Finite Group Databasebeta

← #10594902

Subgroups — #10594902

Order 576.

627 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×C3×Dic6C211yesmaximal, characteristic, complemented
2C2×#3799C211yesmaximal, characteristic, complemented
3#64139C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5#63978C212yesmaximal, complemented
6#63952C212yesmaximal, complemented
7#63907C212yesmaximal, complemented
8#63888C212yesmaximal, complemented
9#63864C211yesmaximal, characteristic, complemented
10#63856C211yesmaximal, characteristic, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5817—31nomaximal
13#5552—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#67C2211yescharacteristic
18C22×C3×Dic3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6—24no—
21C2×C3×Dic6C2211yescharacteristic
22C2×C3×Dic6C2212yes—
23C2×C3×Dic6C2211yescharacteristic
24C2×#626C2212yes—
25C2×#624C2212yes—
26#3799—24no—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29C3×#217C2211yescharacteristic
30C3×(C2×D3)⋊C4C2211yescharacteristic
31C3×Dic3⋊C4C2211yescharacteristic
32C3×C4×Dic3C2211yescharacteristic
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36#3722C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×#239—31no—
41C2×#235—31no—
42C22×Dic6D311yescharacteristic, complemented
43C3×C4×D4—31no—
44#947—32no—
45#947—31no—
46#946—31no—
47#946—32no—
48Q8×Dic3—31no—
49#939—32no—
50#930—31no—
51#930—31no—
52#930—32no—
53#930—61no—
54C4×#67—32no—
55C4×#67D311yescharacteristic, complemented
56#924—31no—
57#896—32no—
58#885—32no—
59#67⋊C4—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
68#638—24no—
69C2×C3⋊Dic3C2312yes—
70C4⋊(C3⋊D3)—22no—
71C3×#67—22no—
72C2×C3×Dic3—24no—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2312yes—
78C3×C4×D3—22no—
79C3×Dic6—22no—
80C3×Dic6—24no—
81C3×Dic6D414yescomplemented
82#626—24no—
83#624—24no—
84#548—91noSylow
85C2×C3×Q8—31no—
86C2×C3×Q8—32no—
87C2×C3×Q8—31no—
88C2×C3×Q8—64no—
89C2×C3×D4—31no—
90C22×C3×C4—21no—
91C22×C3×C4—32no—
92C22×C3×C4—31no—
93C22×C3×C4C2×D311yescharacteristic
94C22×C3×C4C2×D311yescharacteristic
95C2×#67—32no—
96C2×#67—32no—
97C2×#67—32no—
98C2×#67—62no—
99C2×#67C2×D311yescharacteristic
100C22×Dic3C2×D312yes—
101#239—64no—
102#235—64no—
103C2×D12—32no—
104C2×D12—31no—
105C2×D12—31no—
106C2×D12—61no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3C2×D311yescharacteristic
110C2×Dic6—24no—
111C2×Dic6C2×D311yescharacteristic
112C2×Dic6C2×D312yes—
113C2×Dic6C2×D311yescharacteristic
114C3×C4⋊C4—31no—
115C3×C22⋊C4—31no—
116C3×C22⋊C4—31no—
117C3×C42—31no—
118#217—32no—
119#217C2×D311yescharacteristic
120(C2×D3)⋊C4—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4—62no—
128(C2×D3)⋊C4C2×D311yescharacteristic
129C4⋊Dic3—31no—
130C4⋊Dic3—31no—
131C4⋊Dic3—32no—
132C4⋊Dic3—61no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4C2×D311yescharacteristic
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3C2×D311yescharacteristic
140C22×C32—21no—
141C22×C32—21no—
142C22×C32#4112yescomplemented
143C22×C32C2411yescharacteristic
144C2×C3⋊D3—22no—
145C2×C3⋊D3—22no—
146C2×C3⋊D3—22no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C32×C4—21no—
151C32×C4C2×D412yes—
152C3⋊Dic3—22no—
153C3×Dic3—21no—
154C3×Dic3—22no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3C2×D414yes—
158C2×#41—91no—
159C22×Q8—31no—
160#124—91no—
161#124—92no—
162C22⋊Q8—92no—
163C22⋊Q8—91no—
164#121—92no—
165#121—91no—
166C4×Q8—91no—
167C4×D4—31no—
168C4×D4—92no—
169C23×C3—21no—
170C23×C3—31no—
171C23×C3C22×D311yescharacteristic
172C23×C3C22×D311yescharacteristic
173C22×D3—32no—
174C22×D3—32no—
175C22×D3—62no—
176C22×D3C22×D311yescharacteristic
177C3×Q8—34no—
178C3×Q8—64no—
179C3×Q8—62no—
180C3×D4—62no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—32no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
190C2×C3×C4—42no—
191C2×C3×C4—64no—
192C2×C3×C4—62no—
193C2×C3×C4C22×D311yescharacteristic
194C2×C3×C4C22×D311yescharacteristic
195C2×C3×C4C22×D311yescharacteristic
196C2×C3×C4C22×D311yescharacteristic
197#67—22no—
198#67—64no—
199#67—64no—
200#67—64no—
201#67—124no—
202C2×Dic3—24no—
203C2×Dic3—32no—
204C2×Dic3—32no—
205C2×Dic3—62no—
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D311yescharacteristic
209C2×Dic3C22×D312yes—
210C2×Dic3C22×D312yes—
211D12—64no—
212D12—62no—
213D12—62no—
214D12—122no—
215C4×D3—22no—
216C4×D3—64no—
217C4×D3—64no—
218Dic6—22no—
219Dic6—24no—
220Dic6#6714yescomplemented
221C2×C32—22no—
222C2×C32C22×D411yescharacteristic
223C2×C32C2×#4111yescharacteristic
224C2×C322−1+411yescharacteristic
225C3⋊D3—42no—
226C3×D3—41no—
227#41—184no—
228C2×Q8—31no—
229C2×Q8—32no—
230C2×Q8—31no—
231C2×Q8—64no—
232C2×D4—31no—
233C2×D4—92no—
234C2×D4—92no—
235C2×D4—91no—
236C22×C4—32no—
237C22×C4—31no—
238C22×C4—92no—
239C22×C4D3211yescharacteristic
240C4⋊C4—31no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—91no—
244C22⋊C4—31no—
245C22⋊C4—31no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C22⋊C4—92no—
249C22⋊C4—92no—
250C42—31no—
251C42—92no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—61no—
261C22×C3—61no—
262C22×C3—61no—
263C22×C3#23512yescomplemented
264C22×C3#23912yescomplemented
265C22×C3C23×D311yescharacteristic
266C22×C3C23×D311yescharacteristic
267C2×D3—21no—
268C2×D3—21no—
269C2×D3—21no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—122no—
277C2×D3—122no—
278C2×D3—122no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—22no—
282C3×C4—34no—
283C3×C4—41no—
284C3×C4—62no—
285C3×C4—61no—
286C3×C4—61no—
287C3×C4—61no—
288C3×C4D3×D412yes—
289C3×C4C2×#6712yes—
290Dic3—21no—
291Dic3—21no—
292Dic3—21no—
293Dic3—22no—
294Dic3—62no—
295Dic3—62no—
296Dic3—122no—
297Dic3C2×#6714yes—
298C32#54811yescharacteristic, complemented, Sylow
299C23—31no—
300C23—92no—
301C23C2×D3211yescharacteristic
302Q8—34no—
303Q8—64no—
304Q8—62no—
305D4—62no—
306D4—184no—
307D4—184no—
308D4—182no—
309C2×C4—22no—
310C2×C4—32no—
311C2×C4—31no—
312C2×C4—31no—
313C2×C4—31no—
314C2×C4—32no—
315C2×C4—64no—
316C2×C4—62no—
317C2×C4—92no—
318C2×C4—184no—
319C2×C4C2×D3211yescharacteristic
320C2×C4C2×D3211yescharacteristic
321C2×C3—22no—
322C2×C3—22no—
323C2×C3—21no—
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—42no—
327C2×C3—121no—
328C2×C3C2×#23511yescharacteristic
329C2×C3C2×D3×D411yescharacteristic
330C2×C3C2×#23911yescharacteristic
331C2×C3#100711yescharacteristic
332C2×C3#101011yescharacteristic
333C2×C3C22×#6711yescharacteristic
334D3—41no—
335D3—122no—
336D3—122no—
337D3—242no—
338C22—21no—
339C22—21no—
340C22—61no—
341C22—61no—
342C22—61no—
343C22—182no—
344C22—182no—
345C22—182no—
346C22#379912yescomplemented
347C22C22×D3211yescharacteristic
348C4—21no—
349C4—34no—
350C4—61no—
351C4—61no—
352C4—61no—
353C4—62no—
354C4—182no—
355C4D3×#6712yes—
356C3—21no—
357C3#555211yescharacteristic, complemented
358C3#581711yescharacteristic, complemented
359C2—22no—
360C2—121no—
361C2—362no—
362C2#6429711yescharacteristic
363C2C2×#379911yescharacteristic
364C2C2×D3×#6711yescharacteristic
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×C3×Dic6C2semidirect
2. C2×#3799C2semidirect
3. #64139C2semidirect
4. C3×C4×#67C2semidirect
5. #63978C2semidirect
6. #63952C2semidirect
7. #63907C2semidirect
8. #63888C2semidirect
9. #63864C2semidirect
10. #63856C2semidirect
11. Dic3×Dic6C2semidirect
42. C22×Dic6D3semidirect
55. C4×#67D3semidirect
81. C3×Dic6D4semidirect
142. C22×C32#41semidirect
220. Dic6#67semidirect
263. C22×C3#235semidirect
264. C22×C3#239semidirect
298. C32#548semidirect
346. C22#3799semidirect
357. C3#5552semidirect
358. C3#5817semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic62
C2×#37992
#641392
C3×C4×#672
#639782
#639522
#639072
#638882
#638642
#638562
Dic3×Dic62
#58173
#55523