Finite Group Databasebeta

← #10595125

Subgroups — #10595125

Order 576.

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

normal subgroups only

This group has 534 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#64149C211yesmaximal, characteristic, complemented
2C3×#938C211yesmaximal, characteristic, complemented
3C3×D4×Dic3C211yesmaximal, characteristic, complemented
4#63978C212yesmaximal, complemented
5#63975C212yesmaximal, complemented
6#63966C212yesmaximal, complemented
7#63958C212yesmaximal, complemented
8#63909C211yesmaximal, characteristic, complemented
9#63894C211yesmaximal, characteristic, complemented
10#63872C211yesmaximal, characteristic, complemented
11#63855C211yesmaximal, characteristic, complemented
12#5619—31nomaximal
13#5562—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×#638C2212yes—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#67C2212yes—
19C22×C3×Dic3C2212yes—
20C2×C3×C4×D3C2211yescharacteristic
21C2×#626C2212yes—
22C2×#624C2212yes—
23C4×C3⋊Dic3C2211yescharacteristic
24C3×#217C2212yes—
25C3×#217C2212yes—
26C3×C4⋊Dic3C2211yescharacteristic
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29Dic3⋊Dic3C2211yescharacteristic
30Dic3⋊Dic3C2211yescharacteristic
31Dic3⋊Dic3C2212yes—
32Dic3⋊Dic3C2212yes—
33#3722C2212yes—
34#3722C2212yes—
35(C2×D3)⋊Dic3C2211yescharacteristic
36(C2×D3)⋊Dic3C2211yescharacteristic
37C3×#121—31no—
38C3×C4×D4—31no—
39#940—31no—
40#940—31no—
41#940—61no—
42#939—32no—
43#938—32no—
44#938D311yescharacteristic, complemented
45D4×Dic3D311yescharacteristic, complemented
46#930—32no—
47#930—31no—
48#929—32no—
49#926—31no—
50#895—31no—
51#893—31no—
52#890—31no—
53#890—31no—
54#886—32no—
55#884—32no—
56#883—32no—
57#883—32no—
58C4×D12—31no—
59#872—31no—
60C23×C32C2312yes—
61C22×C3⋊D3C2312yes—
62C22×C3×D3C2311yescharacteristic
63C32×D4—22no—
64C2×C32×C4C2311yescharacteristic
65#638—42no—
66#638—42no—
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C4⋊(C3⋊D3)—22no—
70C3×#67—42no—
71C2×C3×Dic3—42no—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2312yes—
77C3×C4×D3—22no—
78#626—42no—
79#624—42no—
80#540—91noSylow
81C2×C3×D4—21no—
82C2×C3×D4—32no—
83C2×C3×D4C2×D311yescharacteristic
84C2×C3×D4C2×D311yescharacteristic
85C22×C3×C4—32no—
86C22×C3×C4—31no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—32no—
92C2×#67—62no—
93C2×#67—62no—
94C2×#67C2×D312yes—
95C22×Dic3C2×D312yes—
96C2×D12—31no—
97C2×D12—31no—
98C2×D12—32no—
99C2×D12—61no—
100C2×C4×D3—32no—
101C2×C4×D3—32no—
102C2×C4×D3C2×D311yescharacteristic
103C3×C4⋊C4—31no—
104C3×C4⋊C4—31no—
105C3×C22⋊C4—32no—
106C3×C22⋊C4—32no—
107C3×C42—31no—
108#217—31no—
109#217—31no—
110#217C2×D312yes—
111#217C2×D312yes—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—32no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118C4⋊Dic3—31no—
119C4⋊Dic3—32no—
120C4⋊Dic3—31no—
121C4⋊Dic3—32no—
122C4⋊Dic3C2×D311yescharacteristic
123C4⋊Dic3C2×D311yescharacteristic
124Dic3⋊C4—31no—
125Dic3⋊C4—31no—
126Dic3⋊C4—32no—
127Dic3⋊C4—32no—
128C4×Dic3—31no—
129C4×Dic3—31no—
130C4×Dic3—61no—
131C4×Dic3C2×D311yescharacteristic
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32C2411yescharacteristic
136C2×C3⋊D3—22no—
137C2×C3⋊D3—22no—
138C2×C3⋊D3—22no—
139C2×C3×D3—21no—
140C2×C3×D3—21no—
141C2×C3×D3—21no—
142C32×C4#4112yes—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3×Dic3—21no—
146C3×Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3—22no—
149C3×Dic3—21no—
150#127—91no—
151#125—91no—
152#123—92no—
153#123—92no—
154#121—31no—
155#121—92no—
156#121—91no—
157#121—92no—
158C4×D4—31no—
159C4×D4—91no—
160#117—91no—
161C23×C3—22no—
162C23×C3—31no—
163C23×C3C22×D312yes—
164C23×C3C22×D312yes—
165C22×D3—32no—
166C22×D3—32no—
167C22×D3—62no—
168C22×D3C22×D311yescharacteristic
169C3×D4—22no—
170C3×D4—22no—
171C3×D4—42no—
172C3×D4—122no—
173C2×C3×C4—21no—
174C2×C3×C4—32no—
175C2×C3×C4—32no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—31no—
179C2×C3×C4—62no—
180C2×C3×C4—122no—
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183#67—42no—
184#67—122no—
185#67—122no—
186#67—122no—
187#67—122no—
188#67—122no—
189#67—124no—
190#67—124no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—42no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D311yescharacteristic
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D312yes—
202C2×Dic3C22×D312yes—
203D12—62no—
204D12—62no—
205D12—122no—
206D12—122no—
207C4×D3—22no—
208C4×D3—122no—
209C4×D3—122no—
210C2×C32—42no—
211C2×C32C2×#4111yescharacteristic
212C2×C32H(5,𝔽2)11yescharacteristic
213C2×C32H(5,𝔽2)11yescharacteristic
214C3⋊D3—42no—
215C3×D3—41no—
216C2×D4—32no—
217C2×D4—92no—
218C2×D4—92no—
219C2×D4—92no—
220C2×D4—91no—
221C2×D4D3211yescharacteristic
222C22×C4—32no—
223C22×C4—31no—
224C22×C4—92no—
225C4⋊C4—31no—
226C4⋊C4—31no—
227C4⋊C4—91no—
228C4⋊C4—91no—
229C4⋊C4—92no—
230C4⋊C4—92no—
231C22⋊C4—32no—
232C22⋊C4—32no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C22⋊C4—91no—
236C22⋊C4—91no—
237C42—31no—
238C42—91no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—22no—
245C22×C3—21no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3—61no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3C23×D311yescharacteristic
253C22×C3C23×D311yescharacteristic
254C2×D3—21no—
255C2×D3—21no—
256C2×D3—21no—
257C2×D3—62no—
258C2×D3—62no—
259C2×D3—62no—
260C2×D3—62no—
261C2×D3—62no—
262C2×D3—62no—
263C2×D3—122no—
264C2×D3—122no—
265C2×D3—122no—
266C3×C4—22no—
267C3×C4—61no—
268C3×C4—61no—
269C3×C4—62no—
270C3×C4—62no—
271C3×C4—61no—
272C3×C4#23512yes—
273C3×C4#23712yes—
274Dic3—21no—
275Dic3—22no—
276Dic3—21no—
277Dic3—22no—
278Dic3—21no—
279Dic3—62no—
280Dic3—62no—
281Dic3—61no—
282Dic3—61no—
283Dic3—61no—
284Dic3—61no—
285C32#54011yescharacteristic, complemented, Sylow
286C23—31no—
287C23—92no—
288C23C2×D3212yes—
289D4—22no—
290D4—122no—
291D4—182no—
292D4—362no—
293D4—362no—
294D4—362no—
295C2×C4—31no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—32no—
299C2×C4—32no—
300C2×C4—62no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—122no—
304C2×C4—362no—
305C2×C4C2×D3211yescharacteristic
306C2×C3—21no—
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—42no—
310C2×C3—42no—
311C2×C3—82no—
312C2×C3—121no—
313C2×C3C2×#23511yescharacteristic
314C2×C3C2×#23711yescharacteristic
315C2×C3#100411yescharacteristic
316C2×C3#100411yescharacteristic
317C2×C3#100911yescharacteristic
318C2×C3#100911yescharacteristic
319D3—41no—
320D3—122no—
321D3—122no—
322D3—242no—
323C22—22no—
324C22—22no—
325C22—22no—
326C22—61no—
327C22—61no—
328C22—61no—
329C22—182no—
330C22—182no—
331C22—182no—
332C22C22×D3211yescharacteristic
333C4—61no—
334C4—62no—
335C4—61no—
336C4—62no—
337C4—61no—
338C4—181no—
339C4—181no—
340C4#380412yes—
341C3—21no—
342C3#556211yescharacteristic, complemented
343C3#561911yescharacteristic, complemented
344C2—42no—
345C2—121no—
346C2—362no—
347C2#6431411yescharacteristic
348C2#6431411yescharacteristic
349C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #64149C2semidirect
2. C3×#938C2semidirect
3. C3×D4×Dic3C2semidirect
4. #63978C2semidirect
5. #63975C2semidirect
6. #63966C2semidirect
7. #63958C2semidirect
8. #63909C2semidirect
9. #63894C2semidirect
10. #63872C2semidirect
11. #63855C2semidirect
44. #938D3semidirect
45. D4×Dic3D3semidirect
285. C32#540semidirect
342. C3#5562semidirect
343. C3#5619semidirect

Maximal subgroups

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

subgroupindex
#641492
C3×#9382
C3×D4×Dic32
#639782
#639752
#639662
#639582
#639092
#638942
#638722
#638552
#56193
#55623