Finite Group Databasebeta

← #86683

Subgroups — #86683

Order 384.

1014 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#547C211yesmaximal, characteristic, complemented
2#5827C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5728C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5665C214yesmaximal, complemented
7#5650C214yesmaximal, complemented
8#5642C212yesmaximal, complemented
9#5625C212yesmaximal, complemented
10#5618C212yesmaximal, complemented
11#5600C212yesmaximal, complemented
12#5557C211yesmaximal, characteristic, complemented
13#5543C211yesmaximal, characteristic, complemented
14#5543C211yesmaximal, characteristic, complemented
15C4×#237C211yesmaximal, characteristic, complemented
16#5513C212yesmaximal, complemented
17C2×#890C212yesmaximal, complemented
18#3515—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2211yescharacteristic
21C3×C4⋊Q8C2211yescharacteristic
22C3×#123C2214yes—
23C3×C22⋊Q8C2212yes—
24C3×#121C2212yes—
25C3×C4×D4C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C2×C3×C4⋊C4C2212yes—
28#946C2212yes—
29Q8×Dic3C2212yes—
30Q8×Dic3C2212yes—
31#944C2212yes—
32#939C2214yes—
33#938C2211yescharacteristic
34#936C2211yescharacteristic
35#935C2214yes—
36D4×Dic3C2214yes—
37D4×Dic3C2214yes—
38D4×Dic3C2211yescharacteristic
39D4×Dic3C2212yes—
40D4×Dic3C2212yes—
41#929C2214yes—
42C4×#67C2212yes—
43C4×#67C2212yes—
44#926C2214yes—
45#926C2212yes—
46C2×C4⋊Dic3C2212yes—
47C2×C4⋊Dic3C2214yes—
48C2×C4⋊Dic3C2212yes—
49#924C2212yes—
50#924C2212yes—
51C2×Dic3⋊C4C2214yes—
52C2×C4×Dic3C2212yes—
53#898C2214yes—
54#897C2212yes—
55#897C2211yescharacteristic
56#892C2211yescharacteristic
57#892C2212yes—
58#890C2214yes—
59#890C2212yes—
60#890C2214yes—
61#890C2212yes—
62#890C2212yes—
63#890C2211yescharacteristic
64#890C2212yes—
65#889C2214yes—
66#889C2214yes—
67#887C2212yes—
68#887C2212yes—
69#887C2211yescharacteristic
70#886C2214yes—
71#883C2212yes—
72#67⋊C4C2212yes—
73#879C2212yes—
74#879C2214yes—
75#879C2214yes—
76#878C2214yes—
77#878C2214yes—
78#877C2214yes—
79#877C2212yes—
80C42×D3C2211yescharacteristic
81#869C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#557—31no—
85#554—31no—
86#554—31no—
87#553—32no—
88#552—34no—
89#552—32no—
90#552—32no—
91#547—32no—
92#547D311yescharacteristic, complemented
93#541—32no—
94#539—34no—
95#533—31no—
96#529—32no—
97#529—31no—
98C2×#125—32no—
99#520—32no—
100C4×#41—31no—
101C3×#41—28no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2314yes—
104C2×C3×D4C2311yescharacteristic
105C2×C3×D4C2311yescharacteristic
106C22×C3×C4C2312yes—
107C22×C3×C4C2314yes—
108C22×C3×C4C2312yes—
109C2×#67C2312yes—
110C2×#67C2312yes—
111C22×Dic3C2312yes—
112C22×Dic3C2314yes—
113C22×Dic3C2314yes—
114C22×Dic3C2312yes—
115#237—24no—
116C2×C4×D3C2311yescharacteristic
117C2×C4×D3C2311yescharacteristic
118C2×C4×D3C2311yescharacteristic
119C2×Dic6C2311yescharacteristic
120C2×Dic6C2311yescharacteristic
121C2×Dic6C2312yes—
122C3×C4⋊C4C2312yes—
123C3×C4⋊C4C2312yes—
124C3×C4⋊C4C2311yescharacteristic
125C3×C4⋊C4C2314yes—
126C3×C4⋊C4C2311yescharacteristic
127C3×C22⋊C4C2312yes—
128C3×C22⋊C4C2314yes—
129C3×C22⋊C4C2312yes—
130C3×C42C2311yescharacteristic
131#217C2312yes—
132#217C2312yes—
133#217C2314yes—
134#217C2314yes—
135#217C2312yes—
136#217C2312yes—
137(C2×D3)⋊C4C2312yes—
138(C2×D3)⋊C4C2312yes—
139(C2×D3)⋊C4C2314yes—
140C4⋊Dic3C2311yescharacteristic
141C4⋊Dic3C2311yescharacteristic
142C4⋊Dic3C2312yes—
143C4⋊Dic3C2312yes—
144C4⋊Dic3C2314yes—
145C4⋊Dic3C2314yes—
146C4⋊Dic3C2312yes—
147C4⋊Dic3C2314yes—
148C4⋊Dic3C2312yes—
149C4⋊Dic3C2314yes—
150Dic3⋊C4C2312yes—
151Dic3⋊C4C2312yes—
152Dic3⋊C4C2314yes—
153Dic3⋊C4C2312yes—
154Dic3⋊C4C2314yes—
155Dic3⋊C4C2314yes—
156Dic3⋊C4C2314yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2312yes—
159C4×Dic3C2312yes—
160C4×Dic3C2312yes—
161C4×Dic3C2311yescharacteristic
162C4×Dic3C2311yescharacteristic
163C4×Dic3C2311yescharacteristic
164C4×Dic3C2312yes—
165C4×Dic3C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2314yes—
168C2×#41—31no—
169C2×#41C2×D312yes—
170C4⋊Q8—32no—
171C4⋊Q8—31no—
172C4⋊Q8C2×D311yescharacteristic
173#126—34no—
174#126—34no—
175#126—32no—
176#126—34no—
177#125—31no—
178#125—32no—
179#125—34no—
180#125—34no—
181#125—32no—
182#125—32no—
183#125—34no—
184#125—34no—
185#125—32no—
186#124—31no—
187#123—34no—
188#123—34no—
189#123—32no—
190#123—34no—
191#123C2×D314yes—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—32no—
195C22⋊Q8—31no—
196C22⋊Q8—32no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8C2×D312yes—
200#121—31no—
201#121—34no—
202#121C2×D312yes—
203C4×Q8—32no—
204C4×Q8—32no—
205C4×Q8—31no—
206C4×Q8—32no—
207C4×Q8—32no—
208C4×Q8—31no—
209C4×Q8—31no—
210C4×D4—31no—
211C4×D4—34no—
212C4×D4—32no—
213C4×D4—34no—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4C2×D311yescharacteristic
219C4×D4C2×D311yescharacteristic
220#117—32no—
221#117—34no—
222#117—31no—
223#117—32no—
224#117—32no—
225#117—34no—
226C2×C4⋊C4—34no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—34no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4C2×D312yes—
231C2×C42—32no—
232C2×C42—31no—
233C23×C3C2412yes—
234C23×C3C2412yes—
235C22×D3C2411yescharacteristic
236C3×Q8—24no—
237C3×D4—28no—
238C3×D4—22no—
239C3×D4#4114yes—
240C2×C3×C4—28no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4C2412yes—
244C2×C3×C4C2414yes—
245C2×C3×C4C2411yescharacteristic
246C2×C3×C4C2412yes—
247C2×C3×C4C2411yescharacteristic
248C2×C3×C4C2411yescharacteristic
249#67—24no—
250#67—42no—
251C2×Dic3—24no—
252C2×Dic3—44no—
253C2×Dic3—44no—
254C2×Dic3—42no—
255C2×Dic3C2411yescharacteristic
256C2×Dic3C2411yescharacteristic
257C2×Dic3C2412yes—
258C2×Dic3C2411yescharacteristic
259C2×Dic3C2412yes—
260C2×Dic3C2412yes—
261C2×Dic3C2412yes—
262C2×Dic3C2414yes—
263C4×D3—22no—
264C4×D3—22no—
265C4×D3—41no—
266Dic6—22no—
267Dic6—22no—
268Dic6—42no—
269#41—28no—
270#41—64no—
271C2×Q8—31no—
272C2×Q8—31no—
273C2×Q8—32no—
274C2×Q8C22×D312yes—
275C2×D4—32no—
276C2×D4—32no—
277C2×D4C22×D311yescharacteristic
278C2×D4C22×D314yes—
279C2×D4C22×D311yescharacteristic
280C22×C4—32no—
281C22×C4—32no—
282C22×C4—34no—
283C22×C4—34no—
284C22×C4—31no—
285C22×C4—31no—
286C22×C4—31no—
287C22×C4C22×D312yes—
288C22×C4C22×D314yes—
289C22×C4C22×D312yes—
290C4⋊C4—31no—
291C4⋊C4—34no—
292C4⋊C4—32no—
293C4⋊C4—32no—
294C4⋊C4—31no—
295C4⋊C4—32no—
296C4⋊C4—34no—
297C4⋊C4—32no—
298C4⋊C4—34no—
299C4⋊C4—34no—
300C4⋊C4—34no—
301C4⋊C4—34no—
302C4⋊C4—34no—
303C4⋊C4—32no—
304C4⋊C4—32no—
305C4⋊C4—32no—
306C4⋊C4—34no—
307C4⋊C4—32no—
308C4⋊C4—34no—
309C4⋊C4C22×D311yescharacteristic
310C4⋊C4C22×D312yes—
311C4⋊C4C22×D314yes—
312C4⋊C4C22×D311yescharacteristic
313C4⋊C4C22×D312yes—
314C22⋊C4—32no—
315C22⋊C4—32no—
316C22⋊C4—32no—
317C22⋊C4—34no—
318C22⋊C4—32no—
319C22⋊C4—34no—
320C22⋊C4—32no—
321C22⋊C4—34no—
322C22⋊C4—32no—
323C22⋊C4C22×D312yes—
324C22⋊C4C22×D314yes—
325C22⋊C4C22×D312yes—
326C42—31no—
327C42—32no—
328C42—31no—
329C42—31no—
330C42—32no—
331C42—32no—
332C42—34no—
333C42—34no—
334C42—32no—
335C42C22×D311yescharacteristic
336C22×C3—22no—
337C22×C3—22no—
338C22×C3—22no—
339C22×C3—22no—
340C22×C3—22no—
341C22×C3C2×#4114yes—
342C22×C3C2511yescharacteristic
343C2×D3—21no—
344C2×D3—21no—
345C2×D3—21no—
346C3×C4—22no—
347C3×C4—24no—
348C3×C4—22no—
349C3×C4—21no—
350C3×C4C2×#4112yes—
351C3×C42−1+412yes—
352Dic3—22no—
353Dic3—21no—
354Dic3—22no—
355Dic3—21no—
356Dic3—24no—
357Dic3—21no—
358Dic3—22no—
359Dic3—22no—
360C23—31no—
361C23C23×D312yes—
362C23C23×D312yes—
363Q8—24no—
364Q8—62no—
365Q8—62no—
366Q8—122no—
367D4—28no—
368D4—22no—
369D4—64no—
370D4—122no—
371D4#23714yes—
372C2×C4—28no—
373C2×C4—24no—
374C2×C4—24no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—34no—
382C2×C4—32no—
383C2×C4—64no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—121no—
387C2×C4—122no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4C23×D312yes—
391C2×C4C23×D314yes—
392C2×C4C23×D311yescharacteristic
393C2×C4C23×D312yes—
394C2×C4C23×D311yescharacteristic
395C2×C4C23×D311yescharacteristic
396C2×C3—24no—
397C2×C3—42no—
398C2×C3C22×#4111yescharacteristic
399C2×C3C2×2−1+411yescharacteristic
400C2×C3#58511yescharacteristic
401D3—41no—
402C22—22no—
403C22—22no—
404C22—22no—
405C22—22no—
406C22—22no—
407C22—61no—
408C22—61no—
409C22—61no—
410C22C2×#23714yes—
411C22C24×D311yescharacteristic
412C4—22no—
413C4—24no—
414C4—22no—
415C4—21no—
416C4—61no—
417C4—62no—
418C4—61no—
419C4—62no—
420C4—61no—
421C4—62no—
422C4—62no—
423C4—64no—
424C4C2×#23712yes—
425C4#101012yes—
426C3#351511yescharacteristic, complemented, Sylow
427C2—24no—
428C2—42no—
429C2—121no—
430C2C22×#23711yescharacteristic
431C2C2×#101011yescharacteristic
432C2#596811yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#547C2
hypercenter#547D3
hypocenterC3#3515
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #547

Complements

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

NH ≅ G/Ntype
1. C3×#547C2semidirect
2. #5827C2semidirect
3. #5824C2semidirect
4. #5728C2semidirect
5. #5721C2semidirect
6. #5665C2semidirect
7. #5650C2semidirect
8. #5642C2semidirect
9. #5625C2semidirect
10. #5618C2semidirect
11. #5600C2semidirect
12. #5557C2semidirect
13. #5543C2semidirect
14. #5543C2semidirect
15. C4×#237C2semidirect
16. #5513C2semidirect
17. C2×#890C2semidirect
92. #547D3semidirect
426. C3#3515semidirect

Maximal subgroups

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

subgroupindex
C3×#5472
#58272
#58242
#57282
#57212
#56652
#56502
#56422
#56252
#56182
#56002
#55572
#55432
C4×#2372
#55132
C2×#8902
#35153