Finite Group Databasebeta

← #10596183

Subgroups — #10596183

Order 576.

588 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#126C211yesmaximal, characteristic, complemented
2#64107C211yesmaximal, characteristic, complemented
3#64106C211yesmaximal, characteristic, complemented
4#64105C211yesmaximal, characteristic, complemented
5#64099C211yesmaximal, characteristic, complemented
6#64098C211yesmaximal, characteristic, complemented
7#64097C211yesmaximal, characteristic, complemented
8#64095C211yesmaximal, characteristic
9#64094C211yesmaximal, characteristic
10#64092C211yesmaximal, characteristic
11#64088C211yesmaximal, characteristic, complemented
12#64087C211yesmaximal, characteristic, complemented
13#64087C211yesmaximal, characteristic, complemented
14#64085C211yesmaximal, characteristic, complemented
15#64078C211yesmaximal, characteristic, complemented
16#5712—34nomaximal
17C2×#638C2211yescharacteristic
18C22×C3⋊Dic3C2211yescharacteristic
19C2×C4×C3⋊D3C2211yescharacteristic
20C2×#634C2211yescharacteristic
21C2×#634C2211yescharacteristic
22C2×#634C2211yescharacteristic
23C32×C4⋊C4C2211yescharacteristic
24C32×C4⋊C4C2211yescharacteristic
25C32×C4⋊C4C2211yescharacteristic
26C32×C22⋊C4C2211yescharacteristic
27C32×C22⋊C4C2211yescharacteristic
28C32×C22⋊C4C2211yescharacteristic
29C32×C42C2211yescharacteristic
30#3757C2211yescharacteristic
31#3757C2211yescharacteristic
32#3752C2211yescharacteristic
33#3752C2211yescharacteristic
34#3752C2211yescharacteristic
35#3752C2211yescharacteristic
36#3752C2211yescharacteristic
37C4⋊(C3⋊Dic3)C2211yescharacteristic
38C4⋊(C3⋊Dic3)C2211yescharacteristic
39C4⋊(C3⋊Dic3)C2211yescharacteristic
40C4⋊(C3⋊Dic3)C2211yescharacteristic
41C4⋊(C3⋊Dic3)C2211yescharacteristic
42C4⋊(C3⋊Dic3)C2211yescharacteristic
43#3750C2211yescharacteristic
44#3750C2211yescharacteristic
45#3750C2211yescharacteristic
46#3750C2211yescharacteristic
47#3750C2211yescharacteristic
48#3750C2211yescharacteristic
49#3750C2211yescharacteristic
50C4×C3⋊Dic3C2211yescharacteristic
51C4×C3⋊Dic3C2211yescharacteristic
52C3×#126D314yescomplemented
53#898—34no—
54#897—34no—
55#896—34no—
56#890—34no—
57#889—34no—
58#888—34no—
59#886—34no—
60#885—34no—
61#883—34no—
62#879—34no—
63#878—34no—
64#878—34no—
65#876—34no—
66#869—34no—
67C23×C32C2311yescharacteristic
68C22×C3⋊D3C2311yescharacteristic
69C2×C32×C4C2311yescharacteristic
70C2×C32×C4C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C2×C32×C4C2311yescharacteristic
73C2×C32×C4C2311yescharacteristic
74C2×C32×C4C2311yescharacteristic
75#638—41no—
76C2×C3⋊Dic3—41no—
77C2×C3⋊Dic3C2311yescharacteristic
78C2×C3⋊Dic3C2311yescharacteristic
79C2×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊Dic3C2311yescharacteristic
82C2×C3⋊Dic3C2311yescharacteristic
83C2×C3⋊Dic3C2311yescharacteristic
84C4×C3⋊D3—41no—
85#634—41no—
86#634—41no—
87#634—41no—
88#563—91noSylow
89C2×#67—34no—
90C22×Dic3—34no—
91C2×C4×D3—34no—
92C2×Dic6—34no—
93C2×Dic6—34no—
94C2×Dic6—34no—
95C3×C4⋊C4C2×D314yes—
96C3×C4⋊C4C2×D314yes—
97C3×C4⋊C4C2×D314yes—
98C3×C22⋊C4C2×D314yes—
99C3×C22⋊C4C2×D314yes—
100C3×C22⋊C4C2×D314yes—
101C3×C42C2×D314yes—
102#217—34no—
103#217—34no—
104(C2×D3)⋊C4—34no—
105(C2×D3)⋊C4—34no—
106(C2×D3)⋊C4—34no—
107(C2×D3)⋊C4—34no—
108(C2×D3)⋊C4—34no—
109C4⋊Dic3—34no—
110C4⋊Dic3—34no—
111C4⋊Dic3—34no—
112C4⋊Dic3—34no—
113C4⋊Dic3—34no—
114C4⋊Dic3—34no—
115Dic3⋊C4—34no—
116Dic3⋊C4—34no—
117Dic3⋊C4—34no—
118Dic3⋊C4—34no—
119Dic3⋊C4—34no—
120Dic3⋊C4—34no—
121Dic3⋊C4—34no—
122C4×Dic3—34no—
123C4×Dic3—34no—
124C22×C32—21no—
125C22×C32—21no—
126C22×C32—21no—
127C22×C32C2411yescharacteristic
128C2×C3⋊D3—21no—
129C2×C3⋊D3—21no—
130C2×C3⋊D3—21no—
131C32×C4—21no—
132C32×C4—21no—
133C32×C4—21no—
134C32×C4—21no—
135C32×C4—21no—
136C32×C4—21no—
137C3⋊Dic3—21no—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3⋊Dic3—21no—
141C3⋊Dic3—21no—
142C3⋊Dic3—21no—
143C3⋊Dic3—21no—
144C4⋊Q8—91no—
145C4⋊Q8—91no—
146#126—91no—
147#126—91no—
148#126—91no—
149#126C3⋊D311yescharacteristic, complemented
150#125—91no—
151#125—91no—
152#124—91no—
153#123—91no—
154#123—91no—
155C22⋊Q8—91no—
156C22⋊Q8—91no—
157C22⋊Q8—91no—
158C22⋊Q8—91no—
159C23×C3C22×D314yes—
160C22×D3—34no—
161C2×C3×C4C22×D314yes—
162C2×C3×C4C22×D314yes—
163C2×C3×C4C22×D314yes—
164C2×C3×C4C22×D314yes—
165C2×C3×C4C22×D314yes—
166C2×C3×C4C22×D314yes—
167#67—124no—
168C2×Dic3—34no—
169C2×Dic3—34no—
170C2×Dic3—34no—
171C2×Dic3—34no—
172C2×Dic3—34no—
173C2×Dic3—34no—
174C2×Dic3—34no—
175C2×Dic3—124no—
176C4×D3—124no—
177Dic6—124no—
178Dic6—124no—
179Dic6—124no—
180C2×C32—41no—
181C2×C32H(5,𝔽2)11yescharacteristic
182C2×C322−1+411yescharacteristic
183C2×C322−1+411yescharacteristic
184C3⋊D3—41no—
185C2×Q8—91no—
186C2×Q8—91no—
187C2×Q8—91no—
188C2×D4—91no—
189C22×C4—91no—
190C22×C4—91no—
191C4⋊C4—91no—
192C4⋊C4—91no—
193C4⋊C4—91no—
194C4⋊C4—91no—
195C4⋊C4—91no—
196C4⋊C4—91no—
197C4⋊C4—91no—
198C4⋊C4—91no—
199C4⋊C4—91no—
200C4⋊C4—91no—
201C4⋊C4—91no—
202C4⋊C4—91no—
203C4⋊C4—91no—
204C4⋊C4C2×C3⋊D311yescharacteristic
205C4⋊C4C2×C3⋊D311yescharacteristic
206C4⋊C4C2×C3⋊D311yescharacteristic
207C22⋊C4—91no—
208C22⋊C4—91no—
209C22⋊C4—91no—
210C22⋊C4—91no—
211C22⋊C4—91no—
212C22⋊C4—91no—
213C22⋊C4—91no—
214C22⋊C4C2×C3⋊D311yescharacteristic
215C22⋊C4C2×C3⋊D311yescharacteristic
216C22⋊C4C2×C3⋊D311yescharacteristic
217C42—91no—
218C42—91no—
219C42C2×C3⋊D311yescharacteristic
220C22×C3—24no—
221C22×C3—24no—
222C22×C3—24no—
223C22×C3C23×D314yes—
224C2×D3—64no—
225C2×D3—64no—
226C2×D3—64no—
227C3×C4—24no—
228C3×C4—24no—
229C3×C4—24no—
230C3×C4—24no—
231C3×C4—24no—
232C3×C4—24no—
233Dic3—64no—
234Dic3—64no—
235Dic3—64no—
236Dic3—64no—
237Dic3—64no—
238Dic3—64no—
239Dic3—64no—
240C32#56311yescharacteristic, complemented, Sylow
241C23—91no—
242C23C22×C3⋊D311yescharacteristic
243Q8—361no—
244Q8—361no—
245Q8—361no—
246D4—361no—
247C2×C4—91no—
248C2×C4—91no—
249C2×C4—91no—
250C2×C4—91no—
251C2×C4—91no—
252C2×C4—91no—
253C2×C4—91no—
254C2×C4—361no—
255C2×C4—361no—
256C2×C4C22×C3⋊D311yescharacteristic
257C2×C4C22×C3⋊D311yescharacteristic
258C2×C4C22×C3⋊D311yescharacteristic
259C2×C4C22×C3⋊D311yescharacteristic
260C2×C4C22×C3⋊D311yescharacteristic
261C2×C4C22×C3⋊D311yescharacteristic
262C2×C3—44no—
263C2×C3#100914yes—
264C2×C3#101014yes—
265C2×C3#101014yes—
266D3—124no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—181no—
271C22—181no—
272C22—181no—
273C22C23×C3⋊D311yescharacteristic
274C4—21no—
275C4—21no—
276C4—21no—
277C4—21no—
278C4—21no—
279C4—21no—
280C4—181no—
281C4—181no—
282C4—181no—
283C4—181no—
284C4—181no—
285C4—181no—
286C4—181no—
287C3#571214yescomplemented
288C2—41no—
289C2—361no—
290C2#6436611yescharacteristic
291C2#6436711yescharacteristic
292C2#6436711yescharacteristic
subgroupG/H
center, FrattiniC22C23×C3⋊D3
commutator, socle, cosocleC22×C32C24
FittingC32×#126C2
prime radicalC2×#638C22
hypercenter#126C3⋊D3
hypocenterC32#563
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ #126

Complements

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

NH ≅ G/Ntype
1. C32×#126C2semidirect
2. #64107C2semidirect
3. #64106C2semidirect
4. #64105C2semidirect
5. #64099C2semidirect
6. #64098C2semidirect
7. #64097C2semidirect
11. #64088C2semidirect
12. #64087C2semidirect
13. #64087C2semidirect
14. #64085C2semidirect
15. #64078C2semidirect
52. C3×#126D3semidirect
149. #126C3⋊D3semidirect
240. C32#563semidirect
287. C3#5712semidirect

Maximal subgroups

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

subgroupindex
C32×#1262
#641072
#641062
#641052
#640992
#640982
#640972
#640952
#640942
#640922
#640882
#640872
#640852
#640782
#57123