Finite Group Databasebeta

← #49124530

Subgroups — #49124530

Order 36224.

528 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 155 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×#22583575C211yesmaximal, characteristic, complemented
2C2×#22583566C211yesmaximal, characteristic
3#31826403C211yesmaximal, characteristic, complemented
4#31826396C212yesmaximal, complemented
5#31826388C212yesmaximal
6#31826386C211yesmaximal, characteristic, complemented
7#31826377C211yesmaximal, characteristic
8C4⋊C4×Dic283C212yesmaximal, complemented
9#31826316C211yesmaximal, characteristic, complemented
10C22⋊C4×Dic283C212yesmaximal
11#399×C283C211yesmaximal, characteristic
12C22×Dic566C2211yescharacteristic
13C23×Dic283C2211yescharacteristic
14#22583575—24no—
15#22583566—24no—
16C2×C4⋊Dic283C2212yes—
17C2×Dic283⋊C4C2212yes—
18C2×Dic283⋊C4C2211yescharacteristic
19C2×Dic283⋊C4C2212yes—
20C2×Dic283⋊C4C2211yescharacteristic
21C2×C22⋊Dic283C2212yes—
22C2×C22⋊Dic283C2211yescharacteristic
23C2×C4×Dic283C2212yes—
24C2×C4×Dic283C2212yes—
25C2×C4×Dic283C2211yescharacteristic
26C2×C4×Dic283C2211yescharacteristic
27C283⋊#95C2212yes—
28C283⋊#95C2212yes—
29C283⋊#95C2211yescharacteristic
30C283⋊#95C2212yes—
31C283⋊#95C2212yes—
32C2×C4⋊C4×C283C2212yes—
33C2×C4⋊C4×C283C2211yescharacteristic
34C2×C22⋊C4×C283C2212yes—
35C2×C22⋊C4×C283C2211yescharacteristic
36#95×C283C2211yescharacteristic
37C2×Dic566—24no—
38C2×Dic566—22no—
39C22×Dic283—22no—
40C22×Dic283—22no—
41C22×Dic283Q814yescomplemented
42C22×Dic283C2311yescharacteristic
43C22×Dic283C2312yes—
44C22×Dic283C2311yescharacteristic
45C22×Dic283C2311yescharacteristic
46C22×Dic283C2312yes—
47C22×Dic283C2311yescharacteristic
48C4⋊Dic283—24no—
49Dic283⋊C4—22no—
50Dic283⋊C4—24no—
51Dic283⋊C4—22no—
52Dic283⋊C4—24no—
53C22⋊Dic283—22no—
54C22⋊Dic283—24no—
55C4×Dic283—22no—
56C4×Dic283—24no—
57C4×Dic283—24no—
58C4×Dic283D414yes—
59C24×C283C2311yescharacteristic
60C22×C4×C283C2311yescharacteristic
61C22×C4×C283C2312yes—
62C22×C4×C283C2311yescharacteristic
63C22×C4×C283C2312yes—
64C4⋊C4×C283—22no—
65C4⋊C4×C283—24no—
66C22⋊C4×C283—22no—
67C22⋊C4×C283—24no—
68Dic566—28no—
69C2×Dic283—22no—
70C2×Dic283—21no—
71C2×Dic283—22no—
72C2×Dic283—21no—
73C2×Dic283—22no—
74C2×Dic283—28no—
75C2×Dic283—21no—
76C2×Dic283—22no—
77C2×Dic283—22no—
78C2×Dic283—22no—
79C2×Dic283—22no—
80C2×Dic283C2×D412yes—
81C2×Dic283C2×D412yes—
82C2×Dic283C2×Q812yes—
83C2×Dic283C2×Q812yes—
84C2×Dic283#4114yes—
85C2×Dic283#4112yes—
86C23×C283—21no—
87C23×C283—22no—
88C23×C283—22no—
89C23×C283—21no—
90C23×C283C2×Q812yes—
91C23×C283C2411yescharacteristic
92C2×C4×C283—22no—
93C2×C4×C283—22no—
94C2×C4×C283—22no—
95C2×C4×C283—21no—
96C2×C4×C283—22no—
97C2×C4×C283—22no—
98C2×C4×C283—22no—
99C2×C4×C283—22no—
100C2×C4×C283C2×D412yes—
101C2×C4×C283#4114yes—
102C22×C283—22no—
103C22×C283—24no—
104C22×C283—41no—
105C22×C283—42no—
106C22×C283—41no—
107C22×C283C22×D411yescharacteristic
108C22×C283C22×Q811yescharacteristic
109C22×C283C2×#4111yescharacteristic
110C22×C283C2×#4112yes—
111C22×C283C2×#4112yes—
112C4×C283—22no—
113C4×C283—24no—
114C4×C283—41no—
115C4×C283—42no—
116Dic283—22no—
117Dic283—24no—
118Dic283—22no—
119Dic283—41no—
120Dic283—42no—
121Dic283C22⋊Q814yes—
122C2×C283—42no—
123C2×C283C2×C22⋊Q811yescharacteristic
124C2×C283#52511yescharacteristic
125C2×C283#52912yes—
126C2×C283#53211yescharacteristic
127C2×C283#53312yes—
128C283#239611yescharacteristic, complemented, Sylow
129#2396—2831nomaximal, Sylow
130C2×C4×Q8—2831no—
131C2×#117—2831no—
132#400—2831no—
133#399D28311yescharacteristic
134#398—2832no—
135#396—2832no—
136#395—2831no—
137#393—2831no—
138C4×C4⋊C4—2832no—
139C4×C4⋊C4—2831no—
140C4×C22⋊C4—2832no—
141C22×Q8—2831no—
142C23×C4—2831no—
143C4×Q8—5664no—
144#117—5664no—
145C2×C4⋊C4—2831no—
146C2×C4⋊C4—2831no—
147C2×C4⋊C4—2832no—
148C2×C4⋊C4—2832no—
149C2×C4⋊C4—2832no—
150C2×C4⋊C4C2×D28311yescharacteristic
151C2×C4⋊C4C2×D28312yes—
152C2×C22⋊C4—2831no—
153C2×C22⋊C4—2832no—
154C2×C22⋊C4C2×D28311yescharacteristic
155C2×C22⋊C4C2×D28312yes—
156C2×C42—2831no—
157C2×C42—2831no—
158C2×C42—2832no—
159C2×C42—2832no—
160#95—2832no—
161#95—2832no—
162#95—2832no—
163#95—2831no—
164#95—2832no—
165#95C2×D28311yescharacteristic
166C24C22×D28311yescharacteristic
167C2×Q8—5664no—
168C2×Q8—5662no—
169C22×C4—2831no—
170C22×C4—2834no—
171C22×C4—2831no—
172C22×C4—2832no—
173C22×C4—2831no—
174C22×C4—2832no—
175C22×C4—2831no—
176C22×C4—5662no—
177C22×C4—5662no—
178C22×C4C22×D28311yescharacteristic
179C22×C4C22×D28312yes—
180C22×C4C22×D28311yescharacteristic
181C22×C4C22×D28312yes—
182C4⋊C4—22no—
183C4⋊C4—24no—
184C4⋊C4—5662no—
185C4⋊C4—5664no—
186C4⋊C4—5664no—
187C4⋊C4—5662no—
188C4⋊C4—5664no—
189C22⋊C4—22no—
190C22⋊C4—24no—
191C22⋊C4—5664no—
192C22⋊C4—5662no—
193C42—2834no—
194C42—5664no—
195C42—5662no—
196C42—5664no—
197C23—21no—
198C23—22no—
199C23—21no—
200C23—22no—
201C23Q8×D28312yes—
202C23C23×D28311yescharacteristic
203Q8—5668no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—2832no—
213C2×C4—2832no—
214C2×C4—2832no—
215C2×C4—2832no—
216C2×C4—2832no—
217C2×C4—2834no—
218C2×C4—5668no—
219C2×C4—5662no—
220C2×C4—5661no—
221C2×C4—5661no—
222C2×C4—5662no—
223C2×C4—5662no—
224C2×C4—5661no—
225C2×C4—5662no—
226C2×C4—5662no—
227C2×C4—5662no—
228C2×C4—5662no—
229C2×C4D4×D28312yes—
230C2×C4#1798539114yes—
231C22—22no—
232C22—24no—
233C22—41no—
234C22—42no—
235C22—41no—
236C22C2×D4×D28311yescharacteristic
237C22C2×Q8×D28311yescharacteristic
238C22#41×D28311yescharacteristic
239C22#41×D28312yes—
240C22C2×#1798539112yes—
241C4—22no—
242C4—24no—
243C4—41no—
244C4—42no—
245C4—2834no—
246C4—5662no—
247C4—5664no—
248C4—5662no—
249C4—11321no—
250C4—11322no—
251C2—42no—
252C2C22⋊Q8×D28311yescharacteristic
253C2#3182697311yescharacteristic
254C2#3182699212yes—
255C2#3182701311yescharacteristic
256C2#3182701712yes—
subgroupG/H
center, FrattiniC23C23×D283
commutator, socle, cosocleC23×C283C24
Fitting#399×C283C2
prime radicalC24×C283C23
hypercenter#399D283
hypocenterC283#2396
LCSG ▷ C23×C283 ▷ C283
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#22583575C2semidirect
3. #31826403C2semidirect
4. #31826396C2semidirect
6. #31826386C2semidirect
8. C4⋊C4×Dic283C2semidirect
9. #31826316C2semidirect
41. C22×Dic283Q8semidirect
128. C283#2396semidirect

Maximal subgroups

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

subgroupindex
C2×#225835752
C2×#225835662
#318264032
#318263962
#318263882
#318263862
#318263772
C4⋊C4×Dic2832
#318263162
C22⋊C4×Dic2832
#399×C2832
#2396283