Finite Group Databasebeta

← #10595217

Subgroups — #10595217

Order 576.

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

normal subgroups only

This group has 527 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
1D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#940C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4#63980C212yesmaximal, complemented
5#63970C212yesmaximal, complemented
6#63969C212yesmaximal, complemented
7#63962C212yesmaximal, complemented
8#63908C211yesmaximal, characteristic, complemented
9#63898C211yesmaximal, characteristic, complemented
10#63869C211yesmaximal, characteristic, complemented
11#63855C211yesmaximal, characteristic, complemented
12#5719—31nomaximal
13#5619—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×C3×#67C2212yes—
17C2×C3×#67C2212yes—
18C2×C3×#67C2212yes—
19C2×C3×D12C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×#625C2212yes—
22C2×D3×Dic3C2212yes—
23#3757C2212yes—
24C4⋊(C3⋊Dic3)C2211yescharacteristic
25C4×C3⋊Dic3C2211yescharacteristic
26C3×#217C2212yes—
27C3×C4⋊Dic3C2211yescharacteristic
28C3×C4×Dic3C2211yescharacteristic
29#3724C2212yes—
30Dic3⋊Dic3C2211yescharacteristic
31Dic3⋊Dic3C2211yescharacteristic
32Dic3⋊Dic3C2212yes—
33(C2×D3)⋊Dic3C2211yescharacteristic
34(C2×D3)⋊Dic3C2211yescharacteristic
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37C3×#127—31no—
38C3×#121—31no—
39#940D311yescharacteristic, complemented
40#939—32no—
41#939—32no—
42#939—32no—
43#938—32no—
44#938—31no—
45#938D311yescharacteristic, complemented
46#935—32no—
47#935—32no—
48D4×Dic3—31no—
49D4×Dic3—31no—
50D4×Dic3—31no—
51D4×Dic3—61no—
52#930—31no—
53#926—31no—
54#893—31no—
55#890—31no—
56#886—32no—
57#883—32no—
58#872—31no—
59#870—31no—
60C23×C32C2312yes—
61C22×C3×D3C2312yes—
62C22×C3×D3C2311yescharacteristic
63C32×D4—22no—
64C2×C32×C4C2311yescharacteristic
65C2×C3⋊Dic3—42no—
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C2×C3⋊Dic3C2311yescharacteristic
69C3×#67—42no—
70C3×#67—42no—
71C3×#67—42no—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C3×D12—22no—
77C3×C4×D3—22no—
78#625—42no—
79D3×Dic3—42no—
80#540—91noSylow
81C2×C3×D4—21no—
82C2×C3×D4—32no—
83C2×C3×D4—32no—
84C2×C3×D4—32no—
85C2×C3×D4—31no—
86C2×C3×D4C2×D311yescharacteristic
87C2×C3×D4C2×D311yescharacteristic
88C22×C3×C4—31no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67C2×D312yes—
92C2×#67C2×D312yes—
93C2×#67C2×D312yes—
94C22×Dic3—32no—
95C22×Dic3—32no—
96C22×Dic3—32no—
97C22×Dic3—62no—
98C2×D12C2×D311yescharacteristic
99C2×C4×D3—32no—
100C2×C4×D3C2×D311yescharacteristic
101C3×C4⋊C4—31no—
102C3×C22⋊C4—32no—
103C3×C42—31no—
104#217—32no—
105#217—32no—
106#217—31no—
107#217—31no—
108#217—32no—
109#217—32no—
110#217—62no—
111#217C2×D312yes—
112(C2×D3)⋊C4—31no—
113(C2×D3)⋊C4—31no—
114(C2×D3)⋊C4—32no—
115(C2×D3)⋊C4—32no—
116C4⋊Dic3—31no—
117C4⋊Dic3—31no—
118C4⋊Dic3—31no—
119C4⋊Dic3—31no—
120C4⋊Dic3—32no—
121C4⋊Dic3—61no—
122C4⋊Dic3C2×D311yescharacteristic
123Dic3⋊C4—32no—
124Dic3⋊C4—32no—
125Dic3⋊C4—31no—
126Dic3⋊C4—31no—
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—21no—
137C2×C3×D3—22no—
138C2×C3×D3—22no—
139C2×C3×D3—21no—
140C2×C3×D3—22no—
141C2×C3×D3—21no—
142C32×C4#4112yes—
143C3⋊Dic3—21no—
144C3⋊Dic3—21no—
145C3⋊Dic3—22no—
146C3×Dic3—21no—
147C3×Dic3—22no—
148C3×Dic3—21no—
149C3×Dic3—21no—
150#127—31no—
151#125—91no—
152#123—92no—
153#123—92no—
154#121—31no—
155#121—91no—
156#121—92no—
157#121—92no—
158C4×D4—91no—
159C4×D4—91no—
160#117—91no—
161C23×C3—22no—
162C23×C3—32no—
163C23×C3—31no—
164C23×C3C22×D312yes—
165C23×C3C22×D312yes—
166C22×D3C22×D312yes—
167C22×D3C22×D311yescharacteristic
168C3×D4—24no—
169C3×D4—22no—
170C3×D4—22no—
171C3×D4—62no—
172C3×D4—122no—
173C3×D4—122no—
174C3×D4—122no—
175C2×C3×C4—21no—
176C2×C3×C4—31no—
177C2×C3×C4—31no—
178C2×C3×C4—32no—
179C2×C3×C4—31no—
180C2×C3×C4—62no—
181C2×C3×C4C22×D311yescharacteristic
182C2×C3×C4C22×D311yescharacteristic
183#67—42no—
184#67—42no—
185#67—42no—
186#67—122no—
187#67—122no—
188C2×Dic3—31no—
189C2×Dic3—31no—
190C2×Dic3—31no—
191C2×Dic3—31no—
192C2×Dic3—32no—
193C2×Dic3—32no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—62no—
197C2×Dic3—122no—
198C2×Dic3—122no—
199C2×Dic3—122no—
200C2×Dic3—124no—
201C2×Dic3C22×D311yescharacteristic
202C2×Dic3C22×D311yescharacteristic
203C2×Dic3C22×D312yes—
204C2×Dic3C22×D311yescharacteristic
205D12—22no—
206C4×D3—22no—
207C4×D3—122no—
208C2×C32—42no—
209C2×C32C2×#4111yescharacteristic
210C2×C32H(5,𝔽2)11yescharacteristic
211C2×C32H(5,𝔽2)11yescharacteristic
212C3×D3—42no—
213C3×D3—41no—
214C2×D4—32no—
215C2×D4—32no—
216C2×D4—32no—
217C2×D4—31no—
218C2×D4—92no—
219C2×D4D3211yescharacteristic
220C22×C4—31no—
221C22×C4—92no—
222C22×C4—92no—
223C4⋊C4—31no—
224C4⋊C4—91no—
225C4⋊C4—91no—
226C4⋊C4—92no—
227C4⋊C4—92no—
228C4⋊C4—91no—
229C22⋊C4—32no—
230C22⋊C4—91no—
231C22⋊C4—91no—
232C22⋊C4—92no—
233C22⋊C4—92no—
234C22⋊C4—92no—
235C42—31no—
236C42—91no—
237C22×C3—24no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—22no—
242C22×C3—22no—
243C22×C3—22no—
244C22×C3—21no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—61no—
248C22×C3—62no—
249C22×C3—62no—
250C22×C3—61no—
251C22×C3—61no—
252C22×C3—62no—
253C22×C3C23×D311yescharacteristic
254C22×C3C23×D311yescharacteristic
255C2×D3—21no—
256C2×D3—22no—
257C2×D3—22no—
258C2×D3—21no—
259C2×D3—22no—
260C2×D3—21no—
261C3×C4—22no—
262C3×C4—61no—
263C3×C4—62no—
264C3×C4—61no—
265C3×C4—61no—
266C3×C4#23712yes—
267C3×C4#23712yes—
268Dic3—21no—
269Dic3—22no—
270Dic3—21no—
271Dic3—21no—
272Dic3—62no—
273Dic3—61no—
274Dic3—61no—
275Dic3—61no—
276Dic3—62no—
277Dic3—61no—
278Dic3—62no—
279Dic3—122no—
280Dic3—121no—
281C32#54011yescharacteristic, complemented, Sylow
282C23—31no—
283C23—32no—
284C23C2×D3212yes—
285D4—22no—
286D4—62no—
287D4—122no—
288D4—122no—
289D4—122no—
290D4—362no—
291C2×C4—31no—
292C2×C4—31no—
293C2×C4—31no—
294C2×C4—32no—
295C2×C4—62no—
296C2×C4—91no—
297C2×C4—91no—
298C2×C4—92no—
299C2×C4—362no—
300C2×C4—362no—
301C2×C4C2×D3211yescharacteristic
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—44no—
306C2×C3—42no—
307C2×C3—42no—
308C2×C3—121no—
309C2×C3—122no—
310C2×C3C2×#23711yescharacteristic
311C2×C3C2×#23711yescharacteristic
312C2×C3#100411yescharacteristic
313C2×C3#100411yescharacteristic
314C2×C3#100411yescharacteristic
315C2×C3#100911yescharacteristic
316D3—42no—
317D3—41no—
318C22—22no—
319C22—22no—
320C22—22no—
321C22—62no—
322C22—61no—
323C22—62no—
324C22—61no—
325C22—61no—
326C22—62no—
327C22C22×D3211yescharacteristic
328C4—62no—
329C4—61no—
330C4—61no—
331C4—61no—
332C4—181no—
333C4—181no—
334C4—182no—
335C4#380512yes—
336C3—21no—
337C3#561911yescharacteristic, complemented
338C3#571911yescharacteristic, complemented
339C2—42no—
340C2—122no—
341C2—121no—
342C2#6431311yescharacteristic
343C2#6431411yescharacteristic
344C2C2×#380511yescharacteristic
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. D4×C3⋊Dic3C2semidirect
2. C3×#940C2semidirect
3. C3×#938C2semidirect
4. #63980C2semidirect
5. #63970C2semidirect
6. #63969C2semidirect
7. #63962C2semidirect
8. #63908C2semidirect
9. #63898C2semidirect
10. #63869C2semidirect
11. #63855C2semidirect
39. #940D3semidirect
45. #938D3semidirect
281. C32#540semidirect
337. C3#5619semidirect
338. C3#5719semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊Dic32
C3×#9402
C3×#9382
#639802
#639702
#639692
#639622
#639082
#638982
#638692
#638552
#57193
#56193