Finite Group Databasebeta

← #10594892

Subgroups — #10594892

Order 576.

697 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×D3C211yesmaximal, characteristic, complemented
2C2×#3795C211yesmaximal, characteristic, complemented
3#64133C211yesmaximal, characteristic, complemented
4C3×#930C211yesmaximal, characteristic, complemented
5#63976C212yesmaximal, complemented
6Dic3×#67C212yesmaximal, complemented
7#63908C211yesmaximal, characteristic, complemented
8#63905C212yesmaximal, complemented
9#63879C212yesmaximal, complemented
10#63857C211yesmaximal, characteristic, complemented
11#63841C211yesmaximal, characteristic, complemented
12#5791—31nomaximal
13#5606—31nomaximal
14C23×C3×D3C2211yescharacteristic
15C22×C32×C4C2211yescharacteristic
16C2×#634C2211yescharacteristic, complemented
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2211yescharacteristic
19C2×C3×D12C2211yescharacteristic, complemented
20C2×C3×C4×D3—24no—
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×C4×D3C2211yescharacteristic
23C2×C3×C4×D3C2211yescharacteristic
24C2×C3×C4×D3C2211yescharacteristic
25C2×#625C2212yes—
26C2×#625C2212yes—
27C2×D3×Dic3C2212yescomplemented
28#3795—24no—
29#3757C2212yes—
30C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
31#3750C2212yescomplemented
32C3×(C2×D3)⋊C4C2212yescomplemented
33C3×C4⋊Dic3C2211yescharacteristic, complemented
34Dic3⋊Dic3C2212yescomplemented
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yescomplemented
38Dic32C2212yescomplemented
39C23×C3×C4—31no—
40C2×#237—31no—
41C2×#235—31no—
42C22×C4×D3D311yescharacteristic, complemented
43C3×#121—31no—
44#953—32no—
45#938—31no—
46#937—32no—
47D4×Dic3—32no—
48#930—31no—
49#930D311yescharacteristic, complemented
50#929—32no—
51C4×#67—32no—
52C4×#67—32no—
53#926—31no—
54#924—31no—
55#924—31no—
56#924—31no—
57#924—61no—
58#897—31no—
59#892—31no—
60#883—32no—
61#67⋊C4—32no—
62C23×C32C2311yescharacteristic
63C22×C3×D3—22no—
64C22×C3×D3—22no—
65C22×C3×D3—22no—
66C22×C3×D3C2312yes—
67C22×C3×D3C2311yescharacteristic
68C22×C3×D3C2311yescharacteristic
69C2×C32×C4—22no—
70C2×C32×C4C2311yescharacteristic
71C2×C32×C4C2311yescharacteristic
72C2×C3⋊Dic3C2312yes—
73C2×C3⋊Dic3C2312yes—
74#634—22no—
75C3×#67—24no—
76C2×C3×Dic3—22no—
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C2×C3×Dic3C2311yescharacteristic
80C3×D12—22no—
81C3×C4×D3—22no—
82C3×C4×D3—22no—
83C3×C4×D3—22no—
84C3×C4×D3D414yescomplemented
85#625—24no—
86#625—24no—
87D3×Dic3—24no—
88#525—91noSylow
89C24×C3—31no—
90C23×D3C2×D311yescharacteristic
91C2×C3×D4—32no—
92C2×C3×D4—31no—
93C22×C3×C4—21no—
94C22×C3×C4—31no—
95C22×C3×C4—31no—
96C22×C3×C4—31no—
97C22×C3×C4—31no—
98C22×C3×C4—31no—
99C22×C3×C4—64no—
100C22×C3×C4C2×D311yescharacteristic
101C22×C3×C4C2×D311yescharacteristic
102C2×#67—32no—
103C2×#67—32no—
104C2×#67—32no—
105C2×#67—32no—
106C2×#67C2×D312yes—
107C22×Dic3—32no—
108C22×Dic3C2×D311yescharacteristic
109#237—64no—
110#235—64no—
111C2×D12C2×D311yescharacteristic, complemented
112C2×C4×D3—24no—
113C2×C4×D3—32no—
114C2×C4×D3C2×D311yescharacteristic
115C2×C4×D3C2×D311yescharacteristic
116C2×C4×D3C2×D311yescharacteristic
117C2×C4×D3C2×D311yescharacteristic
118C2×Dic6—31no—
119C2×Dic6—31no—
120C2×Dic6—61no—
121C3×C4⋊C4—31no—
122C3×C22⋊C4—32no—
123#217—32no—
124#217—32no—
125#217—32no—
126#217—32no—
127#217—32no—
128#217—62no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4C2×D312yescomplemented
133C4⋊Dic3—31no—
134C4⋊Dic3—31no—
135C4⋊Dic3—32no—
136C4⋊Dic3—61no—
137C4⋊Dic3C2×D311yescharacteristic, complemented
138Dic3⋊C4—32no—
139Dic3⋊C4—32no—
140Dic3⋊C4—32no—
141Dic3⋊C4—62no—
142C4×Dic3—32no—
143C4×Dic3—32no—
144C22×C32—21no—
145C22×C32—21no—
146C22×C32#4112yescomplemented
147C22×C32C2411yescharacteristic
148C2×C3×D3—28no—
149C2×C3×D3—21no—
150C2×C3×D3—21no—
151C2×C3×D3—21no—
152C2×C3×D3—21no—
153C2×C3×D3—22no—
154C2×C3×D3—22no—
155C2×C3×D3—22no—
156C2×C3×D3C2×D412yes—
157C2×C3×D3#4112yescomplemented
158C32×C4—21no—
159C32×C4C2×D412yes—
160C3⋊Dic3—22no—
161C3⋊Dic3—22no—
162C3×Dic3—21no—
163C3×Dic3—22no—
164C3×Dic3C2×D412yes—
165C2×#41—91no—
166C23×C4—31no—
167#123—92no—
168C22⋊Q8—91no—
169C22⋊Q8—91no—
170#121—31no—
171#121—91no—
172#120—92no—
173C4×D4—92no—
174C4×D4—92no—
175#117—91no—
176C23×C3—21no—
177C23×C3—31no—
178C23×C3—31no—
179C23×C3—32no—
180C23×C3—62no—
181C23×C3—62no—
182C23×C3—62no—
183C23×C3C22×D311yescharacteristic
184C23×C3C22×D311yescharacteristic
185C22×D3—22no—
186C22×D3—22no—
187C22×D3—22no—
188C22×D3C22×D312yes—
189C22×D3C22×D311yescharacteristic
190C22×D3C22×D311yescharacteristic
191C3×D4—64no—
192C3×D4—62no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—31no—
198C2×C3×C4—34no—
199C2×C3×C4—32no—
200C2×C3×C4—31no—
201C2×C3×C4—42no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4C22×D311yescharacteristic
207C2×C3×C4C22×D311yescharacteristic
208C2×C3×C4C22×D311yescharacteristic
209C2×C3×C4C22×D311yescharacteristic
210#67—24no—
211#67—64no—
212#67—64no—
213#67—64no—
214#67—64no—
215C2×Dic3—22no—
216C2×Dic3—32no—
217C2×Dic3—32no—
218C2×Dic3—32no—
219C2×Dic3—32no—
220C2×Dic3—62no—
221C2×Dic3—62no—
222C2×Dic3—64no—
223C2×Dic3C22×D312yes—
224C2×Dic3C22×D311yescharacteristic
225C2×Dic3C22×D311yescharacteristic
226D12—22no—
227C4×D3—22no—
228C4×D3—22no—
229C4×D3—22no—
230C4×D3—64no—
231C4×D3#6714yescomplemented
232Dic6—62no—
233Dic6—62no—
234Dic6—122no—
235C2×C32—22no—
236C2×C32C22×D411yescharacteristic
237C2×C32C2×#4111yescharacteristic
238C2×C32C2×#4111yescharacteristic
239C3×D3—22no—
240C3×D3—22no—
241C3×D3—42no—
242C24—31no—
243#41—184no—
244C2×Q8—91no—
245C2×D4—32no—
246C2×D4—31no—
247C2×D4—92no—
248C2×D4—92no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4—31no—
254C22×C4—64no—
255C22×C4—92no—
256C22×C4D3211yescharacteristic
257C4⋊C4—31no—
258C4⋊C4—92no—
259C4⋊C4—91no—
260C4⋊C4—92no—
261C22⋊C4—32no—
262C22⋊C4—92no—
263C22⋊C4—92no—
264C22⋊C4—92no—
265C22⋊C4—92no—
266C42—92no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—32no—
274C22×C3—32no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—68no—
278C22×C3—61no—
279C22×C3—61no—
280C22×C3—61no—
281C22×C3—61no—
282C22×C3—62no—
283C22×C3—62no—
284C22×C3—62no—
285C22×C3#23512yescomplemented
286C22×C3#23712yescomplemented
287C22×C3C23×D311yescharacteristic
288C22×C3C23×D311yescharacteristic
289C2×D3—28no—
290C2×D3—21no—
291C2×D3—21no—
292C2×D3—21no—
293C2×D3—21no—
294C2×D3—22no—
295C2×D3—22no—
296C2×D3—22no—
297C2×D3#23512yescomplemented
298C2×D3C2×#6712yes—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—32no—
303C3×C4—41no—
304C3×C4—62no—
305C3×C4—61no—
306C3×C4D3×D412yes—
307C3×C4C2×#6712yes—
308Dic3—22no—
309Dic3—21no—
310Dic3—62no—
311Dic3—62no—
312Dic3—62no—
313Dic3—62no—
314Dic3—122no—
315Dic3—122no—
316Dic3C2×#6712yes—
317C32#52511yescharacteristic, complemented, Sylow
318C23—31no—
319C23—31no—
320C23—32no—
321C23—62no—
322C23—62no—
323C23—62no—
324C23C2×D3211yescharacteristic
325Q8—182no—
326D4—64no—
327D4—62no—
328D4—184no—
329D4—184no—
330C2×C4—22no—
331C2×C4—31no—
332C2×C4—34no—
333C2×C4—32no—
334C2×C4—31no—
335C2×C4—62no—
336C2×C4—62no—
337C2×C4—62no—
338C2×C4—62no—
339C2×C4—92no—
340C2×C4—92no—
341C2×C4—184no—
342C2×C4C2×D3211yescharacteristic
343C2×C4C2×D3211yescharacteristic
344C2×C3—22no—
345C2×C3—22no—
346C2×C3—21no—
347C2×C3—21no—
348C2×C3—21no—
349C2×C3—42no—
350C2×C3—62no—
351C2×C3—62no—
352C2×C3—122no—
353C2×C3C2×#23511yescharacteristic
354C2×C3C2×#23511yescharacteristic
355C2×C3C2×D3×D411yescharacteristic
356C2×C3C2×#23711yescharacteristic
357C2×C3D3×#4111yescharacteristic
358C2×C3C22×#6711yescharacteristic
359D3—22no—
360D3—22no—
361D3—42no—
362C22—21no—
363C22—21no—
364C22—32no—
365C22—32no—
366C22—68no—
367C22—61no—
368C22—61no—
369C22—61no—
370C22—61no—
371C22—62no—
372C22—62no—
373C22—62no—
374C22#379512yescomplemented
375C22C22×D3211yescharacteristic
376C4—21no—
377C4—32no—
378C4—62no—
379C4—61no—
380C4—182no—
381C4—182no—
382C4D3×#6712yes—
383C3—21no—
384C3#560611yescharacteristic, complemented
385C3#579111yescharacteristic, complemented
386C2—22no—
387C2—62no—
388C2—62no—
389C2—122no—
390C2C2×#379511yescharacteristic
391C2D3×#23511yescharacteristic
392C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#525
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×C4×D3C2semidirect
2. C2×#3795C2semidirect
3. #64133C2semidirect
4. C3×#930C2semidirect
5. #63976C2semidirect
6. Dic3×#67C2semidirect
7. #63908C2semidirect
8. #63905C2semidirect
9. #63879C2semidirect
10. #63857C2semidirect
11. #63841C2semidirect
16. C2×#634C22semidirect
19. C2×C3×D12C22semidirect
27. C2×D3×Dic3C22semidirect
30. C4⋊(C3⋊Dic3)C22semidirect
31. #3750C22semidirect
32. C3×(C2×D3)⋊C4C22semidirect
33. C3×C4⋊Dic3C22semidirect
34. Dic3⋊Dic3C22semidirect
37. (C2×D3)⋊Dic3C22semidirect
38. Dic32C22semidirect
42. C22×C4×D3D3semidirect
49. #930D3semidirect
84. C3×C4×D3D4semidirect
111. C2×D12C2×D3semidirect
132. (C2×D3)⋊C4C2×D3semidirect
137. C4⋊Dic3C2×D3semidirect
146. C22×C32#41semidirect
157. C2×C3×D3#41semidirect
231. C4×D3#67semidirect
285. C22×C3#235semidirect
286. C22×C3#237semidirect
297. C2×D3#235semidirect
317. C32#525semidirect
374. C22#3795semidirect
384. C3#5606semidirect
385. C3#5791semidirect

Maximal subgroups

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

subgroupindex
C22×C3×C4×D32
C2×#37952
#641332
C3×#9302
#639762
Dic3×#672
#639082
#639052
#638792
#638572
#638412
#57913
#56063