Finite Group Databasebeta

← #80527

Subgroups — #80527

Order 384.

418 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5681C211yesmaximal, characteristic, complemented
2#5553C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6#5151C211yesmaximal, characteristic, complemented
7#5150C211yesmaximal, characteristic, complemented
8#5058C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4843C211yesmaximal, characteristic, complemented
11#4828C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4788C211yesmaximal, characteristic, complemented
14#4786C211yesmaximal, characteristic, complemented
15#4785C211yesmaximal, characteristic, complemented
16#3399—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×SD16C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C3×#124C2211yescharacteristic
21C3×C4×D4C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24C2×#215C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940C2211yescharacteristic
27#940C2211yescharacteristic
28#940C2211yescharacteristic
29D4×Dic3C2211yescharacteristic
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32C2×#213C2211yescharacteristic
33#930—21no—
34C2×D24C2211yescharacteristic
35C2×D24C2211yescharacteristic
36#893C2211yescharacteristic
37#893C2211yescharacteristic
38#886—21no—
39#885—21no—
40#884—21no—
41#884—21no—
42#883—21no—
43#67⋊C4—21no—
44#874C2211yescharacteristic
45C4×D12C2211yescharacteristic
46C4×Dic6C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#102C2211yescharacteristic
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#803C2211yescharacteristic
60#559—31no—
61#550—31no—
62#496—31no—
63#496—31no—
64#493—31no—
65#463—31no—
66#463D311yescharacteristic, complemented
67#460—31no—
68#459—31no—
69#459—31no—
70#459—31no—
71#446—31no—
72#442—31no—
73#442—31no—
74C4×SD16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C22×Dic3—21no—
86#235—44no—
87C2×D12—21no—
88C2×D12—21no—
89C2×D12C2311yescharacteristic
90C2×D12C2311yescharacteristic
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3—21no—
95C2×Dic6D411yescharacteristic
96C3×SD16—41no—
97C3×D8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4C2311yescharacteristic
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C42C2311yescharacteristic
105#217—21no—
106#215—22no—
107#215—41no—
108#213—22no—
109#213—22no—
110#213—41no—
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—
116C4⋊Dic3—21no—
117C4⋊Dic3D411yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4D411yescharacteristic
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126D24—41no—
127D24—41no—
128C2×#41—61no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16C2×D311yescharacteristic
132C2×D8—31no—
133C2×D8—31no—
134C2×D8—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×D8C2×D311yescharacteristic
138#127—31no—
139#127—31no—
140#127—31no—
141#127—31no—
142#124—31no—
143#124—61no—
144#124C2×D311yescharacteristic
145#123—61no—
146#123—61no—
147#121—61no—
148#121—61no—
149#121—61no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×D4—31no—
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—31no—
156C4×D4—61no—
157C4×D4C2×D311yescharacteristic
158#106—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8C2×D311yescharacteristic
162#103C2×D311yescharacteristic
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#97—31no—
169#97—31no—
170C4×C8—31no—
171C23×C3—21no—
172C23×C3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C3×D4—21no—
179C3×D4—21no—
180C3×D4—21no—
181C3×D4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—42no—
185C2×C3×C4C2×D411yescharacteristic
186C2×C3×C4C2×D411yescharacteristic
187C2×C3×C4C2411yescharacteristic
188#67—42no—
189#67—42no—
190#67—42no—
191#67—81no—
192#67—81no—
193#67—81no—
194C2×Dic3—21no—
195C2×Dic3—81no—
196C2×Dic3C2×D411yescharacteristic
197C2×Dic3C2×D411yescharacteristic
198C2×Dic3C2×D411yescharacteristic
199C2×Dic3C2×D411yescharacteristic
200D12—21no—
201D12—21no—
202D12—21no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—42no—
207D12—81no—
208C4×D3—42no—
209C4×D3—42no—
210C4×D3—81no—
211Dic6—22no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8#4112yescomplemented
216#41—124no—
217C2×Q8—31no—
218C2×Q8C22×D311yescharacteristic
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4C22×D311yescharacteristic
231C2×D4C22×D311yescharacteristic
232C22×C4—21no—
233C22×C4—61no—
234C22×C4—61no—
235C22×C4—61no—
236C22×C4—61no—
237SD16—41no—
238SD16—62no—
239SD16—121no—
240D8—41no—
241D8—62no—
242D8—62no—
243D8—121no—
244D8—121no—
245D8—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4C22×D311yescharacteristic
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—21no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C42—31no—
266C42—31no—
267C42—61no—
268C42C22×D311yescharacteristic
269C22×C3—41no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3C22×D411yescharacteristic
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4C2×#4111yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3AGL(1,ℤ/8)12yes—
296C23—21no—
297C23—21no—
298C23—61no—
299C23—61no—
300C23—61no—
301Q8—21no—
302Q8—21no—
303Q8—62no—
304D4—21no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—61no—
309D4—61no—
310D4—61no—
311D4—61no—
312D4—61no—
313D4—61no—
314D4—122no—
315D4—122no—
316D4—122no—
317D4—122no—
318D4—241no—
319D4—241no—
320D4—241no—
321D4—241no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—31no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—42no—
329C2×C4—61no—
330C2×C4—122no—
331C2×C4—122no—
332C2×C4—241no—
333C2×C4—241no—
334C2×C4D3×D411yescharacteristic
335C2×C4D3×D411yescharacteristic
336C2×C4C23×D311yescharacteristic
337C8—21no—
338C8—21no—
339C8—32no—
340C8—61no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3#55011yescharacteristic
344C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
345C2×C3#57611yescharacteristic
346D3—81no—
347D3—81no—
348D3—81no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—121no—
362C22—121no—
363C22—121no—
364C22C2×D3×D411yescharacteristic
365C4—21no—
366C4—21no—
367C4—32no—
368C4—41no—
369C4—41no—
370C4—61no—
371C4—61no—
372C4—61no—
373C4—121no—
374C4#100411yescharacteristic
375C4D3×#4111yescharacteristic
376C3#339911yescharacteristic, complemented, Sylow
377C2—81no—
378C2—81no—
379C2—241no—
380C2—241no—
381C2—241no—
382C2#559911yescharacteristic
383C2#576911yescharacteristic
384C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3399
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. #5681C2semidirect
2. #5553C2semidirect
3. C3×#463C2semidirect
4. #5176C2semidirect
5. #5163C2semidirect
6. #5151C2semidirect
7. #5150C2semidirect
8. #5058C2semidirect
9. #5014C2semidirect
10. #4843C2semidirect
11. #4828C2semidirect
12. #4819C2semidirect
13. #4788C2semidirect
14. #4786C2semidirect
15. #4785C2semidirect
66. #463D3semidirect
215. C3⋊C8#41semidirect
376. C3#3399semidirect

Maximal subgroups

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

subgroupindex
#56812
#55532
C3×#4632
#51762
#51632
#51512
#51502
#50582
#50142
#48432
#48282
#48192
#47882
#47862
#47852
#33993