Finite Group Databasebeta

← #86228

Subgroups — #86228

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×#532C211yesmaximal, characteristic, complemented
2#5834C212yesmaximal, complemented
3#5824C212yesmaximal, complemented
4#5726C211yesmaximal, characteristic, complemented
5#5721C211yesmaximal, characteristic, complemented
6#5718C211yesmaximal, characteristic, complemented
7#5718C211yesmaximal, characteristic, complemented
8#5680C212yesmaximal, complemented
9#5668C212yesmaximal, complemented
10#5615C214yesmaximal, complemented
11#5614C212yesmaximal, complemented
12#5599C212yesmaximal, complemented
13#5562C212yesmaximal, complemented
14#5546C212yesmaximal, complemented
15#5540C214yesmaximal, complemented
16#5479C211yesmaximal, characteristic, complemented
17C4×#235C211yesmaximal, characteristic, complemented
18#3517—31nomaximal, Sylow
19C2×C3×#41C2212yes—
20C2×#239C2212yes—
21C2×#237C2212yes—
22C2×#237C2212yes—
23C2×#235C2211yescharacteristic
24C3×C4⋊Q8C2211yescharacteristic
25C3×#127C2211yescharacteristic
26C3×#124C2212yes—
27C3×#121C2212yes—
28C3×#121C2212yes—
29C3×C4×D4C2212yes—
30C3×C4×D4C2212yes—
31C2×C3×C42C2211yescharacteristic
32#946C2212yes—
33Q8×Dic3C2212yes—
34#944C2212yes—
35#940C2212yes—
36#940C2212yes—
37#940C2214yes—
38#939C2214yes—
39#939C2214yes—
40#939C2214yes—
41#938C2212yes—
42#938C2212yes—
43#938C2214yes—
44#935C2214yes—
45D4×Dic3C2212yes—
46D4×Dic3C2212yes—
47#930C2211yescharacteristic
48#930C2214yes—
49#929C2212yes—
50#929C2214yes—
51C4×#67C2212yes—
52C4×#67C2212yes—
53C4×#67C2212yes—
54C4×#67C2214yes—
55#926C2211yescharacteristic
56#924C2211yescharacteristic
57C2×Dic3⋊C4C2214yes—
58#895C2212yes—
59#894C2214yes—
60D3×C4⋊C4C2212yes—
61#890C2212yes—
62#889C2214yes—
63#887C2212yes—
64#884C2214yes—
65#884C2214yes—
66#883C2214yes—
67#67⋊C4C2214yes—
68#67⋊C4C2214yes—
69#67⋊C4C2214yes—
70#879C2214yes—
71#877C2214yes—
72#876C2212yes—
73#875C2211yescharacteristic
74C4×D12C2211yescharacteristic
75C4×D12C2211yescharacteristic
76C4×D12C2211yescharacteristic
77#872C2212yes—
78#872C2212yes—
79C42×D3C2212yes—
80#870C2211yescharacteristic
81C4×Dic6C2211yescharacteristic
82C4×Dic6C2211yescharacteristic
83C4×Dic6C2211yescharacteristic
84#557—31no—
85#553—32no—
86#553—34no—
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#532D311yescharacteristic, complemented
97#529—32no—
98#529—31no—
99#520—32no—
100C4×#41—31no—
101C3×#41—44no—
102C2×C3×Q8C2312yes—
103C2×C3×D4C2312yes—
104C2×C3×D4C2312yes—
105C2×C3×D4C2314yes—
106C2×C3×D4C2312yes—
107C22×C3×C4C2311yescharacteristic
108C22×C3×C4C2311yescharacteristic
109C22×C3×C4C2314yes—
110C22×C3×C4C2311yescharacteristic
111C2×#67C2314yes—
112C2×#67C2312yes—
113C2×#67C2314yes—
114C2×#67C2314yes—
115C2×#67C2314yes—
116C2×#67C2312yes—
117C22×Dic3C2314yes—
118C22×Dic3C2314yes—
119#239—44no—
120#237—28no—
121#237—44no—
122#235—24no—
123C2×D12C2311yescharacteristic
124C2×D12C2311yescharacteristic
125C2×D12C2314yes—
126C2×C4×D3C2312yes—
127C2×C4×D3C2312yes—
128C2×C4×D3C2312yes—
129C2×C4×D3C2312yes—
130C2×C4×D3C2314yes—
131C2×Dic6C2311yescharacteristic
132C2×Dic6C2311yescharacteristic
133C3×C4⋊C4C2312yes—
134C3×C4⋊C4C2312yes—
135C3×C22⋊C4C2314yes—
136C3×C22⋊C4C2314yes—
137C3×C42C2311yescharacteristic
138C3×C42C2311yescharacteristic
139C3×C42C2311yescharacteristic
140C3×C42C2311yescharacteristic
141#217C2314yes—
142#217C2314yes—
143#217C2312yes—
144#217C2312yes—
145(C2×D3)⋊C4C2312yes—
146(C2×D3)⋊C4C2312yes—
147(C2×D3)⋊C4C2312yes—
148(C2×D3)⋊C4C2312yes—
149(C2×D3)⋊C4C2312yes—
150(C2×D3)⋊C4C2312yes—
151(C2×D3)⋊C4C2314yes—
152C4⋊Dic3C2311yescharacteristic
153C4⋊Dic3C2311yescharacteristic
154C4⋊Dic3C2311yescharacteristic
155C4⋊Dic3C2311yescharacteristic
156C4⋊Dic3C2314yes—
157Dic3⋊C4C2312yes—
158Dic3⋊C4C2312yes—
159Dic3⋊C4C2312yes—
160Dic3⋊C4C2312yes—
161Dic3⋊C4C2312yes—
162Dic3⋊C4C2312yes—
163Dic3⋊C4C2314yes—
164Dic3⋊C4C2314yes—
165Dic3⋊C4C2314yes—
166C4×Dic3C2312yes—
167C4×Dic3C2312yes—
168C4×Dic3C2312yes—
169C4×Dic3C2312yes—
170C4×Dic3C2314yes—
171C2×#41—32no—
172C2×#41—32no—
173C2×#41—32no—
174C2×#41—31no—
175C2×#41C2×D312yes—
176C4⋊Q8—32no—
177C4⋊Q8C2×D311yescharacteristic
178#127—32no—
179#127—34no—
180#127—32no—
181#127C2×D311yescharacteristic
182#126—34no—
183#126—32no—
184#125—34no—
185#125—32no—
186#125—31no—
187#124—31no—
188#124C2×D312yes—
189#123—34no—
190#123—32no—
191#123—34no—
192#123—34no—
193#123—34no—
194C22⋊Q8—32no—
195C22⋊Q8—31no—
196#121—32no—
197#121—34no—
198#121—34no—
199#121—32no—
200#121—34no—
201#121—34no—
202#121—31no—
203#121—34no—
204#121—32no—
205#121—34no—
206#121—34no—
207#121C2×D312yes—
208#121C2×D312yes—
209C4×Q8—32no—
210C4×Q8—32no—
211C4×Q8—31no—
212C4×Q8—31no—
213C4×Q8—31no—
214C4×D4—32no—
215C4×D4—32no—
216C4×D4—32no—
217C4×D4—32no—
218C4×D4—34no—
219C4×D4—32no—
220C4×D4—34no—
221C4×D4—34no—
222C4×D4—34no—
223C4×D4—31no—
224C4×D4—31no—
225C4×D4—31no—
226C4×D4C2×D312yes—
227C4×D4C2×D312yes—
228#117—31no—
229#117—34no—
230#117—32no—
231#117—32no—
232C2×C4⋊C4—34no—
233C2×C4⋊C4—32no—
234C2×C42—32no—
235C2×C42C2×D311yescharacteristic
236C23×C3C2411yescharacteristic
237C23×C3C2414yes—
238C22×D3C2412yes—
239C22×D3C2412yes—
240C3×Q8—24no—
241C3×D4—24no—
242C3×D4—24no—
243C3×D4—42no—
244C3×D4—44no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—44no—
249C2×C3×C4C2411yescharacteristic
250C2×C3×C4C2411yescharacteristic
251C2×C3×C4C2414yes—
252C2×C3×C4C2411yescharacteristic
253C2×C3×C4C2411yescharacteristic
254C2×C3×C4C2411yescharacteristic
255C2×C3×C4C2411yescharacteristic
256#67—24no—
257#67—28no—
258#67—44no—
259#67—44no—
260#67—44no—
261#67—42no—
262C2×Dic3—28no—
263C2×Dic3—44no—
264C2×Dic3C2412yes—
265C2×Dic3C2412yes—
266C2×Dic3C2412yes—
267C2×Dic3C2412yes—
268C2×Dic3C2412yes—
269C2×Dic3C2412yes—
270D12—22no—
271D12—22no—
272D12—44no—
273C4×D3—24no—
274C4×D3—24no—
275C4×D3—24no—
276C4×D3—42no—
277C4×D3—44no—
278Dic6—22no—
279Dic6#4114yescomplemented
280#41—44no—
281#41—64no—
282#41—68no—
283#41—124no—
284#41—124no—
285C2×Q8—31no—
286C2×Q8—31no—
287C2×Q8C22×D312yes—
288C2×D4—32no—
289C2×D4—34no—
290C2×D4—34no—
291C2×D4—32no—
292C2×D4—34no—
293C2×D4—34no—
294C2×D4—31no—
295C2×D4—31no—
296C2×D4—34no—
297C2×D4C22×D312yes—
298C2×D4C22×D312yes—
299C2×D4C22×D312yes—
300C2×D4C22×D314yes—
301C22×C4—34no—
302C22×C4—34no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—34no—
308C22×C4C22×D311yescharacteristic
309C22×C4C22×D311yescharacteristic
310C22×C4C22×D311yescharacteristic
311C22×C4C22×D314yes—
312C4⋊C4—32no—
313C4⋊C4—31no—
314C4⋊C4—34no—
315C4⋊C4—32no—
316C4⋊C4—34no—
317C4⋊C4—32no—
318C4⋊C4—32no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—32no—
322C4⋊C4—34no—
323C4⋊C4—31no—
324C4⋊C4—32no—
325C4⋊C4—34no—
326C4⋊C4C22×D312yes—
327C4⋊C4C22×D312yes—
328C22⋊C4—34no—
329C22⋊C4—32no—
330C22⋊C4—32no—
331C22⋊C4—34no—
332C22⋊C4—32no—
333C22⋊C4—32no—
334C22⋊C4—32no—
335C22⋊C4—32no—
336C22⋊C4—32no—
337C22⋊C4—34no—
338C22⋊C4—32no—
339C22⋊C4C22×D314yes—
340C22⋊C4C22×D314yes—
341C42—32no—
342C42—32no—
343C42—32no—
344C42—34no—
345C42—32no—
346C42C22×D311yescharacteristic
347C42C22×D311yescharacteristic
348C42C22×D311yescharacteristic
349C42C22×D311yescharacteristic
350C22×C3—24no—
351C22×C3—21no—
352C22×C3—24no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—24no—
356C22×C3C2511yescharacteristic
357C2×D3—22no—
358C2×D3—22no—
359C2×D3—22no—
360C2×D3—22no—
361C2×D3—22no—
362C2×D3—22no—
363C3×C4—21no—
364C3×C4—24no—
365C3×C4—21no—
366C3×C4—21no—
367C3×C4—21no—
368C3×C4C2×#4112yes—
369C3×C4H(5,𝔽2)12yes—
370Dic3—22no—
371Dic3—22no—
372Dic3—22no—
373Dic3—22no—
374Dic3—22no—
375Dic3C2×#4114yes—
376C23—32no—
377C23—32no—
378C23C23×D311yescharacteristic
379C23C23×D314yes—
380Q8—24no—
381Q8—34no—
382Q8—62no—
383D4—24no—
384D4—24no—
385D4—44no—
386D4—42no—
387D4—64no—
388D4—68no—
389D4—62no—
390D4—62no—
391D4—124no—
392D4—124no—
393D4—124no—
394D4—122no—
395D4—124no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—44no—
405C2×C4—41no—
406C2×C4—64no—
407C2×C4—68no—
408C2×C4—64no—
409C2×C4—64no—
410C2×C4—124no—
411C2×C4—122no—
412C2×C4—124no—
413C2×C4C23×D311yescharacteristic
414C2×C4C23×D311yescharacteristic
415C2×C4C23×D314yes—
416C2×C4C23×D311yescharacteristic
417C2×C4C23×D311yescharacteristic
418C2×C4C23×D311yescharacteristic
419C2×C4C23×D311yescharacteristic
420C2×C3—44no—
421C2×C3—41no—
422C2×C3C22×#4111yescharacteristic
423C2×C3C2×H(5,𝔽2)11yescharacteristic
424C2×C3#58511yescharacteristic
425D3—42no—
426D3—42no—
427C22—21no—
428C22—24no—
429C22—21no—
430C22—24no—
431C22—21no—
432C22—24no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22C24×D311yescharacteristic
440C4—21no—
441C4—24no—
442C4—21no—
443C4—21no—
444C4—21no—
445C4—34no—
446C4—62no—
447C4—62no—
448C4—62no—
449C4—62no—
450C4—62no—
451C4#100412yes—
452C4D3×#4112yes—
453C3#351711yescharacteristic, complemented, Sylow
454C2—41no—
455C2—44no—
456C2—122no—
457C2—122no—
458C2C2×#100411yescharacteristic
459C2C2×D3×#4111yescharacteristic
460C2#596411yescharacteristic
subgroupG/H
center, FrattiniC22C24×D3
commutator, socle, cosocleC22×C3C25
FittingC3×#532C2
hypercenter#532D3
hypocenterC3#3517
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. C3×#532C2semidirect
2. #5834C2semidirect
3. #5824C2semidirect
4. #5726C2semidirect
5. #5721C2semidirect
6. #5718C2semidirect
7. #5718C2semidirect
8. #5680C2semidirect
9. #5668C2semidirect
10. #5615C2semidirect
11. #5614C2semidirect
12. #5599C2semidirect
13. #5562C2semidirect
14. #5546C2semidirect
15. #5540C2semidirect
16. #5479C2semidirect
17. C4×#235C2semidirect
96. #532D3semidirect
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×#5322
#58342
#58242
#57262
#57212
#57182
#56802
#56682
#56152
#56142
#55992
#55622
#55462
#55402
#54792
C4×#2352
#35173