Finite Group Databasebeta

← #10595136

Subgroups — #10595136

Order 576.

631 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3804C212yesmaximal, complemented
2#64147C211yesmaximal, characteristic, complemented
3C3×#938C211yesmaximal, characteristic, complemented
4C3×D4×Dic3C211yesmaximal, characteristic, complemented
5#63975C212yesmaximal, complemented
6#63969C212yesmaximal, complemented
7#63964C212yesmaximal, complemented
8#63918C211yesmaximal, characteristic, complemented
9C4×#626C211yesmaximal, characteristic, complemented
10#63900C211yesmaximal, characteristic, complemented
11D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
12#5611—31nomaximal
13#5551—31nomaximal
14C2×C32×D4C2211yescharacteristic
15C2×#638C2212yes—
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×#627C2212yes—
21C2×#626C2211yescharacteristic
22C2×#624C2212yes—
23#3804—28no—
24C2×D3×Dic3C2212yes—
25#3757C2212yes—
26C4⋊(C3⋊Dic3)C2211yescharacteristic
27C3×#217C2212yes—
28C3×#217C2212yes—
29C3×C4⋊Dic3C2211yescharacteristic
30C3×C4⋊Dic3C2211yescharacteristic
31C3×C4×Dic3C2211yescharacteristic
32#3724C2212yes—
33Dic3⋊Dic3C2212yes—
34Dic3⋊Dic3C2211yescharacteristic
35Dic3⋊Dic3C2212yes—
36#3722C2211yescharacteristic
37(C2×D3)⋊Dic3C2211yescharacteristic
38C2×#237—32no—
39C2×#235—32no—
40C3×#121—31no—
41C3×C4×D4—31no—
42#938—31no—
43#938—31no—
44#938—61no—
45#938D311yescharacteristic, complemented
46#935—32no—
47D4×Dic3D311yescharacteristic, complemented
48C4×#67—31no—
49C2×C4⋊Dic3—31no—
50#924—32no—
51D3×C4⋊C4—31no—
52D3×C4⋊C4—31no—
53D3×C4⋊C4—31no—
54#888—31no—
55#883—32no—
56#883—32no—
57#883—32no—
58#878—32no—
59C4×D12—31no—
60#869—31no—
61C23×C32C2312yes—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C32×D4—22no—
65C2×C32×C4C2311yescharacteristic
66#638—24no—
67C2×C3⋊Dic3C2311yescharacteristic
68C2×C3⋊Dic3C2312yes—
69C4×C3⋊D3—22no—
70C3×#67—24no—
71C2×C3×Dic3—24no—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2311yescharacteristic
77C3×C4×D3—22no—
78#627—24no—
79#626D414yescomplemented
80#624—24no—
81D3×Dic3—24no—
82#547—91noSylow
83C2×C3×D4—21no—
84C2×C3×D4—32no—
85C2×C3×D4C2×D311yescharacteristic
86C2×C3×D4C2×D311yescharacteristic
87C22×C3×C4—32no—
88C22×C3×C4—31no—
89C2×#67—32no—
90C2×#67—32no—
91C2×#67—31no—
92C2×#67—62no—
93C2×#67C2×D312yes—
94C22×Dic3—32no—
95C22×Dic3C2×D312yes—
96#237—68no—
97#235—68no—
98C2×D12—31no—
99C2×C4×D3—32no—
100C2×C4×D3—31no—
101C2×C4×D3—31no—
102C2×C4×D3—32no—
103C2×C4×D3—32no—
104C2×C4×D3—61no—
105C2×C4×D3C2×D311yescharacteristic
106C2×Dic6—32no—
107C2×Dic6—32no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—31no—
110C3×C22⋊C4—32no—
111C3×C22⋊C4—32no—
112C3×C42—31no—
113#217—32no—
114#217—32no—
115#217—31no—
116#217—62no—
117#217C2×D312yes—
118#217C2×D312yes—
119(C2×D3)⋊C4—31no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—31no—
122C4⋊Dic3—31no—
123C4⋊Dic3—31no—
124C4⋊Dic3—32no—
125C4⋊Dic3—31no—
126C4⋊Dic3—32no—
127C4⋊Dic3—61no—
128C4⋊Dic3C2×D311yescharacteristic
129C4⋊Dic3C2×D311yescharacteristic
130Dic3⋊C4—31no—
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135C4×Dic3C2×D311yescharacteristic
136C22×C32—22no—
137C22×C32—22no—
138C22×C32—22no—
139C22×C32C2411yescharacteristic
140C2×C3⋊D3—21no—
141C2×C3⋊D3—21no—
142C2×C3⋊D3C2×D412yes—
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3C2×D412yes—
146C32×C4#4112yescomplemented
147C3⋊Dic3—22no—
148C3⋊Dic3—21no—
149C3×Dic3—22no—
150C3×Dic3—22no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3C2×D412yes—
154C2×#41—92no—
155C4⋊Q8—91no—
156#123—92no—
157#123—92no—
158C22⋊Q8—92no—
159#121—31no—
160#121—91no—
161C4×D4—31no—
162C4×D4—91no—
163C2×C4⋊C4—91no—
164C2×C4⋊C4—91no—
165C23×C3—22no—
166C23×C3—31no—
167C23×C3C22×D312yes—
168C23×C3C22×D312yes—
169C22×D3—31no—
170C22×D3—31no—
171C22×D3—61no—
172C22×D3C22×D311yescharacteristic
173C3×D4—22no—
174C3×D4—22no—
175C3×D4—42no—
176C3×D4—64no—
177C2×C3×C4—21no—
178C2×C3×C4—31no—
179C2×C3×C4—32no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—64no—
184C2×C3×C4—62no—
185C2×C3×C4C22×D311yescharacteristic
186C2×C3×C4C22×D311yescharacteristic
187#67—24no—
188#67—34no—
189#67—64no—
190#67—64no—
191#67—124no—
192C2×Dic3—24no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—32no—
196C2×Dic3—32no—
197C2×Dic3—62no—
198C2×Dic3—61no—
199C2×Dic3—64no—
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D311yescharacteristic
202C2×Dic3C22×D312yes—
203C2×Dic3C22×D312yes—
204C2×Dic3C22×D311yescharacteristic
205D12—34no—
206C4×D3—22no—
207C4×D3—64no—
208C4×D3—64no—
209C4×D3—64no—
210C4×D3—62no—
211C4×D3—62no—
212C4×D3—122no—
213Dic6—64no—
214Dic6—64no—
215C2×C32—42no—
216C2×C32C22×D411yescharacteristic
217C2×C32C2×#4111yescharacteristic
218C2×C322−1+411yescharacteristic
219C3⋊D3—22no—
220C3×D3—22no—
221#41—188no—
222C2×Q8—92no—
223C2×D4—32no—
224C2×D4—91no—
225C2×D4—92no—
226C2×D4D3211yescharacteristic
227C22×C4—32no—
228C22×C4—31no—
229C22×C4—92no—
230C22×C4—92no—
231C22×C4—91no—
232C4⋊C4—31no—
233C4⋊C4—31no—
234C4⋊C4—92no—
235C4⋊C4—92no—
236C4⋊C4—91no—
237C4⋊C4—92no—
238C4⋊C4—91no—
239C22⋊C4—32no—
240C22⋊C4—32no—
241C22⋊C4—91no—
242C22⋊C4—91no—
243C22⋊C4—92no—
244C42—31no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—22no—
251C22×C3—21no—
252C22×C3—32no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—42no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3C23×D311yescharacteristic
259C22×C3C23×D311yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3—32no—
263C2×D3—32no—
264C2×D3—62no—
265C2×D3—61no—
266C2×D3—61no—
267C2×D3—61no—
268C2×D3—61no—
269C2×D3—121no—
270C2×D3—121no—
271C2×D3D3×D412yes—
272C3×C4—22no—
273C3×C4—32no—
274C3×C4—61no—
275C3×C4—62no—
276C3×C4—62no—
277C3×C4—61no—
278C3×C4#23512yescomplemented
279C3×C4#23712yescomplemented
280Dic3—22no—
281Dic3—22no—
282Dic3—21no—
283Dic3—21no—
284Dic3—61no—
285Dic3—62no—
286Dic3—62no—
287Dic3—61no—
288Dic3—121no—
289Dic3—122no—
290Dic3D3×D412yes—
291C32#54711yescharacteristic, complemented, Sylow
292C23—31no—
293C23—91no—
294C23C2×D3212yes—
295Q8—184no—
296D4—22no—
297D4—64no—
298D4—94no—
299D4—184no—
300C2×C4—31no—
301C2×C4—31no—
302C2×C4—31no—
303C2×C4—32no—
304C2×C4—32no—
305C2×C4—64no—
306C2×C4—62no—
307C2×C4—91no—
308C2×C4—92no—
309C2×C4—184no—
310C2×C4—184no—
311C2×C4—182no—
312C2×C4C2×D3211yescharacteristic
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—42no—
317C2×C3—42no—
318C2×C3—62no—
319C2×C3—82no—
320C2×C3C2×#23511yescharacteristic
321C2×C3C2×D3×D411yescharacteristic
322C2×C3C2×D3×D411yescharacteristic
323C2×C3C2×#23711yescharacteristic
324C2×C3#101011yescharacteristic
325C2×C3#101011yescharacteristic
326D3—22no—
327D3—62no—
328D3—62no—
329D3—122no—
330C22—22no—
331C22—22no—
332C22—22no—
333C22—32no—
334C22—61no—
335C22—61no—
336C22—92no—
337C22—181no—
338C22—181no—
339C22C22×D3211yescharacteristic
340C4—32no—
341C4—61no—
342C4—62no—
343C4—62no—
344C4—61no—
345C4—182no—
346C4—181no—
347C4#380412yescomplemented
348C3—21no—
349C3#555111yescharacteristic, complemented
350C3#561111yescharacteristic, complemented
351C2—42no—
352C2—62no—
353C2—182no—
354C2#6430911yescharacteristic
355C2D32×D411yescharacteristic
356C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×D4C22
hypercenterC2×D4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#3804C2semidirect
2. #64147C2semidirect
3. C3×#938C2semidirect
4. C3×D4×Dic3C2semidirect
5. #63975C2semidirect
6. #63969C2semidirect
7. #63964C2semidirect
8. #63918C2semidirect
9. C4×#626C2semidirect
10. #63900C2semidirect
11. D3×C4⋊Dic3C2semidirect
45. #938D3semidirect
47. D4×Dic3D3semidirect
79. #626D4semidirect
146. C32×C4#41semidirect
278. C3×C4#235semidirect
279. C3×C4#237semidirect
291. C32#547semidirect
347. C4#3804semidirect
349. C3#5551semidirect
350. C3#5611semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
#641472
C3×#9382
C3×D4×Dic32
#639752
#639692
#639642
#639182
C4×#6262
#639002
D3×C4⋊Dic32
#56113
#55513