Finite Group Databasebeta

← #80355

Subgroups — #80355

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5717C211yesmaximal, characteristic, complemented
2#5558C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5159C211yesmaximal, characteristic, complemented
5#5157C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5071C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4847C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4784C211yesmaximal, characteristic, complemented
13#4777C211yesmaximal, characteristic, complemented
14#4770C211yesmaximal, characteristic, complemented
15D3×#102C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C2×D3×D4—21no—
18C2×D3×D4C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic, complemented
22C3×#127C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24#938—21no—
25#938C2211yescharacteristic
26#938C2211yescharacteristic
27#938C2211yescharacteristic, complemented
28#938C2211yescharacteristic, complemented
29#937—21no—
30#937—21no—
31C2×#213C2211yescharacteristic
32#929—21no—
33C4×#67—21no—
34C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
35C2×D3×C8C2211yescharacteristic, complemented
36#892C2211yescharacteristic, complemented
37D3×C4⋊C4C2211yescharacteristic, complemented
38#884—21no—
39#879—21no—
40C4×D12C2211yescharacteristic
41C4×D12C2211yescharacteristic
42#870C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic, complemented
46#832C2211yescharacteristic, complemented
47#832C2211yescharacteristic, complemented
48#832C2211yescharacteristic, complemented
49#832C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#818C2211yescharacteristic, complemented
56#817C2211yescharacteristic, complemented
57#809C2211yescharacteristic
58#808C2211yescharacteristic, complemented
59#803C2211yescharacteristic, complemented
60#553—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#481—31no—
65#478—31no—
66#469—31no—
67#466—31no—
68#459D311yescharacteristic, complemented
69#447—31no—
70#447—31no—
71#442—31no—
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89D3×D4—22no—
90D3×D4—22no—
91D3×D4—41no—
92D3×D4—41no—
93D3×D4—44no—
94C2×D12D411yescharacteristic
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C3×D8—41no—
101C3×D8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C42C2311yescharacteristic
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#213—22no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115(C2×D3)⋊C4D411yescharacteristic
116C4⋊Dic3D411yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126ΓL(1,𝔽25)—41no—
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×D8—31no—
132C2×D8C2×D311yescharacteristic, complemented
133C2×D8C2×D311yescharacteristic, complemented
134C2×M16—31no—
135C22×C8—31no—
136#127C2×D311yescharacteristic
137#126—61no—
138#125—31no—
139#123—61no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—31no—
144#121—61no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—61no—
151C4×D4C2×D311yescharacteristic, complemented
152#117—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#106—31no—
156#106—31no—
157C4⋊C8C2×D311yescharacteristic, complemented
158#102—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102C2×D311yescharacteristic, complemented
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C22×D3C2×D411yescharacteristic
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—42no—
190C3×D4—81no—
191C2×C3×C4—21no—
192C2×C3×C4—81no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196#67—42no—
197#67—42no—
198#67—42no—
199#67—42no—
200#67—42no—
201#67—81no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207D12—22no—
208D12—21no—
209D12#4112yes—
210C4×D3—21no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—42no—
215C4×D3—81no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221C24—61no—
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—31no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—121no—
236C2×D4—124no—
237C2×D4C22×D311yescharacteristic
238C2×D4C22×D311yescharacteristic
239C2×D4C22×D311yescharacteristic
240C22×C4—21no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—61no—
244C22×C4—61no—
245D8—41no—
246D8—41no—
247D8—62no—
248M16—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—121no—
252C2×C8C22×D311yescharacteristic
253C2×C8C22×D311yescharacteristic
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—31no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—31no—
264C22⋊C4—31no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3C22×D411yescharacteristic
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—21no—
286C2×D3—22no—
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—84no—
292C2×D3—84no—
293C2×D3AGL(1,ℤ/8)12yes—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4C2×#4111yescharacteristic
298C3×C4H(5,𝔽2)11yescharacteristic
299Dic3—21no—
300Dic3—21no—
301Dic3—41no—
302Dic3—41no—
303Dic3—41no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—31no—
308C23—31no—
309C23—61no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—122no—
314C23—241no—
315C23—241no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—32no—
323D4—42no—
324D4—62no—
325D4—61no—
326D4—81no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—122no—
332D4—241no—
333C2×C4—21no—
334C2×C4—31no—
335C2×C4—31no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—81no—
344C2×C4—122no—
345C2×C4—241no—
346C2×C4D3×D411yescharacteristic
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—81no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3#54611yescharacteristic
357C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C2×C3#57611yescharacteristic
359D3—22no—
360D3—41no—
361D3—42no—
362C22—32no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—61no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—62no—
378C22—121no—
379C22—121no—
380C22—244no—
381C22—244no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—21no—
385C4—41no—
386C4—61no—
387C4—61no—
388C4—121no—
389C4—121no—
390C4—121no—
391C4#100411yescharacteristic
392C4D3×#4111yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—62no—
395C2—81no—
396C2—81no—
397C2—81no—
398C2—121no—
399C2—122no—
400C2#561211yescharacteristic
401C2#575711yescharacteristic
402C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #5717C2semidirect
2. #5558C2semidirect
3. C3×#459C2semidirect
4. #5159C2semidirect
5. #5157C2semidirect
6. #5156C2semidirect
7. #5156C2semidirect
8. #5071C2semidirect
9. #5014C2semidirect
10. #4847C2semidirect
11. #4836C2semidirect
12. #4784C2semidirect
13. #4777C2semidirect
14. #4770C2semidirect
15. D3×#102C2semidirect
20. C2×C3×D8C22semidirect
21. C2×C3×D8C22semidirect
23. C3×C4×D4C22semidirect
27. #938C22semidirect
28. #938C22semidirect
34. C2×ΓL(1,𝔽25)C22semidirect
35. C2×D3×C8C22semidirect
36. #892C22semidirect
37. D3×C4⋊C4C22semidirect
42. #870C22semidirect
43. C3×C4⋊C8C22semidirect
44. C3×#102C22semidirect
45. C3×#102C22semidirect
46. #832C22semidirect
47. #832C22semidirect
48. #832C22semidirect
49. #832C22semidirect
50. #832C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #818C22semidirect
56. #817C22semidirect
58. #808C22semidirect
59. #803C22semidirect
68. #459D3semidirect
132. C2×D8C2×D3semidirect
133. C2×D8C2×D3semidirect
151. C4×D4C2×D3semidirect
157. C4⋊C8C2×D3semidirect
166. #102C2×D3semidirect
167. #102C2×D3semidirect
393. C3#3318semidirect

Maximal subgroups

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

subgroupindex
#57172
#55582
C3×#4592
#51592
#51572
#51562
#50712
#50142
#48472
#48362
#47842
#47772
#47702
D3×#1022
#33183