Finite Group Databasebeta

← #86368

Subgroups — #86368

Order 384.

1104 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#537C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3#5832C211yesmaximal, characteristic, complemented
4#5827C211yesmaximal, characteristic, complemented
5Dic3×#41C211yesmaximal, characteristic, complemented
6#5643C212yesmaximal, complemented
7#5641C212yesmaximal, complemented
8#5635C212yesmaximal, complemented
9#5622C212yesmaximal, complemented
10#5616C212yesmaximal, complemented
11#5611C212yesmaximal, complemented
12#5603C212yesmaximal, complemented
13#5601C214yesmaximal, complemented
14#5598C214yesmaximal, complemented
15#5596C212yesmaximal, complemented
16#5504C211yesmaximal, characteristic, complemented
17C2×#888C211yesmaximal, characteristic, complemented
18#3511—31nomaximal, Sylow
19C2×C3×#41C2211yescharacteristic
20C2×C3×#41C2211yescharacteristic
21C2×#237C2214yes—
22C2×#235C2211yescharacteristic
23C22×Dic6C2212yes—
24C3×C22⋊Q8C2212yes—
25C3×C22⋊Q8C2212yes—
26C3×#121C2212yes—
27C3×#121C2214yes—
28C3×#121C2212yes—
29C2×C3×C4⋊C4C2211yescharacteristic
30#947C2211yescharacteristic
31#946C2212yes—
32Q8×Dic3C2211yescharacteristic
33Q8×Dic3C2211yescharacteristic
34Q8×Dic3C2212yes—
35#944C2211yescharacteristic
36#940C2211yescharacteristic
37#939C2212yes—
38#939C2212yes—
39#939C2214yes—
40#939C2214yes—
41#938C2212yes—
42#936C2211yescharacteristic
43#936C2214yes—
44#935C2214yes—
45D4×Dic3C2212yes—
46D4×Dic3C2212yes—
47D4×Dic3C2211yescharacteristic
48D4×Dic3C2211yescharacteristic
49D4×Dic3C2212yes—
50D4×Dic3C2214yes—
51#930C2212yes—
52C4×#67C2212yes—
53C4×#67C2212yes—
54C4×#67C2214yes—
55#926C2212yes—
56#926C2212yes—
57#926C2211yescharacteristic
58C2×C4⋊Dic3C2212yes—
59C2×C4⋊Dic3C2211yescharacteristic
60#924C2212yes—
61C2×Dic3⋊C4C2212yes—
62C2×C4×Dic3C2211yescharacteristic
63C2×C4×Dic3C2212yes—
64#898C2214yes—
65#897C2214yes—
66#893C2212yes—
67#892C2212yes—
68D3×C4⋊C4C2212yes—
69#889C2214yes—
70#888C2212yes—
71#888C2212yes—
72#888C2212yes—
73#888C2212yes—
74#887C2214yes—
75#887C2212yes—
76#885C2214yes—
77#885C2214yes—
78#883C2214yes—
79#67⋊C4C2214yes—
80#879C2214yes—
81#878C2214yes—
82#878C2214yes—
83#877C2214yes—
84#556—32no—
85#555—32no—
86#554—32no—
87#553—34no—
88#548—34no—
89#547—32no—
90#542—32no—
91#542—32no—
92#537—31no—
93#537D311yescharacteristic, complemented
94#533—32no—
95#532—31no—
96#532—32no—
97C2×C4⋊Q8—31no—
98#520—31no—
99#520—31no—
100C4×#41—31no—
101C3×#41—24no—
102C3×#41—24no—
103C2×C3×Q8C2311yescharacteristic
104C2×C3×Q8C2311yescharacteristic
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2311yescharacteristic
107C2×C3×D4C2312yes—
108C2×C3×D4C2312yes—
109C2×C3×D4C2314yes—
110C22×C3×C4C2312yes—
111C22×C3×C4C2312yes—
112C22×C3×C4C2312yes—
113C22×C3×C4C2311yescharacteristic
114C2×#67C2312yes—
115C2×#67C2312yes—
116C2×#67C2312yes—
117C2×#67C2314yes—
118C22×Dic3C2312yes—
119C22×Dic3C2312yes—
120C22×Dic3C2311yescharacteristic
121C22×Dic3C2311yescharacteristic
122C22×Dic3C2314yes—
123C22×Dic3C2312yes—
124#237—48no—
125#235—24no—
126C2×D12C2311yescharacteristic
127C2×C4×D3C2312yes—
128C2×C4×D3C2314yes—
129C2×Dic6—44no—
130C2×Dic6C2314yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2314yes—
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2314yes—
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2314yes—
138#217C2314yes—
139#217C2312yes—
140#217C2312yes—
141#217C2314yes—
142#217C2312yes—
143#217C2312yes—
144#217C2312yes—
145(C2×D3)⋊C4C2314yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2314yes—
148(C2×D3)⋊C4C2312yes—
149C4⋊Dic3C2312yes—
150C4⋊Dic3C2314yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2312yes—
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2311yescharacteristic
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2314yes—
161Dic3⋊C4C2314yes—
162C4×Dic3C2314yes—
163C4×Dic3C2314yes—
164C4×Dic3C2312yes—
165C4×Dic3C2312yes—
166C4×Dic3C2312yes—
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2311yescharacteristic
170C4×Dic3C2311yescharacteristic
171C2×#41—34no—
172C2×#41—31no—
173C2×#41C2×D311yescharacteristic
174C2×#41C2×D311yescharacteristic
175C22×Q8—32no—
176C4⋊Q8—31no—
177C4⋊Q8—32no—
178C4⋊Q8—32no—
179C4⋊Q8—32no—
180C4⋊Q8—32no—
181#127—31no—
182#126—34no—
183#126—34no—
184#125—34no—
185#124—31no—
186#124—31no—
187#124—34no—
188#124—34no—
189#124—34no—
190#123—34no—
191#123—34no—
192C22⋊Q8—32no—
193C22⋊Q8—32no—
194C22⋊Q8—34no—
195C22⋊Q8—34no—
196C22⋊Q8—34no—
197C22⋊Q8C2×D312yes—
198C22⋊Q8C2×D312yes—
199#121—32no—
200#121—32no—
201#121—34no—
202#121—32no—
203#121—34no—
204#121—32no—
205#121C2×D312yes—
206#121C2×D312yes—
207#121C2×D314yes—
208C4×Q8—31no—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—32no—
212C4×Q8—34no—
213C4×D4—31no—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—31no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—32no—
221C4×D4—32no—
222C4×D4—32no—
223C4×D4—34no—
224#117—31no—
225#117—32no—
226#117—32no—
227#117—32no—
228#117—34no—
229C2×C4⋊C4—31no—
230C2×C4⋊C4—32no—
231C2×C4⋊C4—32no—
232C2×C4⋊C4—32no—
233C2×C4⋊C4C2×D311yescharacteristic
234C2×C42—31no—
235C2×C42—32no—
236C23×C3C2412yes—
237C23×C3C2412yes—
238C23×C3C2411yescharacteristic
239C22×D3C2412yes—
240C3×Q8—22no—
241C3×Q8—22no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—44no—
247C2×C3×C4—24no—
248C2×C3×C4—24no—
249C2×C3×C4—42no—
250C2×C3×C4#4114yescomplemented
251C2×C3×C4C2412yes—
252C2×C3×C4C2412yes—
253C2×C3×C4C2414yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—24no—
257#67—24no—
258#67—24no—
259#67—44no—
260C2×Dic3—22no—
261C2×Dic3—24no—
262C2×Dic3—22no—
263C2×Dic3—24no—
264C2×Dic3—42no—
265C2×Dic3—44no—
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268C2×Dic3C2411yescharacteristic
269C2×Dic3C2412yes—
270C2×Dic3C2411yescharacteristic
271C2×Dic3C2414yes—
272C2×Dic3C2411yescharacteristic
273C2×Dic3C2411yescharacteristic
274D12—22no—
275C4×D3—24no—
276C4×D3—44no—
277Dic6—22no—
278Dic6—28no—
279Dic6—44no—
280#41—24no—
281#41—24no—
282#41—64no—
283#41—128no—
284C2×Q8—31no—
285C2×Q8—34no—
286C2×Q8—34no—
287C2×Q8—124no—
288C2×Q8C22×D311yescharacteristic
289C2×Q8C22×D311yescharacteristic
290C2×D4—32no—
291C2×D4—32no—
292C2×D4—32no—
293C2×D4—34no—
294C2×D4—31no—
295C2×D4C22×D312yes—
296C2×D4C22×D312yes—
297C2×D4C22×D311yescharacteristic
298C2×D4C22×D311yescharacteristic
299C2×D4C22×D314yes—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—31no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—34no—
306C22×C4—32no—
307C22×C4—34no—
308C22×C4C22×D312yes—
309C22×C4C22×D312yes—
310C22×C4C22×D312yes—
311C22×C4C22×D311yescharacteristic
312C4⋊C4—31no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4—34no—
317C4⋊C4—34no—
318C4⋊C4—34no—
319C4⋊C4—31no—
320C4⋊C4—32no—
321C4⋊C4—32no—
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4C22×D312yes—
326C4⋊C4C22×D312yes—
327C4⋊C4C22×D314yes—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—34no—
336C22⋊C4—34no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D314yes—
341C42—31no—
342C42—32no—
343C42—31no—
344C42—32no—
345C42—31no—
346C42—31no—
347C42—32no—
348C42—34no—
349C42—34no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3—22no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3C2×#4112yes—
359C22×C3C2511yescharacteristic
360C2×D3—22no—
361C2×D3—22no—
362C2×D3—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—24no—
366C3×C4C2×#4112yes—
367C3×C4C2×#4112yes—
368Dic3—22no—
369Dic3—24no—
370Dic3—21no—
371Dic3—22no—
372Dic3—22no—
373Dic3—21no—
374Dic3H(5,𝔽2)12yes—
375Dic32−1+412yes—
376C23—32no—
377C23C23×D312yes—
378C23C23×D312yes—
379C23C23×D311yescharacteristic
380Q8—22no—
381Q8—22no—
382Q8—68no—
383Q8—62no—
384Q8—124no—
385D4—24no—
386D4—24no—
387D4—22no—
388D4—22no—
389D4—44no—
390D4—64no—
391D4—64no—
392D4—64no—
393D4—62no—
394D4—124no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—31no—
398C2×C4—31no—
399C2×C4—32no—
400C2×C4—31no—
401C2×C4—31no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—34no—
405C2×C4—42no—
406C2×C4—62no—
407C2×C4—64no—
408C2×C4—62no—
409C2×C4—64no—
410C2×C4—64no—
411C2×C4—124no—
412C2×C4—122no—
413C2×C4—124no—
414C2×C4#23714yescomplemented
415C2×C4C23×D312yes—
416C2×C4C23×D312yes—
417C2×C4C23×D314yes—
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D311yescharacteristic
420C2×C3—22no—
421C2×C3—42no—
422C2×C3—42no—
423C2×C3C22×#4111yescharacteristic
424C2×C3C2×H(5,𝔽2)11yescharacteristic
425C2×C3C2×2−1+411yescharacteristic
426D3—42no—
427C22—21no—
428C22—21no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—22no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22C2×#23712yes—
439C22C24×D311yescharacteristic
440C4—22no—
441C4—22no—
442C4—24no—
443C4—32no—
444C4—32no—
445C4—62no—
446C4—62no—
447C4—61no—
448C4—64no—
449C4—62no—
450C4—61no—
451C4C2×#23712yes—
452C4C2×#23712yes—
453C3#351111yescharacteristic, complemented, Sylow
454C2—22no—
455C2—42no—
456C2—42no—
457C2—122no—
458C2C22×#23711yescharacteristic
459C2D3×H(5,𝔽2)11yescharacteristic
460C2D3×2−1+411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#537C2
hypercenter#537D3
hypocenterC3#3511
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #537

Complements

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

NH ≅ G/Ntype
1. C3×#537C2semidirect
2. #5833C2semidirect
3. #5832C2semidirect
4. #5827C2semidirect
5. Dic3×#41C2semidirect
6. #5643C2semidirect
7. #5641C2semidirect
8. #5635C2semidirect
9. #5622C2semidirect
10. #5616C2semidirect
11. #5611C2semidirect
12. #5603C2semidirect
13. #5601C2semidirect
14. #5598C2semidirect
15. #5596C2semidirect
16. #5504C2semidirect
17. C2×#888C2semidirect
93. #537D3semidirect
250. C2×C3×C4#41semidirect
414. C2×C4#237semidirect
453. C3#3511semidirect

Maximal subgroups

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

subgroupindex
C3×#5372
#58332
#58322
#58272
Dic3×#412
#56432
#56412
#56352
#56222
#56162
#56112
#56032
#56012
#55982
#55962
#55042
C2×#8882
#35113