Finite Group Databasebeta

← #80106

Subgroups — #80106

Order 384.

442 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5829C211yesmaximal, characteristic, complemented
2#5611C211yesmaximal, characteristic, complemented
3C3×#449C211yesmaximal, characteristic, complemented
4#5232C211yesmaximal, characteristic, complemented
5#5171C211yesmaximal, characteristic, complemented
6#5165C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5152C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4805C211yesmaximal, characteristic, complemented
11#4794C211yesmaximal, characteristic, complemented
12D3×#102C211yesmaximal, characteristic, complemented
13#4761C211yesmaximal, characteristic, complemented
14#4735C211yesmaximal, characteristic, complemented
15#4728C211yesmaximal, characteristic, complemented
16#3306—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×#237—21no—
19C2×D3×D4C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic
22C3×#121C2211yescharacteristic, complemented
23#947C2211yescharacteristic
24#946C2211yescharacteristic
25#939—21no—
26#938C2211yescharacteristic
27#937—21no—
28#935—21no—
29D4×Dic3C2211yescharacteristic
30D4×Dic3C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#929—21no—
34C4×#67C2211yescharacteristic
35C2×(C2×D3)⋊C4—21no—
36C2×C4⋊Dic3C2211yescharacteristic, complemented
37C2×#208C2211yescharacteristic, complemented
38C2×D3×C8C2211yescharacteristic
39D3×C4⋊C4C2211yescharacteristic
40#888C2211yescharacteristic, complemented
41#883—21no—
42#878—21no—
43C3×#103C2211yescharacteristic, complemented
44C3×#102C2211yescharacteristic
45C3×C22⋊C8C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic, complemented
51#821C2211yescharacteristic
52#821C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#817C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#815C2211yescharacteristic, complemented
57Dic3⋊C8C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59#808C2211yescharacteristic, complemented
60#547—31no—
61#546—31no—
62#489—31no—
63#483—31no—
64#481—31no—
65#476—31no—
66#469—31no—
67#465—31no—
68#463—31no—
69#449D311yescharacteristic, complemented
70#447—31no—
71#442—31no—
72#435—31no—
73#417—31no—
74C2×#102—31no—
75C23×D3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4—21no—
84C22×C3×C4C2311yescharacteristic
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67D411yescharacteristic
89C22×Dic3—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92#237—44no—
93D3×D4—22no—
94D3×D4—41no—
95C2×D12C2311yescharacteristic
96C2×C4×D3—21no—
97C2×C4×D3C2311yescharacteristic
98C2×Dic6—21no—
99C3×SD16—41no—
100C3×D8—22no—
101C2×C3×C8C2311yescharacteristic
102C2×C3×C8C2311yescharacteristic
103C3×C4⋊C4C2311yescharacteristic
104C3×C22⋊C4—21no—
105#217—21no—
106#217—21no—
107#217—21no—
108#217D411yescharacteristic
109#213—41no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118C4⋊Dic3C2311yescharacteristic
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122Dic3⋊C4D411yescharacteristic
123C4×Dic3C2311yescharacteristic
124#208—41no—
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127D3×C8—41no—
128C2×#41—61no—
129C2×#41C2×D311yescharacteristic
130C22×D4—31no—
131C2×SD16C2×D311yescharacteristic, complemented
132C2×D8—31no—
133C2×D8C2×D311yescharacteristic
134C2×M16—31no—
135C22×C8—31no—
136C4⋊Q8—31no—
137#124—31no—
138#123—61no—
139#123—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143#121—31no—
144#121—31no—
145#121—61no—
146#121C2×D311yescharacteristic, complemented
147#120—61no—
148C4×D4—31no—
149C4×D4—31no—
150C4×D4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C22⋊C4—61no—
154#106—31no—
155#106—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#103—31no—
160#103C2×D311yescharacteristic, complemented
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168C22⋊C8—31no—
169C22⋊C8C2×D311yescharacteristic, complemented
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3C2×D411yescharacteristic
174C22×D3—21no—
175C22×D3—42no—
176C22×D3—42no—
177C22×D3—81no—
178C22×D3C2×D411yescharacteristic
179C3×Q8—21no—
180C3×Q8—21no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—42no—
185C3×D4—42no—
186C3×D4#4112yescomplemented
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—42no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—22no—
195#67—42no—
196#67—42no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—42no—
202C2×Dic3—42no—
203C2×Dic3—81no—
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207D12—21no—
208D12—21no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—42no—
212Dic6—42no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—61no—
218#41—22no—
219#41—41no—
220#41—124no—
221C2×Q8—61no—
222C2×Q8C22×D311yescharacteristic
223C2×D4—21no—
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—61no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—121no—
232C2×D4C22×D311yescharacteristic
233C2×D4C22×D311yescharacteristic
234C22×C4—21no—
235C22×C4—31no—
236C22×C4—61no—
237C22×C4—61no—
238C22×C4—61no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241SD16—41no—
242D8—22no—
243D8—121no—
244M16—121no—
245C2×C8—31no—
246C2×C8—31no—
247C2×C8—121no—
248C2×C8C22×D311yescharacteristic
249C2×C8C22×D311yescharacteristic
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—31no—
253C4⋊C4—31no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C42—31no—
270C22×C3—22no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
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—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—84no—
286C2×D3AGL(1,ℤ/8)12yes—
287C3×C4—21no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4C2×#4111yescharacteristic
291C3×C4H(5,𝔽2)11yescharacteristic
292Dic3—21no—
293Dic3—21no—
294Dic3—21no—
295Dic3—41no—
296Dic3—41no—
297Dic3—41no—
298C23—21no—
299C23—21no—
300C23—31no—
301C23—61no—
302C23—122no—
303C23—122no—
304C23—241no—
305C23D3×D411yescharacteristic
306Q8—21no—
307Q8—21no—
308Q8—122no—
309D4—21no—
310D4—21no—
311D4—21no—
312D4—42no—
313D4—42no—
314D4—62no—
315D4—61no—
316D4—61no—
317D4—122no—
318D4—122no—
319D4—241no—
320D4#23712yescomplemented
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—31no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—42no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—122no—
335C2×C4—122no—
336C2×C4—122no—
337C2×C4—241no—
338C2×C4D3×D411yescharacteristic
339C2×C4C23×D311yescharacteristic
340C8—21no—
341C8—21no—
342C8—61no—
343C8—61no—
344C2×C3—42no—
345C2×C3—41no—
346C2×C3—81no—
347C2×C3#54611yescharacteristic
348C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
349C2×C3#57711yescharacteristic
350D3—22no—
351D3—81no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—32no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—61no—
363C22—61no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—244no—
368C22C2×D3×D411yescharacteristic
369C4—21no—
370C4—41no—
371C4—41no—
372C4—61no—
373C4—61no—
374C4—61no—
375C4—121no—
376C4—121no—
377C4—121no—
378C4C2×#23711yescharacteristic
379C4#100411yescharacteristic
380C3#330611yescharacteristic, complemented, Sylow
381C2—41no—
382C2—42no—
383C2—62no—
384C2—81no—
385C2—241no—
386C2#559111yescharacteristic
387C2#577011yescharacteristic
388C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#449C2
socleC22×C3C22×D4
hypercenter#449D3
hypocenterC3#3306
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #5829C2semidirect
2. #5611C2semidirect
3. C3×#449C2semidirect
4. #5232C2semidirect
5. #5171C2semidirect
6. #5165C2semidirect
7. #5156C2semidirect
8. #5152C2semidirect
9. #5039C2semidirect
10. #4805C2semidirect
11. #4794C2semidirect
12. D3×#102C2semidirect
13. #4761C2semidirect
14. #4735C2semidirect
15. #4728C2semidirect
20. C2×C3×SD16C22semidirect
22. C3×#121C22semidirect
30. D4×Dic3C22semidirect
36. C2×C4⋊Dic3C22semidirect
37. C2×#208C22semidirect
40. #888C22semidirect
43. C3×#103C22semidirect
45. C3×C22⋊C8C22semidirect
46. #835C22semidirect
47. #835C22semidirect
50. #832C22semidirect
54. #817C22semidirect
55. #817C22semidirect
56. #815C22semidirect
57. Dic3⋊C8C22semidirect
59. #808C22semidirect
69. #449D3semidirect
131. C2×SD16C2×D3semidirect
146. #121C2×D3semidirect
160. #103C2×D3semidirect
169. C22⋊C8C2×D3semidirect
186. C3×D4#41semidirect
320. D4#237semidirect
380. C3#3306semidirect

Maximal subgroups

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

subgroupindex
#58292
#56112
C3×#4492
#52322
#51712
#51652
#51562
#51522
#50392
#48052
#47942
D3×#1022
#47612
#47352
#47282
#33063