Finite Group Databasebeta

← #10594341

Subgroups — #10594341

Order 576.

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

normal subgroups only

This group has 469 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#64106C211yesmaximal, characteristic, complemented
2C3×#893C211yesmaximal, characteristic, complemented
3C3×#888C211yesmaximal, characteristic, complemented
4#63916C211yesmaximal, characteristic
5#63907C212yesmaximal, complemented
6#63874C211yesmaximal, characteristic, complemented
7#63861C212yesmaximal, complemented
8#63860C211yesmaximal, characteristic, complemented
9#63856C212yesmaximal, complemented
10#63850C212yesmaximal, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5733—31nomaximal
13#5561—31nomaximal
14C2×C4×C3⋊D3C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×D12C2211yescharacteristic
17C2×C3×C4×D3C2212yes—
18C2×C3×Dic6C2212yes—
19C2×#626C2212yes—
20C32×C4⋊C4C2211yescharacteristic
21#3752C2212yes—
22C4⋊(C3⋊Dic3)C2212yes—
23C3×(C2×D3)⋊C4C2212yes—
24C3×C4⋊Dic3C2211yescharacteristic
25C3×Dic3⋊C4C2212yes—
26C3×C4×Dic3C2211yescharacteristic
27C3×C4×Dic3C2211yescharacteristic
28#3724C2212yes—
29#3724C2211yescharacteristic
30#3724C2211yescharacteristic
31#3722C2211yescharacteristic
32#3722C2211yescharacteristic
33(C2×D3)⋊Dic3C2212yes—
34(C2×D3)⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2212yes—
37C3×C4⋊Q8—31no—
38C3×C4×D4—31no—
39#947—32no—
40#946—32no—
41#938—31no—
42#936—31no—
43#929—32no—
44#924—32no—
45#898—32no—
46#897—31no—
47#897—31no—
48#897—61no—
49#895—31no—
50#894—32no—
51#893D311yescharacteristic, complemented
52#888D311yescharacteristic, complemented
53#887—31no—
54#885—32no—
55#879—32no—
56#875—31no—
57#872—31no—
58#872—31no—
59C4×Dic6—31no—
60C22×C3⋊D3C2311yescharacteristic
61C22×C3×D3C2312yes—
62C2×C32×C4C2312yes—
63C2×C32×C4C2311yescharacteristic
64C2×C3⋊Dic3C2312yes—
65C2×C3⋊Dic3C2311yescharacteristic
66C4×C3⋊D3—22no—
67#634—22no—
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2312yes—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2311yescharacteristic
72C2×C3×Dic3C2311yescharacteristic
73C3×D12—22no—
74C3×C4×D3—42no—
75C3×Dic6—42no—
76#626—42no—
77#542—91noSylow
78C2×C3×Q8—32no—
79C2×C3×D4—31no—
80C22×C3×C4—32no—
81C2×#67—32no—
82C2×D12—32no—
83C2×D12C2×D311yescharacteristic
84C2×C4×D3—31no—
85C2×C4×D3—31no—
86C2×C4×D3—61no—
87C2×C4×D3C2×D312yes—
88C2×Dic6—31no—
89C2×Dic6—31no—
90C2×Dic6—61no—
91C2×Dic6C2×D312yes—
92C3×C4⋊C4—21no—
93C3×C4⋊C4—31no—
94C3×C4⋊C4—32no—
95C3×C4⋊C4C2×D311yescharacteristic
96C3×C4⋊C4C2×D311yescharacteristic
97C3×C22⋊C4—32no—
98C3×C42—31no—
99C3×C42—31no—
100#217—32no—
101#217—32no—
102#217—32no—
103(C2×D3)⋊C4—32no—
104(C2×D3)⋊C4—32no—
105(C2×D3)⋊C4—32no—
106(C2×D3)⋊C4—32no—
107(C2×D3)⋊C4—32no—
108(C2×D3)⋊C4—31no—
109(C2×D3)⋊C4—31no—
110(C2×D3)⋊C4—31no—
111(C2×D3)⋊C4—31no—
112(C2×D3)⋊C4—62no—
113(C2×D3)⋊C4C2×D312yes—
114C4⋊Dic3—32no—
115C4⋊Dic3—32no—
116C4⋊Dic3—62no—
117C4⋊Dic3C2×D311yescharacteristic
118Dic3⋊C4—32no—
119Dic3⋊C4—31no—
120Dic3⋊C4—31no—
121Dic3⋊C4—31no—
122Dic3⋊C4—31no—
123Dic3⋊C4—32no—
124Dic3⋊C4C2×D312yes—
125C4×Dic3—32no—
126C4×Dic3—32no—
127C4×Dic3C2×D311yescharacteristic
128C4×Dic3C2×D311yescharacteristic
129C22×C32C2411yescharacteristic
130C2×C3⋊D3—21no—
131C2×C3⋊D3—21no—
132C2×C3⋊D3—21no—
133C2×C3×D3—22no—
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C32×C4—22no—
137C32×C4#4112yes—
138C3⋊Dic3—22no—
139C3⋊Dic3—21no—
140C3×Dic3—22no—
141C3×Dic3—21no—
142C3×Dic3—21no—
143C3×Dic3—21no—
144C3×Dic3—21no—
145C4⋊Q8—31no—
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—91no—
156C23×C3—32no—
157C22×D3—31no—
158C22×D3—31no—
159C22×D3—61no—
160C22×D3C22×D312yes—
161C3×Q8—122no—
162C3×D4—62no—
163C2×C3×C4—21no—
164C2×C3×C4—22no—
165C2×C3×C4—31no—
166C2×C3×C4—32no—
167C2×C3×C4—31no—
168C2×C3×C4—31no—
169C2×C3×C4—31no—
170C2×C3×C4—122no—
171C2×C3×C4C22×D312yes—
172C2×C3×C4C22×D312yes—
173C2×C3×C4C22×D311yescharacteristic
174C2×C3×C4C22×D311yescharacteristic
175#67—122no—
176C2×Dic3—32no—
177C2×Dic3—32no—
178C2×Dic3—31no—
179C2×Dic3—31no—
180C2×Dic3—61no—
181C2×Dic3—62no—
182C2×Dic3C22×D311yescharacteristic
183C2×Dic3C22×D312yes—
184C2×Dic3C22×D311yescharacteristic
185C2×Dic3C22×D311yescharacteristic
186C2×Dic3C22×D311yescharacteristic
187D12—22no—
188D12—122no—
189C4×D3—42no—
190C4×D3—62no—
191C4×D3—62no—
192C4×D3—122no—
193Dic6—42no—
194Dic6—62no—
195Dic6—62no—
196Dic6—122no—
197C2×C32C2×#4111yescharacteristic
198C2×C32H(5,𝔽2)11yescharacteristic
199C2×C322−1+411yescharacteristic
200C3⋊D3—41no—
201C3×D3—42no—
202C2×Q8—32no—
203C2×Q8—91no—
204C2×D4—31no—
205C2×D4—92no—
206C22×C4—32no—
207C22×C4—91no—
208C4⋊C4—31no—
209C4⋊C4—32no—
210C4⋊C4—91no—
211C4⋊C4—91no—
212C4⋊C4—92no—
213C4⋊C4—92no—
214C4⋊C4D3211yescharacteristic
215C22⋊C4—32no—
216C22⋊C4—91no—
217C22⋊C4—91no—
218C22⋊C4—92no—
219C22⋊C4—92no—
220C22⋊C4—92no—
221C22⋊C4—92no—
222C42—31no—
223C42—31no—
224C42—92no—
225C22×C3—21no—
226C22×C3—62no—
227C22×C3—62no—
228C22×C3—62no—
229C22×C3C23×D311yescharacteristic
230C22×C3C23×D311yescharacteristic
231C2×D3—22no—
232C2×D3—22no—
233C2×D3—22no—
234C2×D3—62no—
235C2×D3—61no—
236C2×D3—61no—
237C2×D3—61no—
238C2×D3—61no—
239C2×D3—61no—
240C2×D3—61no—
241C2×D3—121no—
242C2×D3—121no—
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—42no—
247C3×C4—62no—
248C3×C4—61no—
249C3×C4—61no—
250C3×C4—61no—
251C3×C4—61no—
252C3×C4#23512yes—
253C3×C4#23912yes—
254Dic3—21no—
255Dic3—22no—
256Dic3—21no—
257Dic3—21no—
258Dic3—21no—
259Dic3—62no—
260Dic3—61no—
261Dic3—62no—
262Dic3—61no—
263Dic3—122no—
264Dic3—121no—
265C32#54211yescharacteristic, complemented, Sylow
266C23—32no—
267C23—91no—
268Q8—122no—
269Q8—182no—
270D4—62no—
271D4—362no—
272C2×C4—31no—
273C2×C4—31no—
274C2×C4—31no—
275C2×C4—31no—
276C2×C4—32no—
277C2×C4—92no—
278C2×C4—91no—
279C2×C4—122no—
280C2×C4—182no—
281C2×C4C2×D3212yes—
282C2×C4C2×D3211yescharacteristic
283C2×C3—21no—
284C2×C3—21no—
285C2×C3—21no—
286C2×C3—122no—
287C2×C3C2×#23511yescharacteristic
288C2×C3#100411yescharacteristic
289C2×C3#100411yescharacteristic
290C2×C3C2×#23911yescharacteristic
291C2×C3#100711yescharacteristic
292C2×C3#101011yescharacteristic
293D3—42no—
294D3—122no—
295D3—121no—
296D3—121no—
297C22—62no—
298C22—62no—
299C22—62no—
300C22—181no—
301C22—181no—
302C22—181no—
303C22C22×D3211yescharacteristic
304C4—22no—
305C4—61no—
306C4—62no—
307C4—61no—
308C4—61no—
309C4—61no—
310C4—181no—
311C4—182no—
312C4#379912yes—
313C3—21no—
314C3#556111yescharacteristic, complemented
315C3#573311yescharacteristic, complemented
316C2—122no—
317C2—361no—
318C2C2×#379911yescharacteristic
319C2#6431311yescharacteristic
320C2#6431511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
prime radical#63916C2
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. #64106C2semidirect
2. C3×#893C2semidirect
3. C3×#888C2semidirect
5. #63907C2semidirect
6. #63874C2semidirect
7. #63861C2semidirect
8. #63860C2semidirect
9. #63856C2semidirect
10. #63850C2semidirect
11. #63844C2semidirect
51. #893D3semidirect
52. #888D3semidirect
265. C32#542semidirect
314. C3#5561semidirect
315. C3#5733semidirect

Maximal subgroups

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

subgroupindex
#641062
C3×#8932
C3×#8882
#639162
#639072
#638742
#638612
#638602
#638562
#638502
#638442
#57333
#55613