Finite Group Databasebeta

← #27142423

Subgroups — #27142423

Order 12416.

248 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 91 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×#15690734C211yesmaximal, characteristic
2C2×#15690720C211yesmaximal, characteristic
3#20260642C211yesmaximal, characteristic
4#20260631C211yesmaximal, characteristic
5#20260498C211yesmaximal, characteristic
6#20260497C211yesmaximal, characteristic
7C2×#103×C97C211yesmaximal, characteristic
8C22×#11715895C2211yescharacteristic
9C2×C4⋊Dic97C2211yescharacteristic
10C2×C4⋊Dic97C2211yescharacteristic
11C2×#13095490—21no—
12C2×C4×Dic97C2211yescharacteristic
13#15690734C414yescomplemented
14#15690720—22no—
15#15690717—21no—
16#15690717—21no—
17#15690717—21no—
18C22×Q8×C97C2211yescharacteristic
19C22×C8×C97C2211yescharacteristic
20C2×C4⋊C4×C97C2211yescharacteristic
21#103×C97—22no—
22C22×Dic97—21no—
23C22×Dic97—21no—
24C22×Dic97D411yescharacteristic
25C22×Dic97D411yescharacteristic
26C2×#11715895—21no—
27C2×#11715895—21no—
28C2×#11715895C2×C412yes—
29C4⋊Dic97—21no—
30C4⋊Dic97—21no—
31C4⋊Dic97—21no—
32C4⋊Dic97C2×C412yes—
33#13095490—42no—
34C4×Dic97—21no—
35C4×Dic97—21no—
36C2×Q8×C97—21no—
37C2×Q8×C97—21no—
38C2×Q8×C97—21no—
39C2×Q8×C97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97C2×C412yes—
42C22×C4×C97—21no—
43C22×C4×C97—21no—
44C22×C4×C97C2311yescharacteristic
45C2×C8×C97—21no—
46C2×C8×C97—21no—
47C2×C8×C97D412yes—
48C4⋊C4×C97—21no—
49C4⋊C4×C97—21no—
50C23×C97C2×D411yescharacteristic
51Q8×C97—22no—
52Q8×C97—22no—
53Q8×C97—41no—
54Q8×C97—41no—
55C2×C4×C97—22no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97C22×C411yescharacteristic
62C2×C4×C97C2×D411yescharacteristic
63C2×C4×C97#4111yescharacteristic
64C2×C4×C97#4111yescharacteristic
65C2×C4×C97#4111yescharacteristic
66C2×C4×C97#4111yescharacteristic
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—21no—
70C2×Dic97—21no—
71C2×Dic97—21no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—41no—
75C2×Dic97—41no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C8×C97—22no—
80#11715895—22no—
81C22×C97C4×D411yescharacteristic
82C22×C97#12111yescharacteristic
83C22×C97#12411yescharacteristic
84C22×C97AGL(1,ℤ/8)11yescharacteristic
85C22×C97#13711yescharacteristic
86C22×C97#13711yescharacteristic
87C22×C97#13711yescharacteristic
88C4×C97—42no—
89C4×C97—81no—
90C4×C97#11711yescharacteristic
91C4×C97C4×D411yescharacteristic
92C4×C97#12311yescharacteristic
93C4×C97#12611yescharacteristic
94Dic97—42no—
95Dic97—41no—
96Dic97—41no—
97Dic97—81no—
98C2×C97#38811yescharacteristic
99C2×C97#44011yescharacteristic
100C2×C97#44111yescharacteristic
101C2×C97#46811yescharacteristic
102C2×C97#47011yescharacteristic
103C2×C97#48911yescharacteristic
104C2×C97#49111yescharacteristic
105#1935—971nomaximal, Sylow
106C97#193511yescharacteristic, complemented, Sylow
107C2×#103—971no—
108C2×#103D9711yescharacteristic
109C2×#97—971no—
110#391—971no—
111#389—971no—
112#340—971no—
113#340—971no—
114C22×Q8C2×D9711yescharacteristic
115C22×C8—971no—
116C22×C8C2×D9711yescharacteristic
117C2×C4⋊C4—971no—
118C2×C4⋊C4—971no—
119C2×C4⋊C4—1941no—
120C2×C4⋊C4C2×D9711yescharacteristic
121C2×C42—971no—
122#103—22no—
123#103—974no—
124#97—1942no—
125#95—1941no—
126#95—1941no—
127#95—1941no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—21no—
132C2×Q8—21no—
133C2×Q8C4×D9712yes—
134C22×C4—21no—
135C22×C4—21no—
136C22×C4—971no—
137C22×C4—971no—
138C22×C4—1941no—
139C22×C4—1941no—
140C22×C4C22×D9711yescharacteristic
141C2×C8—21no—
142C2×C8—21no—
143C2×C8—972no—
144C2×C8—1941no—
145C2×C8—1941no—
146C2×C8C22⋊D9712yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—972no—
150C4⋊C4—3882no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4—1941no—
154C42—1941no—
155C42—1941no—
156C23D4×D9711yescharacteristic
157Q8—22no—
158Q8—22no—
159Q8—41no—
160Q8—41no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3881no—
168C2×C4—3881no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—3881no—
172C2×C4—1942no—
173C2×C4—1941no—
174C2×C4—1941no—
175C2×C4—1941no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4C2×C4×D9711yescharacteristic
180C2×C4C2×C22⋊D9711yescharacteristic
181C2×C4#1309551011yescharacteristic
182C2×C4#1309551011yescharacteristic
183C2×C4#1309551011yescharacteristic
184C2×C4#1309551111yescharacteristic
185C8—22no—
186C8—1942no—
187C22#1569077011yescharacteristic
188C22#1569078111yescharacteristic
189C22#1569079611yescharacteristic
190C22#1569083711yescharacteristic
191C22#1569083911yescharacteristic
192C22#1569084211yescharacteristic
193C22#1569084211yescharacteristic
194C4—42no—
195C4—81no—
196C4—7761no—
197C4—3881no—
198C4—3881no—
199C4—3882no—
200C4#1569076611yescharacteristic
201C4C4×C22⋊D9711yescharacteristic
202C4#1569079111yescharacteristic
203C4#1569080311yescharacteristic
204C2#2026062311yescharacteristic
205C2#2026083811yescharacteristic
206C2#2026083911yescharacteristic
207C2#2026097811yescharacteristic
208C2#2026099011yescharacteristic
209C2#2026109611yescharacteristic
210C2#2026110211yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC2×#103D97
hypocenterC97#1935
LCSG ▷ C2×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
13. #15690734C4semidirect
106. C97#1935semidirect

Maximal subgroups

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

subgroupindex
C2×#156907342
C2×#156907202
#202606422
#202606312
#202604982
#202604972
C2×#103×C972
#193597