Finite Group Databasebeta

← #10594899

Subgroups — #10594899

Order 576.

617 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×C3×#235C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3#64133C211yesmaximal, characteristic, complemented
4C3×C4×#67C211yesmaximal, characteristic, complemented
5#63970C212yesmaximal, complemented
6#63959C212yesmaximal, complemented
7#63908C211yesmaximal, characteristic, complemented
8#63879C212yesmaximal, complemented
9#63860C211yesmaximal, characteristic, complemented
10#63851C212yesmaximal, complemented
11Dic3×Dic6C211yesmaximal, characteristic, complemented
12#5831—31nomaximal
13#5552—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#634C2211yescharacteristic
16C2×C3×#67C2211yescharacteristic
17C2×C3×#67C2212yes—
18C3×#235—24no—
19C2×C3×D12C2211yescharacteristic
20C2×C3×C4×D3C2211yescharacteristic
21C2×C3×C4×D3C2212yes—
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2212yes—
24C2×#625C2212yes—
25C2×D3×Dic3C2212yes—
26D3×Dic6—24no—
27#3757C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29#3750C2212yes—
30C3×#217C2211yescharacteristic
31C3×(C2×D3)⋊C4C2211yescharacteristic
32C3×Dic3⋊C4C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35(C2×D3)⋊Dic3C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C2×C3×#41—31no—
40C2×D3×Q8—31no—
41C2×#235D311yescharacteristic, complemented
42C22×Dic6—31no—
43C3×C4×D4—31no—
44Q8×Dic3—31no—
45#939—32no—
46#938—32no—
47#938—31no—
48#936—32no—
49#936—31no—
50#930—31no—
51C4×#67—32no—
52C4×#67D311yescharacteristic, complemented
53#924—31no—
54#924—31no—
55#924—32no—
56#924—61no—
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
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2312yes—
70#634—22no—
71C3×#67—22no—
72C3×#67—24no—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2311yescharacteristic
76C2×C3×Dic3C2312yes—
77C3×D12—22no—
78C3×C4×D3—22no—
79C3×C4×D3—24no—
80C3×Dic6D414yescomplemented
81#627—24no—
82#625—24no—
83D3×Dic3—24no—
84#548—91noSylow
85C3×#41—64no—
86C2×C3×Q8—31no—
87C2×C3×D4—32no—
88C2×C3×D4—31no—
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×#67C2×D312yes—
98C2×#67C2×D311yescharacteristic
99C22×Dic3—32no—
100D3×Q8—64no—
101#235—24no—
102C2×D12C2×D311yescharacteristic
103C2×C4×D3—32no—
104C2×C4×D3C2×D312yes—
105C2×C4×D3C2×D311yescharacteristic
106C2×Dic6—31no—
107C2×Dic6—31no—
108C2×Dic6—32no—
109C2×Dic6—32no—
110C2×Dic6—61no—
111C2×Dic6—64no—
112C2×Dic6C2×D311yescharacteristic
113C3×C4⋊C4—31no—
114C3×C22⋊C4—31no—
115C3×C22⋊C4—31no—
116C3×C42—31no—
117#217—32no—
118#217—32no—
119#217—32no—
120#217—32no—
121#217—32no—
122#217—62no—
123#217C2×D311yescharacteristic
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4C2×D311yescharacteristic
128C4⋊Dic3—31no—
129C4⋊Dic3—31no—
130C4⋊Dic3—32no—
131C4⋊Dic3—61no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4—62no—
136Dic3⋊C4C2×D311yescharacteristic
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3C2×D311yescharacteristic
140C22×C32—21no—
141C22×C32—21no—
142C22×C32—21no—
143C22×C32C2411yescharacteristic
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—22no—
147C2×C3×D3—22no—
148C2×C3×D3—22no—
149C2×C3×D3#4112yescomplemented
150C32×C4—21no—
151C32×C4C2×D412yes—
152C3⋊Dic3—22no—
153C3⋊Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—21no—
157C3×Dic3C2×D414yes—
158C2×#41—31no—
159C22×Q8—91no—
160#124—91no—
161#124—92no—
162C22⋊Q8—92no—
163C22⋊Q8—91no—
164#121—91no—
165#121—92no—
166C4×Q8—91no—
167C4×D4—31no—
168C4×D4—92no—
169C23×C3—21no—
170C23×C3—31no—
171C23×C3—32no—
172C23×C3C22×D311yescharacteristic
173C23×C3C22×D311yescharacteristic
174C22×D3C22×D312yes—
175C22×D3C22×D311yescharacteristic
176C3×Q8—34no—
177C3×D4—64no—
178C3×D4—62no—
179C3×D4—62no—
180C2×C3×C4—22no—
181C2×C3×C4—22no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—32no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—31no—
188C2×C3×C4—42no—
189C2×C3×C4—64no—
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—24no—
197#67—64no—
198#67—64no—
199C2×Dic3—32no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—32no—
203C2×Dic3—62no—
204C2×Dic3—62no—
205C2×Dic3—64no—
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D311yescharacteristic
209C2×Dic3C22×D312yes—
210D12—22no—
211C4×D3—22no—
212C4×D3—24no—
213C4×D3—64no—
214Dic6—64no—
215Dic6—62no—
216Dic6—62no—
217Dic6—64no—
218Dic6—122no—
219Dic6#6714yescomplemented
220C2×C32—41no—
221C2×C32C22×D411yescharacteristic
222C2×C32C2×#4111yescharacteristic
223C2×C322−1+411yescharacteristic
224C3×D3—22no—
225C3×D3—42no—
226#41—64no—
227C2×Q8—31no—
228C2×Q8—92no—
229C2×Q8—91no—
230C2×Q8—184no—
231C2×D4—31no—
232C2×D4—32no—
233C2×D4—31no—
234C2×D4—92no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—92no—
238C22×C4D3211yescharacteristic
239C4⋊C4—31no—
240C4⋊C4—92no—
241C4⋊C4—92no—
242C4⋊C4—91no—
243C22⋊C4—31no—
244C22⋊C4—31no—
245C22⋊C4—92no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C22⋊C4—92no—
249C42—31no—
250C42—92no—
251C22×C3—22no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—32no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—61no—
263C22×C3—61no—
264C22×C3—62no—
265C22×C3—62no—
266C22×C3—62no—
267C22×C3C23×D311yescharacteristic
268C22×C3C23×D311yescharacteristic
269C2×D3—21no—
270C2×D3—21no—
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—22no—
274C2×D3#23512yescomplemented
275C3×C4—21no—
276C3×C4—21no—
277C3×C4—22no—
278C3×C4—34no—
279C3×C4—41no—
280C3×C4—61no—
281C3×C4—61no—
282C3×C4—61no—
283C3×C4D3×D412yes—
284C3×C4C2×#6712yes—
285Dic3—21no—
286Dic3—21no—
287Dic3—21no—
288Dic3—62no—
289Dic3—62no—
290Dic3—62no—
291Dic3—62no—
292Dic3—122no—
293Dic3—122no—
294Dic3C2×#6714yes—
295C32#54811yescharacteristic, complemented, Sylow
296C23—31no—
297C23—32no—
298C23C2×D3211yescharacteristic
299Q8—34no—
300Q8—184no—
301Q8—182no—
302D4—64no—
303D4—62no—
304D4—62no—
305D4—184no—
306C2×C4—22no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—31no—
311C2×C4—64no—
312C2×C4—62no—
313C2×C4—92no—
314C2×C4—92no—
315C2×C4—184no—
316C2×C4C2×D3211yescharacteristic
317C2×C4C2×D3211yescharacteristic
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—42no—
322C2×C3—41no—
323C2×C3—41no—
324C2×C3—62no—
325C2×C3—122no—
326C2×C3C2×#23511yescharacteristic
327C2×C3C2×D3×D411yescharacteristic
328C2×C3D3×#4111yescharacteristic
329C2×C3#101011yescharacteristic
330C2×C3#101011yescharacteristic
331C2×C3C22×#6711yescharacteristic
332D3—22no—
333D3—42no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—32no—
338C22—61no—
339C22—61no—
340C22—62no—
341C22—62no—
342C22—62no—
343C22C22×D3211yescharacteristic
344C4—21no—
345C4—34no—
346C4—61no—
347C4—61no—
348C4—61no—
349C4—182no—
350C4—182no—
351C4D3×#6712yes—
352C3—21no—
353C3#555211yescharacteristic, complemented
354C3#583111yescharacteristic, complemented
355C2—41no—
356C2—62no—
357C2—122no—
358C2#6429811yescharacteristic
359C2D3×#23511yescharacteristic
360C2C2×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. C2×C3×#235C2semidirect
2. C2×D3×Dic6C2semidirect
3. #64133C2semidirect
4. C3×C4×#67C2semidirect
5. #63970C2semidirect
6. #63959C2semidirect
7. #63908C2semidirect
8. #63879C2semidirect
9. #63860C2semidirect
10. #63851C2semidirect
11. Dic3×Dic6C2semidirect
41. C2×#235D3semidirect
52. C4×#67D3semidirect
80. C3×Dic6D4semidirect
149. C2×C3×D3#41semidirect
219. Dic6#67semidirect
274. C2×D3#235semidirect
295. C32#548semidirect
353. C3#5552semidirect
354. C3#5831semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#2352
C2×D3×Dic62
#641332
C3×C4×#672
#639702
#639592
#639082
#638792
#638602
#638512
Dic3×Dic62
#58313
#55523