Finite Group Databasebeta

← #10595402

Subgroups — #10595402

Order 576.

618 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×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#3799C211yesmaximal, characteristic, complemented
3#64156C211yesmaximal, characteristic, complemented
4C3×#946C211yesmaximal, characteristic, complemented
5#63909C211yesmaximal, characteristic, complemented
6#63907C212yesmaximal, complemented
7#63906C212yesmaximal, complemented
8#63904C212yesmaximal, complemented
9C4×#625C211yesmaximal, characteristic, complemented
10#63888C212yesmaximal, complemented
11#63843C211yesmaximal, characteristic, complemented
12#5817—31nomaximal
13#5633—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×C4⋊(C3⋊D3)C2211yescharacteristic
16C3×D3×Q8—24no—
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×C4×D3C2212yes—
20C2×C3×Dic6C2212yes—
21C2×C3×Dic6C2211yescharacteristic
22C2×#626C2212yes—
23C2×#626C2212yes—
24C2×#625C2211yescharacteristic
25C2×#624C2212yes—
26#3799—24no—
27#3752C2212yes—
28#3752C2212yes—
29C4×C3⋊Dic3C2211yescharacteristic
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×Dic3⋊C4C2212yes—
33#3724C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36(C2×D3)⋊Dic3C2211yescharacteristic
37(C2×D3)⋊Dic3C2211yescharacteristic
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×#239—31no—
41C2×D3×Q8D311yescharacteristic, complemented
42C2×#235—31no—
43C3×C22⋊Q8—31no—
44#947—31no—
45#947—31no—
46#947—32no—
47#947—61no—
48#946—32no—
49#946D311yescharacteristic, complemented
50Q8×Dic3—31no—
51#930—32no—
52#930—31no—
53C4×#67—31no—
54C4×#67—31no—
55C4×#67—32no—
56#896—32no—
57#895—31no—
58#893—32no—
59#887—31no—
60#885—32no—
61#884—32no—
62C22×C3⋊D3C2312yes—
63C22×C3×D3C2311yescharacteristic
64C22×C3×D3C2311yescharacteristic
65C32×Q8—22no—
66C2×C32×C4C2312yes—
67C2×C32×C4C2311yescharacteristic
68C2×C3⋊Dic3C2311yescharacteristic
69C2×C3⋊Dic3C2311yescharacteristic
70C4⋊(C3⋊D3)—22no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2312yes—
75C3×C4×D3—22no—
76C3×C4×D3—22no—
77C3×C4×D3—24no—
78C3×Dic6—24no—
79C3×Dic6D414yescomplemented
80#626—24no—
81#626—24no—
82#625—22no—
83#624—24no—
84#548—91noSylow
85C2×C3×Q8—21no—
86C2×C3×Q8—31no—
87C2×C3×Q8—32no—
88C2×C3×Q8—64no—
89C2×C3×Q8C2×D311yescharacteristic
90C2×C3×Q8C2×D311yescharacteristic
91C22×C3×C4—32no—
92C22×C3×C4—31no—
93C22×C3×C4—31no—
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—32no—
97C2×#67—32no—
98#239—64no—
99D3×Q8—24no—
100#235—64no—
101C2×D12—32no—
102C2×D12—32no—
103C2×D12—31no—
104C2×D12—31no—
105C2×D12—61no—
106C2×C4×D3—32no—
107C2×C4×D3—32no—
108C2×C4×D3C2×D312yes—
109C2×C4×D3C2×D311yescharacteristic
110C2×C4×D3C2×D311yescharacteristic
111C2×Dic6C2×D312yes—
112C2×Dic6C2×D311yescharacteristic
113C3×C4⋊C4—31no—
114C3×C4⋊C4—32no—
115C3×C22⋊C4—32no—
116#217—31no—
117#217—31no—
118(C2×D3)⋊C4—31no—
119(C2×D3)⋊C4—31no—
120(C2×D3)⋊C4—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—32no—
125(C2×D3)⋊C4—32no—
126(C2×D3)⋊C4—62no—
127(C2×D3)⋊C4—62no—
128(C2×D3)⋊C4C2×D312yes—
129C4⋊Dic3—32no—
130C4⋊Dic3C2×D311yescharacteristic
131Dic3⋊C4—32no—
132Dic3⋊C4—31no—
133Dic3⋊C4—31no—
134Dic3⋊C4C2×D312yes—
135C4×Dic3—31no—
136C4×Dic3—31no—
137C4×Dic3—32no—
138C4×Dic3—32no—
139C4×Dic3—61no—
140C22×C32C2411yescharacteristic
141C2×C3⋊D3—22no—
142C2×C3⋊D3—22no—
143C2×C3⋊D3—22no—
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3#4112yescomplemented
150C32×C4—22no—
151C32×C4C2×D412yes—
152C3⋊Dic3—21no—
153C3⋊Dic3—21no—
154C3×Dic3—21no—
155C3×Dic3—21no—
156C3×Dic3—22no—
157C3×Dic3C2×D414yes—
158C2×#41—91no—
159C22×Q8—31no—
160#124—92no—
161#124—91no—
162C22⋊Q8—31no—
163C22⋊Q8—92no—
164#121—92no—
165#121—91no—
166C4×Q8—91no—
167C4×D4—92no—
168C4×D4—91no—
169C23×C3—31no—
170C23×C3—31no—
171C22×D3—32no—
172C22×D3—32no—
173C22×D3—62no—
174C22×D3C22×D311yescharacteristic
175C22×D3C22×D311yescharacteristic
176C3×Q8—22no—
177C3×Q8—22no—
178C3×Q8—34no—
179C3×Q8—42no—
180C3×Q8—64no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—32no—
184C2×C3×C4—32no—
185C2×C3×C4—31no—
186C2×C3×C4—31no—
187C2×C3×C4—64no—
188C2×C3×C4—62no—
189C2×C3×C4—62no—
190C2×C3×C4C22×D312yes—
191C2×C3×C4C22×D312yes—
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194#67—64no—
195#67—62no—
196#67—62no—
197#67—64no—
198C2×Dic3—31no—
199C2×Dic3—31no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—61no—
204C2×Dic3C22×D312yes—
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D311yescharacteristic
207C2×Dic3C22×D312yes—
208D12—64no—
209D12—64no—
210D12—62no—
211D12—62no—
212D12—122no—
213C4×D3—22no—
214C4×D3—22no—
215C4×D3—24no—
216C4×D3—64no—
217C4×D3—64no—
218Dic6—24no—
219Dic6#6714yescomplemented
220C2×C32C22×D411yescharacteristic
221C2×C32C2×#4111yescharacteristic
222C2×C322−1+411yescharacteristic
223C3⋊D3—42no—
224C3×D3—22no—
225C3×D3—41no—
226#41—184no—
227C2×Q8—31no—
228C2×Q8—32no—
229C2×Q8—64no—
230C2×Q8D3211yescharacteristic
231C2×D4—92no—
232C2×D4—92no—
233C2×D4—91no—
234C2×D4—91no—
235C22×C4—32no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—92no—
239C4⋊C4—32no—
240C4⋊C4—31no—
241C4⋊C4—91no—
242C4⋊C4—92no—
243C22⋊C4—32no—
244C22⋊C4—92no—
245C22⋊C4—91no—
246C22⋊C4—91no—
247C22⋊C4—92no—
248C22⋊C4—92no—
249C42—92no—
250C42—91no—
251C22×C3—21no—
252C22×C3—32no—
253C22×C3—61no—
254C22×C3—61no—
255C22×C3—61no—
256C22×C3—61no—
257C22×C3—61no—
258C22×C3C23×D311yescharacteristic
259C22×C3C23×D311yescharacteristic
260C2×D3—21no—
261C2×D3—21no—
262C2×D3—21no—
263C2×D3—21no—
264C2×D3—21no—
265C2×D3—62no—
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—62no—
271C2×D3—122no—
272C2×D3—122no—
273C2×D3—122no—
274C2×D3#23912yescomplemented
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—34no—
279C3×C4—42no—
280C3×C4—61no—
281C3×C4—62no—
282C3×C4—61no—
283C3×C4D3×D412yes—
284C3×C4C2×#6712yes—
285Dic3—21no—
286Dic3—22no—
287Dic3—21no—
288Dic3—61no—
289Dic3—61no—
290Dic3—61no—
291Dic3—61no—
292Dic3—121no—
293Dic3—121no—
294Dic3C2×#6714yes—
295C32#54811yescharacteristic, complemented, Sylow
296C23—31no—
297C23—31no—
298C23—92no—
299Q8—22no—
300Q8—34no—
301Q8—64no—
302D4—184no—
303D4—184no—
304D4—182no—
305D4—182no—
306C2×C4—32no—
307C2×C4—32no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—64no—
311C2×C4—62no—
312C2×C4—62no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—184no—
316C2×C4C2×D3212yes—
317C2×C4C2×D3211yescharacteristic
318C2×C3—21no—
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—62no—
322C2×C3—121no—
323C2×C3C2×D3×D411yescharacteristic
324C2×C3C2×#23911yescharacteristic
325C2×C3#100711yescharacteristic
326C2×C3#100711yescharacteristic
327C2×C3D3×#4111yescharacteristic
328C2×C3C22×#6711yescharacteristic
329D3—22no—
330D3—41no—
331D3—122no—
332D3—122no—
333D3—242no—
334C22—32no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22—182no—
341C22—182no—
342C22—182no—
343C22C22×D3211yescharacteristic
344C4—22no—
345C4—34no—
346C4—62no—
347C4—61no—
348C4—61no—
349C4—181no—
350C4—181no—
351C4D3×#6712yes—
352C3—21no—
353C3#563311yescharacteristic, complemented
354C3#581711yescharacteristic, complemented
355C2—62no—
356C2—121no—
357C2—362no—
358C2#6431611yescharacteristic
359C2D3×#23911yescharacteristic
360C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
hypercenterC2×Q8D32
hypocenterC32#548
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×D3×Q8C2semidirect
2. C2×#3799C2semidirect
3. #64156C2semidirect
4. C3×#946C2semidirect
5. #63909C2semidirect
6. #63907C2semidirect
7. #63906C2semidirect
8. #63904C2semidirect
9. C4×#625C2semidirect
10. #63888C2semidirect
11. #63843C2semidirect
41. C2×D3×Q8D3semidirect
49. #946D3semidirect
79. C3×Dic6D4semidirect
149. C2×C3×D3#41semidirect
219. Dic6#67semidirect
274. C2×D3#239semidirect
295. C32#548semidirect
353. C3#5633semidirect
354. C3#5817semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×Q82
C2×#37992
#641562
C3×#9462
#639092
#639072
#639062
#639042
C4×#6252
#638882
#638432
#58173
#56333