Finite Group Databasebeta

← #10593983

Subgroups — #10593983

Order 576.

467 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#634C211yesmaximal, characteristic, complemented
2C3×C4×D12C211yesmaximal, characteristic, complemented
3C3×#872C211yesmaximal, characteristic, complemented
4#63918C211yesmaximal, characteristic, complemented
5#63915C212yesmaximal, complemented
6#63908C211yesmaximal, characteristic
7#63905C212yesmaximal, complemented
8#63862C212yesmaximal, complemented
9#63860C211yesmaximal, characteristic, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63851C212yesmaximal, complemented
12#5561—31nomaximal
13#5534—31nomaximal
14C2×#634C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×C4×D3C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×#627C2212yes—
19C2×#625C2212yes—
20C32×C42C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22#3750C2212yes—
23C4×C3⋊Dic3C2212yes—
24C3×(C2×D3)⋊C4C2212yes—
25C3×(C2×D3)⋊C4C2211yescharacteristic
26C3×(C2×D3)⋊C4C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×Dic3⋊C4C2211yescharacteristic
29C3×Dic3⋊C4C2211yescharacteristic
30C3×C4×Dic3C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×C4×D4—31no—
38C3×#117—31no—
39#938—31no—
40#936—31no—
41#930—31no—
42#929—32no—
43#924—31no—
44#924—32no—
45#898—32no—
46#897—31no—
47#896—32no—
48#888—31no—
49#885—32no—
50#885—32no—
51#883—32no—
52#879—32no—
53#875—31no—
54C4×D12D311yescharacteristic, complemented
55#872D311yescharacteristic, complemented
56#869—31no—
57C4×Dic6—31no—
58C4×Dic6—31no—
59C4×Dic6—61no—
60C22×C3×D3C2312yes—
61C22×C3×D3C2311yescharacteristic
62C2×C32×C4C2311yescharacteristic
63C2×C32×C4C2311yescharacteristic
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2312yes—
67#634—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C3×D12—22no—
73C3×C4×D3—22no—
74C3×C4×D3—42no—
75#627—42no—
76#625—42no—
77#542—91noSylow
78C2×C3×D4—31no—
79C22×C3×C4—31no—
80C22×C3×C4—32no—
81C2×#67—32no—
82C2×#67—32no—
83C2×D12C2×D311yescharacteristic
84C2×C4×D3C2×D311yescharacteristic
85C2×C4×D3C2×D312yes—
86C2×Dic6—31no—
87C2×Dic6—31no—
88C2×Dic6—32no—
89C2×Dic6—32no—
90C2×Dic6—61no—
91C3×C4⋊C4—31no—
92C3×C4⋊C4—31no—
93C3×C4⋊C4—31no—
94C3×C22⋊C4—32no—
95C3×C22⋊C4—31no—
96C3×C22⋊C4—31no—
97C3×C42—21no—
98C3×C42—31no—
99C3×C42C2×D311yescharacteristic
100C3×C42C2×D311yescharacteristic
101#217—32no—
102#217—32no—
103#217—32no—
104#217—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4C2×D311yescharacteristic
110(C2×D3)⋊C4C2×D311yescharacteristic
111(C2×D3)⋊C4C2×D312yes—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—32no—
115C4⋊Dic3—32no—
116C4⋊Dic3—61no—
117C4⋊Dic3C2×D311yescharacteristic
118Dic3⋊C4—32no—
119Dic3⋊C4—32no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—62no—
123Dic3⋊C4C2×D311yescharacteristic
124Dic3⋊C4C2×D311yescharacteristic
125C4×Dic3—32no—
126C4×Dic3—32no—
127C4×Dic3—62no—
128C4×Dic3C2×D311yescharacteristic
129C22×C32C2411yescharacteristic
130C2×C3×D3—21no—
131C2×C3×D3—22no—
132C2×C3×D3—21no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C2×C3×D3—21no—
136C32×C4—21no—
137C32×C4—21no—
138C32×C4#4112yes—
139C3⋊Dic3—22no—
140C3⋊Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—22no—
144C3×Dic3—21no—
145C4⋊Q8—91no—
146#126—92no—
147#124—91no—
148#124—92no—
149#123—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152#121—91no—
153C4×Q8—91no—
154C4×D4—31no—
155#117—31no—
156C23×C3—32no—
157C23×C3—31no—
158C22×D3C22×D312yes—
159C22×D3C22×D311yescharacteristic
160C3×D4—62no—
161C2×C3×C4—21no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—31no—
165C2×C3×C4—32no—
166C2×C3×C4—31no—
167C2×C3×C4—31no—
168C2×C3×C4—62no—
169C2×C3×C4—122no—
170C2×C3×C4C22×D311yescharacteristic
171C2×C3×C4C22×D311yescharacteristic
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175C2×C3×C4C22×D311yescharacteristic
176#67—122no—
177#67—122no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—32no—
181C2×Dic3—32no—
182C2×Dic3—62no—
183C2×Dic3—62no—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D311yescharacteristic
186C2×Dic3C22×D312yes—
187C2×Dic3C22×D311yescharacteristic
188D12—22no—
189C4×D3—22no—
190C4×D3—42no—
191Dic6—64no—
192Dic6—62no—
193Dic6—62no—
194Dic6—122no—
195Dic6—122no—
196C2×C32C2×#4111yescharacteristic
197C2×C32H(5,𝔽2)11yescharacteristic
198C2×C322−1+411yescharacteristic
199C3×D3—41no—
200C3×D3—42no—
201C2×Q8—92no—
202C2×Q8—91no—
203C2×D4—31no—
204C2×D4—92no—
205C22×C4—31no—
206C22×C4—32no—
207C4⋊C4—31no—
208C4⋊C4—31no—
209C4⋊C4—31no—
210C4⋊C4—92no—
211C4⋊C4—92no—
212C4⋊C4—91no—
213C4⋊C4—92no—
214C22⋊C4—31no—
215C22⋊C4—31no—
216C22⋊C4—32no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C42—31no—
222C42—92no—
223C42D3211yescharacteristic
224C22×C3—21no—
225C22×C3—61no—
226C22×C3—62no—
227C22×C3—61no—
228C22×C3—62no—
229C22×C3—62no—
230C22×C3—61no—
231C22×C3C23×D311yescharacteristic
232C22×C3C23×D311yescharacteristic
233C2×D3—21no—
234C2×D3—22no—
235C2×D3—22no—
236C2×D3—21no—
237C2×D3—21no—
238C2×D3—22no—
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—21no—
243C3×C4—21no—
244C3×C4—22no—
245C3×C4—41no—
246C3×C4—61no—
247C3×C4—61no—
248C3×C4—62no—
249C3×C4—61no—
250C3×C4#23512yes—
251C3×C4#23512yes—
252Dic3—21no—
253Dic3—22no—
254Dic3—21no—
255Dic3—21no—
256Dic3—64no—
257Dic3—62no—
258Dic3—62no—
259Dic3—62no—
260Dic3—62no—
261Dic3—122no—
262C32#54211yescharacteristic, complemented, Sylow
263C23—31no—
264C23—32no—
265Q8—182no—
266Q8—362no—
267D4—62no—
268D4—362no—
269C2×C4—31no—
270C2×C4—31no—
271C2×C4—32no—
272C2×C4—31no—
273C2×C4—62no—
274C2×C4—92no—
275C2×C4—92no—
276C2×C4—122no—
277C2×C4C2×D3211yescharacteristic
278C2×C4C2×D3211yescharacteristic
279C2×C4C2×D3211yescharacteristic
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—122no—
284C2×C3—121no—
285C2×C3C2×#23511yescharacteristic
286C2×C3C2×#23511yescharacteristic
287C2×C3#100411yescharacteristic
288C2×C3#100911yescharacteristic
289C2×C3#101011yescharacteristic
290C2×C3#101011yescharacteristic
291D3—41no—
292D3—42no—
293C22—61no—
294C22—62no—
295C22—61no—
296C22—62no—
297C22—62no—
298C22—61no—
299C22C22×D3211yescharacteristic
300C4—21no—
301C4—21no—
302C4—61no—
303C4—61no—
304C4—62no—
305C4—61no—
306C4—182no—
307C4—182no—
308C4#379812yes—
309C3—21no—
310C3#553411yescharacteristic, complemented
311C3#556111yescharacteristic, complemented
312C2—121no—
313C2—122no—
314C2#6429811yescharacteristic
315C2C2×#379811yescharacteristic
316C2#6430711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C42C22
prime radical#63908C2
hypercenterC42D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C4×#634C2semidirect
2. C3×C4×D12C2semidirect
3. C3×#872C2semidirect
4. #63918C2semidirect
5. #63915C2semidirect
7. #63905C2semidirect
8. #63862C2semidirect
9. #63860C2semidirect
10. #63857C2semidirect
11. #63851C2semidirect
54. C4×D12D3semidirect
55. #872D3semidirect
262. C32#542semidirect
310. C3#5534semidirect
311. C3#5561semidirect

Maximal subgroups

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

subgroupindex
C4×#6342
C3×C4×D122
C3×#8722
#639182
#639152
#639082
#639052
#638622
#638602
#638572
#638512
#55613
#55343