Finite Group Databasebeta

← #10594834

Subgroups — #10594834

Order 576.

666 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828C211yesmaximal, characteristic, complemented
2C2×#3799C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic, complemented
4C3×#924C211yesmaximal, characteristic, complemented
5#63975C212yesmaximal, complemented
6#63958C212yesmaximal, complemented
7#63894C211yesmaximal, characteristic, complemented
8#63883C212yesmaximal, complemented
9#63866C211yesmaximal, characteristic, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63853C212yesmaximal, complemented
12#5631—31nomaximal
13#5551—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16#3828—24no—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊D3C2212yes—
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#626C2212yes—
24C2×#624C2212yes—
25C2×#624C2212yes—
26#3799—24no—
27C3×#217C2212yes—
28C3×#217C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2212yes—
32C3×Dic3⋊C4C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35Dic3⋊Dic3C2212yes—
36Dic3⋊Dic3C2212yes—
37#3722C2212yes—
38(C2×D3)⋊Dic3C2212yes—
39C2×#239—31no—
40C2×#235—31no—
41C2×#235—31no—
42C2×#235—31no—
43C2×#235—61no—
44C3×C22⋊Q8—31no—
45C3×C4×D4—31no—
46#944—31no—
47#938—32no—
48C4×#67D311yescharacteristic, complemented
49#924—31no—
50#924D311yescharacteristic, complemented
51#897—31no—
52#894—32no—
53#893—31no—
54D3×C4⋊C4—32no—
55D3×C4⋊C4—32no—
56#884—32no—
57#883—32no—
58#883—32no—
59#883—32no—
60C4×D12—31no—
61#869—31no—
62C23×C32C2311yescharacteristic
63C22×C3⋊D3C2312yes—
64C22×C3×D3C2311yescharacteristic
65C2×C32×C4—22no—
66C2×C32×C4C2311yescharacteristic
67C2×C32×C4C2311yescharacteristic
68#638—24no—
69C2×C3⋊Dic3C2312yes—
70C4⋊(C3⋊D3)D414yescomplemented
71C4×C3⋊D3—24no—
72#634—22no—
73C3×#67—22no—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C3×C4×D3—22no—
80C3×Dic6—22no—
81#626—24no—
82#624—24no—
83#624—24no—
84#547—91noSylow
85C2×C3×Q8—31no—
86C2×C3×D4—31no—
87C22×C3×C4—21no—
88C22×C3×C4—31no—
89C22×C3×C4C2×D311yescharacteristic
90C22×C3×C4C2×D311yescharacteristic
91C2×#67—32no—
92C2×#67—32no—
93C2×#67—32no—
94C2×#67—62no—
95C2×#67C2×D311yescharacteristic
96#239—64no—
97#235—64no—
98#235—64no—
99#235—64no—
100#235—68no—
101C2×D12—31no—
102C2×D12—31no—
103C2×D12—32no—
104C2×D12—61no—
105C2×C4×D3—32no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—62no—
112C2×C4×D3C2×D311yescharacteristic
113C2×Dic6—31no—
114C2×Dic6—31no—
115C2×Dic6—61no—
116C2×Dic6C2×D311yescharacteristic
117C3×C4⋊C4—31no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—31no—
120C3×C22⋊C4—31no—
121C3×C22⋊C4—32no—
122C3×C22⋊C4—31no—
123C3×C42—31no—
124#217—32no—
125#217C2×D311yescharacteristic
126#217C2×D312yes—
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4C2×D311yescharacteristic
131C4⋊Dic3—32no—
132C4⋊Dic3—32no—
133C4⋊Dic3—32no—
134C4⋊Dic3C2×D311yescharacteristic
135Dic3⋊C4—32no—
136Dic3⋊C4—32no—
137Dic3⋊C4—32no—
138Dic3⋊C4C2×D312yes—
139Dic3⋊C4C2×D311yescharacteristic
140C4×Dic3C2×D311yescharacteristic
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32C2411yescharacteristic
145C2×C3⋊D3—22no—
146C2×C3⋊D3—22no—
147C2×C3⋊D3C2×D414yes—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C2×C3×D3—21no—
151C32×C4—21no—
152C32×C4C2×D412yes—
153C3⋊Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3—22no—
156C3×Dic3—22no—
157C3×Dic3—21no—
158C3×Dic3#4112yescomplemented
159C2×#41—91no—
160C2×#41—91no—
161C4⋊Q8—91no—
162#123—92no—
163#123—92no—
164C22⋊Q8—31no—
165C22⋊Q8—91no—
166#121—92no—
167C4×D4—31no—
168C4×D4—91no—
169C2×C4⋊C4—92no—
170C23×C3—21no—
171C23×C3—31no—
172C23×C3C22×D311yescharacteristic
173C23×C3C22×D311yescharacteristic
174C22×D3—32no—
175C22×D3—32no—
176C22×D3—62no—
177C22×D3C22×D311yescharacteristic
178C3×Q8—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—31no—
187C2×C3×C4—32no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
190C2×C3×C4—62no—
191C2×C3×C4C22×D311yescharacteristic
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194C2×C3×C4C22×D311yescharacteristic
195#67—22no—
196#67—64no—
197#67—64no—
198#67—64no—
199#67—68no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—62no—
203C2×Dic3C22×D311yescharacteristic
204C2×Dic3C22×D312yes—
205C2×Dic3C22×D312yes—
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D311yescharacteristic
208D12—34no—
209D12—34no—
210D12—64no—
211D12—64no—
212C4×D3—22no—
213C4×D3—64no—
214C4×D3—64no—
215C4×D3—64no—
216C4×D3—64no—
217C4×D3—64no—
218C4×D3—68no—
219C4×D3—64no—
220Dic6—22no—
221Dic6—64no—
222Dic6—62no—
223Dic6—62no—
224C2×C32—41no—
225C2×C32C22×D411yescharacteristic
226C2×C32C2×#4111yescharacteristic
227C2×C322−1+411yescharacteristic
228C3⋊D3—24no—
229C3×D3—41no—
230#41—184no—
231#41—184no—
232C2×Q8—31no—
233C2×Q8—91no—
234C2×D4—31no—
235C2×D4—92no—
236C2×D4—92no—
237C2×D4—91no—
238C22×C4—31no—
239C22×C4—92no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4D3211yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—32no—
245C4⋊C4—31no—
246C4⋊C4—92no—
247C4⋊C4—92no—
248C4⋊C4—92no—
249C22⋊C4—31no—
250C22⋊C4—32no—
251C22⋊C4—31no—
252C22⋊C4—92no—
253C22⋊C4—92no—
254C42—31no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—41no—
264C22×C3—41no—
265C22×C3—61no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3C23×D311yescharacteristic
269C22×C3C23×D311yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—21no—
273C2×D3—34no—
274C2×D3—34no—
275C2×D3—64no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—62no—
280C2×D3—122no—
281C2×D3—122no—
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—32no—
287C3×C4—62no—
288C3×C4—61no—
289C3×C4—62no—
290C3×C4—61no—
291C3×C4D3×D412yes—
292C3×C4D3×D412yes—
293Dic3—22no—
294Dic3—21no—
295Dic3—22no—
296Dic3—21no—
297Dic3—64no—
298Dic3—62no—
299Dic3—62no—
300Dic3#23512yescomplemented
301C32#54711yescharacteristic, complemented, Sylow
302C23—31no—
303C23—92no—
304C23C2×D3211yescharacteristic
305Q8—62no—
306Q8—182no—
307D4—62no—
308D4—94no—
309D4—184no—
310D4—184no—
311C2×C4—22no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—31no—
316C2×C4—32no—
317C2×C4—62no—
318C2×C4—92no—
319C2×C4—184no—
320C2×C4—184no—
321C2×C4—184no—
322C2×C4C2×D3211yescharacteristic
323C2×C4C2×D3211yescharacteristic
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—42no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—121no—
331C2×C3C2×#23511yescharacteristic
332C2×C3C2×D3×D411yescharacteristic
333C2×C3C2×D3×D411yescharacteristic
334C2×C3#100711yescharacteristic
335C2×C3D3×#4111yescharacteristic
336C2×C3#101011yescharacteristic
337D3—41no—
338D3—64no—
339D3—64no—
340D3—124no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—61no—
345C22—61no—
346C22—61no—
347C22—94no—
348C22—182no—
349C22—182no—
350C22C22×D3211yescharacteristic
351C4—21no—
352C4—32no—
353C4—61no—
354C4—62no—
355C4—61no—
356C4—62no—
357C4—182no—
358C4#381112yes—
359C3—21no—
360C3#555111yescharacteristic, complemented
361C3#563111yescharacteristic, complemented
362C2—41no—
363C2—121no—
364C2—184no—
365C2#6429711yescharacteristic
366C2D3×#23511yescharacteristic
367C2C2×#381111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3828C2semidirect
2. C2×#3799C2semidirect
3. C3×C4×#67C2semidirect
4. C3×#924C2semidirect
5. #63975C2semidirect
6. #63958C2semidirect
7. #63894C2semidirect
8. #63883C2semidirect
9. #63866C2semidirect
10. #63857C2semidirect
11. #63853C2semidirect
48. C4×#67D3semidirect
50. #924D3semidirect
70. C4⋊(C3⋊D3)D4semidirect
158. C3×Dic3#41semidirect
300. Dic3#235semidirect
301. C32#547semidirect
360. C3#5551semidirect
361. C3#5631semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
C2×#37992
C3×C4×#672
C3×#9242
#639752
#639582
#638942
#638832
#638662
#638572
#638532
#56313
#55513