Finite Group Databasebeta

← #10594916

Subgroups — #10594916

Order 576.

662 conjugacy classes of subgroups.

The subgroup lattice has 369 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3828C211yesmaximal, characteristic, complemented
2C2×D3×Dic6C211yesmaximal, characteristic, complemented
3C3×C4×#67C211yesmaximal, characteristic, complemented
4C3×#924C211yesmaximal, characteristic, complemented
5#63971C212yesmaximal, complemented
6#63953C212yesmaximal, complemented
7#63909C211yesmaximal, characteristic, complemented
8#63881C212yesmaximal, complemented
9#63865C212yesmaximal, complemented
10#63864C211yesmaximal, characteristic, complemented
11#63844C211yesmaximal, characteristic, complemented
12#5633—31nomaximal
13#5552—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2212yes—
16#3828—24no—
17C2×C4⋊(C3⋊D3)C2211yescharacteristic
18C2×C4×C3⋊D3C2212yes—
19C2×#634C2211yescharacteristic
20C2×C3×#67C2211yescharacteristic
21C2×C3×C4×D3C2211yescharacteristic
22C2×C3×Dic6C2211yescharacteristic
23C2×#627C2212yes—
24C2×#626C2212yes—
25C2×D3×Dic3C2212yes—
26D3×Dic6—24no—
27C3×#217C2212yes—
28C3×#217C2211yescharacteristic
29C3×(C2×D3)⋊C4C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×Dic3⋊C4C2212yes—
32C3×Dic3⋊C4C2211yescharacteristic
33C3×C4×Dic3C2211yescharacteristic
34#3724C2212yes—
35#3722C2212yes—
36#3722C2212yes—
37#3722C2212yes—
38Dic32C2212yes—
39C2×D3×Q8—31no—
40C2×#235—31no—
41C2×#235—31no—
42C2×#235—61no—
43C22×Dic6—31no—
44C3×C22⋊Q8—31no—
45C3×C4×D4—31no—
46#947—31no—
47#939—32no—
48#930—31no—
49C4×#67D311yescharacteristic, complemented
50#924D311yescharacteristic, complemented
51#896—32no—
52#896—32no—
53#895—31no—
54#893—32no—
55#887—31no—
56#885—32no—
57#885—32no—
58#884—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—
71C4×C3⋊D3—24no—
72#634D414yescomplemented
73C3×#67—22no—
74C2×C3×Dic3C2311yescharacteristic
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C2×C3×Dic3C2312yes—
78C2×C3×Dic3C2311yescharacteristic
79C3×C4×D3—22no—
80C3×Dic6—22no—
81#627—24no—
82#626—24no—
83D3×Dic3—24no—
84#548—91noSylow
85C2×C3×Q8—31no—
86C2×C3×D4—31no—
87C22×C3×C4—21no—
88C22×C3×C4—31no—
89C22×C3×C4C2×D311yescharacteristic
90C22×C3×C4C2×D311yescharacteristic
91C2×#67—32no—
92C2×#67—32no—
93C2×#67—32no—
94C2×#67—62no—
95C2×#67C2×D311yescharacteristic
96C22×Dic3—32no—
97D3×Q8—64no—
98#235—64no—
99#235—64no—
100#235—68no—
101C2×D12—32no—
102C2×D12—31no—
103C2×D12—31no—
104C2×D12—61no—
105C2×C4×D3—32no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3—62no—
109C2×C4×D3C2×D311yescharacteristic
110C2×Dic6—31no—
111C2×Dic6—31no—
112C2×Dic6—32no—
113C2×Dic6—32no—
114C2×Dic6—61no—
115C2×Dic6—64no—
116C2×Dic6C2×D311yescharacteristic
117C3×C4⋊C4—31no—
118C3×C4⋊C4—32no—
119C3×C4⋊C4—31no—
120C3×C22⋊C4—32no—
121C3×C22⋊C4—31no—
122C3×C22⋊C4—31no—
123C3×C42—31no—
124#217C2×D311yescharacteristic
125#217C2×D312yes—
126(C2×D3)⋊C4—32no—
127(C2×D3)⋊C4—32no—
128(C2×D3)⋊C4—32no—
129(C2×D3)⋊C4—32no—
130(C2×D3)⋊C4—32no—
131(C2×D3)⋊C4—32no—
132(C2×D3)⋊C4C2×D311yescharacteristic
133C4⋊Dic3C2×D311yescharacteristic
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4C2×D312yes—
137Dic3⋊C4C2×D311yescharacteristic
138C4×Dic3—32no—
139C4×Dic3—32no—
140C4×Dic3C2×D311yescharacteristic
141C22×C32—21no—
142C22×C32—21no—
143C22×C32—21no—
144C22×C32C2411yescharacteristic
145C2×C3⋊D3—22no—
146C2×C3⋊D3—22no—
147C2×C3⋊D3—22no—
148C2×C3×D3—21no—
149C2×C3×D3—21no—
150C2×C3×D3#4112yescomplemented
151C32×C4—21no—
152C32×C4C2×D412yes—
153C3⋊Dic3C2×D414yes—
154C3×Dic3—22no—
155C3×Dic3—21no—
156C3×Dic3—22no—
157C3×Dic3—21no—
158C3×Dic3—21no—
159C2×#41—91no—
160C22×Q8—91no—
161#124—91no—
162#124—92no—
163C22⋊Q8—31no—
164C22⋊Q8—92no—
165#121—92no—
166#121—91no—
167C4×Q8—91no—
168C4×D4—31no—
169C4×D4—92no—
170C23×C3—21no—
171C23×C3—31no—
172C23×C3C22×D311yescharacteristic
173C23×C3C22×D311yescharacteristic
174C22×D3—32no—
175C22×D3—32no—
176C22×D3—62no—
177C22×D3C22×D311yescharacteristic
178C3×Q8—62no—
179C3×D4—62no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—31no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4—31no—
189C2×C3×C4—32no—
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—64no—
197#67—64no—
198#67—64no—
199#67—68no—
200C2×Dic3—32no—
201C2×Dic3—32no—
202C2×Dic3—62no—
203C2×Dic3—64no—
204C2×Dic3C22×D311yescharacteristic
205C2×Dic3C22×D312yes—
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D311yescharacteristic
208C2×Dic3C22×D312yes—
209D12—64no—
210D12—64no—
211D12—62no—
212D12—62no—
213C4×D3—22no—
214C4×D3—64no—
215C4×D3—64no—
216C4×D3—64no—
217C4×D3—68no—
218Dic6—22no—
219Dic6—34no—
220Dic6—34no—
221Dic6—64no—
222Dic6—64no—
223Dic6—64no—
224C2×C32—41no—
225C2×C32C22×D411yescharacteristic
226C2×C32C2×#4111yescharacteristic
227C2×C322−1+411yescharacteristic
228C3⋊D3—42no—
229C3×D3—22no—
230#41—184no—
231C2×Q8—31no—
232C2×Q8—92no—
233C2×Q8—91no—
234C2×Q8—184no—
235C2×D4—31no—
236C2×D4—92no—
237C2×D4—92no—
238C2×D4—91no—
239C22×C4—31no—
240C22×C4—92no—
241C22×C4—92no—
242C22×C4D3211yescharacteristic
243C4⋊C4—31no—
244C4⋊C4—32no—
245C4⋊C4—31no—
246C4⋊C4—92no—
247C22⋊C4—31no—
248C22⋊C4—32no—
249C22⋊C4—31no—
250C22⋊C4—92no—
251C22⋊C4—92no—
252C22⋊C4—92no—
253C42—31no—
254C42—92no—
255C22×C3—22no—
256C22×C3—21no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—32no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3—61no—
267C22×C3—61no—
268C22×C3C23×D311yescharacteristic
269C22×C3C23×D311yescharacteristic
270C2×D3—21no—
271C2×D3—21no—
272C2×D3—64no—
273C2×D3—62no—
274C2×D3—62no—
275C2×D3—62no—
276C2×D3—62no—
277C2×D3—62no—
278C2×D3—62no—
279C2×D3—122no—
280C2×D3—122no—
281C2×D3#23512yescomplemented
282C3×C4—22no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—22no—
286C3×C4—62no—
287C3×C4—61no—
288C3×C4—62no—
289C3×C4—61no—
290C3×C4—61no—
291C3×C4D3×D412yes—
292C3×C4D3×D412yes—
293Dic3—21no—
294Dic3—22no—
295Dic3—21no—
296Dic3—22no—
297Dic3—21no—
298Dic3—34no—
299Dic3—34no—
300Dic3—64no—
301C32#54811yescharacteristic, complemented, Sylow
302C23—31no—
303C23—92no—
304C23C2×D3211yescharacteristic
305Q8—62no—
306Q8—94no—
307Q8—184no—
308D4—62no—
309D4—184no—
310D4—184no—
311D4—182no—
312C2×C4—22no—
313C2×C4—31no—
314C2×C4—31no—
315C2×C4—32no—
316C2×C4—31no—
317C2×C4—32no—
318C2×C4—62no—
319C2×C4—92no—
320C2×C4—184no—
321C2×C4—184no—
322C2×C4C2×D3211yescharacteristic
323C2×C4C2×D3211yescharacteristic
324C2×C3—21no—
325C2×C3—21no—
326C2×C3—21no—
327C2×C3—42no—
328C2×C3—41no—
329C2×C3—41no—
330C2×C3—62no—
331C2×C3C2×#23511yescharacteristic
332C2×C3C2×D3×D411yescharacteristic
333C2×C3C2×D3×D411yescharacteristic
334C2×C3#100711yescharacteristic
335C2×C3D3×#4111yescharacteristic
336C2×C3#101011yescharacteristic
337D3—22no—
338D3—124no—
339D3—122no—
340D3—122no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—32no—
345C22—61no—
346C22—61no—
347C22—182no—
348C22—182no—
349C22—182no—
350C22C22×D3211yescharacteristic
351C4—21no—
352C4—61no—
353C4—61no—
354C4—62no—
355C4—61no—
356C4—62no—
357C4—94no—
358C4#381112yes—
359C3—21no—
360C3#555211yescharacteristic, complemented
361C3#563311yescharacteristic, complemented
362C2—41no—
363C2—62no—
364C2—362no—
365C2#6429711yescharacteristic
366C2D3×#23511yescharacteristic
367C2C2×#381111yescharacteristic
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×#3828C2semidirect
2. C2×D3×Dic6C2semidirect
3. C3×C4×#67C2semidirect
4. C3×#924C2semidirect
5. #63971C2semidirect
6. #63953C2semidirect
7. #63909C2semidirect
8. #63881C2semidirect
9. #63865C2semidirect
10. #63864C2semidirect
11. #63844C2semidirect
49. C4×#67D3semidirect
50. #924D3semidirect
72. #634D4semidirect
150. C2×C3×D3#41semidirect
281. C2×D3#235semidirect
301. C32#548semidirect
360. C3#5552semidirect
361. C3#5633semidirect

Maximal subgroups

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

subgroupindex
C2×#38282
C2×D3×Dic62
C3×C4×#672
C3×#9242
#639712
#639532
#639092
#638812
#638652
#638642
#638442
#56333
#55523