Finite Group Databasebeta

← #86129

Subgroups — #86129

Order 384.

992 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#529C211yesmaximal, characteristic
2#5701C214yesmaximal, complemented
3#5685C212yesmaximal, complemented
4#5671C212yesmaximal, complemented
5#5650C214yesmaximal, complemented
6#5622C212yesmaximal, complemented
7#5600C212yesmaximal
8#5570C211yesmaximal, characteristic, complemented
9#5567C211yesmaximal, characteristic, complemented
10#5542C211yesmaximal, characteristic, complemented
11#5541C212yesmaximal, complemented
12D4×Dic6C211yesmaximal, characteristic, complemented
13D4×Dic6C212yesmaximal, complemented
14#5518C212yesmaximal, complemented
15#5516C212yesmaximal, complemented
16#5470C211yesmaximal, characteristic, complemented
17C2×C4×Dic6C211yesmaximal, characteristic, complemented
18#3523—31nomaximal, Sylow
19C22×Dic6C2211yescharacteristic
20C22×Dic6C2212yes—
21C3×#126C2212yes—
22C3×#125C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×#123C2212yes—
25C3×C22⋊Q8C2211yescharacteristic
26C3×#121C2211yescharacteristic
27C3×C4×Q8C2211yescharacteristic
28C3×C4×D4C2212yes—
29C3×C4×D4C2211yescharacteristic
30C3×#117C2212yes—
31C2×C3×C42C2211yescharacteristic
32Q8×Dic3C2212yes—
33#944C2212yes—
34#936C2212yes—
35#935C2214yes—
36D4×Dic3C2214yes—
37D4×Dic3C2212yes—
38#926C2212yes—
39#926C2212yes—
40C2×C4⋊Dic3C2211yescharacteristic
41#924C2212yes—
42#924C2212yes—
43#924C2212yes—
44#924C2212yes—
45C2×Dic3⋊C4C2212yes—
46C2×C4×Dic3C2212yes—
47#890C2214yes—
48#890C2212yes—
49#889C2214yes—
50#889C2214yes—
51#889C2214yes—
52#889C2214yes—
53#888C2214yes—
54#888C2212yes—
55#887C2214yes—
56#887C2212yes—
57#887C2212yes—
58#887C2214yes—
59#879C2214yes—
60#879C2214yes—
61#879C2214yes—
62#879C2214yes—
63#878C2214yes—
64#878C2214yes—
65#878C2214yes—
66#877C2214yes—
67#877C2214yes—
68#877C2214yes—
69#870C2212yes—
70#870C2212yes—
71#869C2211yescharacteristic
72#869C2211yescharacteristic
73#869C2212yes—
74C4×Dic6C2212yes—
75C4×Dic6C2212yes—
76C4×Dic6C2212yes—
77C4×Dic6C2211yescharacteristic
78C4×Dic6C2211yescharacteristic
79C4×Dic6C2211yescharacteristic
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2212yes—
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84#557—32no—
85#557—31no—
86#557—31no—
87#556—32no—
88#556—31no—
89#552—34no—
90#552—32no—
91D4×Q8—32no—
92D4×Q8—31no—
93#541—34no—
94#541—32no—
95#533—32no—
96#533—31no—
97#529—32no—
98#529D311yescharacteristic
99#519—32no—
100C2×C4×Q8—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2312yes—
103C2×C3×D4C2311yescharacteristic
104C22×C3×C4C2311yescharacteristic
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2312yes—
108C22×Dic3C2314yes—
109C22×Dic3C2312yes—
110C22×Dic3C2312yes—
111C2×Dic6—24no—
112C2×Dic6—28no—
113C2×Dic6C2311yescharacteristic
114C2×Dic6C2312yes—
115C2×Dic6C2312yes—
116C2×Dic6C2312yes—
117C2×Dic6C2314yes—
118C2×Dic6C2311yescharacteristic
119C3×C4⋊C4C2311yescharacteristic
120C3×C4⋊C4C2312yes—
121C3×C4⋊C4C2312yes—
122C3×C4⋊C4C2311yescharacteristic
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2312yes—
125C3×C22⋊C4C2312yes—
126C3×C22⋊C4C2312yes—
127C3×C22⋊C4C2312yes—
128C3×C22⋊C4C2312yes—
129C3×C22⋊C4C2312yes—
130C3×C42C2311yescharacteristic
131C3×C42C2311yescharacteristic
132C3×C42C2311yescharacteristic
133C3×C42C2312yes—
134C3×C42C2311yescharacteristic
135#217C2314yes—
136#217C2314yes—
137#217C2314yes—
138#217C2314yes—
139C4⋊Dic3C2311yescharacteristic
140C4⋊Dic3C2312yes—
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2314yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2314yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2312yes—
161C4×Dic3C2314yes—
162C4×Dic3C2312yes—
163C4×Dic3C2312yes—
164C4×Dic3C2312yes—
165C4×Dic3C2314yes—
166C4×Dic3C2314yes—
167C4×Dic3C2314yes—
168C22×Q8—31no—
169C22×Q8—32no—
170C4⋊Q8—32no—
171C4⋊Q8—32no—
172C4⋊Q8—34no—
173C4⋊Q8—31no—
174C4⋊Q8—31no—
175C4⋊Q8—32no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#126C2×D312yes—
181#125—32no—
182#125—34no—
183#125—34no—
184#125—34no—
185#125—34no—
186#125—34no—
187#125—32no—
188#125—32no—
189#125C2×D311yescharacteristic
190#124—32no—
191#124C2×D311yescharacteristic
192#123—34no—
193#123C2×D312yes—
194C22⋊Q8—32no—
195C22⋊Q8—32no—
196C22⋊Q8—32no—
197C22⋊Q8—32no—
198C22⋊Q8—34no—
199C22⋊Q8—34no—
200C22⋊Q8—34no—
201C22⋊Q8C2×D311yescharacteristic
202#121C2×D311yescharacteristic
203C4×Q8—32no—
204C4×Q8—34no—
205C4×Q8—34no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8—31no—
212C4×Q8—32no—
213C4×Q8—31no—
214C4×Q8—32no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×Q8C2×D311yescharacteristic
219C4×D4—34no—
220C4×D4—32no—
221C4×D4C2×D312yes—
222C4×D4C2×D311yescharacteristic
223#117—32no—
224#117—32no—
225#117—34no—
226#117—34no—
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—31no—
231C2×C42—32no—
232C2×C42C2×D311yescharacteristic
233C23×C3C2412yes—
234C23×C3C2411yescharacteristic
235C3×Q8—22no—
236C3×D4—24no—
237C3×D4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4C2411yescharacteristic
243C2×C3×C4C2411yescharacteristic
244C2×C3×C4C2412yes—
245C2×C3×C4C2412yes—
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×Dic3—24no—
252C2×Dic3—24no—
253C2×Dic3—28no—
254C2×Dic3C2412yes—
255C2×Dic3C2414yes—
256C2×Dic3C2412yes—
257C2×Dic3C2414yes—
258C2×Dic3C2412yes—
259C2×Dic3C2412yes—
260Dic6—22no—
261Dic6—24no—
262Dic6—24no—
263Dic6—24no—
264Dic6—28no—
265Dic6#4114yes—
266C2×Q8—31no—
267C2×Q8—32no—
268C2×Q8—34no—
269C2×Q8—32no—
270C2×Q8—31no—
271C2×Q8—32no—
272C2×Q8—64no—
273C2×Q8—68no—
274C2×Q8C22×D311yescharacteristic
275C2×D4C22×D312yes—
276C2×D4C22×D311yescharacteristic
277C22×C4—32no—
278C22×C4—32no—
279C22×C4—34no—
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D312yes—
284C4⋊C4—32no—
285C4⋊C4—31no—
286C4⋊C4—32no—
287C4⋊C4—31no—
288C4⋊C4—32no—
289C4⋊C4—32no—
290C4⋊C4—32no—
291C4⋊C4—34no—
292C4⋊C4—32no—
293C4⋊C4—34no—
294C4⋊C4—34no—
295C4⋊C4—34no—
296C4⋊C4—32no—
297C4⋊C4—34no—
298C4⋊C4—32no—
299C4⋊C4—32no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—34no—
304C4⋊C4—34no—
305C4⋊C4C22×D312yes—
306C4⋊C4C22×D312yes—
307C4⋊C4C22×D311yescharacteristic
308C4⋊C4C22×D312yes—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D312yes—
311C22⋊C4—34no—
312C22⋊C4—34no—
313C22⋊C4—34no—
314C22⋊C4—34no—
315C22⋊C4C22×D312yes—
316C22⋊C4C22×D312yes—
317C22⋊C4C22×D312yes—
318C22⋊C4C22×D312yes—
319C22⋊C4C22×D312yes—
320C42—32no—
321C42—32no—
322C42—32no—
323C42—32no—
324C42—34no—
325C42—34no—
326C42—34no—
327C42—34no—
328C42C22×D311yescharacteristic
329C42C22×D311yescharacteristic
330C42C22×D311yescharacteristic
331C42C22×D311yescharacteristic
332C42C22×D312yes—
333C22×C3—21no—
334C22×C3—21no—
335C22×C3—22no—
336C22×C3—22no—
337C22×C3—22no—
338C22×C32−1+412yes—
339C22×C3C2511yescharacteristic
340C3×C4—22no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—22no—
346C3×C4—22no—
347C3×C4C2×#4112yes—
348C3×C42−1+412yes—
349Dic3—22no—
350Dic3—24no—
351Dic3—24no—
352Dic3—22no—
353Dic3—22no—
354Dic3C2×#4114yes—
355C23C23×D312yes—
356C23C23×D311yescharacteristic
357Q8—22no—
358Q8—34no—
359Q8—68no—
360Q8—64no—
361Q8—62no—
362Q8—64no—
363Q8—64no—
364D4—24no—
365D4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—34no—
373C2×C4—32no—
374C2×C4—32no—
375C2×C4—34no—
376C2×C4—64no—
377C2×C4—68no—
378C2×C4—64no—
379C2×C4C23×D311yescharacteristic
380C2×C4C23×D311yescharacteristic
381C2×C4C23×D312yes—
382C2×C4C23×D312yes—
383C2×C4C23×D312yes—
384C2×C4C23×D311yescharacteristic
385C2×C4C23×D311yescharacteristic
386C2×C4C23×D311yescharacteristic
387C2×C4C23×D311yescharacteristic
388C2×C3—22no—
389C2×C3—42no—
390C2×C3C22×#4111yescharacteristic
391C2×C3C2×2−1+411yescharacteristic
392C2×C3C2×2−1+411yescharacteristic
393C22—21no—
394C22—21no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22#101012yes—
399C22C24×D311yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—22no—
406C4—22no—
407C4—34no—
408C4—62no—
409C4—64no—
410C4—62no—
411C4—62no—
412C4—64no—
413C4D3×#4112yes—
414C4#101012yes—
415C3#352311yescharacteristic, complemented, Sylow
416C2—22no—
417C2—42no—
418C2C2×D3×#4111yescharacteristic
419C2C2×#101011yescharacteristic
420C2C2×#101011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
prime radicalC3×#121C22
hypercenter#529D3
hypocenterC3#3523
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
2. #5701C2semidirect
3. #5685C2semidirect
4. #5671C2semidirect
5. #5650C2semidirect
6. #5622C2semidirect
8. #5570C2semidirect
9. #5567C2semidirect
10. #5542C2semidirect
11. #5541C2semidirect
12. D4×Dic6C2semidirect
13. D4×Dic6C2semidirect
14. #5518C2semidirect
15. #5516C2semidirect
16. #5470C2semidirect
17. C2×C4×Dic6C2semidirect
415. C3#3523semidirect

Maximal subgroups

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

subgroupindex
C3×#5292
#57012
#56852
#56712
#56502
#56222
#56002
#55702
#55672
#55422
#55412
D4×Dic62
#55182
#55162
#54702
C2×C4×Dic62
#35233