Finite Group Databasebeta

← #10594918

Subgroups — #10594918

Order 576.

658 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#634C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5#63980C212yesmaximal, complemented
6#67⋊Dic3C212yesmaximal, complemented
7#63913C212yesmaximal, complemented
8#63896C212yesmaximal, complemented
9#63860C211yesmaximal, characteristic, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5598—31nomaximal
13#5552—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C22×C3⋊Dic3C2212yes—
16C2×#634—24no—
17C2×#634C2211yescharacteristic
18C2×#634C2212yes—
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×#67C2212yes—
22C2×C3×D12C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×#625C2212yes—
25C2×D3×Dic3C2212yes—
26#3795—24no—
27C3×#217C2211yescharacteristic
28C3×(C2×D3)⋊C4C2211yescharacteristic
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C2×#237—31no—
40C2×#235—31no—
41C22×Dic6—31no—
42C22×Dic6—31no—
43C22×Dic6—61no—
44C3×#121—31no—
45C3×C4×D4—31no—
46#939—32no—
47#939—32no—
48#936—31no—
49D4×Dic3—32no—
50#930D311yescharacteristic, complemented
51C4×#67D311yescharacteristic, complemented
52#924—31no—
53#897—31no—
54#896—32no—
55#887—31no—
56#885—32no—
57#885—32no—
58#67⋊C4—32no—
59#878—32no—
60#875—31no—
61C4×Dic6—31no—
62C23×C32C2311yescharacteristic
63C22×C3×D3C2312yes—
64C22×C3×D3C2311yescharacteristic
65C2×C32×C4—22no—
66C2×C32×C4C2311yescharacteristic
67C2×C32×C4C2311yescharacteristic
68C2×C3⋊Dic3—24no—
69C2×C3⋊Dic3C2312yes—
70C2×C3⋊Dic3C2312yes—
71#634—22no—
72#634—24no—
73#634D414yescomplemented
74C3×#67—22no—
75C3×#67—24no—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C2×C3×Dic3C2311yescharacteristic
80C3×D12—22no—
81C3×C4×D3—22no—
82#625—24no—
83D3×Dic3—24no—
84#548—91noSylow
85C2×C3×D4—32no—
86C2×C3×D4—31no—
87C2×C3×D4—31no—
88C22×C3×C4—21no—
89C22×C3×C4—31no—
90C22×C3×C4C2×D311yescharacteristic
91C22×C3×C4C2×D311yescharacteristic
92C2×#67—32no—
93C2×#67—32no—
94C2×#67C2×D312yes—
95C2×#67C2×D311yescharacteristic
96C22×Dic3—32no—
97C22×Dic3—32no—
98C22×Dic3—32no—
99C22×Dic3—62no—
100#237—64no—
101#235—64no—
102C2×D12C2×D311yescharacteristic
103C2×C4×D3—32no—
104C2×C4×D3C2×D311yescharacteristic
105C2×Dic6—31no—
106C2×Dic6—31no—
107C2×Dic6—31no—
108C2×Dic6—32no—
109C2×Dic6—32no—
110C2×Dic6—31no—
111C2×Dic6—61no—
112C2×Dic6—61no—
113C2×Dic6—62no—
114C2×Dic6—64no—
115C2×Dic6—64no—
116C2×Dic6—68no—
117C3×C4⋊C4—31no—
118C3×C4⋊C4—31no—
119C3×C22⋊C4—31no—
120C3×C22⋊C4—32no—
121C3×C22⋊C4—31no—
122C3×C42—31no—
123#217—32no—
124#217—32no—
125#217—32no—
126#217C2×D311yescharacteristic
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4C2×D311yescharacteristic
131(C2×D3)⋊C4C2×D312yes—
132C4⋊Dic3—32no—
133C4⋊Dic3C2×D311yescharacteristic
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4—32no—
137Dic3⋊C4C2×D311yescharacteristic
138C4×Dic3—32no—
139C4×Dic3—32no—
140C4×Dic3C2×D311yescharacteristic
141C22×C32—21no—
142C22×C32—21no—
143C22×C32#4112yescomplemented
144C22×C32C2411yescharacteristic
145C2×C3×D3—22no—
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—22no—
149C2×C3×D3—22no—
150C2×C3×D3—21no—
151C32×C4—21no—
152C32×C4C2×D412yes—
153C3⋊Dic3—22no—
154C3⋊Dic3C2×D414yes—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3—22no—
158C3×Dic3—21no—
159C2×#41—91no—
160C22×Q8—91no—
161#124—91no—
162#124—92no—
163C22⋊Q8—92no—
164C22⋊Q8—91no—
165#121—31no—
166#121—92no—
167C4×Q8—91no—
168C4×D4—31no—
169C4×D4—92no—
170C23×C3—21no—
171C23×C3—32no—
172C23×C3—31no—
173C23×C3C22×D311yescharacteristic
174C23×C3C22×D311yescharacteristic
175C22×D3C22×D312yes—
176C22×D3C22×D311yescharacteristic
177C3×D4—64no—
178C3×D4—62no—
179C3×D4—62no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—62no—
190C2×C3×C4C22×D311yescharacteristic
191C2×C3×C4C22×D311yescharacteristic
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194#67—22no—
195#67—24no—
196#67—64no—
197#67—64no—
198C2×Dic3—32no—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—62no—
203C2×Dic3—62no—
204C2×Dic3—64no—
205C2×Dic3—64no—
206C2×Dic3—64no—
207C2×Dic3—68no—
208C2×Dic3C22×D311yescharacteristic
209C2×Dic3C22×D312yes—
210C2×Dic3C22×D311yescharacteristic
211C2×Dic3C22×D311yescharacteristic
212D12—22no—
213C4×D3—22no—
214C4×D3—64no—
215Dic6—34no—
216Dic6—34no—
217Dic6—64no—
218Dic6—64no—
219Dic6—64no—
220Dic6—68no—
221Dic6—64no—
222Dic6—62no—
223Dic6—62no—
224C2×C32—22no—
225C2×C32C22×D411yescharacteristic
226C2×C32C2×#4111yescharacteristic
227C2×C322−1+411yescharacteristic
228C3×D3—41no—
229C3×D3—42no—
230#41—184no—
231C2×Q8—91no—
232C2×Q8—92no—
233C2×Q8—91no—
234C2×Q8—184no—
235C2×D4—31no—
236C2×D4—32no—
237C2×D4—31no—
238C2×D4—92no—
239C22×C4—31no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4D3211yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—31no—
245C4⋊C4—92no—
246C4⋊C4—92no—
247C22⋊C4—31no—
248C22⋊C4—31no—
249C22⋊C4—32no—
250C22⋊C4—92no—
251C22⋊C4—92no—
252C22⋊C4—92no—
253C42—31no—
254C42—92no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—22no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—62no—
268C22×C3—61no—
269C22×C3#23512yescomplemented
270C22×C3#23712yescomplemented
271C22×C3C23×D311yescharacteristic
272C22×C3C23×D311yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—21no—
278C2×D3—22no—
279C3×C4—22no—
280C3×C4—21no—
281C3×C4—21no—
282C3×C4—22no—
283C3×C4—62no—
284C3×C4—61no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4D3×D412yes—
288C3×C4D3×D412yes—
289Dic3—21no—
290Dic3—21no—
291Dic3—22no—
292Dic3—21no—
293Dic3—34no—
294Dic3—34no—
295Dic3—64no—
296Dic3—64no—
297Dic3—62no—
298Dic3—62no—
299C32#54811yescharacteristic, complemented, Sylow
300C23—31no—
301C23—32no—
302C23C2×D3211yescharacteristic
303Q8—94no—
304Q8—184no—
305Q8—182no—
306D4—62no—
307D4—64no—
308D4—62no—
309D4—184no—
310C2×C4—22no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—32no—
314C2×C4—31no—
315C2×C4—62no—
316C2×C4—92no—
317C2×C4—92no—
318C2×C4—184no—
319C2×C4—184no—
320C2×C4C2×D3211yescharacteristic
321C2×C4C2×D3211yescharacteristic
322C2×C3—24no—
323C2×C3—22no—
324C2×C3—22no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—21no—
328C2×C3—122no—
329C2×C3—121no—
330C2×C3C2×#23511yescharacteristic
331C2×C3C2×D3×D411yescharacteristic
332C2×C3C2×D3×D411yescharacteristic
333C2×C3C2×#23711yescharacteristic
334C2×C3#101011yescharacteristic
335C2×C3#101011yescharacteristic
336D3—41no—
337D3—42no—
338C22—21no—
339C22—21no—
340C22—61no—
341C22—62no—
342C22—61no—
343C22—62no—
344C22—61no—
345C22—62no—
346C22#379512yescomplemented
347C22C22×D3211yescharacteristic
348C4—21no—
349C4—61no—
350C4—62no—
351C4—61no—
352C4—61no—
353C4—94no—
354C4—182no—
355C4#381112yes—
356C3—21no—
357C3#555211yescharacteristic, complemented
358C3#559811yescharacteristic, complemented
359C2—22no—
360C2—121no—
361C2—122no—
362C2C2×#379511yescharacteristic
363C2#6429811yescharacteristic
364C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C22×#634C2semidirect
2. C2×#3795C2semidirect
3. C3×#930C2semidirect
4. C3×C4×#67C2semidirect
5. #63980C2semidirect
6. #67⋊Dic3C2semidirect
7. #63913C2semidirect
8. #63896C2semidirect
9. #63860C2semidirect
10. #63857C2semidirect
11. #63844C2semidirect
50. #930D3semidirect
51. C4×#67D3semidirect
73. #634D4semidirect
143. C22×C32#41semidirect
269. C22×C3#235semidirect
270. C22×C3#237semidirect
299. C32#548semidirect
346. C22#3795semidirect
357. C3#5552semidirect
358. C3#5598semidirect

Maximal subgroups

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

subgroupindex
C22×#6342
C2×#37952
C3×#9302
C3×C4×#672
#639802
#67⋊Dic32
#639132
#638962
#638602
#638572
#638442
#55983
#55523