Finite Group Databasebeta

← #27142834

Subgroups — #27142834

Order 12416.

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

normal subgroups only

This group has 370 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
1C22×#13095501C211yesmaximal, characteristic
2C2×#15690808C211yesmaximal, characteristic
3C2×#15690735C211yesmaximal, characteristic
4C2×#15690734C211yesmaximal, characteristic
5#20260642C211yesmaximal, characteristic
6#20260489C211yesmaximal, characteristic
7C22×Q16×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095501—22no—
10C2×#13095501—22no—
11C2×#13095501—22no—
12C22×#11715895C2211yescharacteristic
13#15690808—22no—
14#15690808—22no—
15C2×C4⋊Dic97C2211yescharacteristic
16C2×#13095490—22no—
17C2×C4×Dic97C2211yescharacteristic
18#15690735—22no—
19#15690734—22no—
20#15690717—22no—
21C22×Q8×C97C2211yescharacteristic
22C22×Q8×C97C2211yescharacteristic
23C2×Q16×C97—22no—
24C2×Q16×C97—22no—
25C2×Q16×C97—22no—
26C22×C8×C97C2211yescharacteristic
27C2×Dic194—21no—
28C2×Dic194—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194D412yes—
33C22×Dic97—21no—
34C22×Dic97—21no—
35C22×Dic97D412yes—
36#13095501—28no—
37C2×#11715895—21no—
38C2×#11715895—21no—
39C2×#11715895D412yes—
40C4⋊Dic97—21no—
41C4⋊Dic97—21no—
42#13095490—28no—
43C4×Dic97—21no—
44C4×Dic97—21no—
45C2×Q8×C97—21no—
46C2×Q8×C97—21no—
47C2×Q8×C97—21no—
48C2×Q8×C97—21no—
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97—21no—
55C2×Q8×C97D412yes—
56C2×Q8×C97D412yes—
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97C2311yescharacteristic
60Q16×C97—28no—
61C2×C8×C97—21no—
62C2×C8×C97—21no—
63C2×C8×C97D412yes—
64C23×C97C2×D411yescharacteristic
65Q8×C97—22no—
66Q8×C97—22no—
67Q8×C97—22no—
68Q8×C97—22no—
69Q8×C97—41no—
70Q8×C97—41no—
71Q8×C97—41no—
72Q8×C97—41no—
73C2×C4×C97—22no—
74C2×C4×C97—22no—
75C2×C4×C97—41no—
76C2×C4×C97—41no—
77C2×C4×C97—41no—
78C2×C4×C97—41no—
79C2×C4×C97C2×D411yescharacteristic
80C2×C4×C97C2×D411yescharacteristic
81C2×C4×C97C2×D411yescharacteristic
82C2×C4×C97C2×D411yescharacteristic
83C2×C4×C97C2×D411yescharacteristic
84C2×C4×C97#4111yescharacteristic
85C2×Dic97—22no—
86C2×Dic97—22no—
87C2×Dic97—22no—
88C2×Dic97—41no—
89C2×Dic97—41no—
90C2×Dic97—41no—
91C2×Dic97—42no—
92C2×Dic97Q1614yescomplemented
93Dic194—22no—
94Dic194—22no—
95Dic194—41no—
96Dic194—41no—
97C8×C97—22no—
98#11715895—22no—
99C22×C97#12011yescharacteristic
100C22×C97#12111yescharacteristic
101C22×C97#12711yescharacteristic
102C22×C97C2×Q1612yes—
103C22×C97#13712yes—
104C4×C97—42no—
105C4×C97—42no—
106C4×C97#12011yescharacteristic
107C4×C97#12011yescharacteristic
108C4×C97#12111yescharacteristic
109C4×C97#12111yescharacteristic
110Dic97—24no—
111Dic97—42no—
112Dic97—81no—
113C2×C97#39211yescharacteristic
114C2×C97#45112yes—
115C2×C97#46212yes—
116C2×C97#49411yescharacteristic
117C2×C97#49711yescharacteristic
118#2016—971nomaximal, Sylow
119C97#201611yescharacteristic, complemented, Sylow
120C22×Q16—971no—
121C22×Q16D9711yescharacteristic
122C2×C4⋊Q8—971no—
123C2×#103—971no—
124C2×#103—971no—
125#391—971no—
126#336—971no—
127C22×Q8—971no—
128C22×Q8C2×D9711yescharacteristic
129C22×Q8C2×D9711yescharacteristic
130C2×Q16—22no—
131C2×Q16—22no—
132C2×Q16—22no—
133C2×Q16—1942no—
134C2×Q16—1942no—
135C2×Q16—1942no—
136C22×C8—971no—
137C22×C8C2×D9711yescharacteristic
138C4⋊Q8—1942no—
139C4⋊Q8—1942no—
140C2×C4⋊C4—971no—
141C2×C4⋊C4—1942no—
142C2×C42—971no—
143#103—1942no—
144#103—1942no—
145#95—1942no—
146C2×Q8—21no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—21no—
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—972no—
157C2×Q8—1941no—
158C2×Q8—1941no—
159C2×Q8—1941no—
160C2×Q8—1941no—
161C2×Q8—1941no—
162C2×Q8C22⋊D9712yes—
163C2×Q8C22⋊D9712yes—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—972no—
167C22×C4—1941no—
168C22×C4—1941no—
169C22×C4C22×D9711yescharacteristic
170Q16—28no—
171Q16—1948no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—972no—
175C2×C8—1941no—
176C2×C8—1941no—
177C2×C8C22⋊D9712yes—
178C4⋊C4—1948no—
179C4⋊C4—1941no—
180C4⋊C4—1941no—
181C42—1941no—
182C42—1941no—
183C23D4×D9711yescharacteristic
184Q8—22no—
185Q8—22no—
186Q8—22no—
187Q8—22no—
188Q8—41no—
189Q8—41no—
190Q8—41no—
191Q8—41no—
192Q8—3881no—
193Q8—3881no—
194Q8—1942no—
195Q8—1942no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—41no—
199C2×C4—41no—
200C2×C4—41no—
201C2×C4—41no—
202C2×C4—974no—
203C2×C4—3881no—
204C2×C4—3882no—
205C2×C4—3881no—
206C2×C4—3881no—
207C2×C4—1942no—
208C2×C4—1942no—
209C2×C4—1942no—
210C2×C4D4×D9711yescharacteristic
211C2×C4D4×D9711yescharacteristic
212C2×C4C2×C22⋊D9711yescharacteristic
213C2×C4C2×C22⋊D9711yescharacteristic
214C2×C4C2×C22⋊D9711yescharacteristic
215C2×C4#1309551011yescharacteristic
216C8—22no—
217C8—1942no—
218C22#1569077811yescharacteristic
219C22#1569078211yescharacteristic
220C22#1569080611yescharacteristic
221C22Q16×D9712yes—
222C22#1569084212yes—
223C4—42no—
224C4—42no—
225C4—7761no—
226C4—3882no—
227C4—1944no—
228C4#1569077811yescharacteristic
229C4#1569077911yescharacteristic
230C4#1569078111yescharacteristic
231C4#1569078211yescharacteristic
232C2#2026064311yescharacteristic
233C2#2026088912yes—
234C2#2026094312yes—
235C2#2026111311yescharacteristic
236C2#2026112811yescharacteristic
subgroupG/H
centerC23D4×D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC22×Q16×C97C2
socle, prime radicalC23×C97C2×D4
hypercenterC22×Q16D97
hypocenterC97#2016
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
92. C2×Dic97Q16semidirect
119. C97#2016semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×#156908082
C2×#156907352
C2×#156907342
#202606422
#202604892
C22×Q16×C972
#201697