Finite Group Databasebeta

← #86774

Subgroups — #86774

Order 384.

1122 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×#550C211yesmaximal, characteristic, complemented
2#5832C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5719C212yesmaximal, complemented
5#5718C211yesmaximal, characteristic, complemented
6#5718C212yesmaximal, complemented
7#5716C211yesmaximal, characteristic, complemented
8#5615C214yesmaximal, complemented
9#5601C214yesmaximal, complemented
10#5600C212yesmaximal, complemented
11#5599C212yesmaximal, complemented
12#5580C211yesmaximal, characteristic, complemented
13#5567C211yesmaximal, characteristic, complemented
14#5558C212yesmaximal, complemented
15#5551C211yesmaximal, characteristic, complemented
16#5546C212yesmaximal, complemented
17C4×#237C211yesmaximal, characteristic, complemented
18#3517—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#237C2211yescharacteristic
21C2×#237C2212yes—
22C2×#237C2212yes—
23C2×#235C2212yes—
24C3×#127C2211yescharacteristic
25C3×#127C2212yes—
26C3×#121C2212yes—
27C3×#121C2214yes—
28C3×C4×Q8C2211yescharacteristic
29C3×C4×D4C2211yescharacteristic
30C3×C4×D4C2212yes—
31#947C2211yescharacteristic
32#946C2211yescharacteristic
33#944C2212yes—
34#940C2212yes—
35#940C2214yes—
36#939C2214yes—
37#939C2214yes—
38#939C2214yes—
39#939C2214yes—
40#939C2214yes—
41#938C2212yes—
42#938C2212yes—
43#938C2212yes—
44#938C2212yes—
45#936C2212yes—
46#935C2214yes—
47#935C2214yes—
48D4×Dic3C2212yes—
49D4×Dic3C2212yes—
50D4×Dic3C2212yes—
51D4×Dic3C2214yes—
52D4×Dic3C2211yescharacteristic
53#929C2214yes—
54C4×#67C2212yes—
55C4×#67C2214yes—
56C4×#67C2214yes—
57#924C2212yes—
58C2×Dic3⋊C4C2214yes—
59C2×C4×Dic3C2212yes—
60#898C2212yes—
61#895C2211yescharacteristic
62#894C2212yes—
63#892C2211yescharacteristic
64D3×C4⋊C4C2212yes—
65#890C2211yescharacteristic
66#889C2214yes—
67#887C2211yescharacteristic
68#887C2212yes—
69#884C2214yes—
70#883C2214yes—
71#67⋊C4C2212yes—
72#67⋊C4C2214yes—
73#879C2214yes—
74#877C2212yes—
75#877C2214yes—
76C4×D12C2211yescharacteristic
77C4×D12C2212yes—
78#872C2212yes—
79C42×D3C2211yescharacteristic
80#870C2212yes—
81#869C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#557—31no—
85#553—34no—
86#553—32no—
87#550—31no—
88#550—32no—
89#550—32no—
90#550D311yescharacteristic, complemented
91#547—32no—
92#547—31no—
93#540—32no—
94#540—34no—
95#532—32no—
96#532—31no—
97#529—32no—
98#529—31no—
99#520—32no—
100C4×#41—31no—
101C3×#41—44no—
102C2×C3×Q8C2311yescharacteristic
103C2×C3×D4C2312yes—
104C2×C3×D4C2312yes—
105C2×C3×D4C2311yescharacteristic
106C2×C3×D4C2314yes—
107C2×C3×D4C2312yes—
108C2×C3×D4C2314yes—
109C22×C3×C4C2312yes—
110C22×C3×C4C2314yes—
111C2×#67C2312yes—
112C2×#67C2314yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C22×Dic3C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2312yes—
119C22×Dic3C2314yes—
120#237—24no—
121#237—28no—
122#237—44no—
123#235—44no—
124C2×D12C2311yescharacteristic
125C2×D12C2312yes—
126C2×C4×D3C2311yescharacteristic
127C2×C4×D3C2311yescharacteristic
128C2×C4×D3C2312yes—
129C2×C4×D3C2311yescharacteristic
130C2×C4×D3C2314yes—
131C2×Dic6C2312yes—
132C2×Dic6C2311yescharacteristic
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2311yescharacteristic
135C3×C4⋊C4C2312yes—
136C3×C22⋊C4C2314yes—
137C3×C22⋊C4C2312yes—
138C3×C42C2311yescharacteristic
139C3×C42C2312yes—
140#217C2312yes—
141#217C2314yes—
142#217C2314yes—
143#217C2314yes—
144#217C2314yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151C4⋊Dic3C2312yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2312yes—
154C4⋊Dic3C2312yes—
155C4⋊Dic3C2311yescharacteristic
156Dic3⋊C4C2312yes—
157Dic3⋊C4C2314yes—
158Dic3⋊C4C2314yes—
159Dic3⋊C4C2314yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2314yes—
163Dic3⋊C4C2312yes—
164C4×Dic3C2314yes—
165C4×Dic3C2312yes—
166C4×Dic3C2311yescharacteristic
167C4×Dic3C2311yescharacteristic
168C4×Dic3C2311yescharacteristic
169C4×Dic3C2312yes—
170C4×Dic3C2312yes—
171C2×#41—31no—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—32no—
175C2×#41C2×D312yes—
176C4⋊Q8—32no—
177C4⋊Q8—31no—
178#127—32no—
179#127—34no—
180#127C2×D311yescharacteristic
181#127C2×D312yes—
182#126—32no—
183#126—34no—
184#125—31no—
185#125—34no—
186#125—32no—
187#124—31no—
188#124—32no—
189#123—34no—
190#123—34no—
191#123—34no—
192#123—32no—
193#123—34no—
194C22⋊Q8—31no—
195C22⋊Q8—32no—
196#121—32no—
197#121—34no—
198#121—32no—
199#121—34no—
200#121—32no—
201#121—34no—
202#121—32no—
203#121—34no—
204#121—34no—
205#121—31no—
206#121—34no—
207#121C2×D312yes—
208#121C2×D314yes—
209C4×Q8—31no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—32no—
213C4×Q8C2×D311yescharacteristic
214C4×D4—31no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—34no—
222C4×D4—32no—
223C4×D4—34no—
224C4×D4—31no—
225C4×D4—32no—
226C4×D4C2×D311yescharacteristic
227C4×D4C2×D312yes—
228#117—31no—
229#117—32no—
230#117—34no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—32no—
234C2×C42—32no—
235C2×C42—31no—
236C23×C3C2414yes—
237C23×C3C2412yes—
238C22×D3C2411yescharacteristic
239C22×D3C2412yes—
240C3×Q8—22no—
241C3×D4—22no—
242C3×D4—24no—
243C3×D4—24no—
244C3×D4—42no—
245C3×D4—44no—
246C3×D4—44no—
247C2×C3×C4—24no—
248C2×C3×C4—44no—
249C2×C3×C4C2412yes—
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2412yes—
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255#67—24no—
256#67—28no—
257#67—44no—
258#67—44no—
259#67—44no—
260C2×Dic3—24no—
261C2×Dic3—28no—
262C2×Dic3—44no—
263C2×Dic3—42no—
264C2×Dic3C2411yescharacteristic
265C2×Dic3C2411yescharacteristic
266C2×Dic3C2412yes—
267C2×Dic3C2411yescharacteristic
268C2×Dic3C2414yes—
269C2×Dic3C2412yes—
270C2×Dic3C2412yes—
271D12—22no—
272D12—42no—
273C4×D3—22no—
274C4×D3—24no—
275C4×D3—22no—
276C4×D3—41no—
277C4×D3—44no—
278Dic6—24no—
279Dic6#4114yescomplemented
280#41—44no—
281#41—64no—
282#41—68no—
283#41—124no—
284#41—124no—
285C2×Q8—31no—
286C2×Q8—32no—
287C2×Q8C22×D311yescharacteristic
288C2×D4—32no—
289C2×D4—34no—
290C2×D4—34no—
291C2×D4—34no—
292C2×D4—34no—
293C2×D4—31no—
294C2×D4—32no—
295C2×D4C22×D311yescharacteristic
296C2×D4C22×D312yes—
297C2×D4C22×D312yes—
298C2×D4C22×D312yes—
299C2×D4C22×D314yes—
300C2×D4C22×D314yes—
301C22×C4—32no—
302C22×C4—34no—
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—31no—
306C22×C4—31no—
307C22×C4—32no—
308C22×C4—31no—
309C22×C4—34no—
310C22×C4C22×D312yes—
311C22×C4C22×D314yes—
312C4⋊C4—32no—
313C4⋊C4—32no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—32no—
322C4⋊C4—34no—
323C4⋊C4—32no—
324C4⋊C4—34no—
325C4⋊C4C22×D311yescharacteristic
326C4⋊C4C22×D311yescharacteristic
327C4⋊C4C22×D312yes—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—34no—
331C22⋊C4—34no—
332C22⋊C4—34no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—32no—
338C22⋊C4—32no—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D312yes—
341C42—31no—
342C42—32no—
343C42—31no—
344C42—32no—
345C42—31no—
346C42—32no—
347C42—34no—
348C42C22×D311yescharacteristic
349C42C22×D312yes—
350C22×C3—24no—
351C22×C3—22no—
352C22×C3—24no—
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—24no—
356C22×C3C2511yescharacteristic
357C2×D3—21no—
358C2×D3—22no—
359C2×D3—21no—
360C2×D3—22no—
361C2×D3—21no—
362C2×D3—22no—
363C3×C4—22no—
364C3×C4—22no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4C2×#4112yes—
368C3×C4H(5,𝔽2)12yes—
369Dic3—21no—
370Dic3—22no—
371Dic3—24no—
372Dic3—21no—
373Dic3—21no—
374Dic3—22no—
375Dic3C2×#4114yes—
376C23—31no—
377C23—32no—
378C23C23×D312yes—
379C23C23×D314yes—
380Q8—22no—
381Q8—34no—
382Q8—64no—
383D4—24no—
384D4—22no—
385D4—24no—
386D4—44no—
387D4—42no—
388D4—44no—
389D4—64no—
390D4—68no—
391D4—62no—
392D4—122no—
393D4—124no—
394D4—124no—
395D4—124no—
396C2×C4—24no—
397C2×C4—32no—
398C2×C4—31no—
399C2×C4—31no—
400C2×C4—32no—
401C2×C4—31no—
402C2×C4—34no—
403C2×C4—32no—
404C2×C4—44no—
405C2×C4—64no—
406C2×C4—68no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—64no—
410C2×C4—124no—
411C2×C4—121no—
412C2×C4—124no—
413C2×C4—122no—
414C2×C4C23×D312yes—
415C2×C4C23×D312yes—
416C2×C4C23×D311yescharacteristic
417C2×C4C23×D312yes—
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D311yescharacteristic
420C2×C3—44no—
421C2×C3—42no—
422C2×C3C22×#4111yescharacteristic
423C2×C3C2×H(5,𝔽2)11yescharacteristic
424C2×C3#58511yescharacteristic
425D3—41no—
426D3—42no—
427C22—22no—
428C22—24no—
429C22—22no—
430C22—24no—
431C22—22no—
432C22—24no—
433C22—61no—
434C22—62no—
435C22—61no—
436C22—62no—
437C22—61no—
438C22—62no—
439C22C24×D311yescharacteristic
440C4—22no—
441C4—22no—
442C4—21no—
443C4—22no—
444C4—34no—
445C4—61no—
446C4—64no—
447C4—61no—
448C4—62no—
449C4—61no—
450C4—62no—
451C4#100412yes—
452C4D3×#4112yes—
453C3#351711yescharacteristic, complemented, Sylow
454C2—42no—
455C2—44no—
456C2—121no—
457C2—122no—
458C2C2×#100411yescharacteristic
459C2C2×D3×#4111yescharacteristic
460C2#596611yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#550C2
hypercenter#550D3
hypocenterC3#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C3×#550C2semidirect
2. #5832C2semidirect
3. #5824C2semidirect
4. #5719C2semidirect
5. #5718C2semidirect
6. #5718C2semidirect
7. #5716C2semidirect
8. #5615C2semidirect
9. #5601C2semidirect
10. #5600C2semidirect
11. #5599C2semidirect
12. #5580C2semidirect
13. #5567C2semidirect
14. #5558C2semidirect
15. #5551C2semidirect
16. #5546C2semidirect
17. C4×#237C2semidirect
90. #550D3semidirect
279. Dic6#41semidirect
453. C3#3517semidirect

Maximal subgroups

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

subgroupindex
C3×#5502
#58322
#58242
#57192
#57182
#57162
#56152
#56012
#56002
#55992
#55802
#55672
#55582
#55512
#55462
C4×#2372
#35173