Finite Group Databasebeta

← #79505

Subgroups — #79505

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#5565C211yesmaximal, characteristic, complemented
2D4×D12C211yesmaximal, characteristic, complemented
3C3×#435C211yesmaximal, characteristic, complemented
4#5135C211yesmaximal, characteristic, complemented
5#5131C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7C2×#821C211yesmaximal, characteristic, complemented
8#4847C211yesmaximal, characteristic, complemented
9#4843C211yesmaximal, characteristic, complemented
10#4737C211yesmaximal, characteristic, complemented
11#4735C211yesmaximal, characteristic, complemented
12#4732C211yesmaximal, characteristic, complemented
13#4732C211yesmaximal, characteristic, complemented
14#4713C211yesmaximal, characteristic, complemented
15#4711C211yesmaximal, characteristic, complemented
16#3318—31nomaximal, Sylow
17C2×D3×D4—21no—
18C22×D12C2211yescharacteristic
19C22×D12C2211yescharacteristic
20C2×C3×M16C2211yescharacteristic, complemented
21C22×C3×C8C2211yescharacteristic, complemented
22C3×C4×D4C2211yescharacteristic
23#939—21no—
24#930C2211yescharacteristic
25#930C2211yescharacteristic
26#930C2211yescharacteristic, complemented
27#930C2211yescharacteristic, complemented
28#926C2211yescharacteristic, complemented
29C2×C4⋊Dic3C2211yescharacteristic, complemented
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic, complemented
32C2×D24C2211yescharacteristic, complemented
33#898—21no—
34#895—21no—
35#883—21no—
36#882—21no—
37#882—21no—
38#67⋊C4—21no—
39#874C2211yescharacteristic
40C4×D12C2211yescharacteristic
41C4×D12C2211yescharacteristic, complemented
42#870C2211yescharacteristic, complemented
43C3×C4⋊C8C2211yescharacteristic, complemented
44C3×C22⋊C8C2211yescharacteristic, complemented
45C3×C22⋊C8C2211yescharacteristic, complemented
46C3×#97C2211yescharacteristic, complemented
47#821C2211yescharacteristic
48#821C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic, complemented
51#821C2211yescharacteristic, complemented
52#821C2211yescharacteristic, complemented
53#821C2211yescharacteristic, complemented
54#821C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#821C2211yescharacteristic, complemented
57#818C2211yescharacteristic, complemented
58#817C2211yescharacteristic, complemented
59#817C2211yescharacteristic, 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#442—31no—
72#435D311yescharacteristic, complemented
73#418—31no—
74C2×#102—31no—
75C23×D3—21no—
76C23×D3—21no—
77C2×C3×D4D411yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C22×Dic3—21no—
85D3×D4—44no—
86C2×D12—22no—
87C2×D12—22no—
88C2×D12—21no—
89C2×D12—21no—
90C2×D12—21no—
91C2×D12—21no—
92C2×D12—41no—
93C2×D12—41no—
94C2×D12C2311yescharacteristic
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C3×M16—41no—
101C2×C3×C8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C2×C3×C8C2311yescharacteristic
105C2×C3×C8C2311yescharacteristic
106C3×C4⋊C4D411yescharacteristic
107C3×C22⋊C4D411yescharacteristic
108C3×C22⋊C4D411yescharacteristic
109C3×C42C2311yescharacteristic
110#217—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117C4⋊Dic3—21no—
118C4⋊Dic3C2311yescharacteristic
119C4⋊Dic3C2311yescharacteristic
120C4⋊Dic3C2311yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4—21no—
124C4×Dic3—21no—
125D24—22no—
126D24—41no—
127D24—41no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—61no—
131C2×D8—31no—
132C2×D8—31no—
133C2×D8—31no—
134C2×M16C2×D311yescharacteristic, complemented
135C22×C8C2×D311yescharacteristic, complemented
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—31no—
150C4×D4—61no—
151C4×D4C2×D311yescharacteristic
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#102—31no—
167#102—31no—
168C22⋊C8C2×D311yescharacteristic, complemented
169C22⋊C8C2×D311yescharacteristic, complemented
170#97C2×D311yescharacteristic, complemented
171C23×C3C2×D411yescharacteristic
172C23×C3C2×D411yescharacteristic
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—42no—
181C22×D3—81no—
182C22×D3—81no—
183C3×D4—22no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4C2×D411yescharacteristic
189C2×C3×C4C2×D411yescharacteristic
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195#67—42no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—81no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—21no—
209D12—42no—
210D12—42no—
211D12—42no—
212D12—81no—
213D12#4112yes—
214C4×D3—42no—
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—31no—
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—124no—
239C2×D4D1211yescharacteristic
240C22×C4—61no—
241C22×C4—61no—
242C22×C4—61no—
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245D8—62no—
246D8—121no—
247D8—121no—
248M16—41no—
249C2×C8—41no—
250C2×C8C22×D311yescharacteristic
251C2×C8C22×D311yescharacteristic
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⋊C4D1211yescharacteristic
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4D1211yescharacteristic
270C22⋊C4D1211yescharacteristic
271C42—61no—
272C42C22×D311yescharacteristic
273C22×C3—21no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3AGL(1,ℤ/8)12yes—
279C22×C3C22×D411yescharacteristic
280C2×D3—22no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—84no—
293C2×D3—84no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4C2×#4111yescharacteristic
299C3×C4H(5,𝔽2)11yescharacteristic
300Dic3—41no—
301Dic3—41no—
302Dic3—41no—
303Dic3—41no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—122no—
309C23—122no—
310C23—122no—
311C23—122no—
312C23—241no—
313C23—241no—
314C23C2×D1211yescharacteristic
315C23C2×D1211yescharacteristic
316D4—22no—
317D4—32no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—61no—
324D4—61no—
325D4—122no—
326D4—122no—
327D4—122no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—241no—
332D4—241no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4—241no—
344C2×C4C2×D1211yescharacteristic
345C2×C4C2×D1211yescharacteristic
346C2×C4C2×D1211yescharacteristic
347C2×C4C2×D1211yescharacteristic
348C2×C4C23×D311yescharacteristic
349C8—21no—
350C8—21no—
351C8—21no—
352C8—21no—
353C2×C3—22no—
354C2×C3—41no—
355C2×C3#54611yescharacteristic
356C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
357C2×C3#57611yescharacteristic
358D3—42no—
359D3—81no—
360D3—81no—
361D3—81no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—62no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—121no—
378C22—121no—
379C22—244no—
380C22—244no—
381C22#90812yes—
382C22C22×D1211yescharacteristic
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—121no—
388C4—121no—
389C4—121no—
390C4—121no—
391C4#100411yescharacteristic
392C4D3×#4111yescharacteristic
393C3#331811yescharacteristic, complemented, Sylow
394C2—22no—
395C2—41no—
396C2—122no—
397C2—241no—
398C2—241no—
399C2—241no—
400C2#555411yescharacteristic
401C2C2×#90811yescharacteristic
402C2#575211yescharacteristic
subgroupG/H
centerC22C22×D12
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#435C2
socleC22×C3C22×D4
hypercenter#435D3
hypocenterC3#3318
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #435

Complements

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

NH ≅ G/Ntype
1. #5565C2semidirect
2. D4×D12C2semidirect
3. C3×#435C2semidirect
4. #5135C2semidirect
5. #5131C2semidirect
6. #5115C2semidirect
7. C2×#821C2semidirect
8. #4847C2semidirect
9. #4843C2semidirect
10. #4737C2semidirect
11. #4735C2semidirect
12. #4732C2semidirect
13. #4732C2semidirect
14. #4713C2semidirect
15. #4711C2semidirect
20. C2×C3×M16C22semidirect
21. C22×C3×C8C22semidirect
26. #930C22semidirect
27. #930C22semidirect
28. #926C22semidirect
29. C2×C4⋊Dic3C22semidirect
31. C2×D24C22semidirect
32. C2×D24C22semidirect
41. C4×D12C22semidirect
42. #870C22semidirect
43. C3×C4⋊C8C22semidirect
44. C3×C22⋊C8C22semidirect
45. C3×C22⋊C8C22semidirect
46. C3×#97C22semidirect
50. #821C22semidirect
51. #821C22semidirect
52. #821C22semidirect
53. #821C22semidirect
54. #821C22semidirect
55. #821C22semidirect
56. #821C22semidirect
57. #818C22semidirect
58. #817C22semidirect
59. #817C22semidirect
72. #435D3semidirect
134. C2×M16C2×D3semidirect
135. C22×C8C2×D3semidirect
157. C4⋊C8C2×D3semidirect
168. C22⋊C8C2×D3semidirect
169. C22⋊C8C2×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
#55652
D4×D122
C3×#4352
#51352
#51312
#51152
C2×#8212
#48472
#48432
#47372
#47352
#47322
#47132
#47112
#33183