Finite Group Databasebeta

← #27142413

Subgroups — #27142413

Order 12416.

256 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 95 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×#15690735C211yesmaximal, characteristic
2C2×#15690733C211yesmaximal, characteristic
3C2×C8×Dic97C211yesmaximal, characteristic
4#20260640C211yesmaximal, characteristic
5#20260634C211yesmaximal, characteristic
6#20260497C211yesmaximal, characteristic
7#340×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C22×#11715895C2211yescharacteristic
10C2×C4⋊Dic97—21no—
11C2×C4⋊Dic97C2211yescharacteristic
12C2×C4×Dic97C2211yescharacteristic
13#15690735—22no—
14#15690733C414yescomplemented
15C8×Dic97—22no—
16#15690717—21no—
17#15690717—21no—
18#15690717—21no—
19C22×C8×C97C2211yescharacteristic
20C2×C4⋊C4×C97C2211yescharacteristic
21C2×C4⋊C4×C97C2211yescharacteristic
22C2×Dic194—21no—
23C2×Dic194—21no—
24C2×Dic194—21no—
25C2×Dic194—21no—
26C2×Dic194—21no—
27C2×Dic194C2×C412yes—
28C22×Dic97—21no—
29C22×Dic97—21no—
30C22×Dic97D411yescharacteristic
31C22×Dic97D411yescharacteristic
32C2×#11715895—21no—
33C2×#11715895—21no—
34C2×#11715895C2×C412yes—
35C4⋊Dic97—21no—
36C4⋊Dic97—21no—
37C4⋊Dic97—42no—
38C4×Dic97—21no—
39C4×Dic97—21no—
40C22×C4×C97—21no—
41C22×C4×C97—21no—
42C22×C4×C97C2311yescharacteristic
43C2×C8×C97—21no—
44C2×C8×C97—21no—
45C2×C8×C97D412yes—
46C4⋊C4×C97—21no—
47C4⋊C4×C97—21no—
48C4⋊C4×C97—21no—
49C4⋊C4×C97C2×C412yes—
50C23×C97C2×D411yescharacteristic
51C2×C4×C97—22no—
52C2×C4×C97—41no—
53C2×C4×C97—41no—
54C2×C4×C97—41no—
55C2×C4×C97—41no—
56C2×C4×C97—41no—
57C2×C4×C97C22×C411yescharacteristic
58C2×C4×C97C2×D411yescharacteristic
59C2×C4×C97#4111yescharacteristic
60C2×C4×C97#4111yescharacteristic
61C2×C4×C97#4111yescharacteristic
62C2×C4×C97#4111yescharacteristic
63C2×Dic97—21no—
64C2×Dic97—21no—
65C2×Dic97—21no—
66C2×Dic97—21no—
67C2×Dic97—22no—
68C2×Dic97—41no—
69C2×Dic97—41no—
70C2×Dic97—41no—
71C2×Dic97—41no—
72C2×Dic97—41no—
73C2×Dic97SD1612yes—
74C2×Dic97Q1612yes—
75Dic194—22no—
76Dic194—22no—
77Dic194—41no—
78Dic194—41no—
79C8×C97—22no—
80#11715895—22no—
81C22×C97C4×D411yescharacteristic
82C22×C97#12111yescharacteristic
83C22×C97#12411yescharacteristic
84C22×C97C2×SD1611yescharacteristic
85C22×C97C2×Q1611yescharacteristic
86C22×C97#13511yescharacteristic
87C22×C97#13511yescharacteristic
88C4×C97—42no—
89C4×C97—81no—
90C4×C97#11711yescharacteristic
91C4×C97C4×D411yescharacteristic
92C4×C97#12311yescharacteristic
93C4×C97#12611yescharacteristic
94Dic97—22no—
95Dic97—22no—
96Dic97—42no—
97Dic97—81no—
98C2×C97#38811yescharacteristic
99C2×C97C4×SD1611yescharacteristic
100C2×C97C4×Q1611yescharacteristic
101C2×C97#46511yescharacteristic
102C2×C97#46711yescharacteristic
103C2×C97#48711yescharacteristic
104C2×C97#48811yescharacteristic
105#1933—971nomaximal, Sylow
106C97#193311yescharacteristic, complemented, Sylow
107C2×#103—971no—
108C2×#103—971no—
109C2×C4×C8—971no—
110#391—971no—
111#389—971no—
112#340—971no—
113#340D9711yescharacteristic
114C22×Q8—971no—
115C22×C8—971no—
116C22×C8C2×D9711yescharacteristic
117C2×C4⋊C4—971no—
118C2×C4⋊C4—1941no—
119C2×C4⋊C4C2×D9711yescharacteristic
120C2×C4⋊C4C2×D9711yescharacteristic
121C2×C42—971no—
122#103—974no—
123#103—1942no—
124C4×C8—1942no—
125#95—1941no—
126#95—1941no—
127#95—1941no—
128C2×Q8—972no—
129C2×Q8—1941no—
130C2×Q8—1941no—
131C2×Q8—1941no—
132C2×Q8—1941no—
133C2×Q8—1941no—
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×C8D38812yes—
147C4⋊C4—21no—
148C4⋊C4—21no—
149C4⋊C4—21no—
150C4⋊C4—3882no—
151C4⋊C4—1941no—
152C4⋊C4—1941no—
153C4⋊C4C4×D9712yes—
154C42—1941no—
155C42—1941no—
156C23D4×D9711yescharacteristic
157Q8—3881no—
158Q8—3881no—
159Q8—1942no—
160Q8—1942no—
161C2×C4—22no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—972no—
168C2×C4—972no—
169C2×C4—3881no—
170C2×C4—3881no—
171C2×C4—3881no—
172C2×C4—3881no—
173C2×C4—3881no—
174C2×C4—1942no—
175C2×C4—1941no—
176C2×C4—1941no—
177C2×C4—1941no—
178C2×C4—1941no—
179C2×C4C2×C4×D9711yescharacteristic
180C2×C4C2×D38811yescharacteristic
181C2×C4#1309551011yescharacteristic
182C2×C4#1309551011yescharacteristic
183C2×C4#1309551111yescharacteristic
184C2×C4#1309551211yescharacteristic
185C8—22no—
186C8—1942no—
187C22#1569077011yescharacteristic
188C22#1569078511yescharacteristic
189C22#1569079611yescharacteristic
190C22SD16×D9711yescharacteristic
191C22Q16×D9711yescharacteristic
192C22#1569082911yescharacteristic
193C22#1569083111yescharacteristic
194C4—42no—
195C4—81no—
196C4—7761no—
197C4—3882no—
198C4—1942no—
199C4—1942no—
200C4#1569076811yescharacteristic
201C4C4×D38811yescharacteristic
202C4#1569079411yescharacteristic
203C4#1569080511yescharacteristic
204C2#2026062711yescharacteristic
205C2#2026082611yescharacteristic
206C2#2026082711yescharacteristic
207C2#2026096511yescharacteristic
208C2#2026097711yescharacteristic
209C2#2026108411yescharacteristic
210C2#2026108811yescharacteristic
subgroupG/H
centerC23D4×D97
commutatorC2×C4×C97C22×C4
FrattiniC22×C4C22×D97
Fitting#340×C97C2
socle, prime radicalC23×C97C2×D4
cosocleC22×C4×C97C23
hypercenter#340D97
hypocenterC97#1933
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
14. #15690733C4semidirect
106. C97#1933semidirect

Maximal subgroups

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

subgroupindex
C2×#156907352
C2×#156907332
C2×C8×Dic972
#202606402
#202606342
#202604972
#340×C972
#193397