Finite Group Databasebeta

← #27142744

Subgroups — #27142744

Order 12416.

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

normal subgroups only

This group has 392 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, complemented
2C2×#15690789C211yesmaximal, characteristic
3C2×#15690734C211yesmaximal, characteristic, complemented
4C2×#15690733C211yesmaximal, characteristic, complemented
5C2×C22⋊#11715895C211yesmaximal, characteristic
6#20260497C211yesmaximal, characteristic, complemented
7C2×C22⋊Q8×C97C211yesmaximal, characteristic
8C22×Dic194C2211yescharacteristic
9C2×#13095501—22no—
10C2×#13095501—22no—
11C2×#13095501—22no—
12C22×#11715895C2211yescharacteristic
13C22×#11715895C2211yescharacteristic
14#15690789—22no—
15#15690789—22no—
16C2×C4⋊Dic97C2211yescharacteristic
17C2×#13095490—21no—
18C2×C22⋊Dic97—21no—
19#15690734—22no—
20#15690733—22no—
21C22⋊#11715895—22no—
22C22×Q8×C97C2211yescharacteristic
23C23×C4×C97C2211yescharacteristic
24C22⋊Q8×C97—22no—
25C22⋊Q8×C97—22no—
26C2×C4⋊C4×C97—21no—
27C2×C4⋊C4×C97C2211yescharacteristic
28C2×C22⋊C4×C97—21no—
29C2×Dic194—21no—
30C2×Dic194—21no—
31C2×Dic194—21no—
32C2×Dic194—21no—
33C2×Dic194—21no—
34C2×Dic194D412yes—
35C22×Dic97—21no—
36C22×Dic97—21no—
37#13095501—28no—
38C2×#11715895—21no—
39C2×#11715895—21no—
40C2×#11715895—21no—
41C2×#11715895—21no—
42C2×#11715895—21no—
43C2×#11715895D412yes—
44C4⋊Dic97—21no—
45C4⋊Dic97D412yes—
46#13095490—24no—
47C22⋊Dic97—24no—
48C24×C97D411yescharacteristic
49C2×Q8×C97—21no—
50C2×Q8×C97—21no—
51C2×Q8×C97—21no—
52C2×Q8×C97—21no—
53C2×Q8×C97—21no—
54C2×Q8×C97D412yes—
55C22×C4×C97—21no—
56C22×C4×C97—21no—
57C22×C4×C97—21no—
58C22×C4×C97—21no—
59C22×C4×C97—21no—
60C22×C4×C97—21no—
61C22×C4×C97—21no—
62C22×C4×C97—21no—
63C22×C4×C97D411yescharacteristic
64C22×C4×C97C2311yescharacteristic
65C4⋊C4×C97—24no—
66C4⋊C4×C97—21no—
67C4⋊C4×C97D412yes—
68C22⋊C4×C97—24no—
69C23×C97—21no—
70C23×C97—21no—
71C23×C97—21no—
72C23×C97—21no—
73C23×C97—21no—
74C23×C97—21no—
75C23×C97Q1612yescomplemented
76C23×C97C2×D411yescharacteristic
77Q8×C97—22no—
78Q8×C97—22no—
79Q8×C97—41no—
80Q8×C97—41no—
81C2×C4×C97—22no—
82C2×C4×C97—22no—
83C2×C4×C97—21no—
84C2×C4×C97—21no—
85C2×C4×C97—21no—
86C2×C4×C97—41no—
87C2×C4×C97—41no—
88C2×C4×C97—41no—
89C2×C4×C97—41no—
90C2×C4×C97—41no—
91C2×C4×C97—41no—
92C2×C4×C97—41no—
93C2×C4×C97—41no—
94C2×C4×C97C2×D411yescharacteristic
95C2×C4×C97C2×D411yescharacteristic
96C2×C4×C97C2×D411yescharacteristic
97C2×C4×C97C2×D411yescharacteristic
98C2×C4×C97C2×D411yescharacteristic
99C2×C4×C97#4111yescharacteristic
100C2×Dic97—22no—
101C2×Dic97—22no—
102C2×Dic97—41no—
103C2×Dic97—41no—
104C2×Dic97—41no—
105C2×Dic97—41no—
106Dic194—22no—
107Dic194—22no—
108Dic194—41no—
109Dic194—41no—
110#11715895—22no—
111#11715895—41no—
112C22×C97—22no—
113C22×C97—22no—
114C22×C97—22no—
115C22×C97—41no—
116C22×C97—41no—
117C22×C97—41no—
118C22×C97—41no—
119C22×C97#12011yescharacteristic
120C22×C97#12011yescharacteristic
121C22×C97#12111yescharacteristic
122C22×C97C2×Q1611yescharacteristic
123C22×C97#13511yescharacteristic
124C22×C97#13711yescharacteristic
125C22×C97#13711yescharacteristic
126C4×C97—41no—
127C4×C97—42no—
128C4×C97—42no—
129C4×C97#12011yescharacteristic
130C4×C97#12111yescharacteristic
131C4×C97#12111yescharacteristic
132C4×C97#12711yescharacteristic
133Dic97—42no—
134Dic97—42no—
135C2×C97—42no—
136C2×C97#39211yescharacteristic
137C2×C97#45111yescharacteristic
138C2×C97#45111yescharacteristic
139C2×C97#45211yescharacteristic
140C2×C97#45211yescharacteristic
141C2×C97#46711yescharacteristic
142C2×C97#47011yescharacteristic
143#2001—971nomaximal, Sylow
144C97#200111yescharacteristic, complemented, Sylow
145C22×Q16—971no—
146C2×C22⋊Q8—971no—
147C2×C22⋊Q8D9711yescharacteristic
148C2×#103—971no—
149C2×#103—971no—
150C2×C22⋊C8—971no—
151#340—971no—
152C22×Q8—971no—
153C22×Q8C2×D9711yescharacteristic
154C23×C4C2×D9711yescharacteristic
155C2×Q16—1942no—
156C2×Q16—1942no—
157C2×Q16—1942no—
158C22×C8—971no—
159C22×C8—971no—
160C22⋊Q8—22no—
161C22⋊Q8—22no—
162C22⋊Q8—1942no—
163C22⋊Q8—1942no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—971no—
166C2×C4⋊C4—1941no—
167C2×C4⋊C4C2×D9711yescharacteristic
168C2×C22⋊C4—21no—
169C2×C22⋊C4—1941no—
170#103—1942no—
171#103—1942no—
172C22⋊C8—1942no—
173C24C22⋊D9711yescharacteristic
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—972no—
180C2×Q8—1941no—
181C2×Q8—1941no—
182C2×Q8—1941no—
183C2×Q8—1941no—
184C2×Q8—1941no—
185C2×Q8C22⋊D9712yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—1941no—
195C22×C4—1941no—
196C22×C4C22⋊D9711yescharacteristic
197C22×C4C22×D9711yescharacteristic
198Q16—1948no—
199C2×C8—972no—
200C2×C8—1941no—
201C2×C8—1941no—
202C2×C8—1941no—
203C2×C8—1941no—
204C2×C8—1941no—
205C4⋊C4—24no—
206C4⋊C4—21no—
207C4⋊C4—972no—
208C4⋊C4—1944no—
209C4⋊C4—1941no—
210C4⋊C4C22⋊D9712yes—
211C22⋊C4—24no—
212C22⋊C4—1944no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#1309550112yescomplemented
220C23C2×C22⋊D9711yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—3881no—
226Q8—3881no—
227Q8—1942no—
228Q8—1942no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—41no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—41no—
238C2×C4—41no—
239C2×C4—41no—
240C2×C4—41no—
241C2×C4—41no—
242C2×C4—3881no—
243C2×C4—3881no—
244C2×C4—3881no—
245C2×C4—3881no—
246C2×C4—1942no—
247C2×C4—1942no—
248C2×C4D4×D9711yescharacteristic
249C2×C4D4×D9711yescharacteristic
250C2×C4D4×D9711yescharacteristic
251C2×C4C2×C22⋊D9711yescharacteristic
252C2×C4C2×C22⋊D9711yescharacteristic
253C2×C4#1309551011yescharacteristic
254C8—3881no—
255C8—1942no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—41no—
260C22—41no—
261C22—41no—
262C22—41no—
263C22#1569077811yescharacteristic
264C22#1569077911yescharacteristic
265C22#1569078211yescharacteristic
266C22C2×#1309550111yescharacteristic
267C22#1569083011yescharacteristic
268C22#1569084011yescharacteristic
269C22#1569084111yescharacteristic
270C4—42no—
271C4—41no—
272C4—42no—
273C4—3882no—
274C4—3882no—
275C4#1569077811yescharacteristic
276C4#1569078111yescharacteristic
277C4#1569078211yescharacteristic
278C4#1569080611yescharacteristic
279C2—42no—
280C2#2026064311yescharacteristic
281C2#2026089011yescharacteristic
282C2#2026089111yescharacteristic
283C2#2026089511yescharacteristic
284C2#2026089711yescharacteristic
285C2#2026097411yescharacteristic
286C2#2026099111yescharacteristic
subgroupG/H
centerC23C2×C22⋊D97
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
FittingC2×C22⋊Q8×C97C2
socleC23×C97C2×D4
prime radicalC24×C97D4
hypercenterC2×C22⋊Q8D97
hypocenterC97#2001
LCSG ▷ C22×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13095501C2semidirect
3. C2×#15690734C2semidirect
4. C2×#15690733C2semidirect
6. #20260497C2semidirect
75. C23×C97Q16semidirect
144. C97#2001semidirect
219. C23#13095501semidirect

Maximal subgroups

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

subgroupindex
C22×#130955012
C2×#156907892
C2×#156907342
C2×#156907332
C2×C22⋊#117158952
#202604972
C2×C22⋊Q8×C972
#200197