Finite Group Databasebeta

← #10594928

Subgroups — #10594928

Order 576.

631 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3804C212yesmaximal, complemented
2C4×#638C211yesmaximal, characteristic, complemented
3C3×#930C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊Dic3C211yesmaximal, characteristic, complemented
5#63961C212yesmaximal, complemented
6#63919C211yesmaximal, characteristic, complemented
7#63916C211yesmaximal, characteristic, complemented
8#63900C211yesmaximal, characteristic, complemented
9#63899C212yesmaximal, complemented
10#63898C211yesmaximal, characteristic, complemented
11#63854C212yesmaximal, complemented
12#5611—31nomaximal
13#5507—31nomaximal
14C22×C32×C4C2211yescharacteristic
15C2×#638C2211yescharacteristic
16C2×C4×C3⋊D3C2211yescharacteristic
17C2×C3×#67C2212yes—
18C22×C3×Dic3C2212yes—
19C2×C3×D12C2211yescharacteristic
20C2×#627C2212yes—
21C2×#626C2212yes—
22C2×#624C2212yes—
23#3804—28no—
24C2×D3×Dic3C2212yes—
25#3757C2211yescharacteristic
26#3752C2211yescharacteristic
27#3750C2211yescharacteristic
28C4×C3⋊Dic3C2211yescharacteristic
29C3×(C2×D3)⋊C4C2212yes—
30C3×C4⋊Dic3C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×C4⋊Dic3C2211yescharacteristic
33C3×C4⋊Dic3C2211yescharacteristic
34#3724C2212yes—
35#3724C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38C2×#237—32no—
39C2×#235—32no—
40C3×#121—31no—
41C2×C3×C4⋊C4—31no—
42#938—31no—
43#935—32no—
44D4×Dic3—31no—
45#930D311yescharacteristic, complemented
46#929—32no—
47C4×#67—31no—
48C4×#67—31no—
49C4×#67—61no—
50C2×C4⋊Dic3D311yescharacteristic, complemented
51#896—32no—
52#895—31no—
53#894—32no—
54#893—31no—
55D3×C4⋊C4—31no—
56D3×C4⋊C4—31no—
57#888—31no—
58#888—31no—
59#883—32no—
60#878—32no—
61C23×C32C2311yescharacteristic
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2312yes—
64C2×C32×C4—22no—
65C2×C32×C4C2311yescharacteristic
66C2×C32×C4C2311yescharacteristic
67#638D414yescomplemented
68C2×C3⋊Dic3C2311yescharacteristic
69C2×C3⋊Dic3C2311yescharacteristic
70C2×C3⋊Dic3C2311yescharacteristic
71C4×C3⋊D3—22no—
72C3×#67—24no—
73C2×C3×Dic3—24no—
74C2×C3×Dic3C2312yes—
75C2×C3×Dic3C2312yes—
76C2×C3×Dic3C2312yes—
77C3×D12—22no—
78#627—24no—
79#626—24no—
80#624—24no—
81D3×Dic3—24no—
82#547—91noSylow
83C2×C3×D4—32no—
84C2×C3×D4—31no—
85C22×C3×C4—21no—
86C22×C3×C4—32no—
87C22×C3×C4C2×D311yescharacteristic
88C22×C3×C4C2×D311yescharacteristic
89C2×#67—31no—
90C2×#67—31no—
91C2×#67—32no—
92C2×#67—61no—
93C2×#67C2×D312yes—
94C22×Dic3—32no—
95C22×Dic3C2×D312yes—
96#237—68no—
97#235—68no—
98C2×D12—32no—
99C2×D12C2×D311yescharacteristic
100C2×C4×D3—32no—
101C2×C4×D3—31no—
102C2×C4×D3—31no—
103C2×C4×D3—32no—
104C2×C4×D3—32no—
105C2×C4×D3—61no—
106C2×Dic6—32no—
107C2×Dic6—32no—
108C3×C4⋊C4—31no—
109C3×C4⋊C4—31no—
110C3×C4⋊C4—31no—
111C3×C4⋊C4—32no—
112C3×C22⋊C4—32no—
113#217—31no—
114#217—31no—
115#217—32no—
116#217—32no—
117#217—61no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—31no—
122(C2×D3)⋊C4—61no—
123(C2×D3)⋊C4C2×D312yes—
124C4⋊Dic3C2×D312yes—
125C4⋊Dic3C2×D311yescharacteristic
126C4⋊Dic3C2×D311yescharacteristic
127C4⋊Dic3C2×D311yescharacteristic
128Dic3⋊C4—31no—
129Dic3⋊C4—31no—
130Dic3⋊C4—32no—
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—61no—
135C4×Dic3—31no—
136C4×Dic3—31no—
137C4×Dic3—61no—
138C22×C32—21no—
139C22×C32—21no—
140C22×C32C2×D412yes—
141C22×C32C2411yescharacteristic
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3⋊D3C2×D412yes—
145C2×C3×D3—22no—
146C2×C3×D3—22no—
147C2×C3×D3—22no—
148C32×C4—21no—
149C32×C4#4112yescomplemented
150C3⋊Dic3—21no—
151C3⋊Dic3—21no—
152C3⋊Dic3C2×D412yes—
153C3×Dic3—22no—
154C3×Dic3—22no—
155C3×Dic3—22no—
156C2×#41—92no—
157C4⋊Q8—91no—
158#123—92no—
159#123—92no—
160C22⋊Q8—92no—
161#121—31no—
162#121—91no—
163C4×D4—91no—
164C4×D4—91no—
165C2×C4⋊C4—31no—
166C2×C4⋊C4—91no—
167C23×C3—21no—
168C23×C3—32no—
169C23×C3C22×D311yescharacteristic
170C23×C3C22×D311yescharacteristic
171C22×D3—31no—
172C22×D3—31no—
173C22×D3—61no—
174C22×D3C22×D312yes—
175C3×D4—64no—
176C3×D4—62no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—32no—
182C2×C3×C4—32no—
183C2×C3×C4—32no—
184C2×C3×C4—42no—
185C2×C3×C4—64no—
186C2×C3×C4C22×D311yescharacteristic
187C2×C3×C4C22×D311yescharacteristic
188C2×C3×C4C22×D311yescharacteristic
189C2×C3×C4C22×D311yescharacteristic
190#67—24no—
191#67—34no—
192#67—34no—
193#67—64no—
194#67—64no—
195C2×Dic3—24no—
196C2×Dic3—31no—
197C2×Dic3—31no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3—61no—
205C2×Dic3—64no—
206C2×Dic3C22×D312yes—
207C2×Dic3C22×D312yes—
208C2×Dic3C22×D312yes—
209D12—22no—
210D12—64no—
211C4×D3—64no—
212C4×D3—64no—
213C4×D3—64no—
214C4×D3—62no—
215C4×D3—62no—
216C4×D3—122no—
217Dic6—64no—
218Dic6—64no—
219C2×C32—22no—
220C2×C32C22×D411yescharacteristic
221C2×C32C2×#4111yescharacteristic
222C2×C322−1+411yescharacteristic
223C3⋊D3—22no—
224C3×D3—42no—
225#41—188no—
226C2×Q8—92no—
227C2×D4—32no—
228C2×D4—31no—
229C2×D4—92no—
230C2×D4—91no—
231C22×C4—32no—
232C22×C4—92no—
233C22×C4—92no—
234C22×C4—91no—
235C22×C4D3211yescharacteristic
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—32no—
239C4⋊C4—31no—
240C4⋊C4—92no—
241C4⋊C4—92no—
242C4⋊C4—91no—
243C22⋊C4—32no—
244C22⋊C4—92no—
245C22⋊C4—92no—
246C22⋊C4—91no—
247C22⋊C4—91no—
248C42—91no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—22no—
255C22×C3—41no—
256C22×C3—41no—
257C22×C3—62no—
258C22×C3—62no—
259C22×C3—62no—
260C22×C3D3×D412yes—
261C22×C3D3×D412yes—
262C22×C3C23×D311yescharacteristic
263C22×C3C23×D311yescharacteristic
264C2×D3—22no—
265C2×D3—22no—
266C2×D3—22no—
267C2×D3—32no—
268C2×D3—32no—
269C2×D3—62no—
270C2×D3—61no—
271C2×D3—61no—
272C2×D3—61no—
273C2×D3—61no—
274C2×D3—121no—
275C2×D3—121no—
276C3×C4—21no—
277C3×C4—21no—
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—62no—
281C3×C4—62no—
282C3×C4—62no—
283C3×C4#23512yescomplemented
284C3×C4#23712yescomplemented
285Dic3—22no—
286Dic3—22no—
287Dic3—22no—
288Dic3—32no—
289Dic3—32no—
290Dic3—62no—
291Dic3—61no—
292Dic3—61no—
293Dic3—61no—
294Dic3—61no—
295Dic3—121no—
296Dic3—121no—
297C32#54711yescharacteristic, complemented, Sylow
298C23—32no—
299C23—91no—
300C23C2×D3211yescharacteristic
301Q8—184no—
302D4—64no—
303D4—62no—
304D4—94no—
305D4—184no—
306C2×C4—22no—
307C2×C4—32no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—64no—
311C2×C4—91no—
312C2×C4—91no—
313C2×C4—91no—
314C2×C4—184no—
315C2×C4—184no—
316C2×C4—182no—
317C2×C4C2×D3211yescharacteristic
318C2×C4C2×D3211yescharacteristic
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—21no—
324C2×C3—42no—
325C2×C3—122no—
326C2×C3C2×#23511yescharacteristic
327C2×C3C2×D3×D411yescharacteristic
328C2×C3C2×D3×D411yescharacteristic
329C2×C3C2×#23711yescharacteristic
330C2×C3#100711yescharacteristic
331C2×C3#101011yescharacteristic
332D3—42no—
333D3—62no—
334D3—62no—
335D3—122no—
336C22—21no—
337C22—21no—
338C22—62no—
339C22—62no—
340C22—62no—
341C22—92no—
342C22—181no—
343C22—181no—
344C22#380212yes—
345C22C22×D3211yescharacteristic
346C4—21no—
347C4—62no—
348C4—62no—
349C4—62no—
350C4—92no—
351C4—181no—
352C4—181no—
353C4#380412yescomplemented
354C3—21no—
355C3#550711yescharacteristic, complemented
356C3#561111yescharacteristic, complemented
357C2—22no—
358C2—122no—
359C2—182no—
360C2#6429711yescharacteristic
361C2C2×#380211yescharacteristic
362C2C2×#380411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC22×C32×C4C22
hypercenterC22×C4D32
hypocenterC32#547
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#3804C2semidirect
2. C4×#638C2semidirect
3. C3×#930C2semidirect
4. C2×C3×C4⋊Dic3C2semidirect
5. #63961C2semidirect
6. #63919C2semidirect
7. #63916C2semidirect
8. #63900C2semidirect
9. #63899C2semidirect
10. #63898C2semidirect
11. #63854C2semidirect
45. #930D3semidirect
50. C2×C4⋊Dic3D3semidirect
67. #638D4semidirect
149. C32×C4#41semidirect
283. C3×C4#235semidirect
284. C3×C4#237semidirect
297. C32#547semidirect
353. C4#3804semidirect
355. C3#5507semidirect
356. C3#5611semidirect

Maximal subgroups

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

subgroupindex
C2×#38042
C4×#6382
C3×#9302
C2×C3×C4⋊Dic32
#639612
#639192
#639162
#639002
#638992
#638982
#638542
#56113
#55073