Finite Group Databasebeta

← #86138

Subgroups — #86138

Order 384.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 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#5651C214yesmaximal, complemented
6#5625C212yesmaximal, complemented
7#5603C212yesmaximal
8#5567C211yesmaximal, characteristic, complemented
9Q8×Dic6C211yesmaximal, characteristic, complemented
10#5543C211yesmaximal, characteristic, complemented
11#5542C211yesmaximal, characteristic, complemented
12#5542C212yesmaximal, complemented
13#5541C212yesmaximal, complemented
14#5519C212yesmaximal, complemented
15#5518C212yesmaximal, complemented
16#5470C211yesmaximal, characteristic, complemented
17#5470C211yesmaximal, characteristic, complemented
18#3555—31nomaximal, Sylow
19C3×#126C2212yes—
20C3×#125C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×#123C2212yes—
23C3×C22⋊Q8C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×C4×Q8C2211yescharacteristic
26C3×C4×D4C2212yes—
27C3×C4×D4C2211yescharacteristic
28C3×#117C2212yes—
29C2×C3×C42C2211yescharacteristic
30Q8×Dic3C2212yes—
31#944C2212yes—
32#936C2214yes—
33#936C2212yes—
34#935C2214yes—
35D4×Dic3C2212yes—
36#926C2212yes—
37#926C2212yes—
38#926C2212yes—
39C2×C4⋊Dic3C2212yes—
40#924C2212yes—
41#924C2212yes—
42#924C2211yescharacteristic
43#924C2211yescharacteristic
44#924C2212yes—
45#924C2212yes—
46#924C2212yes—
47#890C2212yes—
48#890C2214yes—
49#889C2214yes—
50#889C2214yes—
51#889C2214yes—
52#889C2214yes—
53#888C2214yes—
54#888C2214yes—
55#888C2212yes—
56#887C2214yes—
57#887C2212yes—
58#887C2212yes—
59#879C2214yes—
60#879C2214yes—
61#879C2214yes—
62#879C2214yes—
63#879C2214yes—
64#879C2214yes—
65#878C2214yes—
66#878C2214yes—
67#878C2214yes—
68#877C2214yes—
69#870C2212yes—
70#870C2212yes—
71#870C2212yes—
72#869C2211yescharacteristic
73#869C2211yescharacteristic
74#869C2211yescharacteristic
75#869C2211yescharacteristic
76#869C2212yes—
77C4×Dic6C2212yes—
78C4×Dic6C2212yes—
79C4×Dic6C2212yes—
80C4×Dic6C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2212yes—
83C4×Dic6C2212yes—
84#563—34no—
85Q82—31no—
86#557—32no—
87#557—31no—
88#556—32no—
89#556—32no—
90#556—31no—
91#554—31no—
92#552—32no—
93#544—32no—
94#542—32no—
95#541—34no—
96#541—32no—
97#541—32no—
98#533—31no—
99#533—31no—
100#529D311yescharacteristic
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2312yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C22×Dic3C2314yes—
109C2×Dic6C2311yescharacteristic
110C2×Dic6C2311yescharacteristic
111C2×Dic6C2312yes—
112C2×Dic6C2312yes—
113C2×Dic6C2312yes—
114C2×Dic6C2314yes—
115C3×C4⋊C4C2311yescharacteristic
116C3×C4⋊C4C2312yes—
117C3×C4⋊C4C2311yescharacteristic
118C3×C4⋊C4C2312yes—
119C3×C4⋊C4C2312yes—
120C3×C4⋊C4C2312yes—
121C3×C22⋊C4C2312yes—
122C3×C22⋊C4C2312yes—
123C3×C22⋊C4C2312yes—
124C3×C22⋊C4C2312yes—
125C3×C22⋊C4C2312yes—
126C3×C42C2311yescharacteristic
127C3×C42C2311yescharacteristic
128C3×C42C2312yes—
129C3×C42C2311yescharacteristic
130C3×C42C2311yescharacteristic
131#217C2314yes—
132#217C2314yes—
133#217C2314yes—
134#217C2312yes—
135#217C2314yes—
136#217C2312yes—
137C4⋊Dic3C2311yescharacteristic
138C4⋊Dic3C2312yes—
139C4⋊Dic3C2311yescharacteristic
140C4⋊Dic3C2312yes—
141C4⋊Dic3C2312yes—
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2312yes—
145C4⋊Dic3C2312yes—
146C4⋊Dic3C2314yes—
147C4⋊Dic3C2314yes—
148Dic3⋊C4C2312yes—
149Dic3⋊C4C2312yes—
150Dic3⋊C4C2314yes—
151Dic3⋊C4C2314yes—
152Dic3⋊C4C2312yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160C4×Dic3C2314yes—
161C4×Dic3C2312yes—
162C4×Dic3C2312yes—
163C4×Dic3C2314yes—
164C4×Dic3C2314yes—
165C4×Dic3C2314yes—
166C4⋊Q8—32no—
167C4⋊Q8—34no—
168C4⋊Q8—32no—
169C4⋊Q8—34no—
170C4⋊Q8—31no—
171C4⋊Q8—31no—
172C4⋊Q8—31no—
173C4⋊Q8—31no—
174C4⋊Q8—32no—
175#126—34no—
176#126—34no—
177#126—34no—
178#126—34no—
179#126—34no—
180#126—34no—
181#126C2×D312yes—
182#125—34no—
183#125—34no—
184#125—34no—
185#125—32no—
186#125—34no—
187#125—34no—
188#125—32no—
189#125—32no—
190#125—32no—
191#125C2×D311yescharacteristic
192#124—32no—
193#124—34no—
194#124C2×D311yescharacteristic
195#123—34no—
196#123C2×D312yes—
197C22⋊Q8—32no—
198C22⋊Q8—31no—
199C22⋊Q8—31no—
200C22⋊Q8—32no—
201C22⋊Q8—32no—
202C22⋊Q8—32no—
203C22⋊Q8—32no—
204C22⋊Q8—34no—
205C22⋊Q8—34no—
206C22⋊Q8—34no—
207C22⋊Q8C2×D311yescharacteristic
208#121C2×D311yescharacteristic
209C4×Q8—32no—
210C4×Q8—34no—
211C4×Q8—32no—
212C4×Q8—32no—
213C4×Q8—32no—
214C4×Q8—31no—
215C4×Q8—32no—
216C4×Q8—32no—
217C4×Q8—31no—
218C4×Q8—32no—
219C4×Q8—32no—
220C4×Q8C2×D311yescharacteristic
221C4×D4—32no—
222C4×D4C2×D312yes—
223C4×D4C2×D311yescharacteristic
224#117—32no—
225#117—32no—
226#117—32no—
227#117—34no—
228#117C2×D312yes—
229C2×C4⋊C4—32no—
230C2×C42C2×D311yescharacteristic
231C23×C3C2411yescharacteristic
232C23×C3C2412yes—
233C3×Q8—22no—
234C3×D4—22no—
235C3×D4—42no—
236C2×C3×C4—24no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—41no—
240C2×C3×C4C2411yescharacteristic
241C2×C3×C4C2412yes—
242C2×C3×C4C2412yes—
243C2×C3×C4C2412yes—
244C2×C3×C4C2411yescharacteristic
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2411yescharacteristic
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249C2×Dic3—44no—
250C2×Dic3C2414yes—
251C2×Dic3C2412yes—
252C2×Dic3C2412yes—
253C2×Dic3C2412yes—
254C2×Dic3C2414yes—
255C2×Dic3C2412yes—
256Dic6—22no—
257Dic6—22no—
258Dic6—24no—
259Dic6—44no—
260Dic6—42no—
261Dic6—42no—
262C2×Q8—31no—
263C2×Q8—31no—
264C2×Q8—32no—
265C2×Q8—32no—
266C2×Q8—32no—
267C2×Q8—34no—
268C2×Q8C22×D311yescharacteristic
269C2×D4C22×D311yescharacteristic
270C2×D4C22×D312yes—
271C22×C4—34no—
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D312yes—
275C22×C4C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—32no—
278C4⋊C4—34no—
279C4⋊C4—32no—
280C4⋊C4—32no—
281C4⋊C4—31no—
282C4⋊C4—32no—
283C4⋊C4—34no—
284C4⋊C4—32no—
285C4⋊C4—32no—
286C4⋊C4—32no—
287C4⋊C4—32no—
288C4⋊C4—34no—
289C4⋊C4—34no—
290C4⋊C4—34no—
291C4⋊C4—34no—
292C4⋊C4—34no—
293C4⋊C4—32no—
294C4⋊C4—32no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—34no—
298C4⋊C4—34no—
299C4⋊C4C22×D312yes—
300C4⋊C4C22×D312yes—
301C4⋊C4C22×D311yescharacteristic
302C4⋊C4C22×D312yes—
303C4⋊C4C22×D311yescharacteristic
304C4⋊C4C22×D312yes—
305C22⋊C4—32no—
306C22⋊C4—34no—
307C22⋊C4—32no—
308C22⋊C4—34no—
309C22⋊C4—34no—
310C22⋊C4—34no—
311C22⋊C4C22×D312yes—
312C22⋊C4C22×D312yes—
313C22⋊C4C22×D312yes—
314C22⋊C4C22×D312yes—
315C22⋊C4C22×D312yes—
316C42—34no—
317C42—32no—
318C42—32no—
319C42—34no—
320C42—34no—
321C42—34no—
322C42C22×D311yescharacteristic
323C42C22×D311yescharacteristic
324C42C22×D311yescharacteristic
325C42C22×D311yescharacteristic
326C42C22×D312yes—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—22no—
330C22×C3—22no—
331C22×C3—22no—
332C22×C3—21no—
333C22×C3C2511yescharacteristic
334C3×C4—21no—
335C3×C4—22no—
336C3×C4—21no—
337C3×C4—22no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—22no—
341C3×C42−1+412yes—
342C3×C42−1+412yes—
343Dic3—22no—
344Dic3—22no—
345Dic3—24no—
346Dic3—22no—
347Dic3—24no—
348Dic3—22no—
349C23C23×D311yescharacteristic
350C23C23×D312yes—
351Q8—22no—
352Q8—62no—
353Q8—62no—
354Q8—64no—
355Q8—122no—
356Q8—122no—
357Q8—124no—
358D4—22no—
359D4—42no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—22no—
363C2×C4—34no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4—34no—
368C2×C4—32no—
369C2×C4—41no—
370C2×C4—124no—
371C2×C4C23×D311yescharacteristic
372C2×C4C23×D312yes—
373C2×C4C23×D312yes—
374C2×C4C23×D312yes—
375C2×C4C23×D311yescharacteristic
376C2×C4C23×D311yescharacteristic
377C2×C4C23×D311yescharacteristic
378C2×C4C23×D311yescharacteristic
379C2×C4C23×D311yescharacteristic
380C2×C3—41no—
381C2×C3—42no—
382C2×C3C2×2−1+411yescharacteristic
383C2×C3C2×2−1+411yescharacteristic
384C2×C3#58511yescharacteristic
385C22—21no—
386C22—21no—
387C22—22no—
388C22—22no—
389C22—21no—
390C22—22no—
391C22C24×D311yescharacteristic
392C4—21no—
393C4—22no—
394C4—21no—
395C4—22no—
396C4—21no—
397C4—21no—
398C4—22no—
399C4—62no—
400C4—62no—
401C4—64no—
402C4—62no—
403C4—64no—
404C4—62no—
405C4#101012yes—
406C4#101012yes—
407C3#355511yescharacteristic, complemented, Sylow
408C2—41no—
409C2—42no—
410C2C2×#101011yescharacteristic
411C2C2×#101011yescharacteristic
412C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
prime radicalC3×#121C22
hypercenter#529D3
hypocenterC3#3555
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. #5651C2semidirect
6. #5625C2semidirect
8. #5567C2semidirect
9. Q8×Dic6C2semidirect
10. #5543C2semidirect
11. #5542C2semidirect
12. #5542C2semidirect
13. #5541C2semidirect
14. #5519C2semidirect
15. #5518C2semidirect
16. #5470C2semidirect
17. #5470C2semidirect
407. C3#3555semidirect

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
#56512
#56252
#56032
#55672
Q8×Dic62
#55432
#55422
#55412
#55192
#55182
#54702
#35553