Finite Group Databasebeta

← #86264

Subgroups — #86264

Order 384.

1104 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#533C211yesmaximal, characteristic, complemented
2#5728C211yesmaximal, characteristic, complemented
3#5725C211yesmaximal, characteristic, complemented
4#5725C211yesmaximal, characteristic, complemented
5D3×C4⋊Q8C211yesmaximal, characteristic, complemented
6#5695C212yesmaximal, complemented
7#5690C212yesmaximal, complemented
8#5643C212yesmaximal, complemented
9#5633C212yesmaximal, complemented
10#5631C212yesmaximal, complemented
11#5629C212yesmaximal, complemented
12#5582C212yesmaximal, complemented
13#5576C212yesmaximal, complemented
14#5574C212yesmaximal, complemented
15#5568C212yesmaximal, complemented
16#5532C212yesmaximal, complemented
17#5530C212yesmaximal, complemented
18#5477C211yesmaximal, characteristic, complemented
19C4×#235C211yesmaximal, characteristic, complemented
20#3511—31nomaximal, Sylow
21C2×#239C2212yes—
22C2×#239C2212yes—
23C2×D3×Q8C2212yes—
24C2×#235C2211yescharacteristic
25C2×#235C2212yes—
26C3×C4⋊Q8C2211yescharacteristic
27C3×C4⋊Q8C2211yescharacteristic
28C3×#125C2212yes—
29C3×C22⋊Q8C2212yes—
30C3×C22⋊Q8C2212yes—
31C3×C4×Q8C2212yes—
32C3×C4×Q8C2212yes—
33C3×#117C2212yes—
34C2×C3×C42C2211yescharacteristic
35#947C2212yes—
36#947C2212yes—
37#946C2212yes—
38Q8×Dic3C2212yes—
39#944C2212yes—
40#930C2212yes—
41#930C2212yes—
42C4×#67C2211yescharacteristic
43C4×#67C2211yescharacteristic
44C4×#67C2211yescharacteristic
45C4×#67C2211yescharacteristic
46#926C2211yescharacteristic
47#898C2214yes—
48#897C2214yes—
49#897C2212yes—
50#896C2214yes—
51#896C2214yes—
52#895C2212yes—
53#895C2212yes—
54#895C2212yes—
55#895C2214yes—
56#895C2212yes—
57#894C2214yes—
58#893C2214yes—
59#893C2212yes—
60#893C2212yes—
61#893C2212yes—
62#893C2212yes—
63#892C2212yes—
64D3×C4⋊C4C2214yes—
65D3×C4⋊C4C2212yes—
66D3×C4⋊C4C2212yes—
67#890C2212yes—
68#888C2212yes—
69#888C2212yes—
70#887C2212yes—
71#885C2214yes—
72#884C2214yes—
73#884C2214yes—
74#883C2214yes—
75#876C2214yes—
76#875C2212yes—
77#875C2214yes—
78#874C2211yescharacteristic
79C4×D12C2212yes—
80C4×D12C2211yescharacteristic
81C4×D12C2211yescharacteristic
82C4×D12C2212yes—
83C4×D12C2214yes—
84#872C2211yescharacteristic
85#872C2211yescharacteristic
86#872C2214yes—
87C42×D3C2211yescharacteristic
88C42×D3C2211yescharacteristic
89#869C2211yescharacteristic
90C4×Dic6C2211yescharacteristic
91C4×Dic6C2211yescharacteristic
92#556—32no—
93#555—32no—
94#554—32no—
95#553—32no—
96#553—32no—
97#548—32no—
98#548—32no—
99#547—32no—
100#542—32no—
101#542—32no—
102#537—32no—
103#533—31no—
104#533D311yescharacteristic, complemented
105#532—31no—
106#532—31no—
107#532—31no—
108C2×C4⋊Q8—31no—
109#520—32no—
110C4×#41—31no—
111C2×C3×Q8C2312yes—
112C2×C3×Q8C2312yes—
113C22×C3×C4C2311yescharacteristic
114C22×C3×C4C2311yescharacteristic
115C22×C3×C4C2311yescharacteristic
116C2×#67C2311yescharacteristic
117C2×#67C2311yescharacteristic
118C2×#67C2314yes—
119#239—28no—
120#239—44no—
121D3×Q8—44no—
122#235—24no—
123#235—44no—
124C2×D12C2311yescharacteristic
125C2×D12C2312yes—
126C2×D12C2312yes—
127C2×D12C2314yes—
128C2×D12C2312yes—
129C2×D12C2314yes—
130C2×C4×D3C2311yescharacteristic
131C2×C4×D3C2311yescharacteristic
132C2×C4×D3C2311yescharacteristic
133C2×C4×D3C2311yescharacteristic
134C2×C4×D3C2311yescharacteristic
135C2×C4×D3C2314yes—
136C2×C4×D3C2314yes—
137C2×C4×D3C2314yes—
138C2×C4×D3C2314yes—
139C2×C4×D3C2311yescharacteristic
140C2×Dic6C2311yescharacteristic
141C2×Dic6C2312yes—
142C2×Dic6C2314yes—
143C3×C4⋊C4C2312yes—
144C3×C4⋊C4C2312yes—
145C3×C4⋊C4C2312yes—
146C3×C4⋊C4C2312yes—
147C3×C4⋊C4C2314yes—
148C3×C4⋊C4C2314yes—
149C3×C22⋊C4C2314yes—
150C3×C42C2311yescharacteristic
151C3×C42C2311yescharacteristic
152C3×C42C2311yescharacteristic
153C3×C42C2311yescharacteristic
154C3×C42C2314yes—
155#217C2311yescharacteristic
156#217C2311yescharacteristic
157(C2×D3)⋊C4C2311yescharacteristic
158(C2×D3)⋊C4C2314yes—
159(C2×D3)⋊C4C2311yescharacteristic
160(C2×D3)⋊C4C2311yescharacteristic
161(C2×D3)⋊C4C2311yescharacteristic
162(C2×D3)⋊C4C2314yes—
163(C2×D3)⋊C4C2314yes—
164(C2×D3)⋊C4C2314yes—
165(C2×D3)⋊C4C2314yes—
166(C2×D3)⋊C4C2314yes—
167(C2×D3)⋊C4C2314yes—
168C4⋊Dic3C2311yescharacteristic
169C4⋊Dic3C2311yescharacteristic
170C4⋊Dic3C2312yes—
171C4⋊Dic3C2312yes—
172C4⋊Dic3C2314yes—
173Dic3⋊C4C2311yescharacteristic
174Dic3⋊C4C2311yescharacteristic
175Dic3⋊C4C2311yescharacteristic
176Dic3⋊C4C2311yescharacteristic
177Dic3⋊C4C2314yes—
178Dic3⋊C4C2314yes—
179Dic3⋊C4C2314yes—
180C4×Dic3C2311yescharacteristic
181C4×Dic3C2311yescharacteristic
182C4×Dic3C2311yescharacteristic
183C4×Dic3C2311yescharacteristic
184C4×Dic3C2311yescharacteristic
185C4×Dic3C2311yescharacteristic
186C4×Dic3C2314yes—
187C2×#41—32no—
188C2×#41—32no—
189C2×#41—31no—
190C2×#41—32no—
191C22×Q8—32no—
192C4⋊Q8—32no—
193C4⋊Q8—32no—
194C4⋊Q8—32no—
195C4⋊Q8—31no—
196C4⋊Q8C2×D311yescharacteristic
197C4⋊Q8C2×D311yescharacteristic
198#127—31no—
199#126—34no—
200#126—34no—
201#125—32no—
202#125C2×D312yes—
203#124—32no—
204#124—32no—
205#124—34no—
206#124—32no—
207#124—34no—
208#123—34no—
209#123—34no—
210C22⋊Q8—32no—
211C22⋊Q8—32no—
212C22⋊Q8—34no—
213C22⋊Q8—34no—
214C22⋊Q8—34no—
215C22⋊Q8C2×D312yes—
216C22⋊Q8C2×D312yes—
217#121—32no—
218#121—32no—
219#121—32no—
220#121—32no—
221#121—32no—
222#121—32no—
223#121—34no—
224#121—34no—
225#121—34no—
226C4×Q8—32no—
227C4×Q8—32no—
228C4×Q8—31no—
229C4×Q8—31no—
230C4×Q8C2×D312yes—
231C4×Q8C2×D312yes—
232C4×D4—31no—
233C4×D4—31no—
234C4×D4—31no—
235C4×D4—31no—
236C4×D4—32no—
237C4×D4—34no—
238C4×D4—32no—
239C4×D4—32no—
240C4×D4—32no—
241C4×D4—31no—
242C4×D4—31no—
243C4×D4—32no—
244C4×D4—34no—
245C4×D4—32no—
246#117—31no—
247#117—32no—
248#117—31no—
249#117—31no—
250#117—34no—
251#117C2×D312yes—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—34no—
254C2×C4⋊C4—32no—
255C2×C42—31no—
256C2×C42—31no—
257C2×C42C2×D311yescharacteristic
258C23×C3C2411yescharacteristic
259C22×D3C2414yes—
260C22×D3C2411yescharacteristic
261C22×D3C2411yescharacteristic
262C3×Q8—24no—
263C3×Q8—24no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4C2414yes—
268C2×C3×C4C2411yescharacteristic
269C2×C3×C4C2411yescharacteristic
270C2×C3×C4C2414yes—
271C2×C3×C4C2411yescharacteristic
272C2×C3×C4C2411yescharacteristic
273C2×C3×C4C2411yescharacteristic
274C2×C3×C4C2411yescharacteristic
275#67—22no—
276#67—22no—
277#67—44no—
278C2×Dic3C2411yescharacteristic
279C2×Dic3C2411yescharacteristic
280C2×Dic3C2411yescharacteristic
281C2×Dic3C2411yescharacteristic
282C2×Dic3C2411yescharacteristic
283C2×Dic3C2414yes—
284C2×Dic3C2411yescharacteristic
285D12—22no—
286D12—28no—
287D12—24no—
288D12—24no—
289D12—24no—
290D12—44no—
291C4×D3—22no—
292C4×D3—22no—
293C4×D3—22no—
294C4×D3—22no—
295C4×D3—22no—
296C4×D3—28no—
297C4×D3—44no—
298C4×D3—44no—
299C4×D3—44no—
300C4×D3#4114yescomplemented
301Dic6—24no—
302Dic6—22no—
303Dic6—44no—
304#41—64no—
305#41—68no—
306#41—124no—
307#41—124no—
308C2×Q8—31no—
309C2×Q8—32no—
310C2×Q8—34no—
311C2×Q8—124no—
312C2×Q8C22×D312yes—
313C2×Q8C22×D312yes—
314C2×D4—31no—
315C2×D4—31no—
316C2×D4—34no—
317C2×D4—32no—
318C2×D4—34no—
319C2×D4—31no—
320C2×D4—32no—
321C2×D4—32no—
322C2×D4—34no—
323C22×C4—31no—
324C22×C4—31no—
325C22×C4—31no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—34no—
330C22×C4—34no—
331C22×C4—34no—
332C22×C4—34no—
333C22×C4C22×D311yescharacteristic
334C22×C4C22×D311yescharacteristic
335C22×C4C22×D311yescharacteristic
336C4⋊C4—31no—
337C4⋊C4—34no—
338C4⋊C4—32no—
339C4⋊C4—32no—
340C4⋊C4—34no—
341C4⋊C4—34no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—34no—
348C4⋊C4C22×D312yes—
349C4⋊C4C22×D312yes—
350C4⋊C4C22×D314yes—
351C4⋊C4C22×D312yes—
352C4⋊C4C22×D312yes—
353C4⋊C4C22×D314yes—
354C22⋊C4—31no—
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—34no—
359C22⋊C4—34no—
360C22⋊C4—31no—
361C22⋊C4—31no—
362C22⋊C4—34no—
363C22⋊C4—34no—
364C22⋊C4—34no—
365C22⋊C4—34no—
366C22⋊C4—34no—
367C22⋊C4C22×D314yes—
368C42—31no—
369C42—31no—
370C42—31no—
371C42—31no—
372C42—31no—
373C42—31no—
374C42—34no—
375C42C22×D311yescharacteristic
376C42C22×D311yescharacteristic
377C42C22×D311yescharacteristic
378C42C22×D311yescharacteristic
379C42C22×D314yes—
380C22×C3—21no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3C2511yescharacteristic
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—24no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—24no—
390C2×D3—21no—
391C2×D3—24no—
392C2×D3C2×#4112yes—
393C3×C4—21no—
394C3×C4—24no—
395C3×C4—24no—
396C3×C4—21no—
397C3×C4—21no—
398C3×C4C2×#4112yes—
399C3×C4H(5,𝔽2)12yes—
400C3×C42−1+412yes—
401Dic3—21no—
402Dic3—24no—
403Dic3—21no—
404Dic3—21no—
405Dic3—21no—
406Dic3—21no—
407Dic3C2×#4112yes—
408C23—31no—
409C23—31no—
410C23—34no—
411C23C23×D311yescharacteristic
412Q8—24no—
413Q8—24no—
414Q8—62no—
415Q8—64no—
416Q8—124no—
417D4—62no—
418D4—68no—
419D4—62no—
420D4—62no—
421D4—64no—
422D4—64no—
423D4—64no—
424D4—124no—
425D4—124no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—31no—
430C2×C4—34no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—34no—
437C2×C4—68no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—124no—
444C2×C4—124no—
445C2×C4—124no—
446C2×C4C23×D314yes—
447C2×C4C23×D311yescharacteristic
448C2×C4C23×D311yescharacteristic
449C2×C4C23×D311yescharacteristic
450C2×C4C23×D314yes—
451C2×C4C23×D311yescharacteristic
452C2×C4C23×D311yescharacteristic
453C2×C4C23×D311yescharacteristic
454C2×C3—41no—
455C2×C3C22×#4111yescharacteristic
456C2×C3C2×H(5,𝔽2)11yescharacteristic
457C2×C3C2×2−1+411yescharacteristic
458D3—22no—
459D3—44no—
460D3—41no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—32no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—64no—
469C22—64no—
470C22—61no—
471C22—61no—
472C22—64no—
473C22C24×D311yescharacteristic
474C4—21no—
475C4—21no—
476C4—24no—
477C4—21no—
478C4—24no—
479C4—32no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—64no—
486C4#100712yes—
487C4D3×#4112yes—
488C4#100912yes—
489C3#351111yescharacteristic, complemented, Sylow
490C2—41no—
491C2—62no—
492C2—121no—
493C2—124no—
494C2C2×#100711yescharacteristic
495C2C2×D3×#4111yescharacteristic
496C2C2×#100911yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#533C2
hypercenter#533D3
hypocenterC3#3511
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #533

Complements

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

NH ≅ G/Ntype
1. C3×#533C2semidirect
2. #5728C2semidirect
3. #5725C2semidirect
4. #5725C2semidirect
5. D3×C4⋊Q8C2semidirect
6. #5695C2semidirect
7. #5690C2semidirect
8. #5643C2semidirect
9. #5633C2semidirect
10. #5631C2semidirect
11. #5629C2semidirect
12. #5582C2semidirect
13. #5576C2semidirect
14. #5574C2semidirect
15. #5568C2semidirect
16. #5532C2semidirect
17. #5530C2semidirect
18. #5477C2semidirect
19. C4×#235C2semidirect
104. #533D3semidirect
300. C4×D3#41semidirect
489. C3#3511semidirect

Maximal subgroups

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

subgroupindex
C3×#5332
#57282
#57252
D3×C4⋊Q82
#56952
#56902
#56432
#56332
#56312
#56292
#55822
#55762
#55742
#55682
#55322
#55302
#54772
C4×#2352
#35113