Finite Group Databasebeta

← #10595397

Subgroups — #10595397

Order 576.

577 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×#3798C211yesmaximal, characteristic
3#64155C211yesmaximal, characteristic, complemented
4C3×#946C211yesmaximal, characteristic, complemented
5#63907C212yesmaximal, complemented
6#63905C212yesmaximal, complemented
7#63871C211yesmaximal, characteristic, complemented
8#63856C211yesmaximal, characteristic, complemented
9#63850C212yesmaximal, complemented
10#63841C211yesmaximal, characteristic, complemented
11#63839C212yesmaximal, complemented
12#5816—31nomaximal
13#5628—31nomaximal
14C2×C32×Q8C2211yescharacteristic
15C2×C4×C3⋊D3C2211yescharacteristic
16C3×D3×Q8—24no—
17C2×C3×C4×D3C2211yescharacteristic
18C2×C3×C4×D3C2212yes—
19C2×C3×C4×D3C2211yescharacteristic
20C2×C3×Dic6C2212yes—
21C2×C3×Dic6C2211yescharacteristic
22C2×#627C2211yescharacteristic
23C2×#626C2211yescharacteristic
24C2×#626C2211yescharacteristic
25C2×#625C2211yescharacteristic
26#3798—24no—
27#3752C2212yes—
28C4⋊(C3⋊Dic3)C2211yescharacteristic
29#3750C2212yes—
30C3×(C2×D3)⋊C4C2212yes—
31C3×C4⋊Dic3C2211yescharacteristic
32C3×Dic3⋊C4C2212yes—
33#3724C2212yes—
34Dic3⋊Dic3C2212yes—
35#3722C2212yes—
36(C2×D3)⋊Dic3C2212yes—
37(C2×D3)⋊Dic3C2212yes—
38Dic32C2212yes—
39C22×C3×Q8—31no—
40C2×D3×Q8D311yescharacteristic, complemented
41C2×#235—31no—
42C2×#235—31no—
43C3×C22⋊Q8—31no—
44#947—32no—
45#946—31no—
46#946—31no—
47#946—31no—
48#946—31no—
49#946—61no—
50#946D311yescharacteristic, complemented
51#944—32no—
52#929—32no—
53#929—32no—
54#926—31no—
55#924—31no—
56#897—31no—
57#894—32no—
58#892—31no—
59#888—32no—
60#885—32no—
61#883—32no—
62C22×C3⋊D3C2311yescharacteristic
63C22×C3×D3C2311yescharacteristic
64C22×C3×D3C2311yescharacteristic
65C32×Q8—22no—
66C2×C32×C4C2312yes—
67C2×C32×C4C2311yescharacteristic
68C2×C3⋊Dic3C2312yes—
69C2×C3⋊Dic3C2311yescharacteristic
70C4×C3⋊D3—22no—
71C2×C3×Dic3C2312yes—
72C2×C3×Dic3C2312yes—
73C2×C3×Dic3C2311yescharacteristic
74C2×C3×Dic3C2311yescharacteristic
75C3×C4×D3—24no—
76C3×C4×D3—22no—
77C3×C4×D3D414yes—
78C3×Dic6—22no—
79C3×Dic6—24no—
80#627—22no—
81#626—22no—
82#626—22no—
83#625—22no—
84#536—91noSylow
85C2×C3×Q8—21no—
86C2×C3×Q8—32no—
87C2×C3×Q8—31no—
88C2×C3×Q8—64no—
89C2×C3×Q8C2×D311yescharacteristic
90C2×C3×Q8C2×D311yescharacteristic
91C22×C3×C4—31no—
92C22×C3×C4—31no—
93C22×C3×C4—32no—
94C2×#67—31no—
95C2×#67—31no—
96C2×#67—31no—
97C2×#67—31no—
98D3×Q8—24no—
99#235—64no—
100#235—64no—
101C2×D12—31no—
102C2×D12—31no—
103C2×C4×D3—31no—
104C2×C4×D3—31no—
105C2×C4×D3—61no—
106C2×C4×D3C2×D312yes—
107C2×C4×D3C2×D311yescharacteristic
108C2×C4×D3C2×D311yescharacteristic
109C2×Dic6—31no—
110C2×Dic6—31no—
111C2×Dic6C2×D312yes—
112C2×Dic6C2×D311yescharacteristic
113C3×C4⋊C4—32no—
114C3×C4⋊C4—31no—
115C3×C22⋊C4—32no—
116#217—32no—
117#217—32no—
118(C2×D3)⋊C4—32no—
119(C2×D3)⋊C4—32no—
120(C2×D3)⋊C4—32no—
121(C2×D3)⋊C4—32no—
122(C2×D3)⋊C4—32no—
123(C2×D3)⋊C4—32no—
124(C2×D3)⋊C4—62no—
125(C2×D3)⋊C4C2×D312yes—
126C4⋊Dic3—31no—
127C4⋊Dic3—31no—
128C4⋊Dic3—32no—
129C4⋊Dic3—61no—
130C4⋊Dic3C2×D311yescharacteristic
131Dic3⋊C4—32no—
132Dic3⋊C4—32no—
133Dic3⋊C4—32no—
134Dic3⋊C4—32no—
135Dic3⋊C4—32no—
136Dic3⋊C4—62no—
137Dic3⋊C4C2×D312yes—
138C4×Dic3—32no—
139C4×Dic3—32no—
140C22×C32C2411yescharacteristic
141C2×C3⋊D3—21no—
142C2×C3⋊D3—21no—
143C2×C3⋊D3—21no—
144C2×C3×D3—21no—
145C2×C3×D3—21no—
146C2×C3×D3—21no—
147C2×C3×D3—21no—
148C2×C3×D3—21no—
149C2×C3×D3C2×D412yes—
150C32×C4—22no—
151C32×C4C2×D412yes—
152C3⋊Dic3—21no—
153C3⋊Dic3—22no—
154C3×Dic3—21no—
155C3×Dic3—22no—
156C3×Dic3—22no—
157C3×Dic3C2×D412yes—
158C2×#41—91no—
159C22×Q8—31no—
160C4⋊Q8—92no—
161#124—92no—
162#123—92no—
163#123—92no—
164C22⋊Q8—31no—
165C22⋊Q8—91no—
166C22⋊Q8—91no—
167C22⋊Q8—91no—
168#117—91no—
169C23×C3—31no—
170C23×C3—31no—
171C22×D3—31no—
172C22×D3—31no—
173C22×D3—61no—
174C22×D3C22×D311yescharacteristic
175C22×D3C22×D311yescharacteristic
176C3×Q8—22no—
177C3×Q8—22no—
178C3×Q8—42no—
179C3×Q8—64no—
180C3×Q8—62no—
181C2×C3×C4—21no—
182C2×C3×C4—22no—
183C2×C3×C4—31no—
184C2×C3×C4—34no—
185C2×C3×C4—32no—
186C2×C3×C4—32no—
187C2×C3×C4—31no—
188C2×C3×C4—64no—
189C2×C3×C4—62no—
190C2×C3×C4C22×D312yes—
191C2×C3×C4C22×D312yes—
192C2×C3×C4C22×D311yescharacteristic
193C2×C3×C4C22×D311yescharacteristic
194#67—62no—
195#67—62no—
196#67—62no—
197#67—62no—
198C2×Dic3—32no—
199C2×Dic3—32no—
200C2×Dic3—31no—
201C2×Dic3—31no—
202C2×Dic3—61no—
203C2×Dic3—62no—
204C2×Dic3C22×D312yes—
205C2×Dic3C22×D311yescharacteristic
206C2×Dic3C22×D312yes—
207C2×Dic3C22×D311yescharacteristic
208D12—62no—
209D12—62no—
210C4×D3—24no—
211C4×D3—22no—
212C4×D3—62no—
213C4×D3—62no—
214C4×D3—122no—
215C4×D3#6714yes—
216Dic6—22no—
217Dic6—24no—
218Dic6—62no—
219Dic6—62no—
220C2×C32C22×D411yescharacteristic
221C2×C322−1+411yescharacteristic
222C2×C322−1+411yescharacteristic
223C3⋊D3—41no—
224C3×D3—22no—
225C3×D3—41no—
226#41—184no—
227C2×Q8—31no—
228C2×Q8—32no—
229C2×Q8—64no—
230C2×Q8—91no—
231C2×Q8D3211yescharacteristic
232C2×D4—91no—
233C2×D4—91no—
234C2×D4—91no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—32no—
238C22×C4—91no—
239C4⋊C4—31no—
240C4⋊C4—32no—
241C4⋊C4—92no—
242C4⋊C4—92no—
243C4⋊C4—91no—
244C4⋊C4—92no—
245C22⋊C4—32no—
246C22⋊C4—92no—
247C22⋊C4—92no—
248C22⋊C4—92no—
249C22⋊C4—92no—
250C42—92no—
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—61no—
267C2×D3—61no—
268C2×D3—61no—
269C2×D3—61no—
270C2×D3—61no—
271C2×D3—61no—
272C2×D3—121no—
273C2×D3—121no—
274C2×D3C2×#6712yes—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—22no—
278C3×C4—32no—
279C3×C4—42no—
280C3×C4—62no—
281C3×C4—62no—
282C3×C4—61no—
283C3×C4D3×D412yes—
284C3×C4C2×#6712yes—
285Dic3—21no—
286Dic3—22no—
287Dic3—22no—
288Dic3—61no—
289Dic3—62no—
290Dic3—62no—
291Dic3—61no—
292Dic3—121no—
293Dic3—122no—
294Dic3C2×#6712yes—
295C32#53611yescharacteristic, complemented, Sylow
296C23—31no—
297C23—31no—
298C23—91no—
299Q8—22no—
300Q8—64no—
301Q8—62no—
302Q8—182no—
303D4—182no—
304D4—182no—
305D4—182no—
306C2×C4—31no—
307C2×C4—34no—
308C2×C4—32no—
309C2×C4—32no—
310C2×C4—31no—
311C2×C4—64no—
312C2×C4—62no—
313C2×C4—92no—
314C2×C4—91no—
315C2×C4—182no—
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×C3#100711yescharacteristic
325C2×C3#100711yescharacteristic
326C2×C3#100711yescharacteristic
327C2×C3#101011yescharacteristic
328C2×C3C22×#6711yescharacteristic
329D3—22no—
330D3—41no—
331D3—122no—
332D3—121no—
333D3—121no—
334C22—32no—
335C22—61no—
336C22—61no—
337C22—61no—
338C22—61no—
339C22—61no—
340C22—181no—
341C22—181no—
342C22—181no—
343C22C22×D3211yescharacteristic
344C4—22no—
345C4—32no—
346C4—62no—
347C4—61no—
348C4—62no—
349C4—182no—
350C4—181no—
351C4D3×#6712yes—
352C3—21no—
353C3#562811yescharacteristic, complemented
354C3#581611yescharacteristic, complemented
355C2—62no—
356C2—121no—
357C2—361no—
358C2#6431511yescharacteristic
359C2#6431611yescharacteristic
360C2C2×D3×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D32
commutator, socle, cosocleC22×C32C24
FittingC2×C32×Q8C22
prime radicalC2×#3798C2
hypercenterC2×Q8D32
hypocenterC32#536
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
3. #64155C2semidirect
4. C3×#946C2semidirect
5. #63907C2semidirect
6. #63905C2semidirect
7. #63871C2semidirect
8. #63856C2semidirect
9. #63850C2semidirect
10. #63841C2semidirect
11. #63839C2semidirect
40. C2×D3×Q8D3semidirect
50. #946D3semidirect
295. C32#536semidirect
353. C3#5628semidirect
354. C3#5816semidirect

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×#37982
#641552
C3×#9462
#639072
#639052
#638712
#638562
#638502
#638412
#638392
#58163
#56283