Finite Group Databasebeta

← #27142001

Subgroups — #27142001

Order 12416.

298 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 298 conjugacy classes of subgroups — too many to draw. Only its 101 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×#15690775C211yesmaximal, characteristic
2C2×#15690735C211yesmaximal, characteristic
3C2×#15690733C211yesmaximal, characteristic
4#20260631C211yesmaximal, characteristic
5#20260497C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C2×#106×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10#15690775—22no—
11#15690775—22no—
12C2×C4⋊Dic97C2211yescharacteristic
13C2×#13095490—21no—
14C2×#13095490—21no—
15C2×C4×Dic97—21no—
16C2×C4×Dic97C2211yescharacteristic
17#15690735C414yescomplemented
18#15690733—22no—
19#15690717—21no—
20C22×C8×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×C4⋊C4×C97C2211yescharacteristic
23#106×C97—22no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194C2×C412yes—
28C2×Dic194D412yes—
29C2×Dic194Q812yes—
30C22×Dic97—21no—
31C22×Dic97—21no—
32C22×Dic97D411yescharacteristic
33C22×Dic97D411yescharacteristic
34C2×#11715895—21no—
35C2×#11715895—21no—
36C2×#11715895—21no—
37C4⋊Dic97—21no—
38C4⋊Dic97C2×C412yes—
39#13095490—24no—
40#13095490—42no—
41C4×Dic97—24no—
42C4×Dic97—21no—
43C4×Dic97—21no—
44C22×C4×C97—21no—
45C22×C4×C97—21no—
46C22×C4×C97C2311yescharacteristic
47C2×C8×C97—21no—
48C2×C8×C97—21no—
49C2×C8×C97C2×C412yes—
50C4⋊C4×C97—21no—
51C4⋊C4×C97—21no—
52C4⋊C4×C97—21no—
53C4⋊C4×C97D412yes—
54C23×C97C2×D411yescharacteristic
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×C97C2×D411yescharacteristic
64C2×C4×C97C2×Q811yescharacteristic
65C2×C4×C97#4111yescharacteristic
66C2×C4×C97#4111yescharacteristic
67C2×Dic97—22no—
68C2×Dic97—21no—
69C2×Dic97—22no—
70C2×Dic97—22no—
71C2×Dic97—21no—
72C2×Dic97—22no—
73C2×Dic97—21no—
74C2×Dic97—21no—
75C2×Dic97—21no—
76C2×Dic97—41no—
77C2×Dic97—41no—
78C2×Dic97SD1612yes—
79Dic194—22no—
80Dic194—22no—
81Dic194—22no—
82Dic194C4⋊C414yescomplemented
83C8×C97—22no—
84#11715895—41no—
85C22×C97C4×D411yescharacteristic
86C22×C97#12011yescharacteristic
87C22×C97C22⋊Q811yescharacteristic
88C22×C97C2×SD1611yescharacteristic
89C22×C97#13511yescharacteristic
90C22×C97#13711yescharacteristic
91C22×C97#13711yescharacteristic
92C4×C97—42no—
93C4×C97—81no—
94C4×C97C2×C4⋊C411yescharacteristic
95C4×C97C4×D411yescharacteristic
96C4×C97C22⋊Q811yescharacteristic
97C4×C97#12311yescharacteristic
98Dic97—24no—
99Dic97—22no—
100Dic97—42no—
101Dic97—41no—
102C2×C97#38511yescharacteristic
103C2×C97C4×SD1611yescharacteristic
104C2×C97#44111yescharacteristic
105C2×C97#44811yescharacteristic
106C2×C97#45211yescharacteristic
107C2×C97#47511yescharacteristic
108C2×C97#47911yescharacteristic
109#1865—971nomaximal, Sylow
110C97#186511yescharacteristic, complemented, Sylow
111C2×C4×Q8—971no—
112C2×#106D9711yescharacteristic
113C2×#103—971no—
114C2×#103—971no—
115#389—971no—
116#340—971no—
117#336—971no—
118C22×Q8—971no—
119C22×C8—971no—
120C22×C8C2×D9711yescharacteristic
121C4×Q8—1942no—
122C4×Q8—1942no—
123C2×C4⋊C4—971no—
124C2×C4⋊C4—1941no—
125C2×C4⋊C4—1941no—
126C2×C4⋊C4C2×D9711yescharacteristic
127C2×C4⋊C4C2×D9711yescharacteristic
128C2×C42—971no—
129C2×C42—1941no—
130#106—22no—
131#103—974no—
132#103—1942no—
133#95—1941no—
134C2×Q8—972no—
135C2×Q8—972no—
136C2×Q8—972no—
137C2×Q8—1941no—
138C2×Q8—1941no—
139C2×Q8—1941no—
140C22×C4—21no—
141C22×C4—21no—
142C22×C4—971no—
143C22×C4—971no—
144C22×C4—1941no—
145C22×C4—1941no—
146C22×C4C22×D9711yescharacteristic
147C2×C8—21no—
148C2×C8—21no—
149C2×C8—1941no—
150C2×C8—1941no—
151C2×C8—1941no—
152C2×C8C4×D9712yes—
153C4⋊C4—21no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—972no—
157C4⋊C4—3882no—
158C4⋊C4—1944no—
159C4⋊C4—1941no—
160C4⋊C4C22⋊D9712yes—
161C42—1944no—
162C42—1941no—
163C42—1941no—
164C23D4×D9711yescharacteristic
165Q8—974no—
166Q8—1942no—
167Q8—1942no—
168Q8—1942no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—41no—
175C2×C4—972no—
176C2×C4—3881no—
177C2×C4—3881no—
178C2×C4—1942no—
179C2×C4—1941no—
180C2×C4—1941no—
181C2×C4—1941no—
182C2×C4—1942no—
183C2×C4—1941no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4—1941no—
187C2×C4C2×C4×D9711yescharacteristic
188C2×C4D4×D9711yescharacteristic
189C2×C4C2×C22⋊D9711yescharacteristic
190C2×C4Q8×D9711yescharacteristic
191C2×C4#1309551111yescharacteristic
192C2×C4#1309551211yescharacteristic
193C8—22no—
194C8—3881no—
195C22#1569077211yescharacteristic
196C22#1569077811yescharacteristic
197C22#1569078811yescharacteristic
198C22SD16×D9711yescharacteristic
199C22#1569083111yescharacteristic
200C22#1569083911yescharacteristic
201C22#1569083911yescharacteristic
202C4—42no—
203C4—81no—
204C4—3881no—
205C4—3882no—
206C4—1942no—
207C4—1944no—
208C4C4⋊C4×D9711yescharacteristic
209C4C4×C22⋊D9711yescharacteristic
210C4#1569078611yescharacteristic
211C4#1569079511yescharacteristic
212C2#2026060611yescharacteristic
213C2#2026082211yescharacteristic
214C2#2026084111yescharacteristic
215C2#2026087211yescharacteristic
216C2#2026089411yescharacteristic
217C2#2026101911yescharacteristic
218C2#2026104511yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
FittingC2×#106×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenterC2×#106D97
hypocenterC97#1865
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#106

Complements

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

NH ≅ G/Ntype
17. #15690735C4semidirect
82. Dic194C4⋊C4semidirect
110. C97#1865semidirect

Maximal subgroups

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

subgroupindex
C2×#156907752
C2×#156907352
C2×#156907332
#202606312
#202604972
#202604892
C2×#106×C972
#186597