Finite Group Databasebeta

← #79882

Subgroups — #79882

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#5558C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3C3×#442C211yesmaximal, characteristic, complemented
4#5156C211yesmaximal, characteristic, complemented
5#5156C211yesmaximal, characteristic, complemented
6#5015C211yesmaximal, characteristic, complemented
7#5009C211yesmaximal, characteristic, complemented
8#4866C211yesmaximal, characteristic, complemented
9#4863C211yesmaximal, characteristic, complemented
10#4782C211yesmaximal, characteristic, complemented
11#4775C211yesmaximal, characteristic, complemented
12#4770C211yesmaximal, characteristic, complemented
13D3×#102C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C2×D3×D4C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C22×D12—21no—
20C2×C3×D8C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#938C2211yescharacteristic, complemented
24#938C2211yescharacteristic, complemented
25C2×#213C2211yescharacteristic, complemented
26C2×#213C2211yescharacteristic, complemented
27#930—21no—
28#929—21no—
29C4×#67—21no—
30C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
31C2×D3×C8C2211yescharacteristic, complemented
32#895C2211yescharacteristic
33#895C2211yescharacteristic
34#892C2211yescharacteristic, complemented
35D3×C4⋊C4C2211yescharacteristic, complemented
36#884—21no—
37#882—21no—
38#882—21no—
39#879—21no—
40#874C2211yescharacteristic
41C4×D12C2211yescharacteristic
42#870C2211yescharacteristic, complemented
43C3×#106C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#809C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#808C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59C4⋊(C3⋊C8)C2211yescharacteristic, 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#459—31no—
69#447—31no—
70#447—31no—
71#442D311yescharacteristic, complemented
72#435—31no—
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85D3×D4—22no—
86D3×D4—22no—
87D3×D4—41no—
88D3×D4—41no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—21no—
93C2×D12—44no—
94C2×D12D411yescharacteristic
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×C4×D3C2311yescharacteristic
100C3×D8—22no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C42C2311yescharacteristic
108#217—21no—
109#217—21no—
110#213—41no—
111#213—41no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4D411yescharacteristic
116(C2×D3)⋊C4D411yescharacteristic
117C4⋊Dic3D411yescharacteristic
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×D8—31no—
133C2×D8C2×D311yescharacteristic
134C2×M16—31no—
135C22×C8—31no—
136#127—31no—
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—61no—
150C4×D4C2×D311yescharacteristic
151C4×D4C2×D311yescharacteristic, complemented
152#117—31no—
153C2×C4⋊C4—31no—
154#107—31no—
155#106—31no—
156#106C2×D311yescharacteristic, complemented
157C4⋊C8—31no—
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
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97C2×D311yescharacteristic, complemented
171C23×C3—21no—
172C23×C3—21no—
173C22×D3—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#4112yes—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—42no—
190C2×C3×C4—81no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—42no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202D12—22no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—42no—
208D12—42no—
209D12—42no—
210D12—81no—
211C4×D3—21no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—81no—
216C3×C8—21no—
217C3×C8—21no—
218C3⋊C8—21no—
219C3⋊C8—21no—
220C24—61no—
221C24—61no—
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—62no—
230C2×D4—62no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—124no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4C22×D311yescharacteristic
239C2×D4C22×D311yescharacteristic
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—61no—
245D8—22no—
246D8—121no—
247D8—121no—
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—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4C22×D311yescharacteristic
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—31no—
265C22⋊C4—31no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—22no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3C22×D411yescharacteristic
280C2×D3—21no—
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
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×C4—41no—
298C3×C4C2×#4111yescharacteristic
299C3×C4H(5,𝔽2)11yescharacteristic
300Dic3—21no—
301Dic3—21no—
302Dic3—41no—
303Dic3—41no—
304C23—21no—
305C23—21no—
306C23—31no—
307C23—31no—
308C23—61no—
309C23—61no—
310C23—122no—
311C23—122no—
312C23—122no—
313C23—122no—
314C23—241no—
315C23—241no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—62no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—61no—
324D4—122no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—241no—
331D4—241no—
332D4#23512yes—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—42no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—81no—
345C2×C4—241no—
346C2×C4D3×D411yescharacteristic
347C2×C4D3×D411yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—61no—
352C8—61no—
353C2×C3—42no—
354C2×C3—81no—
355C2×C3#54611yescharacteristic
356C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
357C2×C3#57611yescharacteristic
358D3—22no—
359D3—41no—
360D3—81no—
361D3—81no—
362C22—22no—
363C22—32no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—61no—
370C22—61no—
371C22—61no—
372C22—61no—
373C22—61no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—244no—
381C22—244no—
382C22C2×D3×D411yescharacteristic
383C4—21no—
384C4—21no—
385C4—41no—
386C4—41no—
387C4—61no—
388C4—61no—
389C4—121no—
390C4—121no—
391C4C2×#23511yescharacteristic
392C4#100911yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—42no—
395C2—62no—
396C2—81no—
397C2—121no—
398C2—241no—
399C2—241no—
400C2#555011yescharacteristic
401C2D3×AGL(1,ℤ/8)11yescharacteristic
402C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. #5558C2semidirect
2. D4×D12C2semidirect
3. C3×#442C2semidirect
4. #5156C2semidirect
5. #5156C2semidirect
6. #5015C2semidirect
7. #5009C2semidirect
8. #4866C2semidirect
9. #4863C2semidirect
10. #4782C2semidirect
11. #4775C2semidirect
12. #4770C2semidirect
13. D3×#102C2semidirect
14. #4713C2semidirect
15. #4706C2semidirect
22. C3×C4×D4C22semidirect
23. #938C22semidirect
24. #938C22semidirect
25. C2×#213C22semidirect
26. C2×#213C22semidirect
30. C2×ΓL(1,𝔽25)C22semidirect
31. C2×D3×C8C22semidirect
34. #892C22semidirect
35. D3×C4⋊C4C22semidirect
42. #870C22semidirect
43. C3×#106C22semidirect
45. C3×#102C22semidirect
46. C3×#97C22semidirect
49. #821C22semidirect
50. #821C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #809C22semidirect
56. #809C22semidirect
57. #808C22semidirect
58. #807C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
71. #442D3semidirect
151. C4×D4C2×D3semidirect
156. #106C2×D3semidirect
167. #102C2×D3semidirect
170. #97C2×D3semidirect
393. C3#3318semidirect

Maximal subgroups

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

subgroupindex
#55582
D4×D122
C3×#4422
#51562
#50152
#50092
#48662
#48632
#47822
#47752
#47702
D3×#1022
#47132
#47062
#33183