Finite Group Databasebeta

← #10594496

Subgroups — #10594496

Order 576.

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

normal subgroups only

This group has 508 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#64099C211yesmaximal, characteristic, complemented
2C3×#895C211yesmaximal, characteristic, complemented
3C3×#893C211yesmaximal, characteristic, complemented
4#63914C211yesmaximal, characteristic, complemented
5#63908C212yesmaximal, complemented
6#63908C211yesmaximal, characteristic, complemented
7#63905C212yesmaximal, complemented
8Dic3×D12C211yesmaximal, characteristic, complemented
9#63891C212yesmaximal, complemented
10#63885C212yesmaximal, complemented
11#63855C211yesmaximal, characteristic, complemented
12#5619—31nomaximal
13#5562—31nomaximal
14C2×C3×D12C2211yescharacteristic
15C2×C3×D12C2211yescharacteristic
16C2×C3×D12C2212yes—
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×#625C2212yes—
20C2×#625C2212yes—
21C2×#625C2212yes—
22C2×D3×Dic3C2212yes—
23C32×C4⋊C4C2211yescharacteristic
24C4⋊(C3⋊Dic3)C2212yes—
25C4⋊(C3⋊Dic3)C2211yescharacteristic
26#3750C2212yes—
27C4×C3⋊Dic3C2211yescharacteristic
28C3×(C2×D3)⋊C4C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4×Dic3C2211yescharacteristic
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2211yescharacteristic
33(C2×D3)⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×#121—31no—
38C3×C4×D4—31no—
39#940—31no—
40#940—31no—
41#939—32no—
42#938—31no—
43#938—32no—
44D4×Dic3—31no—
45#930—32no—
46#930—31no—
47#929—32no—
48#926—31no—
49#895D311yescharacteristic, complemented
50#893D311yescharacteristic, complemented
51#890—31no—
52#890—31no—
53#890—61no—
54#886—32no—
55#884—32no—
56#883—32no—
57#883—32no—
58C4×D12—31no—
59#872—31no—
60C22×C3×D3C2311yescharacteristic
61C22×C3×D3C2312yes—
62C22×C3×D3C2312yes—
63C2×C32×C4C2312yes—
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3C2312yes—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3×Dic3C2311yescharacteristic
69C2×C3×Dic3C2311yescharacteristic
70C2×C3×Dic3C2311yescharacteristic
71C3×D12—22no—
72C3×D12—22no—
73C3×D12—42no—
74C3×C4×D3—22no—
75C3×C4×D3—42no—
76#625—42no—
77#625—42no—
78#625—42no—
79D3×Dic3—42no—
80#540—91noSylow
81C2×C3×D4—31no—
82C2×C3×D4—31no—
83C2×C3×D4—32no—
84C22×C3×C4—31no—
85C22×C3×C4—32no—
86C2×#67—32no—
87C2×#67—32no—
88C2×#67—32no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—32no—
92C22×Dic3—32no—
93C2×D12C2×D311yescharacteristic
94C2×D12C2×D311yescharacteristic
95C2×D12C2×D312yes—
96C2×C4×D3—32no—
97C2×C4×D3C2×D311yescharacteristic
98C2×C4×D3C2×D312yes—
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2×D311yescharacteristic
101C3×C4⋊C4C2×D311yescharacteristic
102C3×C22⋊C4—32no—
103C3×C22⋊C4—32no—
104C3×C42—31no—
105#217—32no—
106#217—32no—
107#217—31no—
108#217—31no—
109(C2×D3)⋊C4—31no—
110(C2×D3)⋊C4—31no—
111(C2×D3)⋊C4—32no—
112(C2×D3)⋊C4—32no—
113(C2×D3)⋊C4C2×D312yes—
114(C2×D3)⋊C4C2×D312yes—
115C4⋊Dic3—31no—
116C4⋊Dic3—32no—
117C4⋊Dic3—32no—
118C4⋊Dic3—31no—
119C4⋊Dic3—31no—
120C4⋊Dic3—31no—
121C4⋊Dic3—61no—
122C4⋊Dic3—62no—
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4—31no—
126Dic3⋊C4—31no—
127Dic3⋊C4—62no—
128C4×Dic3—31no—
129C4×Dic3—31no—
130C4×Dic3—61no—
131C4×Dic3C2×D311yescharacteristic
132C22×C32C2411yescharacteristic
133C2×C3×D3—21no—
134C2×C3×D3—22no—
135C2×C3×D3—22no—
136C2×C3×D3—22no—
137C2×C3×D3—21no—
138C2×C3×D3—22no—
139C2×C3×D3—22no—
140C2×C3×D3—22no—
141C2×C3×D3—21no—
142C32×C4—22no—
143C32×C4#4112yes—
144C3⋊Dic3—22no—
145C3⋊Dic3—21no—
146C3⋊Dic3—21no—
147C3×Dic3—21no—
148C3×Dic3—21no—
149C3×Dic3—21no—
150#127—91no—
151#125—91no—
152#123—92no—
153#123—92no—
154#121—31no—
155#121—92no—
156#121—92no—
157#121—91no—
158C4×D4—31no—
159C4×D4—91no—
160#117—91no—
161C23×C3—31no—
162C23×C3—32no—
163C23×C3—32no—
164C22×D3C22×D311yescharacteristic
165C22×D3C22×D312yes—
166C22×D3C22×D312yes—
167C3×D4—62no—
168C3×D4—62no—
169C3×D4—122no—
170C2×C3×C4—21no—
171C2×C3×C4—22no—
172C2×C3×C4—31no—
173C2×C3×C4—31no—
174C2×C3×C4—31no—
175C2×C3×C4—62no—
176C2×C3×C4—122no—
177C2×C3×C4C22×D312yes—
178C2×C3×C4C22×D312yes—
179C2×C3×C4C22×D311yescharacteristic
180C2×C3×C4C22×D311yescharacteristic
181#67—122no—
182#67—122no—
183#67—122no—
184#67—122no—
185#67—122no—
186#67—122no—
187C2×Dic3—31no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—32no—
192C2×Dic3—32no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—62no—
196C2×Dic3—122no—
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D311yescharacteristic
200D12—22no—
201D12—22no—
202D12—42no—
203C4×D3—22no—
204C4×D3—42no—
205C4×D3—122no—
206C2×C32C2×#4111yescharacteristic
207C2×C32H(5,𝔽2)11yescharacteristic
208C2×C32H(5,𝔽2)11yescharacteristic
209C3×D3—42no—
210C3×D3—41no—
211C3×D3—42no—
212C2×D4—31no—
213C2×D4—31no—
214C2×D4—32no—
215C2×D4—92no—
216C2×D4—92no—
217C2×D4—92no—
218C22×C4—31no—
219C22×C4—32no—
220C22×C4—92no—
221C4⋊C4—91no—
222C4⋊C4—91no—
223C4⋊C4—91no—
224C4⋊C4—92no—
225C4⋊C4—92no—
226C4⋊C4D3211yescharacteristic
227C22⋊C4—32no—
228C22⋊C4—32no—
229C22⋊C4—91no—
230C22⋊C4—91no—
231C22⋊C4—92no—
232C22⋊C4—92no—
233C42—31no—
234C42—91no—
235C22×C3—21no—
236C22×C3—61no—
237C22×C3—62no—
238C22×C3—62no—
239C22×C3—62no—
240C22×C3—62no—
241C22×C3—61no—
242C22×C3—62no—
243C22×C3—61no—
244C22×C3—62no—
245C22×C3C23×D311yescharacteristic
246C22×C3C23×D311yescharacteristic
247C2×D3—22no—
248C2×D3—21no—
249C2×D3—22no—
250C2×D3—22no—
251C2×D3—21no—
252C2×D3—22no—
253C2×D3—22no—
254C2×D3—22no—
255C2×D3—21no—
256C3×C4—22no—
257C3×C4—22no—
258C3×C4—22no—
259C3×C4—42no—
260C3×C4—61no—
261C3×C4—61no—
262C3×C4—61no—
263C3×C4#23512yes—
264C3×C4#23712yes—
265Dic3—21no—
266Dic3—21no—
267Dic3—21no—
268Dic3—61no—
269Dic3—61no—
270Dic3—62no—
271Dic3—62no—
272Dic3—61no—
273Dic3—61no—
274Dic3—121no—
275Dic3—122no—
276Dic3—121no—
277C32#54011yescharacteristic, complemented, Sylow
278C23—32no—
279C23—32no—
280C23—31no—
281D4—62no—
282D4—62no—
283D4—122no—
284D4—362no—
285D4—362no—
286D4—362no—
287C2×C4—31no—
288C2×C4—31no—
289C2×C4—31no—
290C2×C4—62no—
291C2×C4—91no—
292C2×C4—91no—
293C2×C4—92no—
294C2×C4—122no—
295C2×C4—362no—
296C2×C4C2×D3212yes—
297C2×C4C2×D3211yescharacteristic
298C2×C3—21no—
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—121no—
302C2×C3—122no—
303C2×C3—122no—
304C2×C3C2×#23511yescharacteristic
305C2×C3C2×#23711yescharacteristic
306C2×C3#100411yescharacteristic
307C2×C3#100411yescharacteristic
308C2×C3#100911yescharacteristic
309C2×C3#100911yescharacteristic
310D3—42no—
311D3—41no—
312D3—42no—
313C22—61no—
314C22—62no—
315C22—62no—
316C22—62no—
317C22—61no—
318C22—62no—
319C22—62no—
320C22—62no—
321C22—61no—
322C22C22×D3211yescharacteristic
323C4—22no—
324C4—61no—
325C4—61no—
326C4—61no—
327C4—181no—
328C4—182no—
329C4—181no—
330C4#379512yes—
331C3—21no—
332C3#556211yescharacteristic, complemented
333C3#561911yescharacteristic, complemented
334C2—121no—
335C2—122no—
336C2—122no—
337C2C2×#379511yescharacteristic
338C2#6431311yescharacteristic
339C2#6432011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#540
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #64099C2semidirect
2. C3×#895C2semidirect
3. C3×#893C2semidirect
4. #63914C2semidirect
5. #63908C2semidirect
6. #63908C2semidirect
7. #63905C2semidirect
8. Dic3×D12C2semidirect
9. #63891C2semidirect
10. #63885C2semidirect
11. #63855C2semidirect
49. #895D3semidirect
50. #893D3semidirect
277. C32#540semidirect
332. C3#5562semidirect
333. C3#5619semidirect

Maximal subgroups

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

subgroupindex
#640992
C3×#8952
C3×#8932
#639142
#639082
#639052
Dic3×D122
#638912
#638852
#638552
#56193
#55623