156C2×Q8—1971no—
157C2×Q8—1971no—
158C2×Q8C22×D19712yes—
159C2×Q8C22×D19711yescharacteristic
160Q16—24no—
161Q16—3942no—
162C2×C8—1971no—
163C2×C8—1971no—
164C2×C8C22×D19712yes—
165C4⋊C4—21no—
166C4⋊C4—21no—
167C4⋊C4—21no—
168C4⋊C4—3942no—
169C4⋊C4—3942no—
170C4⋊C4—3942no—
171C4⋊C4—3942no—
172C4⋊C4—3942no—
173C4⋊C4—3941no—
174C4⋊C4—3941no—
175C4⋊C4—1972no—
176C4⋊C4—1971no—
177C4⋊C4—1972no—
178C4⋊C4—1971no—
179C4⋊C4C22×D19711yescharacteristic
180C42—21no—
181C42—3942no—
182C42—3942no—
183C42—3941no—
184C42—1972no—
185C42C22×D19711yescharacteristic
186Q8—21no—
187Q8—22no—
188Q8—22no—
189Q8—15761no—
190Q8—15762no—
191Q8—3941no—
192Q8—3941no—
193Q8—1974no—
194Q8#1569135712yes—
195C2×C4—21no—
196C2×C4—21no—
197C2×C4—22no—
198C2×C4—3941no—
199C2×C4—3941no—
200C2×C4—3942no—
201C2×C4—1972no—
202C2×C4—1972no—
203C2×C4D4×D19711yescharacteristic
204C2×C4D4×D19711yescharacteristic
205C2×C4C23×D19711yescharacteristic
206C8—22no—
207C8—3941no—
208C8—3941no—
209C22C2×D4×D19711yescharacteristic
210C4—21no—
211C4—22no—
212C4—41no—
213C4—42no—
214C4—7882no—
215C4—7881no—
216C4—7881no—
217C4—3942no—
218C4—1974no—
219C4Q16×D19712yes—
220C4C2×#1569135711yescharacteristic
221C4#2026337311yescharacteristic
222C2#2716173511yescharacteristic
223C2C2×Q16×D19711yescharacteristic
224C2#2716188811yescharacteristic
subgroupG/H
centerC22C2×D4×D197
commutator, cosocleC2×C4×C197C24
FrattiniC2×C4C23×D197
Fitting#462×C197C2
socle, prime radicalC22×C197C22×D4
hypercenter#462D197
hypocenterC197#3363
LCSG ▷ C2×C4×C197 ▷ C2×C197 ▷ C197
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
77. Dic394Q16semidirect
112. C197#3363semidirect

Maximal subgroups

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

subgroupindex
Q8×Dic3942
#271618042
#271614292
#271614282
#271613752
#271613722
#271612872
#271612032
Q16×Dic1972
C4×#156913482
#462×C1972
#3363197