Finite Group Databasebeta

← #10595403

Subgroups — #10595403

Order 576.

597 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Q8C211yesmaximal, characteristic, complemented
2C2×#3797C211yesmaximal, characteristic, complemented
3#64155C211yesmaximal, characteristic, complemented
4C3×#944C211yesmaximal, characteristic, complemented
5#63904C212yesmaximal
6C4×#626C211yesmaximal, characteristic
7#63888C212yesmaximal, complemented
8#63856C211yesmaximal, characteristic, complemented
9#63851C212yesmaximal, complemented
10Dic3×Dic6C211yesmaximal, characteristic, complemented
11#63839C212yesmaximal, complemented
12Q8×#67—31nomaximal
13#5727—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C3×D3×Q8—24no—
17C2×C3×C4×D3C2212yes—
18C2×C3×C4×D3C2211yescharacteristic
19C2×C3×Dic6C2211yescharacteristic
20C2×C3×Dic6C2212yes—
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2212yes—
23C2×#626C2211yescharacteristic
24C2×#624C2212yes—
25#3797—24no—
26#3752C2212yes—
27C4⋊(C3⋊Dic3)C2211yescharacteristic
28#3750C2212yes—
29C3×Dic3⋊C4C2212yes—
30C3×Dic3⋊C4C2212yes—
31C3×C4×Dic3C2211yescharacteristic
32#3724C2212yes—
33Dic3⋊Dic3C2211yescharacteristic
34Dic3⋊Dic3C2212yes—
35#3722C2211yescharacteristic
36(C2×D3)⋊Dic3C2211yescharacteristic
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×D3×Q8—31no—
41C2×D3×Q8—31no—
42C2×D3×Q8D311yescharacteristic, complemented
43C3×C4⋊Q8—31no—
44#946—31no—
45#946—31no—
46#946—31no—
47#946—32no—
48#946—61no—
49Q8×Dic3—31no—
50#944—32no—
51#944D311yescharacteristic, complemented
52C4×#67—32no—
53C4×#67—31no—
54#924—31no—
55#924—32no—
56#896—32no—
57#896—32no—
58#893—32no—
59#888—32no—
60C4×D12—31no—
61C4×Dic6—31no—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C32×Q8—22no—
65C2×C32×C4C2312yes—
66C2×C32×C4C2311yescharacteristic
67C2×C3⋊Dic3C2312yes—
68C2×C3⋊Dic3C2311yescharacteristic
69C4×C3⋊D3—22no—
70C2×C3×Dic3C2311yescharacteristic
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2311yescharacteristic
73C2×C3×Dic3C2312yes—
74C2×C3×Dic3C2311yescharacteristic
75C3×C4×D3—22no—
76C3×C4×D3—24no—
77C3×Dic6—22no—
78C3×Dic6—24no—
79C3×Dic6D414yescomplemented
80#627—24no—
81#626Q814yescomplemented
82#624—24no—
83D4×Q8—91noSylow
84C2×C3×Q8—21no—
85C2×C3×Q8—31no—
86C2×C3×Q8—32no—
87C2×C3×Q8—31no—
88C2×C3×Q8—64no—
89C2×C3×Q8C2×D311yescharacteristic
90C2×C3×Q8C2×D311yescharacteristic
91C22×C3×C4—32no—
92C22×C3×C4—31no—
93C2×#67—31no—
94D3×Q8—24no—
95D3×Q8—64no—
96D3×Q8—64no—
97C2×D12—31no—
98C2×C4×D3—31no—
99C2×C4×D3—31no—
100C2×C4×D3—32no—
101C2×C4×D3—32no—
102C2×C4×D3—61no—
103C2×C4×D3C2×D311yescharacteristic
104C2×C4×D3C2×D312yes—
105C2×Dic6—32no—
106C2×Dic6—32no—
107C2×Dic6C2×D312yes—
108C2×Dic6C2×D311yescharacteristic
109C2×Dic6C2×D311yescharacteristic
110C3×C4⋊C4—32no—
111C3×C4⋊C4—32no—
112C3×C42—31no—
113#217—32no—
114#217—31no—
115(C2×D3)⋊C4—31no—
116(C2×D3)⋊C4—32no—
117(C2×D3)⋊C4—32no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—31no—
120(C2×D3)⋊C4—31no—
121(C2×D3)⋊C4—62no—
122C4⋊Dic3—31no—
123C4⋊Dic3—31no—
124C4⋊Dic3—31no—
125C4⋊Dic3—32no—
126C4⋊Dic3—61no—
127Dic3⋊C4—32no—
128Dic3⋊C4—32no—
129Dic3⋊C4—32no—
130Dic3⋊C4—32no—
131Dic3⋊C4—31no—
132Dic3⋊C4—32no—
133Dic3⋊C4—62no—
134Dic3⋊C4C2×D312yes—
135Dic3⋊C4C2×D312yes—
136C4×Dic3—32no—
137C4×Dic3—32no—
138C4×Dic3C2×D311yescharacteristic
139C22×C32C2411yescharacteristic
140C2×C3⋊D3—21no—
141C2×C3⋊D3—21no—
142C2×C3⋊D3C2×Q812yescomplemented
143C2×C3×D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3C2×Q812yescomplemented
146C32×C4—22no—
147C32×C4C2×D412yes—
148C3⋊Dic3—21no—
149C3⋊Dic3—22no—
150C3×Dic3—22no—
151C3×Dic3—21no—
152C3×Dic3—21no—
153C3×Dic3C2×D414yes—
154C3×Dic3C2×Q812yescomplemented
155C22×Q8—31no—
156C22×Q8—91no—
157C4⋊Q8—31no—
158C4⋊Q8—92no—
159C22⋊Q8—92no—
160C22⋊Q8—92no—
161C22⋊Q8—91no—
162C22⋊Q8—91no—
163C4×Q8—91no—
164C4×D4—91no—
165C4×D4—92no—
166C23×C3—31no—
167C22×D3—31no—
168C22×D3—31no—
169C22×D3—61no—
170C22×D3C22×D311yescharacteristic
171C3×Q8—22no—
172C3×Q8—22no—
173C3×Q8—34no—
174C3×Q8—42no—
175C3×Q8—64no—
176C3×Q8—62no—
177C2×C3×C4—21no—
178C2×C3×C4—22no—
179C2×C3×C4—31no—
180C2×C3×C4—32no—
181C2×C3×C4—31no—
182C2×C3×C4—31no—
183C2×C3×C4—32no—
184C2×C3×C4—64no—
185C2×C3×C4—62no—
186C2×C3×C4C22×D312yes—
187C2×C3×C4C22×D312yes—
188C2×C3×C4C22×D311yescharacteristic
189C2×C3×C4C22×D311yescharacteristic
190#67—34no—
191C2×Dic3—32no—
192C2×Dic3—32no—
193C2×Dic3—31no—
194C2×Dic3—31no—
195C2×Dic3—61no—
196C2×Dic3—62no—
197C2×Dic3C22×D311yescharacteristic
198C2×Dic3C22×D311yescharacteristic
199C2×Dic3C22×D312yes—
200C2×Dic3C22×D311yescharacteristic
201C2×Dic3C22×D312yes—
202D12—34no—
203C4×D3—22no—
204C4×D3—24no—
205C4×D3—64no—
206C4×D3—62no—
207C4×D3—62no—
208C4×D3—64no—
209C4×D3—122no—
210Dic6—22no—
211Dic6—24no—
212Dic6—64no—
213Dic6—64no—
214Dic6#6714yescomplemented
215C2×C32C22×D411yescharacteristic
216C2×C32C22×Q811yescharacteristic
217C2×C322−1+411yescharacteristic
218C3⋊D3—22no—
219C3×D3—22no—
220C2×Q8—31no—
221C2×Q8—31no—
222C2×Q8—32no—
223C2×Q8—64no—
224C2×Q8—92no—
225C2×Q8—184no—
226C2×Q8D3211yescharacteristic
227C2×D4—91no—
228C22×C4—32no—
229C22×C4—31no—
230C22×C4—92no—
231C22×C4—91no—
232C4⋊C4—32no—
233C4⋊C4—32no—
234C4⋊C4—91no—
235C4⋊C4—92no—
236C4⋊C4—92no—
237C4⋊C4—91no—
238C4⋊C4—92no—
239C22⋊C4—91no—
240C22⋊C4—91no—
241C22⋊C4—92no—
242C22⋊C4—92no—
243C42—31no—
244C42—92no—
245C22×C3—21no—
246C22×C3—32no—
247C22×C3—61no—
248C22×C3—61no—
249C22×C3C23×D311yescharacteristic
250C22×C3C23×D311yescharacteristic
251C2×D3—21no—
252C2×D3—21no—
253C2×D3—32no—
254C2×D3—32no—
255C2×D3—62no—
256C2×D3—61no—
257C2×D3—61no—
258C2×D3—61no—
259C2×D3—61no—
260C2×D3—121no—
261C2×D3—121no—
262C2×D3D3×Q812yescomplemented
263C3×C4—22no—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—34no—
267C3×C4—32no—
268C3×C4—42no—
269C3×C4—61no—
270C3×C4—61no—
271C3×C4—62no—
272C3×C4D3×D412yes—
273C3×C4C2×#6712yes—
274Dic3—22no—
275Dic3—21no—
276Dic3—21no—
277Dic3—61no—
278Dic3—62no—
279Dic3—62no—
280Dic3—61no—
281Dic3—121no—
282Dic3—122no—
283Dic3D3×Q812yescomplemented
284Dic3C2×#6714yes—
285C32D4×Q811yescharacteristic, complemented, Sylow
286C23—31no—
287C23—91no—
288Q8—22no—
289Q8—34no—
290Q8—64no—
291Q8—62no—
292Q8—184no—
293D4—94no—
294C2×C4—31no—
295C2×C4—32no—
296C2×C4—31no—
297C2×C4—32no—
298C2×C4—31no—
299C2×C4—64no—
300C2×C4—62no—
301C2×C4—92no—
302C2×C4—91no—
303C2×C4—184no—
304C2×C4—182no—
305C2×C4C2×D3212yes—
306C2×C4C2×D3211yescharacteristic
307C2×C3—21no—
308C2×C3—21no—
309C2×C3—21no—
310C2×C3—62no—
311C2×C3C2×D3×D411yescharacteristic
312C2×C3C2×D3×Q811yescharacteristic
313C2×C3C2×D3×Q811yescharacteristic
314C2×C3#100711yescharacteristic
315C2×C3#100711yescharacteristic
316C2×C3C22×#6711yescharacteristic
317D3—22no—
318D3—62no—
319D3—62no—
320D3—122no—
321C22—32no—
322C22—61no—
323C22—61no—
324C22—92no—
325C22—181no—
326C22—181no—
327C22C22×D3211yescharacteristic
328C4—22no—
329C4—32no—
330C4—34no—
331C4—61no—
332C4—61no—
333C4—62no—
334C4—182no—
335C4—181no—
336C4D3×#6712yes—
337C3—21no—
338C3#572711yescharacteristic, complemented
339C3Q8×#6711yescharacteristic, complemented
340C2—62no—
341C2—182no—
342C2#6431611yescharacteristic
343C2D32×Q811yescharacteristic
344C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC2×#626C22
hypercenterC2×Q8D32
hypocenterC32D4×Q8
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×#3797C2semidirect
3. #64155C2semidirect
4. C3×#944C2semidirect
7. #63888C2semidirect
8. #63856C2semidirect
9. #63851C2semidirect
10. Dic3×Dic6C2semidirect
11. #63839C2semidirect
42. C2×D3×Q8D3semidirect
51. #944D3semidirect
79. C3×Dic6D4semidirect
81. #626Q8semidirect
142. C2×C3⋊D3C2×Q8semidirect
145. C2×C3×D3C2×Q8semidirect
154. C3×Dic3C2×Q8semidirect
214. Dic6#67semidirect
262. C2×D3D3×Q8semidirect
283. Dic3D3×Q8semidirect
285. C32D4×Q8semidirect
338. C3#5727semidirect
339. C3Q8×#67semidirect

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×#37972
#641552
C3×#9442
#639042
C4×#6262
#638882
#638562
#638512
Dic3×Dic62
#638392
Q8×#673
#57273