Finite Group Databasebeta

← #27142885

Subgroups — #27142885

Order 12416.

258 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 85 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×#15690801C211yesmaximal, characteristic
2C2×#15690734C211yesmaximal, characteristic
3#20260642C211yesmaximal, characteristic
4#20260498C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C2×#103×C97C211yesmaximal, characteristic
8C22×#11715895C2211yescharacteristic
9#15690801—22no—
10#15690801—22no—
11C2×C4⋊Dic97—21no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×C4⋊Dic97C2211yescharacteristic
14C2×#13095490—21no—
15C2×C4×Dic97C2211yescharacteristic
16#15690734—22no—
17#15690717—21no—
18#15690717—21no—
19C22×Q8×C97C2211yescharacteristic
20C22×C8×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22#103×C97—22no—
23C22×Dic97—21no—
24C22×Dic97—21no—
25C22×Dic97D411yescharacteristic
26C22×Dic97D411yescharacteristic
27C2×#11715895—21no—
28C2×#11715895—21no—
29C2×#11715895—21no—
30C4⋊Dic97—24no—
31C4⋊Dic97—21no—
32C4⋊Dic97—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97D412yes—
35#13095490—24no—
36C4×Dic97—21no—
37C4×Dic97—21no—
38C2×Q8×C97—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97Q812yes—
44C22×C4×C97—21no—
45C22×C4×C97—21no—
46C22×C4×C97C2311yescharacteristic
47C2×C8×C97—21no—
48C2×C8×C97—21no—
49C2×C8×C97—21no—
50C4⋊C4×C97—21no—
51C4⋊C4×C97D412yes—
52C23×C97C2×D411yescharacteristic
53Q8×C97—22no—
54Q8×C97—22no—
55Q8×C97—41no—
56Q8×C97—41no—
57C2×C4×C97—22no—
58C2×C4×C97—22no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D411yescharacteristic
64C2×C4×C97C2×D411yescharacteristic
65C2×C4×C97C2×Q811yescharacteristic
66C2×C4×C97#4111yescharacteristic
67C2×C4×C97#4111yescharacteristic
68C2×C4×C97#4111yescharacteristic
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—22no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C8×C97—41no—
82#11715895—41no—
83C22×C97#12011yescharacteristic
84C22×C97C22⋊Q811yescharacteristic
85C22×C97#12411yescharacteristic
86C22×C97#13511yescharacteristic
87C22×C97#13511yescharacteristic
88C22×C97#13711yescharacteristic
89C22×C97#13711yescharacteristic
90C4×C97—42no—
91C4×C97—42no—
92C4×C97C22⋊Q811yescharacteristic
93C4×C97C22⋊Q811yescharacteristic
94C4×C97#12411yescharacteristic
95C4×C97#12411yescharacteristic
96Dic97—41no—
97Dic97—41no—
98Dic97—42no—
99Dic97—81no—
100C2×C97#39311yescharacteristic
101C2×C97#45211yescharacteristic
102C2×C97#45211yescharacteristic
103C2×C97#47911yescharacteristic
104C2×C97#47911yescharacteristic
105C2×C97#48811yescharacteristic
106C2×C97#49111yescharacteristic
107#2026—971nomaximal, Sylow
108C97#202611yescharacteristic, complemented, Sylow
109C2×#125—971no—
110C2×#103—971no—
111C2×#103D9711yescharacteristic
112#391—971no—
113#340—971no—
114#340—971no—
115#336—971no—
116C22×Q8C2×D9711yescharacteristic
117C22×C8—971no—
118C22×C8C2×D9711yescharacteristic
119#125—1942no—
120#125—1942no—
121C2×C4⋊C4—971no—
122C2×C4⋊C4—971no—
123C2×C4⋊C4—1941no—
124C2×C4⋊C4—1941no—
125C2×C4⋊C4C2×D9711yescharacteristic
126C2×C42—971no—
127#103—22no—
128#103—1942no—
129#95—1941no—
130#95—1941no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8—21no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8Dic19412yes—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—971no—
140C22×C4—971no—
141C22×C4—1941no—
142C22×C4—1941no—
143C22×C4C22×D9711yescharacteristic
144C2×C8—21no—
145C2×C8—21no—
146C2×C8—21no—
147C2×C8—1941no—
148C2×C8—1941no—
149C2×C8—1941no—
150C4⋊C4—21no—
151C4⋊C4—972no—
152C4⋊C4—1944no—
153C4⋊C4—1944no—
154C4⋊C4—1941no—
155C4⋊C4—1941no—
156C4⋊C4—1941no—
157C4⋊C4C22⋊D9712yes—
158C42—1941no—
159C42—1941no—
160C23D4×D9711yescharacteristic
161Q8—22no—
162Q8—22no—
163Q8—41no—
164Q8—41no—
165C2×C4—22no—
166C2×C4—22no—
167C2×C4—41no—
168C2×C4—41no—
169C2×C4—41no—
170C2×C4—41no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3881no—
174C2×C4—3881no—
175C2×C4—3881no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1942no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4—1941no—
183C2×C4D4×D9711yescharacteristic
184C2×C4C2×C22⋊D9711yescharacteristic
185C2×C4C2×Dic19411yescharacteristic
186C2×C4#1309551011yescharacteristic
187C2×C4#1309551011yescharacteristic
188C2×C4#1309551111yescharacteristic
189C8—41no—
190C8—3881no—
191C22#1569077811yescharacteristic
192C22#1569078711yescharacteristic
193C22#1569079611yescharacteristic
194C22#1569083111yescharacteristic
195C22#1569083311yescharacteristic
196C22#1569083911yescharacteristic
197C22#1569084211yescharacteristic
198C4—42no—
199C4—42no—
200C4—7761no—
201C4—3881no—
202C4—3882no—
203C4—3881no—
204C4#1569078711yescharacteristic
205C4#1569078911yescharacteristic
206C4#1569079611yescharacteristic
207C4#1569079711yescharacteristic
208C2#2026064711yescharacteristic
209C2#2026089411yescharacteristic
210C2#2026089611yescharacteristic
211C2#2026104011yescharacteristic
212C2#2026104411yescharacteristic
213C2#2026109111yescharacteristic
214C2#2026110211yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socle, prime radicalC23×C97C2×D4
hypercenterC2×#103D97
hypocenterC97#2026
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
108. C97#2026semidirect

Maximal subgroups

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

subgroupindex
C2×#156908012
C2×#156907342
#202606422
#202604982
#202604972
#202604892
C2×#103×C972
#202697