Finite Group Databasebeta

← #80340

Subgroups — #80340

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5715C211yesmaximal, characteristic, complemented
2#5562C211yesmaximal, characteristic, complemented
3C3×#459C211yesmaximal, characteristic, complemented
4#5158C211yesmaximal, characteristic, complemented
5#5156C211yesmaximal, characteristic, complemented
6#5154C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5072C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4768C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3220—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic, complemented
21C3×#127C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic, complemented
23#940C2211yescharacteristic
24#940C2211yescharacteristic, complemented
25#939—21no—
26#939—21no—
27#938C2211yescharacteristic
28#937—21no—
29#937—21no—
30#936C2211yescharacteristic
31C2×#214C2211yescharacteristic, complemented
32C2×#213C2211yescharacteristic
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic, complemented
35#930—21no—
36#929—21no—
37C2×D24C2211yescharacteristic
38C2×#204C2211yescharacteristic, complemented
39#895C2211yescharacteristic
40#890C2211yescharacteristic, complemented
41#884—21no—
42#883—21no—
43#875C2211yescharacteristic
44C4×D12C2211yescharacteristic, complemented
45#872C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic
48C3×#102C2211yescharacteristic, complemented
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56C8×Dic3C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#540—31no—
61#535—31no—
62#496—31no—
63#493—31no—
64#490—31no—
65#488—31no—
66#463—31no—
67#463—31no—
68#459—31no—
69#459D311yescharacteristic, complemented
70#449—31no—
71#449—31no—
72#447—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C22×Dic3—21no—
90#237—41no—
91#237—41no—
92D3×D4—41no—
93D3×D4—41no—
94C2×D12—21no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3C2311yescharacteristic
99C2×Dic6C2311yescharacteristic
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#214—41no—
111#213—41no—
112#213—41no—
113#213—41no—
114(C2×D3)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4D411yescharacteristic
119(C2×D3)⋊C4D411yescharacteristic
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123Dic3⋊C4D411yescharacteristic
124Dic3⋊C4D411yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×D8—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×D8—31no—
138C2×D8C2×D311yescharacteristic
139C2×D8C2×D311yescharacteristic, complemented
140#127—31no—
141#127—31no—
142#127C2×D311yescharacteristic
143#125—31no—
144#124—31no—
145#124—31no—
146#123—61no—
147#123—61no—
148#121—31no—
149#121—31no—
150#121—61no—
151#121—61no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×D4—31no—
157C4×D4C2×D311yescharacteristic, complemented
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic, complemented
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—81no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—81no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—42no—
196#67—42no—
197#67—81no—
198#67—81no—
199#67—81no—
200#67—81no—
201#67—81no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—42no—
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208D12—21no—
209D12—21no—
210D12—21no—
211D12—21no—
212D12—81no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—81no—
216Dic6—21no—
217Dic6—21no—
218C3×C8—21no—
219C3×C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C24—61no—
223#41—121no—
224#41—121no—
225C2×Q8—31no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—31no—
229C2×D4—31no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4C22×D311yescharacteristic
241C2×D4C22×D311yescharacteristic
242C2×D4C22×D311yescharacteristic
243C22×C4—21no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247SD16—121no—
248SD16—121no—
249D8—41no—
250D8—41no—
251D8—121no—
252D8—121no—
253D8—121no—
254D8—121no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8C22×D311yescharacteristic
258C2×C8C22×D311yescharacteristic
259C4⋊C4—31no—
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4C22×D311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—31no—
267C22⋊C4—31no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C42—31no—
276C42C22×D311yescharacteristic
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3C22×D411yescharacteristic
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—84no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4H(5,𝔽2)11yescharacteristic
301C3×C4H(5,𝔽2)11yescharacteristic
302Dic3—21no—
303Dic3—21no—
304Dic3—21no—
305Dic3—41no—
306Dic3—41no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—31no—
311C23—61no—
312C23—61no—
313C23—122no—
314C23—241no—
315C23—241no—
316Q8—61no—
317Q8—61no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—61no—
325D4—61no—
326D4—61no—
327D4—61no—
328D4—81no—
329D4—81no—
330D4—122no—
331D4—122no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337D4—241no—
338C2×C4—21no—
339C2×C4—31no—
340C2×C4—31no—
341C2×C4—31no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—81no—
347C2×C4—122no—
348C2×C4—241no—
349C2×C4D3×D411yescharacteristic
350C2×C4D3×D411yescharacteristic
351C2×C4C23×D311yescharacteristic
352C8—21no—
353C8—21no—
354C8—61no—
355C8—61no—
356C2×C3—81no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3#53511yescharacteristic
360C2×C3#57611yescharacteristic
361C2×C3#57611yescharacteristic
362D3—41no—
363D3—81no—
364D3—81no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—121no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—244no—
384C22C2×D3×D411yescharacteristic
385C4—21no—
386C4—21no—
387C4—41no—
388C4—61no—
389C4—61no—
390C4—61no—
391C4—121no—
392C4—121no—
393C4#100411yescharacteristic
394C4#100911yescharacteristic
395C3#322011yescharacteristic, complemented, Sylow
396C2—81no—
397C2—81no—
398C2—81no—
399C2—121no—
400C2—241no—
401C2—241no—
402C2#560711yescharacteristic
403C2#575711yescharacteristic
404C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#459C2
socleC22×C3C22×D4
hypercenter#459D3
hypocenterC3#3220
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. #5715C2semidirect
2. #5562C2semidirect
3. C3×#459C2semidirect
4. #5158C2semidirect
5. #5156C2semidirect
6. #5154C2semidirect
7. #5151C2semidirect
8. #5072C2semidirect
9. #5016C2semidirect
10. #4844C2semidirect
11. #4840C2semidirect
12. #4781C2semidirect
13. #4773C2semidirect
14. #4768C2semidirect
15. #4764C2semidirect
20. C2×C3×D8C22semidirect
22. C3×C4×D4C22semidirect
24. #940C22semidirect
31. C2×#214C22semidirect
34. C2×#213C22semidirect
38. C2×#204C22semidirect
40. #890C22semidirect
44. C4×D12C22semidirect
46. C3×C4⋊C8C22semidirect
48. C3×#102C22semidirect
50. #832C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #459D3semidirect
139. C2×D8C2×D3semidirect
157. C4×D4C2×D3semidirect
160. C4⋊C8C2×D3semidirect
168. #102C2×D3semidirect
395. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#57152
#55622
C3×#4592
#51582
#51562
#51542
#51512
#50722
#50162
#48442
#48402
#47812
#47732
#47682
#47642
#32203