Finite Group Databasebeta

← #10594182

Subgroups — #10594182

Order 576.

742 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#626C211yesmaximal, characteristic
2C22⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C3×D3×C22⋊C4C211yesmaximal, characteristic, complemented
4C3×#877C211yesmaximal, characteristic, complemented
5#63974C211yesmaximal, characteristic, complemented
6D3×#217C211yesmaximal, characteristic, complemented
7#63955C211yesmaximal, characteristic, complemented
8#63904C212yesmaximal, complemented
9C4×#626C212yesmaximal, complemented
10#63881C212yesmaximal, complemented
11#63880C212yesmaximal, complemented
12#5521—31nomaximal
13#5486—31nomaximal
14C23×C3⋊D3C2211yescharacteristic
15C23×C3×D3C2211yescharacteristic
16C2×C4×C3⋊D3C2212yescomplemented
17C22×C3×Dic3C2211yescharacteristic
18C2×C3×C4×D3C2212yescomplemented
19C2×#626C418yescomplemented
20C2×#626C2211yescharacteristic
21C2×#626C2211yescharacteristic
22C2×#626C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#624C2212yescomplemented
25C2×D3×Dic3C2212yescomplemented
26C32×C22⋊C4C2211yescharacteristic
27#3757C2211yescharacteristic
28#3752C2212yescomplemented
29C3×#217C2211yescharacteristic
30C3×#217C2211yescharacteristic
31C3×(C2×D3)⋊C4C2212yescomplemented
32C3×Dic3⋊C4C2212yescomplemented
33C3×C4×Dic3C2212yescomplemented
34Dic3⋊Dic3C2212yescomplemented
35#3722C2212yescomplemented
36(C2×D3)⋊Dic3C2212yescomplemented
37Dic32C2212yescomplemented
38C22×#67—31no—
39C22×D12—31no—
40C3×#117—31no—
41C2×C3×C22⋊C4—31no—
42C2×#217—31no—
43C4×#67—32no—
44C4×#67—32no—
45#926—31no—
46#893—32no—
47#893—32no—
48#67⋊C4—32no—
49#67⋊C4—32no—
50D3×C22⋊C4—31no—
51D3×C22⋊C4—31no—
52D3×C22⋊C4—31no—
53D3×C22⋊C4—31no—
54D3×C22⋊C4—31no—
55D3×C22⋊C4—61no—
56D3×C22⋊C4D311yescharacteristic, complemented
57#877—31no—
58#877D311yescharacteristic, complemented
59C4×D12—32no—
60C4×D12—32no—
61C23×C32C2311yescharacteristic
62C22×C3⋊D3—42no—
63C22×C3⋊D3C2×C414yescomplemented
64C22×C3⋊D3C2311yescharacteristic
65C22×C3⋊D3C2311yescharacteristic
66C22×C3×D3—42no—
67C22×C3×D3C2×C414yescomplemented
68C22×C3×D3C2311yescharacteristic
69C22×C3×D3C2311yescharacteristic
70C2×C32×C4C2312yes—
71C2×C3⋊Dic3C2312yes—
72C4×C3⋊D3—42no—
73C2×C3×Dic3C2×C414yescomplemented
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2311yescharacteristic
78C3×C4×D3—42no—
79#626C2×C414yescomplemented
80#626C2×C414yescomplemented
81#626C2×C414yescomplemented
82#626C2×C414yescomplemented
83#624—42no—
84D3×Dic3—42no—
85#518—91noSylow
86C24×C3—31no—
87C23×D3—31no—
88C23×D3—31no—
89C23×D3—61no—
90C23×D3C2×D311yescharacteristic
91C22×C3×C4—31no—
92C22×C3×C4—32no—
93C2×#67—38no—
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—31no—
97C2×#67—31no—
98C22×Dic3—32no—
99C22×Dic3C2×D311yescharacteristic
100C2×D12—38no—
101C2×D12—31no—
102C2×D12—31no—
103C2×D12—31no—
104C2×D12—31no—
105C2×C4×D3—32no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—62no—
111C2×C4×D3C2×D312yescomplemented
112C3×C4⋊C4—32no—
113C3×C22⋊C4—21no—
114C3×C22⋊C4—31no—
115C3×C22⋊C4—31no—
116C3×C22⋊C4—32no—
117C3×C22⋊C4C2×D311yescharacteristic
118C3×C22⋊C4C2×D311yescharacteristic
119C3×C42—32no—
120#217—31no—
121#217—31no—
122#217—32no—
123#217—61no—
124#217C2×D311yescharacteristic
125#217C2×D311yescharacteristic
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—62no—
132(C2×D3)⋊C4C2×D312yescomplemented
133C4⋊Dic3—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4C2×D312yescomplemented
136C4×Dic3—32no—
137C4×Dic3—32no—
138C4×Dic3C2×D312yescomplemented
139C22×C32—22no—
140C22×C32C22×C412yes—
141C22×C32C2411yescharacteristic
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3—44no—
145C2×C3⋊D3C22×C412yes—
146C2×C3⋊D3C22×C412yes—
147C2×C3×D3—22no—
148C2×C3×D3—22no—
149C2×C3×D3—44no—
150C2×C3×D3C22×C412yes—
151C2×C3×D3C22×C412yes—
152C32×C4—22no—
153C3⋊Dic3—22no—
154C3×Dic3—22no—
155C3×Dic3—22no—
156C3×Dic3C22×C412yes—
157C3×Dic3C22×C412yes—
158C22×D4—91no—
159C4×D4—92no—
160C4×D4—92no—
161C4×D4—92no—
162C4×D4—92no—
163#117—31no—
164#117—91no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—91no—
167C2×C22⋊C4—91no—
168C2×C22⋊C4—91no—
169C23×C3—21no—
170C23×C3—31no—
171C23×C3—31no—
172C23×C3—34no—
173C23×C3—122no—
174C23×C3C22×D311yescharacteristic
175C23×C3C22×D311yescharacteristic
176C22×D3—31no—
177C22×D3—31no—
178C22×D3—31no—
179C22×D3—31no—
180C22×D3—34no—
181C22×D3—34no—
182C22×D3—42no—
183C22×D3—61no—
184C22×D3—64no—
185C22×D3—61no—
186C22×D3—122no—
187C22×D3—122no—
188C22×D3—124no—
189C22×D3C4×D314yescomplemented
190C22×D3C22×D311yescharacteristic
191C22×D3C22×D311yescharacteristic
192C2×C3×C4—22no—
193C2×C3×C4—34no—
194C2×C3×C4—31no—
195C2×C3×C4—31no—
196C2×C3×C4—32no—
197C2×C3×C4—32no—
198C2×C3×C4—122no—
199C2×C3×C4C22×D312yes—
200C2×C3×C4C22×D312yes—
201#67—34no—
202#67—34no—
203#67—34no—
204#67—34no—
205C2×Dic3—32no—
206C2×Dic3—32no—
207C2×Dic3—62no—
208C2×Dic3—122no—
209C2×Dic3C4×D314yescomplemented
210C2×Dic3C22×D312yes—
211C2×Dic3C22×D312yes—
212C2×Dic3C22×D311yescharacteristic
213C2×Dic3C22×D311yescharacteristic
214D12—34no—
215D12—34no—
216D12—34no—
217D12—34no—
218C4×D3—42no—
219C4×D3—122no—
220C4×D3—122no—
221C4×D3—122no—
222C4×D3—122no—
223C4×D3—122no—
224C4×D3—124no—
225C2×C32—22no—
226C2×C32C23×C411yescharacteristic
227C2×C32H(5,𝔽2)12yes—
228C3⋊D3—22no—
229C3⋊D3—22no—
230C3×D3—22no—
231C3×D3—22no—
232C24—31no—
233C24—91no—
234C2×D4—91no—
235C2×D4—91no—
236C2×D4—91no—
237C2×D4—91no—
238C2×D4—98no—
239C22×C4—31no—
240C22×C4—32no—
241C22×C4—92no—
242C22×C4—92no—
243C22×C4—92no—
244C4⋊C4—32no—
245C4⋊C4—92no—
246C22⋊C4—31no—
247C22⋊C4—31no—
248C22⋊C4—32no—
249C22⋊C4—92no—
250C22⋊C4—92no—
251C22⋊C4—91no—
252C22⋊C4—92no—
253C22⋊C4D3211yescharacteristic
254C42—32no—
255C42—92no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—21no—
259C22×C3—22no—
260C22×C3—32no—
261C22×C3—32no—
262C22×C3—42no—
263C22×C3—62no—
264C22×C3—62no—
265C22×C3—124no—
266C22×C3C2×C4×D312yes—
267C22×C3C2×C4×D312yes—
268C22×C3C23×D311yescharacteristic
269C22×C3C23×D311yescharacteristic
270C2×D3—22no—
271C2×D3—22no—
272C2×D3—32no—
273C2×D3—32no—
274C2×D3—32no—
275C2×D3—32no—
276C2×D3—44no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—64no—
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—62no—
283C2×D3—62no—
284C2×D3—122no—
285C2×D3—124no—
286C2×D3—124no—
287C2×D3—128no—
288C2×D3C2×C4×D312yes—
289C2×D3C2×C4×D312yes—
290C3×C4—22no—
291C3×C4—22no—
292C3×C4—32no—
293C3×C4—32no—
294C3×C4—42no—
295C3×C4—62no—
296C3×C4—62no—
297Dic3—22no—
298Dic3—22no—
299Dic3—62no—
300Dic3—62no—
301Dic3—122no—
302Dic3C2×C4×D312yes—
303Dic3C2×C4×D312yes—
304C32#51811yescharacteristic, complemented, Sylow
305C23—31no—
306C23—31no—
307C23—34no—
308C23—91no—
309C23—91no—
310C23—94no—
311C23—122no—
312C23—362no—
313C23C2×D3211yescharacteristic
314D4—94no—
315D4—94no—
316D4—94no—
317D4—94no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—34no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—92no—
324C2×C4—122no—
325C2×C4—362no—
326C2×C4—362no—
327C2×C4—362no—
328C2×C4C2×D3212yes—
329C2×C3—22no—
330C2×C3—22no—
331C2×C3—21no—
332C2×C3—22no—
333C2×C3—42no—
334C2×C3—62no—
335C2×C3—62no—
336C2×C3C22×C4×D311yescharacteristic
337C2×C3C22×C4×D311yescharacteristic
338C2×C3#100412yes—
339C2×C3#100912yes—
340D3—22no—
341D3—22no—
342D3—64no—
343D3—62no—
344D3—62no—
345D3—62no—
346D3—62no—
347D3—122no—
348C22—22no—
349C22—32no—
350C22—32no—
351C22—62no—
352C22—62no—
353C22—92no—
354C22—92no—
355C22—124no—
356C22—182no—
357C22—182no—
358C22—364no—
359C22C4×D3212yes—
360C22C22×D3211yescharacteristic
361C4—22no—
362C4—32no—
363C4—32no—
364C4—62no—
365C4—62no—
366C4—182no—
367C3—21no—
368C3#548611yescharacteristic, complemented
369C3#552111yescharacteristic, complemented
370C2—22no—
371C2—62no—
372C2—62no—
373C2—182no—
374C2—182no—
375C2C2×C4×D3211yescharacteristic
376C2#6431412yes—
subgroupG/H
center, FrattiniC22C22×D32
commutatorC2×C32C23×C4
FittingC32×C22⋊C4C22
socle, cosocleC22×C32C24
prime radicalC22×#626C2
hypercenterC22⋊C4D32
hypocenterC32#518
LCSG ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22⋊C4×C3⋊D3C2semidirect
3. C3×D3×C22⋊C4C2semidirect
4. C3×#877C2semidirect
5. #63974C2semidirect
6. D3×#217C2semidirect
7. #63955C2semidirect
8. #63904C2semidirect
9. C4×#626C2semidirect
10. #63881C2semidirect
11. #63880C2semidirect
16. C2×C4×C3⋊D3C22semidirect
18. C2×C3×C4×D3C22semidirect
19. C2×#626C4semidirect
24. C2×#624C22semidirect
25. C2×D3×Dic3C22semidirect
28. #3752C22semidirect
31. C3×(C2×D3)⋊C4C22semidirect
32. C3×Dic3⋊C4C22semidirect
33. C3×C4×Dic3C22semidirect
34. Dic3⋊Dic3C22semidirect
35. #3722C22semidirect
36. (C2×D3)⋊Dic3C22semidirect
37. Dic32C22semidirect
56. D3×C22⋊C4D3semidirect
58. #877D3semidirect
63. C22×C3⋊D3C2×C4semidirect
67. C22×C3×D3C2×C4semidirect
73. C2×C3×Dic3C2×C4semidirect
79. #626C2×C4semidirect
80. #626C2×C4semidirect
81. #626C2×C4semidirect
82. #626C2×C4semidirect
111. C2×C4×D3C2×D3semidirect
132. (C2×D3)⋊C4C2×D3semidirect
135. Dic3⋊C4C2×D3semidirect
138. C4×Dic3C2×D3semidirect
189. C22×D3C4×D3semidirect
209. C2×Dic3C4×D3semidirect
304. C32#518semidirect
368. C3#5486semidirect
369. C3#5521semidirect

Maximal subgroups

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

subgroupindex
C22×#6262
C22⋊C4×C3⋊D32
C3×D3×C22⋊C42
C3×#8772
#639742
D3×#2172
#639552
#639042
C4×#6262
#638812
#638802
#55213
#54863