Finite Group Databasebeta

← #80391

Subgroups — #80391

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5715C211yesmaximal, characteristic
2#5583C211yesmaximal, characteristic, complemented
3C3×#460C211yesmaximal, characteristic, complemented
4#5175C211yesmaximal, characteristic, complemented
5#5173C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5167C211yesmaximal, characteristic, complemented
8#5075C211yesmaximal, characteristic, complemented
9#5028C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4840C211yesmaximal, characteristic, complemented
12#4783C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4768C211yesmaximal, characteristic, complemented
15#4764C211yesmaximal, characteristic, complemented
16#3222—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C3×#127C2211yescharacteristic
22C3×C4×Q8C2211yescharacteristic, complemented
23#946C2211yescharacteristic
24#944C2211yescharacteristic, complemented
25C2×#216C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27#940C2211yescharacteristic
28#939—21no—
29#939—21no—
30#937—21no—
31#937—21no—
32#936C2211yescharacteristic
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic, complemented
35C2×Dic12C2211yescharacteristic, complemented
36C2×#204C2211yescharacteristic
37#898—21no—
38#897C2211yescharacteristic
39#890C2211yescharacteristic, complemented
40#889—21no—
41#876—21no—
42#875C2211yescharacteristic
43#872C2211yescharacteristic
44#870—21no—
45C4×Dic6C2211yescharacteristic, complemented
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#541—31no—
61#535—31no—
62#497—31no—
63#492—31no—
64#490—31no—
65#488—31no—
66#464—31no—
67#464—31no—
68#460—31no—
69#460D311yescharacteristic, complemented
70#452—31no—
71#452—31no—
72#450—31no—
73#450—31no—
74#433—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4—21no—
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#237—41no—
87#237—41no—
88D3×D4—41no—
89D3×D4—41no—
90C2×D12C2311yescharacteristic
91C2×C4×D3C2311yescharacteristic
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×SD16—41no—
95C3×SD16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C42—21no—
101C3×C42C2311yescharacteristic
102#217—21no—
103#217—21no—
104#216—41no—
105#215—41no—
106#214—41no—
107#214—41no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123Dic3⋊C4D411yescharacteristic
124C4×Dic3—21no—
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4—31no—
132C2×Q16—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16C2×D311yescharacteristic
139C2×SD16C2×D311yescharacteristic, complemented
140C4⋊Q8—31no—
141#127—31no—
142#127C2×D311yescharacteristic
143#126—61no—
144#126—61no—
145#125—31no—
146#125—61no—
147#125—61no—
148#124—31no—
149#124—31no—
150C22⋊Q8—31no—
151C22⋊Q8—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×Q8C2×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—
175C22×D3—21no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3—81no—
179C22×D3C2×D411yescharacteristic
180C3×Q8—21no—
181C3×Q8—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—81no—
187C3×D4—81no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—42no—
195#67—81no—
196#67—81no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—42no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206C4×D3—21no—
207C4×D3—21no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—21no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216C24—61no—
217#41—121no—
218#41—121no—
219C2×Q8—31no—
220C2×Q8—31no—
221C2×Q8C22×D311yescharacteristic
222C2×D4—21no—
223C2×D4—21no—
224C2×D4—31no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4C22×D311yescharacteristic
232C2×D4C22×D311yescharacteristic
233C22×C4—31no—
234C22×C4—61no—
235Q16—121no—
236Q16—121no—
237SD16—41no—
238SD16—41no—
239SD16—121no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—31no—
260C22⋊C4—31no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—21no—
268C42—31no—
269C42—61no—
270C42C22×D311yescharacteristic
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3C22×D411yescharacteristic
278C2×D3—21no—
279C2×D3—21no—
280C2×D3—21no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—84no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4H(5,𝔽2)11yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—21no—
298C23—21no—
299C23—31no—
300C23—61no—
301C23—122no—
302C23—241no—
303C23—241no—
304Q8—21no—
305Q8—21no—
306Q8—61no—
307Q8—61no—
308Q8—61no—
309Q8—61no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—61no—
315D4—61no—
316D4—81no—
317D4—81no—
318D4—122no—
319D4—122no—
320D4—241no—
321D4—241no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—122no—
333C2×C4D3×D411yescharacteristic
334C2×C4D3×D411yescharacteristic
335C2×C4C23×D311yescharacteristic
336C8—21no—
337C8—21no—
338C8—61no—
339C8—61no—
340C2×C3—81no—
341C2×C3—81no—
342C2×C3#53511yescharacteristic
343C2×C3#57711yescharacteristic
344C2×C3#57711yescharacteristic
345D3—41no—
346D3—81no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—61no—
354C22—61no—
355C22—61no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—244no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—21no—
363C4—41no—
364C4—41no—
365C4—61no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100911yescharacteristic
373C3#322211yescharacteristic, complemented, Sylow
374C2—81no—
375C2—81no—
376C2—121no—
377C2—241no—
378C2#560711yescharacteristic
379C2#576211yescharacteristic
380C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#460C2
socleC22×C3C22×D4
prime radical#5715C2
hypercenter#460D3
hypocenterC3#3222
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #460

Complements

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

NH ≅ G/Ntype
2. #5583C2semidirect
3. C3×#460C2semidirect
4. #5175C2semidirect
5. #5173C2semidirect
6. #5169C2semidirect
7. #5167C2semidirect
8. #5075C2semidirect
9. #5028C2semidirect
10. #4845C2semidirect
11. #4840C2semidirect
12. #4783C2semidirect
13. #4776C2semidirect
14. #4768C2semidirect
15. #4764C2semidirect
20. C2×C3×SD16C22semidirect
22. C3×C4×Q8C22semidirect
24. #944C22semidirect
25. C2×#216C22semidirect
34. C2×#214C22semidirect
35. C2×Dic12C22semidirect
39. #890C22semidirect
45. C4×Dic6C22semidirect
46. C3×C4⋊C8C22semidirect
48. C3×#102C22semidirect
50. #832C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. C8×Dic3C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #460D3semidirect
139. C2×SD16C2×D3semidirect
157. C4×Q8C2×D3semidirect
160. C4⋊C8C2×D3semidirect
168. #102C2×D3semidirect
373. C3#3222semidirect

Maximal subgroups

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

subgroupindex
#57152
#55832
C3×#4602
#51752
#51732
#51692
#51672
#50752
#50282
#48452
#48402
#47832
#47762
#47682
#47642
#32223