Finite Group Databasebeta

← #10594518

Subgroups — #10594518

Order 576.

667 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
1C2×#3799C212yesmaximal, complemented
2#64104C211yesmaximal, characteristic, complemented
3C3×#893C211yesmaximal, characteristic, complemented
4C3×#888C211yesmaximal, characteristic, complemented
5#63916C211yesmaximal, characteristic, complemented
6#63911C211yesmaximal, characteristic, complemented
7#63910C212yesmaximal, complemented
8#63907C212yesmaximal, complemented
9#63904C212yesmaximal, complemented
10#63894C211yesmaximal, characteristic, complemented
11C4×#624C211yesmaximal, characteristic, complemented
12#5725—31nomaximal
13#5548—31nomaximal
14C2×C4⋊(C3⋊D3)C2212yes—
15C2×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×D12C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×C3×Dic6C2212yes—
20C2×#626C2212yes—
21C2×#626C2212yes—
22C2×#626C2212yes—
23C2×#624C2212yes—
24C2×#624C2211yescharacteristic
25C2×#624C2211yescharacteristic
26#3799—28no—
27C32×C4⋊C4C2211yescharacteristic
28#3752C2212yes—
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2212yes—
32C3×C4×Dic3C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2211yescharacteristic
37Dic32C2211yescharacteristic
38C2×#239—32no—
39C2×#235—32no—
40C3×C4⋊Q8—31no—
41C3×C4×D4—31no—
42#947—32no—
43#940—31no—
44#938—31no—
45C4×#67—32no—
46#895—31no—
47#895—31no—
48#895—31no—
49#895—32no—
50#895—61no—
51#893—32no—
52#893—31no—
53#893D311yescharacteristic, complemented
54#888D311yescharacteristic, complemented
55#885—32no—
56#884—32no—
57#874—31no—
58C4×D12—31no—
59C42×D3—31no—
60C42×D3—31no—
61C22×C3⋊D3C2312yes—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2312yes—
64C2×C32×C4C2312yes—
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C4⋊(C3⋊D3)—24no—
68C4⋊(C3⋊D3)—22no—
69C4×C3⋊D3—22no—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C3×D12—22no—
76C3×C4×D3—24no—
77C3×Dic6—24no—
78#626—24no—
79#626—24no—
80#626—24no—
81#624—24no—
82#624—22no—
83#624D414yescomplemented
84#532—91noSylow
85C2×C3×Q8—32no—
86C2×C3×D4—31no—
87C22×C3×C4—32no—
88C2×#67—32no—
89C2×#67—32no—
90C2×#67—32no—
91#239—68no—
92#235—68no—
93C2×D12—32no—
94C2×D12—32no—
95C2×D12—32no—
96C2×D12—31no—
97C2×D12—31no—
98C2×D12—32no—
99C2×D12—32no—
100C2×D12—61no—
101C2×D12—62no—
102C2×D12C2×D311yescharacteristic
103C2×C4×D3—31no—
104C2×C4×D3—31no—
105C2×C4×D3—31no—
106C2×C4×D3—31no—
107C2×C4×D3—31no—
108C2×C4×D3—31no—
109C2×C4×D3—32no—
110C2×C4×D3—32no—
111C2×C4×D3—61no—
112C2×C4×D3C2×D312yes—
113C2×Dic6C2×D312yes—
114C3×C4⋊C4—21no—
115C3×C4⋊C4—32no—
116C3×C4⋊C4—31no—
117C3×C4⋊C4C2×D311yescharacteristic
118C3×C4⋊C4C2×D311yescharacteristic
119C3×C22⋊C4—32no—
120C3×C42—31no—
121C3×C42—31no—
122#217—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—32no—
128(C2×D3)⋊C4—62no—
129(C2×D3)⋊C4C2×D312yes—
130C4⋊Dic3C2×D311yescharacteristic
131Dic3⋊C4C2×D312yes—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—31no—
136C4×Dic3C2×D311yescharacteristic
137C4×Dic3C2×D311yescharacteristic
138C22×C32C2411yescharacteristic
139C2×C3⋊D3—21no—
140C2×C3⋊D3—21no—
141C2×C3⋊D3—22no—
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3⋊D3C2×D412yes—
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3—22no—
148C32×C4—22no—
149C32×C4#4112yescomplemented
150C3⋊Dic3—21no—
151C3×Dic3—22no—
152C3×Dic3—21no—
153C3×Dic3C2×D412yes—
154C3×Dic3C2×D412yes—
155C3×Dic3#4112yescomplemented
156C2×#41—92no—
157C4⋊Q8—31no—
158#127—91no—
159#124—92no—
160#121—92no—
161#121—91no—
162#121—91no—
163C4×D4—31no—
164C4×D4—91no—
165C4×D4—92no—
166C2×C42—91no—
167C23×C3—32no—
168C22×D3—31no—
169C22×D3—31no—
170C22×D3—32no—
171C22×D3—32no—
172C22×D3—61no—
173C22×D3—62no—
174C22×D3C22×D312yes—
175C3×Q8—64no—
176C3×D4—62no—
177C2×C3×C4—21no—
178C2×C3×C4—22no—
179C2×C3×C4—31no—
180C2×C3×C4—31no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—64no—
185C2×C3×C4C22×D312yes—
186C2×C3×C4C22×D312yes—
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189#67—64no—
190#67—64no—
191#67—64no—
192C2×Dic3—31no—
193C2×Dic3—31no—
194C2×Dic3—61no—
195C2×Dic3C22×D311yescharacteristic
196C2×Dic3C22×D312yes—
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D311yescharacteristic
200D12—22no—
201D12—64no—
202D12—64no—
203D12—64no—
204D12—64no—
205D12—64no—
206D12—62no—
207D12—62no—
208D12—122no—
209D12—124no—
210C4×D3—24no—
211C4×D3—34no—
212C4×D3—34no—
213C4×D3—62no—
214C4×D3—64no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—64no—
218C4×D3—62no—
219C4×D3—122no—
220Dic6—24no—
221C2×C32C22×D411yescharacteristic
222C2×C32C2×#4111yescharacteristic
223C2×C32C2×#4111yescharacteristic
224C3⋊D3—22no—
225C3⋊D3—42no—
226C3×D3—42no—
227#41—188no—
228C2×Q8—32no—
229C2×D4—31no—
230C2×D4—92no—
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—91no—
234C2×D4—92no—
235C22×C4—32no—
236C22×C4—91no—
237C22×C4—91no—
238C22×C4—92no—
239C22×C4—91no—
240C4⋊C4—32no—
241C4⋊C4—31no—
242C4⋊C4D3211yescharacteristic
243C22⋊C4—32no—
244C22⋊C4—92no—
245C22⋊C4—92no—
246C22⋊C4—92no—
247C42—31no—
248C42—31no—
249C42—91no—
250C42—91no—
251C22×C3—21no—
252C22×C3—62no—
253C22×C3—62no—
254C22×C3—62no—
255C22×C3C23×D311yescharacteristic
256C22×C3C23×D311yescharacteristic
257C2×D3—22no—
258C2×D3—22no—
259C2×D3—22no—
260C2×D3—32no—
261C2×D3—32no—
262C2×D3—62no—
263C2×D3—61no—
264C2×D3—61no—
265C2×D3—61no—
266C2×D3—61no—
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—62no—
272C2×D3—62no—
273C2×D3—121no—
274C2×D3—121no—
275C2×D3—122no—
276C2×D3—122no—
277C2×D3—122no—
278C3×C4—22no—
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—32no—
282C3×C4—32no—
283C3×C4—32no—
284C3×C4—42no—
285C3×C4—62no—
286C3×C4—61no—
287C3×C4#23512yescomplemented
288C3×C4#23912yescomplemented
289Dic3—22no—
290Dic3—21no—
291Dic3—61no—
292Dic3—61no—
293Dic3—121no—
294Dic3D3×D412yes—
295Dic3D3×D412yes—
296Dic3#23912yescomplemented
297C32#53211yescharacteristic, complemented, Sylow
298C23—32no—
299C23—91no—
300C23—92no—
301Q8—64no—
302D4—62no—
303D4—184no—
304D4—184no—
305D4—184no—
306D4—184no—
307D4—182no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—31no—
312C2×C4—32no—
313C2×C4—64no—
314C2×C4—94no—
315C2×C4—91no—
316C2×C4—184no—
317C2×C4—182no—
318C2×C4—182no—
319C2×C4C2×D3212yes—
320C2×C4C2×D3211yescharacteristic
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—122no—
325C2×C3C2×#23511yescharacteristic
326C2×C3C2×D3×D411yescharacteristic
327C2×C3C2×D3×D411yescharacteristic
328C2×C3C2×#23911yescharacteristic
329C2×C3C2×#23911yescharacteristic
330C2×C3D3×#4111yescharacteristic
331D3—42no—
332D3—62no—
333D3—62no—
334D3—122no—
335D3—122no—
336D3—122no—
337D3—242no—
338C22—62no—
339C22—62no—
340C22—62no—
341C22—92no—
342C22—181no—
343C22—181no—
344C22—182no—
345C22—182no—
346C22—182no—
347C22C22×D3211yescharacteristic
348C4—22no—
349C4—32no—
350C4—32no—
351C4—32no—
352C4—62no—
353C4—61no—
354C4—181no—
355C4#379912yescomplemented
356C3—21no—
357C3#554811yescharacteristic, complemented
358C3#572511yescharacteristic, complemented
359C2—122no—
360C2—182no—
361C2—362no—
362C2C2×#379911yescharacteristic
363C2D32×D411yescharacteristic
364C2D3×#23911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC32×C4⋊C4C22
hypercenterC4⋊C4D32
hypocenterC32#532
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#3799C2semidirect
2. #64104C2semidirect
3. C3×#893C2semidirect
4. C3×#888C2semidirect
5. #63916C2semidirect
6. #63911C2semidirect
7. #63910C2semidirect
8. #63907C2semidirect
9. #63904C2semidirect
10. #63894C2semidirect
11. C4×#624C2semidirect
53. #893D3semidirect
54. #888D3semidirect
83. #624D4semidirect
149. C32×C4#41semidirect
155. C3×Dic3#41semidirect
287. C3×C4#235semidirect
288. C3×C4#239semidirect
296. Dic3#239semidirect
297. C32#532semidirect
355. C4#3799semidirect
357. C3#5548semidirect
358. C3#5725semidirect

Maximal subgroups

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

subgroupindex
C2×#37992
#641042
C3×#8932
C3×#8882
#639162
#639112
#639102
#639072
#639042
#638942
C4×#6242
#57253
#55483