Finite Group Databasebeta

← #10591345

Subgroups — #10591345

Order 576.

338 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 338 conjugacy classes of subgroups — too many to draw. Only its 104 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×C3×C4⋊Dic3C211yesmaximal, characteristic
2C2×#3724C211yesmaximal, characteristic
3C2×#3724C211yesmaximal, characteristic
4C2×Dic3⋊Dic3C211yesmaximal, characteristic
5#63658C211yesmaximal, characteristic
6C3×C3⋊#95C211yesmaximal, characteristic
7#63579C211yesmaximal, characteristic
8#4982—31nomaximal
9#4657—31nomaximal
10C22×C32×C4C2211yescharacteristic
11C22×C3⋊Dic3C2211yescharacteristic
12C22×C3⋊Dic3C2211yescharacteristic
13C22×C3×Dic3C2211yescharacteristic
14C22×C3×Dic3C2211yescharacteristic
15C22×C3×Dic3C2211yescharacteristic
16C22×C3×Dic3C2211yescharacteristic
17C3×C4⋊Dic3—22no—
18#3724—22no—
19#3724—22no—
20Dic3⋊Dic3—22no—
21C2×C3×C4⋊C4—31no—
22C2×C4⋊Dic3—31no—
23C2×C4⋊Dic3D311yescharacteristic
24C2×Dic3⋊C4—31no—
25C2×Dic3⋊C4—31no—
26C2×Dic3⋊C4—31no—
27C2×Dic3⋊C4—31no—
28C2×Dic3⋊C4—31no—
29C3×#95—31no—
30C3⋊#95—31no—
31C3⋊#95—31no—
32C3⋊#95—31no—
33C3⋊#95—31no—
34C3⋊#95—61no—
35C3⋊#95D311yescharacteristic
36C23×C32C2311yescharacteristic
37C2×C32×C4—21no—
38C2×C32×C4—21no—
39C2×C32×C4Q812yes—
40C2×C3⋊Dic3—21no—
41C2×C3⋊Dic3—21no—
42C2×C3⋊Dic3—21no—
43C2×C3⋊Dic3—21no—
44C2×C3⋊Dic3D412yes—
45C2×C3⋊Dic3Q812yes—
46C2×C3×Dic3—21no—
47C2×C3×Dic3—21no—
48C2×C3×Dic3—21no—
49C2×C3×Dic3—21no—
50C2×C3×Dic3—21no—
51C2×C3×Dic3—21no—
52C2×C3×Dic3—21no—
53C2×C3×Dic3—21no—
54C2×C3×Dic3—21no—
55C2×C3×Dic3—21no—
56C2×C3×Dic3—21no—
57C2×C3×Dic3D412yes—
58#398—91noSylow
59C22×C3×C4—21no—
60C22×C3×C4—31no—
61C22×C3×C4—31no—
62C22×C3×C4—31no—
63C22×C3×C4—31no—
64C22×C3×C4C2×D311yescharacteristic
65C22×C3×C4C2×D311yescharacteristic
66C22×Dic3—31no—
67C22×Dic3—31no—
68C22×Dic3—31no—
69C22×Dic3—31no—
70C22×Dic3—61no—
71C22×Dic3—61no—
72C22×Dic3C2×D311yescharacteristic
73C22×Dic3C2×D311yescharacteristic
74C22×Dic3C2×D311yescharacteristic
75C22×Dic3C2×D311yescharacteristic
76C3×C4⋊C4—62no—
77C4⋊Dic3—22no—
78C4⋊Dic3—62no—
79Dic3⋊C4—62no—
80Dic3⋊C4—62no—
81Dic3⋊C4—62no—
82Dic3⋊C4—62no—
83Dic3⋊C4—62no—
84C22×C32C2×D411yescharacteristic
85C22×C32C2×D411yescharacteristic
86C22×C32C2×Q811yescharacteristic
87C22×C32C2×Q811yescharacteristic
88C22×C32#4111yescharacteristic
89C22×C32#4111yescharacteristic
90C22×C32#4111yescharacteristic
91C32×C4—22no—
92C3⋊Dic3—22no—
93C3⋊Dic3—22no—
94C3×Dic3—22no—
95C3×Dic3—41no—
96C3×Dic3—41no—
97C3×Dic3—41no—
98C2×C4⋊C4—31no—
99C2×C4⋊C4—91no—
100C2×C4⋊C4—91no—
101C2×C4⋊C4—91no—
102#95—31no—
103#95—91no—
104#95—91no—
105C23×C3—21no—
106C23×C3C22×D311yescharacteristic
107C23×C3C22×D311yescharacteristic
108C2×C3×C4—21no—
109C2×C3×C4—21no—
110C2×C3×C4—21no—
111C2×C3×C4—21no—
112C2×C3×C4—22no—
113C2×C3×C4—32no—
114C2×C3×C4—41no—
115C2×C3×C4—41no—
116C2×C3×C4—61no—
117C2×C3×C4—61no—
118C2×C3×C4—61no—
119C2×C3×C4—61no—
120C2×C3×C4—61no—
121C2×C3×C4—61no—
122C2×C3×C4—61no—
123C2×C3×C4—61no—
124C2×C3×C4—61no—
125C2×C3×C4—61no—
126C2×C3×C4—61no—
127C2×C3×C4Dic612yes—
128C2×C3×C4Dic612yes—
129C2×Dic3—21no—
130C2×Dic3—21no—
131C2×Dic3—21no—
132C2×Dic3—21no—
133C2×Dic3—21no—
134C2×Dic3—21no—
135C2×Dic3—21no—
136C2×Dic3—21no—
137C2×Dic3—21no—
138C2×Dic3—21no—
139C2×Dic3—21no—
140C2×Dic3—32no—
141C2×Dic3—32no—
142C2×Dic3—32no—
143C2×Dic3—32no—
144C2×Dic3—62no—
145C2×Dic3—62no—
146C2×Dic3—61no—
147C2×Dic3—61no—
148C2×Dic3—61no—
149C2×Dic3—61no—
150C2×Dic3—61no—
151C2×Dic3—61no—
152C2×Dic3—61no—
153C2×Dic3—61no—
154C2×Dic3—121no—
155C2×Dic3—121no—
156C2×Dic3—121no—
157C2×Dic3—121no—
158C2×Dic3#6712yes—
159C2×C32#12111yescharacteristic
160C2×C32C22⋊Q811yescharacteristic
161C2×C32C22⋊Q811yescharacteristic
162C2×C32#12311yescharacteristic
163C2×C32#12511yescharacteristic
164C2×C32#12511yescharacteristic
165C2×C32C4⋊Q811yescharacteristic
166C22×C4—31no—
167C22×C4—31no—
168C22×C4—31no—
169C22×C4—31no—
170C22×C4—91no—
171C22×C4—91no—
172C22×C4D3211yescharacteristic
173C4⋊C4—62no—
174C4⋊C4—182no—
175C4⋊C4—182no—
176C4⋊C4—182no—
177C22×C3—21no—
178C22×C3—21no—
179C22×C3—21no—
180C22×C3—21no—
181C22×C3—21no—
182C22×C3—21no—
183C22×C3—21no—
184C22×C3C2×Dic611yescharacteristic
185C22×C3C2×Dic611yescharacteristic
186C22×C3#23511yescharacteristic
187C22×C3#23511yescharacteristic
188C22×C3#23511yescharacteristic
189C22×C3D3×D411yescharacteristic
190C22×C3D3×D411yescharacteristic
191C22×C3D3×D411yescharacteristic
192C22×C3#23711yescharacteristic
193C22×C3#23711yescharacteristic
194C22×C3D3×Q811yescharacteristic
195C22×C3D3×Q811yescharacteristic
196C22×C3#23911yescharacteristic
197C22×C3C2×#6711yescharacteristic
198C3×C4—22no—
199C3×C4—22no—
200C3×C4—42no—
201C3×C4—62no—
202C3×C4—121no—
203C3×C4—121no—
204C3×C4—121no—
205Dic3—22no—
206Dic3—41no—
207Dic3—41no—
208Dic3—41no—
209Dic3—62no—
210Dic3—62no—
211Dic3—62no—
212Dic3—62no—
213Dic3—122no—
214Dic3—122no—
215C32#39811yescharacteristic, complemented, Sylow
216C23C2×D3211yescharacteristic
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—32no—
220C2×C4—61no—
221C2×C4—61no—
222C2×C4—61no—
223C2×C4—61no—
224C2×C4—61no—
225C2×C4—61no—
226C2×C4—61no—
227C2×C4—61no—
228C2×C4—61no—
229C2×C4—61no—
230C2×C4—61no—
231C2×C4—92no—
232C2×C4—92no—
233C2×C4—181no—
234C2×C4—181no—
235C2×C4—181no—
236C2×C4—181no—
237C2×C4#62712yes—
238C2×C3—21no—
239C2×C3—21no—
240C2×C3—21no—
241C2×C3—21no—
242C2×C3—21no—
243C2×C3—21no—
244C2×C3—21no—
245C2×C3#87011yescharacteristic
246C2×C3#87811yescharacteristic
247C2×C3#88311yescharacteristic
248C2×C3#88411yescharacteristic
249C2×C3#88811yescharacteristic
250C2×C3#88811yescharacteristic
251C2×C3#88911yescharacteristic
252C2×C3#88911yescharacteristic
253C2×C3#89011yescharacteristic
254C2×C3#89411yescharacteristic
255C2×C3#89611yescharacteristic
256C2×C3#92411yescharacteristic
257C2×C3#93911yescharacteristic
258C2×C3#94611yescharacteristic
259C22#379511yescharacteristic
260C22#379711yescharacteristic
261C22#379911yescharacteristic
262C22#380211yescharacteristic
263C22#380411yescharacteristic
264C22C2×#62711yescharacteristic
265C22D3×#6711yescharacteristic
266C4—22no—
267C4—62no—
268C4—121no—
269C4—121no—
270C4—121no—
271C4—182no—
272C4—182no—
273C3—21no—
274C3#465711yescharacteristic, complemented
275C3#498211yescharacteristic, complemented
276C2#6384711yescharacteristic
277C2#6385411yescharacteristic
278C2#6386911yescharacteristic
279C2#6388511yescharacteristic
280C2#6388811yescharacteristic
281C2#6391911yescharacteristic
282C2#6396411yescharacteristic
subgroupG/H
center, FrattiniC23C2×D32
commutator, socle, prime radical, cosocleC23×C32C23
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
215. C32#398semidirect
274. C3#4657semidirect
275. C3#4982semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4⋊Dic32
C2×#37242
C2×Dic3⋊Dic32
#636582
C3×C3⋊#952
#635792
#49823
#46573