Finite Group Databasebeta

← #27142875

Subgroups — #27142875

Order 12416.

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

normal subgroups only

This group has 290 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
1C2×#15690809C211yesmaximal, characteristic
2C2×#15690734C211yesmaximal, characteristic
3#20260642C211yesmaximal, characteristic
4#20260498C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C2×#103×C97C211yesmaximal, characteristic
8C22×Dic194—21no—
9C22×#11715895C2211yescharacteristic
10#15690809—22no—
11#15690809—22no—
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—
23C2×Dic194—24no—
24C2×Dic194—42no—
25C2×Dic194—42no—
26C22×Dic97—21no—
27C22×Dic97—21no—
28C22×Dic97D411yescharacteristic
29C22×Dic97D411yescharacteristic
30C2×#11715895—21no—
31C2×#11715895—21no—
32C2×#11715895—21no—
33C4⋊Dic97—21no—
34C4⋊Dic97—21no—
35C4⋊Dic97—21no—
36C4⋊Dic97D412yes—
37#13095490—24no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C2×Q8×C97—21no—
41C2×Q8×C97—21no—
42C2×Q8×C97—21no—
43C2×Q8×C97—21no—
44C2×Q8×C97—21no—
45C2×Q8×C97Q812yescomplemented
46C22×C4×C97—21no—
47C22×C4×C97—21no—
48C22×C4×C97C2311yescharacteristic
49C2×C8×C97—21no—
50C2×C8×C97—21no—
51C2×C8×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97D412yes—
54C23×C97C2×D411yescharacteristic
55Q8×C97—22no—
56Q8×C97—22no—
57Q8×C97—41no—
58Q8×C97—41no—
59C2×C4×C97—22no—
60C2×C4×C97—22no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97—41no—
64C2×C4×C97—41no—
65C2×C4×C97C2×D411yescharacteristic
66C2×C4×C97C2×D411yescharacteristic
67C2×C4×C97C2×Q811yescharacteristic
68C2×C4×C97#4111yescharacteristic
69C2×C4×C97#4111yescharacteristic
70C2×C4×C97#4111yescharacteristic
71C2×Dic97—21no—
72C2×Dic97—21no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—22no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97—41no—
79C2×Dic97—41no—
80C2×Dic97—41no—
81C2×Dic97SD1612yes—
82C2×Dic97Q1612yes—
83Dic194—48no—
84C8×C97—41no—
85#11715895—41no—
86C22×C97#12011yescharacteristic
87C22×C97C22⋊Q811yescharacteristic
88C22×C97#12411yescharacteristic
89C22×C97C2×SD1611yescharacteristic
90C22×C97C2×Q1611yescharacteristic
91C22×C97AGL(1,ℤ/8)11yescharacteristic
92C22×C97#13711yescharacteristic
93C4×C97—42no—
94C4×C97—42no—
95C4×C97C22⋊Q811yescharacteristic
96C4×C97C22⋊Q811yescharacteristic
97C4×C97#12411yescharacteristic
98C4×C97#12411yescharacteristic
99Dic97—22no—
100Dic97—22no—
101Dic97—42no—
102Dic97—81no—
103C2×C97#39311yescharacteristic
104C2×C97#45011yescharacteristic
105C2×C97#45111yescharacteristic
106C2×C97#47511yescharacteristic
107C2×C97#47711yescharacteristic
108C2×C97#48711yescharacteristic
109C2×C97#48911yescharacteristic
110#2024—971nomaximal, Sylow
111C97#202411yescharacteristic, complemented, Sylow
112C2×C4⋊Q8—971no—
113C2×#103—971no—
114C2×#103D9711yescharacteristic
115#391—971no—
116#340—971no—
117#340—971no—
118#336—971no—
119C22×Q8—1941no—
120C22×Q8C2×D9711yescharacteristic
121C22×C8—971no—
122C22×C8C2×D9711yescharacteristic
123C4⋊Q8—1942no—
124C4⋊Q8—1942no—
125C2×C4⋊C4—971no—
126C2×C4⋊C4—971no—
127C2×C4⋊C4—1941no—
128C2×C4⋊C4C2×D9711yescharacteristic
129C2×C42—971no—
130#103—22no—
131#103—1942no—
132#95—1941no—
133#95—1941no—
134C2×Q8—21no—
135C2×Q8—21no—
136C2×Q8—21no—
137C2×Q8—21no—
138C2×Q8—21no—
139C2×Q8—3882no—
140C2×Q8—3882no—
141C2×Q8—1944no—
142C2×Q8Dic19412yescomplemented
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—971no—
146C22×C4—971no—
147C22×C4—1941no—
148C22×C4—1941no—
149C22×C4C22×D9711yescharacteristic
150C2×C8—21no—
151C2×C8—21no—
152C2×C8—21no—
153C2×C8—1941no—
154C2×C8—1941no—
155C2×C8—1941no—
156C4⋊C4—21no—
157C4⋊C4—972no—
158C4⋊C4—1944no—
159C4⋊C4—1941no—
160C4⋊C4—1941no—
161C4⋊C4—1941no—
162C4⋊C4C22⋊D9712yes—
163C42—1941no—
164C42—1941no—
165C23D4×D9711yescharacteristic
166Q8—22no—
167Q8—22no—
168Q8—41no—
169Q8—41no—
170Q8—3888no—
171C2×C4—22no—
172C2×C4—22no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—41no—
176C2×C4—41no—
177C2×C4—972no—
178C2×C4—972no—
179C2×C4—3881no—
180C2×C4—3881no—
181C2×C4—3881no—
182C2×C4—3881no—
183C2×C4—3881no—
184C2×C4—1942no—
185C2×C4—1941no—
186C2×C4—1941no—
187C2×C4—1941no—
188C2×C4—1941no—
189C2×C4D4×D9711yescharacteristic
190C2×C4C2×C22⋊D9711yescharacteristic
191C2×C4C2×Dic19411yescharacteristic
192C2×C4#1309551011yescharacteristic
193C2×C4#1309551011yescharacteristic
194C2×C4#1309551111yescharacteristic
195C8—41no—
196C8—3881no—
197C22#1569077811yescharacteristic
198C22#1569078711yescharacteristic
199C22#1569079611yescharacteristic
200C22SD16×D9711yescharacteristic
201C22Q16×D9711yescharacteristic
202C22#1569083711yescharacteristic
203C22#1569084211yescharacteristic
204C4—42no—
205C4—42no—
206C4—7761no—
207C4—3882no—
208C4—1942no—
209C4—1942no—
210C4#1569078711yescharacteristic
211C4#1569078911yescharacteristic
212C4#1569079611yescharacteristic
213C4#1569079711yescharacteristic
214C2#2026064711yescharacteristic
215C2#2026088411yescharacteristic
216C2#2026088911yescharacteristic
217C2#2026101811yescharacteristic
218C2#2026102811yescharacteristic
219C2#2026108411yescharacteristic
220C2#2026109611yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×#103×C97C2
socle, prime radicalC23×C97C2×D4
hypercenterC2×#103D97
hypocenterC97#2024
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
45. C2×Q8×C97Q8semidirect
111. C97#2024semidirect
142. C2×Q8Dic194semidirect

Maximal subgroups

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

subgroupindex
C2×#156908092
C2×#156907342
#202606422
#202604982
#202604972
#202604892
C2×#103×C972
#202497