Finite Group Databasebeta

← #10594223

Subgroups — #10594223

Order 576.

518 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 122 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
1#64086C211yesmaximal, characteristic, complemented
2C3×#884C212yesmaximal, complemented
3#63980C211yesmaximal, characteristic, complemented
4#63980C211yesmaximal, characteristic, complemented
5#63970C212yesmaximal, complemented
6#63914C211yesmaximal, characteristic, complemented
7#63905C212yesmaximal, complemented
8#63902C211yesmaximal, characteristic, complemented
9C4×#625C211yesmaximal, characteristic, complemented
10#63884C212yesmaximal, complemented
11#63870C211yesmaximal, characteristic, complemented
12#5615—32nomaximal
13C22×C3⋊Dic3C2211yescharacteristic
14C2×C3×#67C2212yes—
15C2×C3×#67C2212yes—
16C2×C3×D12C2212yes—
17C2×C3×C4×D3C2212yes—
18C2×#625C2211yescharacteristic
19C2×#625C2211yescharacteristic
20C2×#625C2212yes—
21C2×D3×Dic3C2212yes—
22C32×C22⋊C4C2211yescharacteristic
23#3757C2211yescharacteristic
24#3750C2211yescharacteristic
25#3750C2211yescharacteristic
26C4×C3⋊Dic3C2211yescharacteristic
27C4×C3⋊Dic3C2211yescharacteristic
28C3×(C2×D3)⋊C4C2212yes—
29C3×Dic3⋊C4C2212yes—
30#3724C2211yescharacteristic
31#3724C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36C3×#121—32no—
37#940—32no—
38#939—32no—
39#939—32no—
40#939—32no—
41#938—32no—
42#935—32no—
43#929—32no—
44C4×#67—32no—
45#894—32no—
46#889—32no—
47#884D312yescomplemented
48#883—32no—
49#67⋊C4—32no—
50#877—32no—
51#877—61no—
52C23×C32C2311yescharacteristic
53C22×C3×D3C2312yes—
54C22×C3×D3C2312yes—
55C2×C32×C4C2311yescharacteristic
56C2×C32×C4C2311yescharacteristic
57C2×C3⋊Dic3—22no—
58C2×C3⋊Dic3C2311yescharacteristic
59C2×C3⋊Dic3C2311yescharacteristic
60C2×C3⋊Dic3C2311yescharacteristic
61C2×C3⋊Dic3C2311yescharacteristic
62C3×#67—42no—
63C3×#67—42no—
64C2×C3×Dic3C2312yes—
65C2×C3×Dic3C2312yes—
66C3×D12—42no—
67C3×C4×D3—42no—
68#625—22no—
69#625—22no—
70#625—42no—
71D3×Dic3—42no—
72#540—91noSylow
73C2×C3×D4—32no—
74C2×C3×D4—32no—
75C2×C3×D4—32no—
76C22×C3×C4—32no—
77C2×#67—32no—
78C2×#67—32no—
79C2×#67—32no—
80C2×#67—32no—
81C2×#67C2×D312yes—
82C2×#67C2×D312yes—
83C22×Dic3—32no—
84C22×Dic3—32no—
85C22×Dic3—61no—
86C2×D12C2×D312yes—
87C2×C4×D3—32no—
88C2×C4×D3C2×D312yes—
89C3×C4⋊C4—32no—
90C3×C22⋊C4—21no—
91C3×C22⋊C4—32no—
92C3×C22⋊C4C2×D312yes—
93#217—32no—
94#217—32no—
95#217—32no—
96#217—32no—
97#217—61no—
98(C2×D3)⋊C4—32no—
99(C2×D3)⋊C4—32no—
100(C2×D3)⋊C4—32no—
101(C2×D3)⋊C4C2×D312yes—
102C4⋊Dic3—32no—
103Dic3⋊C4—32no—
104Dic3⋊C4—32no—
105Dic3⋊C4—32no—
106Dic3⋊C4—32no—
107Dic3⋊C4—32no—
108Dic3⋊C4—61no—
109Dic3⋊C4—61no—
110Dic3⋊C4C2×D312yes—
111C4×Dic3—32no—
112C4×Dic3—32no—
113C4×Dic3—61no—
114C4×Dic3—61no—
115C22×C32—21no—
116C22×C32—21no—
117C22×C32—21no—
118C22×C32C2411yescharacteristic
119C2×C3×D3—22no—
120C2×C3×D3—22no—
121C2×C3×D3—22no—
122C2×C3×D3—22no—
123C2×C3×D3—22no—
124C2×C3×D3—22no—
125C32×C4—21no—
126C32×C4—21no—
127C3⋊Dic3—21no—
128C3⋊Dic3—21no—
129C3⋊Dic3—21no—
130C3⋊Dic3#4112yes—
131C3×Dic3—22no—
132C3×Dic3—22no—
133#127—91no—
134#125—91no—
135#123—92no—
136#123—92no—
137#121—32no—
138#121—92no—
139#121—91no—
140#121—91no—
141C4×D4—91no—
142C4×D4—91no—
143#117—91no—
144C23×C3—21no—
145C23×C3—32no—
146C23×C3—32no—
147C23×C3C22×D312yes—
148C22×D3C22×D312yes—
149C22×D3C22×D312yes—
150C3×D4—122no—
151C3×D4—122no—
152C3×D4—122no—
153C2×C3×C4—21no—
154C2×C3×C4—21no—
155C2×C3×C4—32no—
156C2×C3×C4—32no—
157C2×C3×C4—122no—
158C2×C3×C4C22×D312yes—
159C2×C3×C4C22×D312yes—
160#67—42no—
161#67—42no—
162#67—64no—
163#67—64no—
164#67—122no—
165#67—122no—
166C2×Dic3—32no—
167C2×Dic3—32no—
168C2×Dic3—32no—
169C2×Dic3—32no—
170C2×Dic3—61no—
171C2×Dic3—61no—
172C2×Dic3—61no—
173C2×Dic3—61no—
174C2×Dic3—64no—
175C2×Dic3—64no—
176C2×Dic3—122no—
177C2×Dic3C22×D312yes—
178C2×Dic3C22×D312yes—
179D12—42no—
180C4×D3—42no—
181C4×D3—122no—
182C2×C32—41no—
183C2×C32C2×#4111yescharacteristic
184C2×C32H(5,𝔽2)11yescharacteristic
185C2×C32H(5,𝔽2)11yescharacteristic
186C3×D3—42no—
187C3×D3—42no—
188C2×D4—32no—
189C2×D4—32no—
190C2×D4—32no—
191C2×D4—91no—
192C2×D4—91no—
193C2×D4—92no—
194C22×C4—32no—
195C22×C4—92no—
196C22×C4—91no—
197C4⋊C4—32no—
198C4⋊C4—91no—
199C4⋊C4—91no—
200C4⋊C4—92no—
201C4⋊C4—91no—
202C4⋊C4—91no—
203C22⋊C4—32no—
204C22⋊C4—92no—
205C22⋊C4—92no—
206C22⋊C4—92no—
207C22⋊C4—91no—
208C22⋊C4D3211yescharacteristic
209C42—91no—
210C42—91no—
211C22×C3—22no—
212C22×C3—22no—
213C22×C3—22no—
214C22×C3—22no—
215C22×C3—21no—
216C22×C3—41no—
217C22×C3—41no—
218C22×C3—62no—
219C22×C3—62no—
220C22×C3—62no—
221C22×C3—62no—
222C22×C3—62no—
223C22×C3—62no—
224C22×C3C23×D312yes—
225C2×D3—22no—
226C2×D3—22no—
227C2×D3—22no—
228C2×D3—22no—
229C2×D3—22no—
230C2×D3—22no—
231C3×C4—22no—
232C3×C4—22no—
233C3×C4—41no—
234C3×C4—41no—
235C3×C4—62no—
236C3×C4—62no—
237Dic3—22no—
238Dic3—22no—
239Dic3—34no—
240Dic3—62no—
241Dic3—62no—
242Dic3—62no—
243Dic3—62no—
244Dic3—62no—
245Dic3—121no—
246Dic3—121no—
247C32#54011yescharacteristic, complemented, Sylow
248C23—32no—
249C23—32no—
250C23C2×D3211yescharacteristic
251D4—122no—
252D4—122no—
253D4—122no—
254D4—182no—
255D4—182no—
256D4—362no—
257C2×C4—32no—
258C2×C4—32no—
259C2×C4—91no—
260C2×C4—91no—
261C2×C4—91no—
262C2×C4—91no—
263C2×C4—122no—
264C2×C4—182no—
265C2×C4—362no—
266C2×C4C2×D3211yescharacteristic
267C2×C4C2×D3211yescharacteristic
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—21no—
271C2×C3—42no—
272C2×C3—42no—
273C2×C3—122no—
274C2×C3—122no—
275C2×C3#100412yes—
276C2×C3#100412yes—
277C2×C3D3×#4112yes—
278D3—42no—
279D3—42no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—62no—
284C22—62no—
285C22—62no—
286C22—62no—
287C22—62no—
288C22—62no—
289C22C22×D3211yescharacteristic
290C4—21no—
291C4—21no—
292C4—62no—
293C4—62no—
294C4—92no—
295C4—181no—
296C4—181no—
297C4—181no—
298C3—21no—
299C3#561512yescomplemented
300C2—41no—
301C2—122no—
302C2—122no—
303C2#6430611yescharacteristic
304C2#6431311yescharacteristic
305C2#6431311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C22⋊C4C22
hypercenterC22⋊C4D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. #64086C2semidirect
2. C3×#884C2semidirect
3. #63980C2semidirect
4. #63980C2semidirect
5. #63970C2semidirect
6. #63914C2semidirect
7. #63905C2semidirect
8. #63902C2semidirect
9. C4×#625C2semidirect
10. #63884C2semidirect
11. #63870C2semidirect
47. #884D3semidirect
247. C32#540semidirect
299. C3#5615semidirect

Maximal subgroups

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

subgroupindex
#640862
C3×#8842
#639802
#639702
#639142
#639052
#639022
C4×#6252
#638842
#638702
#56153