Finite Group Databasebeta

← #10594369

Subgroups — #10594369

Order 576.

667 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3796C212yesmaximal, complemented
2#64102C211yesmaximal, characteristic, complemented
3C3×#895C211yesmaximal, characteristic, complemented
4C3×#888C211yesmaximal, characteristic, complemented
5#63911C211yesmaximal, characteristic, complemented
6#63909C211yesmaximal, characteristic, complemented
7#63907C212yesmaximal, complemented
8#63898C211yesmaximal, characteristic, complemented
9#63886C212yesmaximal, complemented
10#63881C212yesmaximal, complemented
11C4×D3×Dic3C211yesmaximal, characteristic, complemented
12#5725—31nomaximal
13#5596—31nomaximal
14C2×C4⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊D3C2212yes—
16C2×C3×D12C2212yes—
17C2×C3×D12C2211yescharacteristic
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×Dic6C2212yes—
20C2×#626C2212yes—
21C2×#626C2212yes—
22C2×#626C2212yes—
23C2×D3×Dic3C2212yes—
24C2×D3×Dic3C2211yescharacteristic
25C2×D3×Dic3C2211yescharacteristic
26#3796—28no—
27C32×C4⋊C4C2211yescharacteristic
28#3752C2212yes—
29C4×C3⋊Dic3C2211yescharacteristic
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×Dic3⋊C4C2212yes—
33C3×C4×Dic3C2211yescharacteristic
34#3722C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36Dic32C2211yescharacteristic
37Dic32C2211yescharacteristic
38C2×#239—32no—
39C2×#237—32no—
40C3×C4⋊Q8—31no—
41C3×#121—31no—
42#947—32no—
43#940—31no—
44#939—32no—
45D4×Dic3—31no—
46#930—31no—
47C2×C4×Dic3—31no—
48#895—32no—
49#895—31no—
50#895D311yescharacteristic, complemented
51#893—31no—
52#893—31no—
53#893—31no—
54#893—32no—
55#893—61no—
56#888D311yescharacteristic, complemented
57#885—32no—
58#67⋊C4—32no—
59#874—31no—
60C42×D3—31no—
61C22×C3⋊D3C2312yes—
62C22×C3×D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C2×C32×C4C2312yes—
65C2×C32×C4C2311yescharacteristic
66C2×C3⋊Dic3C2311yescharacteristic
67C2×C3⋊Dic3C2311yescharacteristic
68C4⋊(C3⋊D3)—22no—
69C4×C3⋊D3—24no—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C3×D12—24no—
75C3×D12—22no—
76C3×C4×D3—22no—
77C3×Dic6—24no—
78#626—24no—
79#626—24no—
80#626—24no—
81D3×Dic3—24no—
82D3×Dic3—22no—
83D3×Dic3D414yescomplemented
84#532—91noSylow
85C2×C3×Q8—32no—
86C2×C3×D4—31no—
87C2×C3×D4—32no—
88C22×C3×C4—31no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—32no—
92C22×Dic3—31no—
93C22×Dic3—32no—
94C22×Dic3—31no—
95#239—68no—
96#237—68no—
97C2×D12—32no—
98C2×D12—32no—
99C2×D12—32no—
100C2×D12—31no—
101C2×D12—31no—
102C2×D12—61no—
103C2×D12C2×D311yescharacteristic
104C2×D12C2×D312yes—
105C2×C4×D3—31no—
106C2×C4×D3—31no—
107C2×C4×D3—32no—
108C2×C4×D3—32no—
109C2×C4×D3—32no—
110C2×C4×D3—62no—
111C2×C4×D3C2×D311yescharacteristic
112C2×Dic6C2×D312yes—
113C3×C4⋊C4—21no—
114C3×C4⋊C4—32no—
115C3×C4⋊C4—31no—
116C3×C4⋊C4C2×D311yescharacteristic
117C3×C4⋊C4C2×D311yescharacteristic
118C3×C22⋊C4—32no—
119C3×C42—31no—
120#217—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—62no—
127(C2×D3)⋊C4C2×D312yes—
128C4⋊Dic3C2×D311yescharacteristic
129Dic3⋊C4C2×D312yes—
130C4×Dic3—31no—
131C4×Dic3—31no—
132C4×Dic3—31no—
133C4×Dic3—31no—
134C4×Dic3—31no—
135C4×Dic3—31no—
136C4×Dic3—61no—
137C4×Dic3C2×D311yescharacteristic
138C22×C32C2411yescharacteristic
139C2×C3⋊D3—22no—
140C2×C3⋊D3—22no—
141C2×C3⋊D3—22no—
142C2×C3×D3—21no—
143C2×C3×D3—21no—
144C2×C3×D3—22no—
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3C2×D412yes—
148C32×C4—22no—
149C32×C4#4112yescomplemented
150C3⋊Dic3—21no—
151C3⋊Dic3C2×D412yes—
152C3×Dic3—22no—
153C3×Dic3—21no—
154C3×Dic3C2×D412yes—
155C3×Dic3#4112yescomplemented
156C2×#41—92no—
157C4⋊Q8—31no—
158#127—91no—
159#124—92no—
160#121—31no—
161#121—91no—
162#121—92no—
163C4×D4—92no—
164C4×D4—91no—
165C4×D4—91no—
166C2×C42—91no—
167C23×C3—31no—
168C23×C3—32no—
169C22×D3—32no—
170C22×D3—32no—
171C22×D3—62no—
172C22×D3C22×D312yes—
173C22×D3C22×D311yescharacteristic
174C3×Q8—64no—
175C3×D4—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—32no—
183C2×C3×C4—62no—
184C2×C3×C4C22×D312yes—
185C2×C3×C4C22×D312yes—
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188#67—64no—
189#67—64no—
190#67—64no—
191C2×Dic3—31no—
192C2×Dic3—31no—
193C2×Dic3—34no—
194C2×Dic3—31no—
195C2×Dic3—31no—
196C2×Dic3—61no—
197C2×Dic3—61no—
198C2×Dic3—64no—
199C2×Dic3—62no—
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D312yes—
202C2×Dic3C22×D311yescharacteristic
203C2×Dic3C22×D311yescharacteristic
204D12—24no—
205D12—22no—
206D12—64no—
207D12—64no—
208D12—64no—
209D12—64no—
210D12—62no—
211D12—62no—
212C4×D3—22no—
213C4×D3—34no—
214C4×D3—64no—
215C4×D3—64no—
216C4×D3—64no—
217C4×D3—62no—
218C4×D3—124no—
219Dic6—24no—
220C2×C32C22×D411yescharacteristic
221C2×C32C2×#4111yescharacteristic
222C2×C32C2×#4111yescharacteristic
223C3⋊D3—42no—
224C3×D3—22no—
225C3×D3—42no—
226#41—188no—
227C2×Q8—32no—
228C2×D4—31no—
229C2×D4—32no—
230C2×D4—92no—
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—91no—
234C22×C4—31no—
235C22×C4—91no—
236C22×C4—91no—
237C22×C4—92no—
238C22×C4—92no—
239C4⋊C4—32no—
240C4⋊C4—31no—
241C4⋊C4D3211yescharacteristic
242C22⋊C4—32no—
243C22⋊C4—92no—
244C22⋊C4—92no—
245C22⋊C4—92no—
246C42—31no—
247C42—91no—
248C42—91no—
249C42—91no—
250C22×C3—21no—
251C22×C3—32no—
252C22×C3—61no—
253C22×C3—61no—
254C22×C3—62no—
255C22×C3—62no—
256C22×C3—62no—
257C22×C3C23×D311yescharacteristic
258C22×C3C23×D311yescharacteristic
259C2×D3—21no—
260C2×D3—21no—
261C2×D3—22no—
262C2×D3—22no—
263C2×D3—22no—
264C2×D3—64no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3D3×D412yes—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—32no—
278C3×C4—32no—
279C3×C4—42no—
280C3×C4—62no—
281C3×C4—61no—
282C3×C4#23712yescomplemented
283C3×C4#23912yescomplemented
284Dic3—22no—
285Dic3—21no—
286Dic3—32no—
287Dic3—32no—
288Dic3—62no—
289Dic3—61no—
290Dic3—61no—
291Dic3—121no—
292Dic3D3×D412yes—
293Dic3#23912yescomplemented
294C32#53211yescharacteristic, complemented, Sylow
295C23—31no—
296C23—32no—
297C23—92no—
298Q8—64no—
299D4—64no—
300D4—62no—
301D4—184no—
302D4—184no—
303D4—184no—
304D4—182no—
305C2×C4—31no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—32no—
309C2×C4—62no—
310C2×C4—94no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—184no—
314C2×C4—182no—
315C2×C4—184no—
316C2×C4C2×D3212yes—
317C2×C4C2×D3211yescharacteristic
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—62no—
322C2×C3—122no—
323C2×C3C2×D3×D411yescharacteristic
324C2×C3C2×D3×D411yescharacteristic
325C2×C3C2×#23711yescharacteristic
326C2×C3C2×#23911yescharacteristic
327C2×C3C2×#23911yescharacteristic
328C2×C3D3×#4111yescharacteristic
329D3—22no—
330D3—42no—
331D3—124no—
332D3—122no—
333D3—122no—
334C22—32no—
335C22—61no—
336C22—61no—
337C22—62no—
338C22—62no—
339C22—62no—
340C22—182no—
341C22—182no—
342C22—182no—
343C22C22×D3211yescharacteristic
344C4—22no—
345C4—32no—
346C4—32no—
347C4—61no—
348C4—62no—
349C4—92no—
350C4—181no—
351C4#379612yescomplemented
352C3—21no—
353C3#559611yescharacteristic, complemented
354C3#572511yescharacteristic, complemented
355C2—62no—
356C2—122no—
357C2—362no—
358C2C2×#379611yescharacteristic
359C2D32×D411yescharacteristic
360C2D3×#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×#3796C2semidirect
2. #64102C2semidirect
3. C3×#895C2semidirect
4. C3×#888C2semidirect
5. #63911C2semidirect
6. #63909C2semidirect
7. #63907C2semidirect
8. #63898C2semidirect
9. #63886C2semidirect
10. #63881C2semidirect
11. C4×D3×Dic3C2semidirect
50. #895D3semidirect
56. #888D3semidirect
83. D3×Dic3D4semidirect
149. C32×C4#41semidirect
155. C3×Dic3#41semidirect
282. C3×C4#237semidirect
283. C3×C4#239semidirect
293. Dic3#239semidirect
294. C32#532semidirect
351. C4#3796semidirect
353. C3#5596semidirect
354. C3#5725semidirect

Maximal subgroups

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

subgroupindex
C2×#37962
#641022
C3×#8952
C3×#8882
#639112
#639092
#639072
#638982
#638862
#638812
C4×D3×Dic32
#57253
#55963