Finite Group Databasebeta

← #10594498

Subgroups — #10594498

Order 576.

466 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 128 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#64097C211yesmaximal, characteristic, complemented
2C3×#895C211yesmaximal, characteristic
3C3×#887C211yesmaximal, characteristic, complemented
4#63896C212yesmaximal, complemented
5Dic3×D12C211yesmaximal, characteristic, complemented
6#63884C212yesmaximal, complemented
7#63863C211yesmaximal, characteristic, complemented
8#63860C212yesmaximal, complemented
9#63859C212yesmaximal, complemented
10#63856C211yesmaximal, characteristic, complemented
11#63855C211yesmaximal, characteristic, complemented
12#5603—31nomaximal
13#5582—31nomaximal
14C2×#634C2212yes—
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2212yes—
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×Dic6C2211yescharacteristic
19C2×D3×Dic3C2212yes—
20C32×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊Dic3)C2211yescharacteristic
22#3750C2212yes—
23C4×C3⋊Dic3C2211yescharacteristic
24C3×(C2×D3)⋊C4C2212yes—
25C3×Dic3⋊C4C2212yes—
26C3×C4×Dic3C2211yescharacteristic
27C3×C4×Dic3C2212yes—
28#3724C2212yes—
29Dic3⋊Dic3C2211yescharacteristic
30Dic3⋊Dic3C2211yescharacteristic
31(C2×D3)⋊Dic3C2211yescharacteristic
32(C2×D3)⋊Dic3C2211yescharacteristic
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×#121—31no—
38C3×C4×Q8—31no—
39#947—31no—
40#946—31no—
41#936—31no—
42#936—32no—
43#935—32no—
44D4×Dic3—31no—
45#926—31no—
46#924—31no—
47#896—32no—
48#895D311yescharacteristic
49#894—32no—
50#888—31no—
51#888—31no—
52#888—61no—
53#887D311yescharacteristic, complemented
54#879—32no—
55#878—32no—
56#876—32no—
57#875—32no—
58C4×D12—31no—
59#872—31no—
60C22×C3×D3C2311yescharacteristic
61C22×C3×D3C2312yes—
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2311yescharacteristic
65C2×C3⋊Dic3C2311yescharacteristic
66C2×C3⋊Dic3C2312yes—
67#634—42no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C3×D12—22no—
73C3×D12—42no—
74C3×C4×D3—22no—
75C3×Dic6—22no—
76D3×Dic3—42no—
77#542—91noSylow
78C2×C3×Q8—31no—
79C2×C3×D4—31no—
80C2×C3×D4—32no—
81C22×C3×C4—31no—
82C22×Dic3—32no—
83C2×D12C2×D311yescharacteristic
84C2×D12C2×D312yes—
85C2×C4×D3—32no—
86C2×C4×D3C2×D311yescharacteristic
87C2×Dic6—32no—
88C2×Dic6—32no—
89C2×Dic6—62no—
90C2×Dic6C2×D311yescharacteristic
91C3×C4⋊C4—21no—
92C3×C4⋊C4—32no—
93C3×C4⋊C4C2×D311yescharacteristic
94C3×C4⋊C4C2×D311yescharacteristic
95C3×C22⋊C4—32no—
96C3×C42—31no—
97C3×C42—32no—
98#217—32no—
99#217—32no—
100#217—32no—
101#217—32no—
102#217—31no—
103#217—31no—
104(C2×D3)⋊C4—31no—
105(C2×D3)⋊C4—31no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—32no—
109(C2×D3)⋊C4—32no—
110(C2×D3)⋊C4C2×D312yes—
111C4⋊Dic3—31no—
112C4⋊Dic3—31no—
113C4⋊Dic3—31no—
114C4⋊Dic3—31no—
115C4⋊Dic3—61no—
116Dic3⋊C4—32no—
117Dic3⋊C4—32no—
118Dic3⋊C4—31no—
119Dic3⋊C4—31no—
120Dic3⋊C4—32no—
121Dic3⋊C4—32no—
122Dic3⋊C4—62no—
123Dic3⋊C4C2×D312yes—
124C4×Dic3—31no—
125C4×Dic3—31no—
126C4×Dic3—61no—
127C4×Dic3C2×D312yes—
128C4×Dic3C2×D311yescharacteristic
129C22×C32C2411yescharacteristic
130C2×C3×D3—21no—
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—21no—
134C2×C3×D3—21no—
135C2×C3×D3—22no—
136C32×C4—22no—
137C32×C4#4112yes—
138C3⋊Dic3—21no—
139C3⋊Dic3—21no—
140C3⋊Dic3—22no—
141C3×Dic3—22no—
142C3×Dic3—21no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C4⋊Q8—91no—
146#126—92no—
147#124—92no—
148#124—91no—
149#123—92no—
150C22⋊Q8—92no—
151C22⋊Q8—91no—
152#121—31no—
153C4×Q8—31no—
154C4×D4—91no—
155#117—91no—
156C23×C3—31no—
157C23×C3—32no—
158C22×D3C22×D311yescharacteristic
159C22×D3C22×D312yes—
160C3×Q8—62no—
161C3×D4—62no—
162C3×D4—122no—
163C2×C3×C4—21no—
164C2×C3×C4—22no—
165C2×C3×C4—31no—
166C2×C3×C4—31no—
167C2×C3×C4—32no—
168C2×C3×C4—31no—
169C2×C3×C4—62no—
170C2×C3×C4C22×D312yes—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D311yescharacteristic
173C2×C3×C4C22×D311yescharacteristic
174C2×Dic3—31no—
175C2×Dic3—31no—
176C2×Dic3—31no—
177C2×Dic3—31no—
178C2×Dic3—32no—
179C2×Dic3—32no—
180C2×Dic3—61no—
181C2×Dic3—61no—
182C2×Dic3—62no—
183C2×Dic3—122no—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D312yes—
186C2×Dic3C22×D311yescharacteristic
187C2×Dic3C22×D311yescharacteristic
188D12—22no—
189D12—42no—
190C4×D3—22no—
191C4×D3—122no—
192Dic6—22no—
193Dic6—122no—
194Dic6—122no—
195Dic6—124no—
196C2×C32C2×#4111yescharacteristic
197C2×C32H(5,𝔽2)11yescharacteristic
198C2×C322−1+411yescharacteristic
199C3×D3—41no—
200C3×D3—42no—
201C2×Q8—31no—
202C2×Q8—92no—
203C2×D4—31no—
204C2×D4—32no—
205C22×C4—31no—
206C22×C4—92no—
207C4⋊C4—32no—
208C4⋊C4—91no—
209C4⋊C4—91no—
210C4⋊C4—92no—
211C4⋊C4—91no—
212C4⋊C4—92no—
213C4⋊C4D3211yescharacteristic
214C22⋊C4—32no—
215C22⋊C4—91no—
216C22⋊C4—91no—
217C22⋊C4—92no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C42—31no—
222C42—32no—
223C42—91no—
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—22no—
241C3×C4—22no—
242C3×C4—42no—
243C3×C4—61no—
244C3×C4—62no—
245C3×C4—61no—
246C3×C4—61no—
247C3×C4#23512yes—
248C3×C4#23712yes—
249Dic3—22no—
250Dic3—21no—
251Dic3—21no—
252Dic3—21no—
253Dic3—62no—
254Dic3—62no—
255Dic3—61no—
256Dic3—62no—
257Dic3—61no—
258Dic3—61no—
259Dic3—62no—
260Dic3—61no—
261Dic3—122no—
262C32#54211yescharacteristic, complemented, Sylow
263C23—32no—
264C23—31no—
265Q8—62no—
266Q8—362no—
267D4—62no—
268D4—122no—
269C2×C4—31no—
270C2×C4—32no—
271C2×C4—31no—
272C2×C4—31no—
273C2×C4—62no—
274C2×C4—91no—
275C2×C4—91no—
276C2×C4—92no—
277C2×C4—362no—
278C2×C4C2×D3212yes—
279C2×C4C2×D3211yescharacteristic
280C2×C3—21no—
281C2×C3—21no—
282C2×C3—21no—
283C2×C3—121no—
284C2×C3—122no—
285C2×C3C2×#23511yescharacteristic
286C2×C3C2×#23711yescharacteristic
287C2×C3#100411yescharacteristic
288C2×C3#100711yescharacteristic
289C2×C3#100911yescharacteristic
290C2×C3#101011yescharacteristic
291D3—41no—
292D3—42no—
293C22—61no—
294C22—62no—
295C22—62no—
296C22—61no—
297C22—62no—
298C22—61no—
299C22C22×D3211yescharacteristic
300C4—22no—
301C4—61no—
302C4—62no—
303C4—61no—
304C4—61no—
305C4—182no—
306C4—181no—
307C4—181no—
308C4#379512yes—
309C3—21no—
310C3#558211yescharacteristic, complemented
311C3#560311yescharacteristic, complemented
312C2—121no—
313C2—122no—
314C2C2×#379511yescharacteristic
315C2#6431411yescharacteristic
316C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radicalC3×#895C2
hypercenterC4⋊C4D32
hypocenterC32#542
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64097C2semidirect
3. C3×#887C2semidirect
4. #63896C2semidirect
5. Dic3×D12C2semidirect
6. #63884C2semidirect
7. #63863C2semidirect
8. #63860C2semidirect
9. #63859C2semidirect
10. #63856C2semidirect
11. #63855C2semidirect
53. #887D3semidirect
262. C32#542semidirect
310. C3#5582semidirect
311. C3#5603semidirect

Maximal subgroups

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

subgroupindex
#640972
C3×#8952
C3×#8872
#638962
Dic3×D122
#638842
#638632
#638602
#638592
#638562
#638552
#56033
#55823