Finite Group Databasebeta

← #86126

Subgroups — #86126

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×#529C211yesmaximal, characteristic, complemented
2#5709C214yesmaximal, complemented
3#5691C212yesmaximal, complemented
4#5668C212yesmaximal, complemented
5#5663C214yesmaximal, complemented
6#5631C212yesmaximal, complemented
7#5599C212yesmaximal, complemented
8#5577C211yesmaximal, characteristic, complemented
9#5567C211yesmaximal, characteristic, complemented
10#5565C212yesmaximal, complemented
11#5555C211yesmaximal, characteristic, complemented
12#5553C211yesmaximal, characteristic, complemented
13#5553C212yesmaximal, complemented
14#5535C212yesmaximal, complemented
15#5523C212yesmaximal, complemented
16#5477C211yesmaximal, characteristic, complemented
17C4×#235C211yesmaximal, characteristic, complemented
18#3517—31nomaximal, Sylow
19C2×#239C2212yes—
20C2×#237C2212yes—
21C2×#235C2211yescharacteristic
22C2×#235C2212yes—
23C2×#235C2212yes—
24C3×#126C2212yes—
25C3×#125C2211yescharacteristic
26C3×#124C2211yescharacteristic
27C3×#123C2212yes—
28C3×C22⋊Q8C2211yescharacteristic
29C3×#121C2211yescharacteristic
30C3×C4×Q8C2211yescharacteristic
31C3×C4×D4C2211yescharacteristic
32C3×C4×D4C2212yes—
33C3×#117C2212yes—
34C2×C3×C42C2211yescharacteristic
35#944C2212yes—
36#940C2214yes—
37#940C2212yes—
38#939C2214yes—
39#930C2212yes—
40#930C2212yes—
41#930C2212yes—
42#930C2212yes—
43C4×#67C2212yes—
44C4×#67C2212yes—
45#926C2211yescharacteristic
46C2×C4⋊Dic3C2212yes—
47#898C2214yes—
48#897C2212yes—
49#895C2214yes—
50#895C2212yes—
51#894C2214yes—
52#894C2214yes—
53#893C2214yes—
54#893C2212yes—
55#892C2214yes—
56D3×C4⋊C4C2214yes—
57#889C2214yes—
58#887C2212yes—
59#886C2214yes—
60#884C2214yes—
61#884C2214yes—
62#884C2214yes—
63#884C2214yes—
64#884C2214yes—
65#883C2214yes—
66#67⋊C4C2214yes—
67#67⋊C4C2214yes—
68#67⋊C4C2214yes—
69#875C2212yes—
70#874C2211yescharacteristic
71#874C2212yes—
72C4×D12C2212yes—
73C4×D12C2211yescharacteristic
74C4×D12C2211yescharacteristic
75C4×D12C2212yes—
76C4×D12C2212yes—
77#872C2212yes—
78C42×D3C2212yes—
79#870C2212yes—
80#869C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2212yes—
84#557—31no—
85#553—34no—
86#553—32no—
87#550—32no—
88#550—31no—
89#550—31no—
90#550—32no—
91#547—32no—
92#547—31no—
93#540—34no—
94#540—32no—
95#532—32no—
96#532—31no—
97#529—32no—
98#529D311yescharacteristic, complemented
99#520—32no—
100C4×#41—31no—
101C2×C3×Q8C2311yescharacteristic
102C2×C3×D4C2311yescharacteristic
103C2×C3×D4C2312yes—
104C22×C3×C4C2312yes—
105C22×C3×C4C2311yescharacteristic
106C22×C3×C4C2311yescharacteristic
107C22×C3×C4C2311yescharacteristic
108C2×#67C2314yes—
109C2×#67C2312yes—
110C2×#67C2314yes—
111C2×#67C2314yes—
112C2×#67C2314yes—
113C22×Dic3C2314yes—
114#239—44no—
115#237—44no—
116#235—24no—
117#235—28no—
118#235—44no—
119C2×D12C2311yescharacteristic
120C2×D12C2312yes—
121C2×D12C2312yes—
122C2×D12C2312yes—
123C2×D12C2314yes—
124C2×D12C2314yes—
125C2×C4×D3C2312yes—
126C2×C4×D3C2312yes—
127C2×C4×D3C2314yes—
128C2×C4×D3C2312yes—
129C2×C4×D3C2314yes—
130C2×C4×D3C2314yes—
131C2×Dic6C2312yes—
132C2×Dic6C2311yescharacteristic
133C3×C4⋊C4C2311yescharacteristic
134C3×C4⋊C4C2312yes—
135C3×C4⋊C4C2312yes—
136C3×C4⋊C4C2312yes—
137C3×C4⋊C4C2311yescharacteristic
138C3×C4⋊C4C2312yes—
139C3×C22⋊C4C2312yes—
140C3×C22⋊C4C2312yes—
141C3×C22⋊C4C2312yes—
142C3×C22⋊C4C2312yes—
143C3×C22⋊C4C2312yes—
144C3×C42C2311yescharacteristic
145C3×C42C2312yes—
146C3×C42C2311yescharacteristic
147C3×C42C2311yescharacteristic
148C3×C42C2311yescharacteristic
149#217C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2312yes—
152(C2×D3)⋊C4C2314yes—
153(C2×D3)⋊C4C2314yes—
154(C2×D3)⋊C4C2314yes—
155(C2×D3)⋊C4C2314yes—
156(C2×D3)⋊C4C2314yes—
157C4⋊Dic3C2312yes—
158C4⋊Dic3C2311yescharacteristic
159C4⋊Dic3C2311yescharacteristic
160C4⋊Dic3C2312yes—
161C4⋊Dic3C2314yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2312yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166Dic3⋊C4C2314yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2314yes—
170C4×Dic3C2312yes—
171C2×#41—32no—
172C2×#41—32no—
173C2×#41—31no—
174C2×#41—32no—
175C2×#41—32no—
176C4⋊Q8—32no—
177C4⋊Q8—31no—
178#127—32no—
179#127—34no—
180#127—31no—
181#127—32no—
182#126—34no—
183#126C2×D312yes—
184#125—34no—
185#125—32no—
186#125C2×D311yescharacteristic
187#124—32no—
188#124C2×D311yescharacteristic
189#123—34no—
190#123—34no—
191#123—34no—
192#123—34no—
193#123C2×D312yes—
194C22⋊Q8—32no—
195C22⋊Q8C2×D311yescharacteristic
196#121—34no—
197#121—32no—
198#121—32no—
199#121—32no—
200#121—32no—
201#121—32no—
202#121—34no—
203#121—34no—
204#121—34no—
205#121—34no—
206#121—34no—
207#121—34no—
208#121C2×D311yescharacteristic
209C4×Q8—32no—
210C4×Q8—31no—
211C4×Q8—31no—
212C4×Q8—32no—
213C4×Q8C2×D311yescharacteristic
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—34no—
218C4×D4—34no—
219C4×D4—34no—
220C4×D4—34no—
221C4×D4—31no—
222C4×D4—31no—
223C4×D4—32no—
224C4×D4—32no—
225C4×D4—32no—
226C4×D4C2×D311yescharacteristic
227C4×D4C2×D312yes—
228#117—31no—
229#117—34no—
230#117—32no—
231#117C2×D312yes—
232C2×C4⋊C4—32no—
233C2×C4⋊C4—34no—
234C2×C42—32no—
235C2×C42C2×D311yescharacteristic
236C23×C3C2411yescharacteristic
237C23×C3C2412yes—
238C22×D3C2412yes—
239C22×D3C2414yes—
240C3×Q8—22no—
241C3×D4—22no—
242C3×D4—42no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—41no—
247C2×C3×C4C2412yes—
248C2×C3×C4C2411yescharacteristic
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2412yes—
251C2×C3×C4C2411yescharacteristic
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2412yes—
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—28no—
257#67—24no—
258#67—44no—
259#67—44no—
260#67—44no—
261C2×Dic3—44no—
262C2×Dic3C2412yes—
263C2×Dic3C2412yes—
264C2×Dic3C2414yes—
265C2×Dic3C2412yes—
266D12—22no—
267D12—24no—
268D12—24no—
269D12—42no—
270D12—44no—
271D12—44no—
272C4×D3—24no—
273C4×D3—24no—
274C4×D3—28no—
275C4×D3—42no—
276C4×D3—44no—
277C4×D3—44no—
278Dic6—24no—
279Dic6#4114yescomplemented
280#41—64no—
281#41—68no—
282#41—124no—
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×D4—32no—
296C2×D4—32no—
297C2×D4—34no—
298C2×D4—34no—
299C2×D4C22×D311yescharacteristic
300C2×D4C22×D312yes—
301C22×C4—34no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—34no—
305C22×C4—32no—
306C22×C4—34no—
307C22×C4—34no—
308C22×C4C22×D311yescharacteristic
309C22×C4C22×D311yescharacteristic
310C22×C4C22×D312yes—
311C22×C4C22×D311yescharacteristic
312C4⋊C4—32no—
313C4⋊C4—31no—
314C4⋊C4—34no—
315C4⋊C4—34no—
316C4⋊C4—32no—
317C4⋊C4—32no—
318C4⋊C4—31no—
319C4⋊C4—34no—
320C4⋊C4—32no—
321C4⋊C4—34no—
322C4⋊C4C22×D312yes—
323C4⋊C4C22×D312yes—
324C4⋊C4C22×D311yescharacteristic
325C4⋊C4C22×D312yes—
326C4⋊C4C22×D311yescharacteristic
327C4⋊C4C22×D312yes—
328C22⋊C4—32no—
329C22⋊C4—34no—
330C22⋊C4—32no—
331C22⋊C4—32no—
332C22⋊C4—34no—
333C22⋊C4—34no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4C22×D312yes—
337C22⋊C4C22×D312yes—
338C22⋊C4C22×D312yes—
339C22⋊C4C22×D312yes—
340C22⋊C4C22×D312yes—
341C42—32no—
342C42—32no—
343C42—32no—
344C42—34no—
345C42C22×D311yescharacteristic
346C42C22×D311yescharacteristic
347C42C22×D311yescharacteristic
348C42C22×D311yescharacteristic
349C42C22×D312yes—
350C22×C3—21no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3C2511yescharacteristic
357C2×D3—22no—
358C2×D3—24no—
359C2×D3—22no—
360C2×D3—24no—
361C2×D3—22no—
362C2×D3—24no—
363C3×C4—21no—
364C3×C4—22no—
365C3×C4—21no—
366C3×C4—22no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—22no—
370C3×C4C2×#4112yes—
371C3×C4H(5,𝔽2)12yes—
372Dic3—24no—
373Dic3—22no—
374Dic3—22no—
375Dic3C2×#4114yes—
376C23—32no—
377C23—34no—
378C23C23×D311yescharacteristic
379C23C23×D312yes—
380Q8—22no—
381Q8—34no—
382Q8—64no—
383D4—22no—
384D4—42no—
385D4—64no—
386D4—68no—
387D4—62no—
388D4—64no—
389D4—64no—
390D4—124no—
391D4—122no—
392D4—124no—
393D4—124no—
394D4—124no—
395D4—124no—
396C2×C4—22no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—34no—
403C2×C4—41no—
404C2×C4—64no—
405C2×C4—68no—
406C2×C4—64no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4—122no—
410C2×C4—124no—
411C2×C4C23×D311yescharacteristic
412C2×C4C23×D311yescharacteristic
413C2×C4C23×D312yes—
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D311yescharacteristic
416C2×C4C23×D312yes—
417C2×C4C23×D312yes—
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D311yescharacteristic
420C2×C3—42no—
421C2×C3—41no—
422C2×C3C22×#4111yescharacteristic
423C2×C3C2×H(5,𝔽2)11yescharacteristic
424C2×C3#58511yescharacteristic
425D3—42no—
426D3—44no—
427C22—21no—
428C22—22no—
429C22—21no—
430C22—22no—
431C22—21no—
432C22—22no—
433C22—62no—
434C22—64no—
435C22—62no—
436C22—64no—
437C22—62no—
438C22—64no—
439C22C24×D311yescharacteristic
440C4—21no—
441C4—22no—
442C4—21no—
443C4—22no—
444C4—21no—
445C4—21no—
446C4—22no—
447C4—34no—
448C4—62no—
449C4—64no—
450C4—62no—
451C4D3×#4112yes—
452C4#100912yes—
453C3#351711yescharacteristic, complemented, Sylow
454C2—41no—
455C2—42no—
456C2—122no—
457C2—124no—
458C2C2×D3×#4111yescharacteristic
459C2C2×#100911yescharacteristic
460C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#529C2
hypercenter#529D3
hypocenterC3#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #529

Complements

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

NH ≅ G/Ntype
1. C3×#529C2semidirect
2. #5709C2semidirect
3. #5691C2semidirect
4. #5668C2semidirect
5. #5663C2semidirect
6. #5631C2semidirect
7. #5599C2semidirect
8. #5577C2semidirect
9. #5567C2semidirect
10. #5565C2semidirect
11. #5555C2semidirect
12. #5553C2semidirect
13. #5553C2semidirect
14. #5535C2semidirect
15. #5523C2semidirect
16. #5477C2semidirect
17. C4×#235C2semidirect
98. #529D3semidirect
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×#5292
#57092
#56912
#56682
#56632
#56312
#55992
#55772
#55672
#55652
#55552
#55532
#55352
#55232
#54772
C4×#2352
#35173