Finite Group Databasebeta

← #83386

Subgroups — #83386

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#535C211yesmaximal, characteristic, complemented
2#5833C211yesmaximal, characteristic, complemented
3#5234C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5217C211yesmaximal, characteristic, complemented
6#5042C211yesmaximal, characteristic, complemented
7#5041C211yesmaximal, characteristic, complemented
8#5039C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#5037C211yesmaximal, characteristic, complemented
11#5037C211yesmaximal, characteristic, complemented
12#5034C211yesmaximal, characteristic, complemented
13#5033C211yesmaximal, characteristic, complemented
14#5003C211yesmaximal, characteristic, complemented
15#4996C211yesmaximal, characteristic, complemented
16#3198—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic, complemented
20C3×#127—21no—
21C3×#124—21no—
22C3×#121C2211yescharacteristic
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic
25C3×#121C2211yescharacteristic
26C3×#120—21no—
27C3×#120—21no—
28C3×#117C2211yescharacteristic
29#946—21no—
30#939—21no—
31#939—21no—
32#938—21no—
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#213C2211yescharacteristic
36C2×#213C2211yescharacteristic
37#930C2211yescharacteristic
38#930C2211yescharacteristic, complemented
39C2×C4⋊Dic3C2211yescharacteristic, complemented
40#924C2211yescharacteristic
41#924C2211yescharacteristic, complemented
42C2×#208C2211yescharacteristic, complemented
43C22×C3⋊C8C2211yescharacteristic, complemented
44#832C2211yescharacteristic
45#832C2211yescharacteristic
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#830C2211yescharacteristic
49#830C2211yescharacteristic
50#830C2211yescharacteristic, complemented
51#830C2211yescharacteristic, complemented
52#810C2211yescharacteristic, complemented
53#810C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#808C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#537—31no—
61#535D311yescharacteristic, complemented
62#485—31no—
63#482—31no—
64#481—31no—
65#480—31no—
66#469—31no—
67#468—31no—
68#466—31no—
69#465—31no—
70#450—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74#408—31no—
75C24×C3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8D411yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—41no—
85C2×C3×D4—41no—
86C2×C3×D4D411yescharacteristic
87C2×C3×D4D411yescharacteristic
88C2×C3×D4D411yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C2×C3×D4C2311yescharacteristic
91C22×C3×C4C2311yescharacteristic
92C22×C3×C4C2311yescharacteristic
93C22×C3×C4C2311yescharacteristic
94C2×#67—21no—
95C2×#67—21no—
96C2×#67—21no—
97C22×Dic3—21no—
98#235—41no—
99#235—41no—
100C2×D12C2311yescharacteristic
101C2×C4×D3—21no—
102C2×Dic6C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C42—21no—
111#217—21no—
112#217—21no—
113#214—41no—
114#214—41no—
115#213—41no—
116#213—41no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124#208—41no—
125C2×C3⋊C8—41no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130C2×#41—31no—
131C2×#41C2×D311yescharacteristic
132C22×D4C2×D311yescharacteristic
133C2×SD16—31no—
134C2×SD16—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×M16—31no—
138C22×C8—31no—
139#127—21no—
140#124—21no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—61no—
147#121—61no—
148#121—61no—
149#121C2×D311yescharacteristic
150#121C2×D311yescharacteristic
151#121C2×D311yescharacteristic
152#121C2×D311yescharacteristic
153#120—21no—
154#120—21no—
155#117C2×D311yescharacteristic
156C2×C4⋊C4—31no—
157#107—31no—
158#107—31no—
159#106—31no—
160#106—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—42no—
176C23×C3—81no—
177C23×C3—81no—
178C23×C3C2×D411yescharacteristic
179C23×C3C2×D411yescharacteristic
180C23×C3C2×D411yescharacteristic
181C22×D3—21no—
182C3×Q8—41no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—41no—
188C3×D4—41no—
189C3×D4—41no—
190C3×D4—42no—
191C3×D4—81no—
192C3×D4—81no—
193C3×D4—81no—
194C3×D4—81no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2411yescharacteristic
207#67—42no—
208#67—81no—
209#67—81no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—81no—
214D12—21no—
215D12—21no—
216C4×D3—42no—
217Dic6—21no—
218Dic6—21no—
219C3⋊C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C24—21no—
224#41—41no—
225#41—41no—
226#41—121no—
227#41—121no—
228C2×Q8—31no—
229C2×Q8#6711yescharacteristic
230C2×D4—21no—
231C2×D4—21no—
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—21no—
235C2×D4—31no—
236C2×D4—41no—
237C2×D4—41no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4#6711yescharacteristic
242C2×D4#6711yescharacteristic
243C2×D4#6711yescharacteristic
244C2×D4C22×D311yescharacteristic
245C2×D4C22×D311yescharacteristic
246C22×C4—61no—
247C22×C4—61no—
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251SD16—121no—
252SD16—121no—
253D8—121no—
254D8—121no—
255M16—121no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—31no—
259C2×C8—31no—
260C2×C8—121no—
261C4⋊C4—31no—
262C4⋊C4—31no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4C22×D311yescharacteristic
267C4⋊C4C22×D311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—21no—
271C22⋊C4—21no—
272C22⋊C4—21no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—84no—
294C22×C3C22×D411yescharacteristic
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4H(5,𝔽2)11yescharacteristic
304C3×C4H(5,𝔽2)11yescharacteristic
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—61no—
312C23—81no—
313C23—81no—
314C23C2×#6711yescharacteristic
315C23C2×#6711yescharacteristic
316C23C2×#6711yescharacteristic
317Q8—41no—
318Q8—61no—
319Q8—61no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—42no—
328D4—61no—
329D4—61no—
330D4—81no—
331D4—81no—
332D4—81no—
333D4—81no—
334D4—122no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—122no—
349C2×C4—241no—
350C2×C4C2×#6711yescharacteristic
351C2×C4C2×#6711yescharacteristic
352C2×C4C2×#6711yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—61no—
355C8—61no—
356C8—61no—
357C8—61no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—81no—
362C2×C3—81no—
363C2×C3#53411yescharacteristic
364C2×C3#57611yescharacteristic
365C2×C3#57711yescharacteristic
366D3—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—41no—
377C22—41no—
378C22—41no—
379C22—41no—
380C22—41no—
381C22—41no—
382C22—84no—
383C22—121no—
384C22—121no—
385C22—121no—
386C22C22×#6711yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—121no—
393C4—121no—
394C4—121no—
395C4#100411yescharacteristic
396C4#100411yescharacteristic
397C3#319811yescharacteristic, complemented, Sylow
398C2—41no—
399C2—41no—
400C2—41no—
401C2—81no—
402C2—81no—
403C2—241no—
404C2#580411yescharacteristic
405C2#583511yescharacteristic
406C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5833C2semidirect
3. #5234C2semidirect
4. #5218C2semidirect
5. #5217C2semidirect
6. #5042C2semidirect
7. #5041C2semidirect
8. #5039C2semidirect
9. #5038C2semidirect
10. #5037C2semidirect
11. #5037C2semidirect
12. #5034C2semidirect
13. #5033C2semidirect
14. #5003C2semidirect
15. #4996C2semidirect
19. C2×#235C22semidirect
38. #930C22semidirect
39. C2×C4⋊Dic3C22semidirect
41. #924C22semidirect
42. C2×#208C22semidirect
43. C22×C3⋊C8C22semidirect
50. #830C22semidirect
51. #830C22semidirect
52. #810C22semidirect
53. #810C22semidirect
54. #809C22semidirect
55. #809C22semidirect
56. #808C22semidirect
57. #808C22semidirect
58. #807C22semidirect
59. #807C22semidirect
61. #535D3semidirect
397. C3#3198semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#58332
#52342
#52182
#52172
#50422
#50412
#50392
#50382
#50372
#50342
#50332
#50032
#49962
#31983