Finite Group Databasebeta

← #81399

Subgroups — #81399

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
1#5608C211yesmaximal, characteristic, complemented
2#5525C211yesmaximal, characteristic, complemented
3C3×#482C211yesmaximal, characteristic, complemented
4#5036C211yesmaximal, characteristic, complemented
5#4996C211yesmaximal, characteristic, complemented
6#4886C211yesmaximal, characteristic, complemented
7#4882C211yesmaximal, characteristic, complemented
8#4864C211yesmaximal, characteristic, complemented
9#4863C211yesmaximal, characteristic, complemented
10#4782C211yesmaximal, characteristic, complemented
11#4781C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4731C211yesmaximal, characteristic, complemented
15#4727C211yesmaximal, characteristic, complemented
16#3198—31nomaximal, Sylow
17C2×#237C2211yescharacteristic, complemented
18C2×#235C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C3×#121C2211yescharacteristic
21C3×#117C2211yescharacteristic
22#939—21no—
23#938—21no—
24#938C2211yescharacteristic, complemented
25C2×#213C2211yescharacteristic
26C2×#213C2211yescharacteristic
27#924C2211yescharacteristic
28C2×#204C2211yescharacteristic
29C2×#204C2211yescharacteristic
30C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
31C2×D3×C8C2211yescharacteristic, complemented
32#897C2211yescharacteristic, complemented
33#895C2211yescharacteristic
34#895C2211yescharacteristic
35#895C2211yescharacteristic
36#895C2211yescharacteristic
37D3×C4⋊C4C2211yescharacteristic, complemented
38#884—21no—
39#882—21no—
40#882—21no—
41#878—21no—
42#875—21no—
43#874—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#106C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic, complemented
48C3×C22⋊C8C2211yescharacteristic
49#830C2211yescharacteristic
50#821C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54#810C2211yescharacteristic, complemented
55#810C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#537—31no—
61#535—31no—
62#485—31no—
63#482D311yescharacteristic, complemented
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—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67D411yescharacteristic
82C2×#67D411yescharacteristic
83C22×Dic3—21no—
84#237—41no—
85#237—41no—
86#235—41no—
87#235—41no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—41no—
94C2×D12—41no—
95C2×D12D411yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×Dic6D411yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C3×C4⋊C4C2311yescharacteristic
106C3×C4⋊C4C2311yescharacteristic
107C3×C4⋊C4C2311yescharacteristic
108C3×C22⋊C4—21no—
109C3×C22⋊C4—21no—
110C3×C42—21no—
111#217—21no—
112#217—21no—
113#213—41no—
114#213—41no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127#204—41no—
128ΓL(1,𝔽25)—41no—
129D3×C8—41no—
130C2×#41—31no—
131C2×#41—31no—
132C22×D4—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×M16—31no—
138C22×C8—31no—
139#127—61no—
140#124—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143C22⋊Q8—61no—
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121C2×D311yescharacteristic
153#120—61no—
154#120—61no—
155#117C2×D311yescharacteristic
156C2×C4⋊C4—31no—
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic, complemented
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8C2×D311yescharacteristic
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C22×D3C2×D411yescharacteristic
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—81no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—41no—
193#67—41no—
194#67—42no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—42no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—41no—
206D12—42no—
207D12—81no—
208D12—81no—
209D12—81no—
210D12—81no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—81no—
216Dic6—21no—
217Dic6—21no—
218Dic6—41no—
219C3×C8—21no—
220C3×C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C24—61no—
224#41—121no—
225#41—121no—
226#41—121no—
227#41—121no—
228C2×Q8—31no—
229C2×Q8—31no—
230C2×D4—21no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4C22×D311yescharacteristic
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C22×D311yescharacteristic
251SD16—121no—
252SD16—121no—
253D8—121no—
254D8—121no—
255M16—121no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—121no—
259C2×C8C22×D311yescharacteristic
260C2×C8C22×D311yescharacteristic
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4—61no—
265C4⋊C4C22×D311yescharacteristic
266C4⋊C4C22×D311yescharacteristic
267C4⋊C4C22×D311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
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—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—84no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4H(5,𝔽2)11yescharacteristic
303C3×C4H(5,𝔽2)11yescharacteristic
304Dic3—21no—
305Dic3—21no—
306Dic3—41no—
307Dic3—41no—
308C23—21no—
309C23—31no—
310C23—31no—
311C23—61no—
312C23—61no—
313C23—122no—
314C23—241no—
315C23—241no—
316C23D3×D411yescharacteristic
317Q8—61no—
318Q8—61no—
319Q8—121no—
320D4—21no—
321D4—21no—
322D4—61no—
323D4—61no—
324D4—61no—
325D4—61no—
326D4—81no—
327D4—121no—
328D4—122no—
329D4—121no—
330D4—121no—
331D4—122no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—122no—
351C2×C4—241no—
352C2×C4D3×D411yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—21no—
355C8—21no—
356C8—61no—
357C8—61no—
358C2×C3—41no—
359C2×C3—81no—
360C2×C3#53411yescharacteristic
361C2×C3#57611yescharacteristic
362C2×C3#57711yescharacteristic
363D3—41no—
364D3—41no—
365D3—81no—
366D3—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—61no—
392C4—61no—
393C4—121no—
394C4—121no—
395C4#100911yescharacteristic
396C4#100911yescharacteristic
397C3#319811yescharacteristic, complemented, Sylow
398C2—41no—
399C2—81no—
400C2—121no—
401C2—121no—
402C2—241no—
403C2—241no—
404C2#565311yescharacteristic
405C2#577011yescharacteristic
406C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#482C2
socleC22×C3C22×D4
hypercenter#482D3
hypocenterC3#3198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #482

Complements

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

NH ≅ G/Ntype
1. #5608C2semidirect
2. #5525C2semidirect
3. C3×#482C2semidirect
4. #5036C2semidirect
5. #4996C2semidirect
6. #4886C2semidirect
7. #4882C2semidirect
8. #4864C2semidirect
9. #4863C2semidirect
10. #4782C2semidirect
11. #4781C2semidirect
12. #4781C2semidirect
13. #4775C2semidirect
14. #4731C2semidirect
15. #4727C2semidirect
17. C2×#237C22semidirect
24. #938C22semidirect
30. C2×ΓL(1,𝔽25)C22semidirect
31. C2×D3×C8C22semidirect
32. #897C22semidirect
37. D3×C4⋊C4C22semidirect
44. C3×#107C22semidirect
45. C3×#106C22semidirect
46. C3×#102C22semidirect
47. C3×#102C22semidirect
52. (C2×D3)⋊C8C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. #810C22semidirect
55. #810C22semidirect
58. #808C22semidirect
59. #807C22semidirect
63. #482D3semidirect
158. #107C2×D3semidirect
160. #106C2×D3semidirect
167. #102C2×D3semidirect
168. #102C2×D3semidirect
397. C3#3198semidirect

Maximal subgroups

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

subgroupindex
#56082
#55252
C3×#4822
#50362
#49962
#48862
#48822
#48642
#48632
#47822
#47812
#47752
#47312
#47272
#31983