Finite Group Databasebeta

← #83495

Subgroups — #83495

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#546C211yesmaximal, characteristic, complemented
2#5565C211yesmaximal, characteristic, complemented
3#5238C211yesmaximal, characteristic, complemented
4#5233C211yesmaximal, characteristic, complemented
5#5217C211yesmaximal, characteristic, complemented
6C2×#832C211yesmaximal, characteristic, complemented
7#5057C211yesmaximal, characteristic, complemented
8#5056C211yesmaximal, characteristic, complemented
9#5048C211yesmaximal, characteristic, complemented
10#5044C211yesmaximal, characteristic, complemented
11#5038C211yesmaximal, characteristic, complemented
12#5035C211yesmaximal, characteristic, complemented
13C4×#213C211yesmaximal, characteristic, complemented
14D4×C3⋊C8C211yesmaximal, characteristic, complemented
15#5009C211yesmaximal, characteristic, complemented
16#3297—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C3×#124C2211yescharacteristic
20C3×#123—21no—
21C3×C22⋊Q8C2211yescharacteristic
22C3×#121—21no—
23C3×#121C2211yescharacteristic
24C3×#120—21no—
25C3×C4×D4C2211yescharacteristic
26C3×C4×D4C2211yescharacteristic
27C2×C3×C22⋊C4—21no—
28C2×#215C2211yescharacteristic, complemented
29#939—21no—
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930C2211yescharacteristic
33#930C2211yescharacteristic, complemented
34#926C2211yescharacteristic, complemented
35C2×C4⋊Dic3C2211yescharacteristic
36C22×C3⋊C8C2211yescharacteristic
37C22×C3⋊C8C2211yescharacteristic, complemented
38#898—21no—
39#883—21no—
40#67⋊C4—21no—
41C4×D12C2211yescharacteristic, complemented
42#870C2211yescharacteristic, complemented
43#835C2211yescharacteristic, complemented
44#835C2211yescharacteristic, complemented
45#835C2211yescharacteristic, complemented
46#832C2211yescharacteristic
47#832C2211yescharacteristic
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#830C2211yescharacteristic
53#830C2211yescharacteristic, complemented
54#809C2211yescharacteristic, complemented
55#808C2211yescharacteristic, complemented
56#807C2211yescharacteristic, complemented
57#807C2211yescharacteristic, complemented
58C4⋊(C3⋊C8)C2211yescharacteristic, complemented
59C4×C3⋊C8C2211yescharacteristic, complemented
60#553—31no—
61#546D311yescharacteristic, complemented
62#488—31no—
63#484—31no—
64#482—31no—
65#478—31no—
66#466—31no—
67#465—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71C4×D8—31no—
72C8×D4—31no—
73#416—31no—
74C2×#102—31no—
75C24×C3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—41no—
84C2×C3×D4D411yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C2×C3×D4C2311yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—21no—
92C2×#67—21no—
93C22×Dic3—21no—
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C3×C4⋊C4—21no—
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4—21no—
104C3×C22⋊C4D411yescharacteristic
105C3×C22⋊C4D411yescharacteristic
106C3×C42C2311yescharacteristic
107#217—21no—
108#215—41no—
109#213—22no—
110#213—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120C4×Dic3—21no—
121C2×C3⋊C8—41no—
122C2×C3⋊C8—41no—
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×#41C2×D311yescharacteristic
128C22×D4C2×D311yescharacteristic
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C22×C8—31no—
133C22×C8—31no—
134#126—61no—
135#125—31no—
136#124C2×D311yescharacteristic
137#123—21no—
138#123—61no—
139C22⋊Q8C2×D311yescharacteristic
140#121—21no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121C2×D311yescharacteristic
145#120—21no—
146C4×D4—31no—
147C4×D4—61no—
148C4×D4C2×D311yescharacteristic
149C4×D4C2×D311yescharacteristic
150#117—31no—
151C2×C4⋊C4—31no—
152C2×C22⋊C4—21no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103—31no—
159#103—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—81no—
175C23×C3C2×D411yescharacteristic
176C23×C3C2×D411yescharacteristic
177C22×D3—21no—
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—22no—
181C3×D4—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—42no—
185C3×D4—42no—
186C3×D4—81no—
187C3×D4#4112yes—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—42no—
195C2×C3×C4—81no—
196C2×C3×C4C2×D411yescharacteristic
197C2×C3×C4C2×D411yescharacteristic
198C2×C3×C4C2×D411yescharacteristic
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2411yescharacteristic
201#67—42no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—81no—
207D12—21no—
208D12—21no—
209C4×D3—81no—
210C3⋊C8—21no—
211C3⋊C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—21no—
215#41—22no—
216#41—41no—
217C2×Q8C22×D311yescharacteristic
218C2×D4—22no—
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—21no—
222C2×D4—31no—
223C2×D4—41no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4#6711yescharacteristic
227C2×D4C22×D311yescharacteristic
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—61no—
232C22×C4—61no—
233C22×C4C22×D311yescharacteristic
234C22×C4C22×D311yescharacteristic
235SD16—121no—
236D8—62no—
237D8—121no—
238C2×C8—31no—
239C2×C8—31no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8—121no—
243C2×C8—121no—
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4#6711yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—21no—
255C22⋊C4—21no—
256C22⋊C4—21no—
257C22⋊C4—21no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4#6711yescharacteristic
263C22⋊C4#6711yescharacteristic
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—21no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—22no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—84no—
278C22×C3#13512yes—
279C22×C3C22×D411yescharacteristic
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C3×C4—21no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—41no—
288C3×C4—41no—
289C3×C4C2×#4111yescharacteristic
290C3×C4H(5,𝔽2)11yescharacteristic
291Dic3—41no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—21no—
296C23—42no—
297C23—42no—
298C23—61no—
299C23—81no—
300C23C2×#6711yescharacteristic
301C23C2×#6711yescharacteristic
302Q8—21no—
303Q8—21no—
304D4—22no—
305D4—21no—
306D4—21no—
307D4—21no—
308D4—42no—
309D4—42no—
310D4—61no—
311D4—61no—
312D4—81no—
313D4—122no—
314D4—241no—
315D4#23712yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—42no—
323C2×C4—61no—
324C2×C4—61no—
325C2×C4—61no—
326C2×C4—81no—
327C2×C4—241no—
328C2×C4—241no—
329C2×C4C2×#6711yescharacteristic
330C2×C4C2×#6711yescharacteristic
331C2×C4C2×#6711yescharacteristic
332C2×C4C2×#6711yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—61no—
335C8—61no—
336C8—61no—
337C8—61no—
338C2×C3—22no—
339C2×C3—41no—
340C2×C3—42no—
341C2×C3—81no—
342C2×C3#54611yescharacteristic
343C2×C3C2×#13511yescharacteristic
344C2×C3#57611yescharacteristic
345D3—81no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—22no—
350C22—21no—
351C22—21no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—84no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22#95012yes—
362C22C22×#6711yescharacteristic
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—41no—
368C4—41no—
369C4—121no—
370C4—121no—
371C4—121no—
372C4C2×#23711yescharacteristic
373C4#100411yescharacteristic
374C3#329711yescharacteristic, complemented, Sylow
375C2—22no—
376C2—41no—
377C2—42no—
378C2—81no—
379C2—241no—
380C2#580611yescharacteristic
381C2C2×#95011yescharacteristic
382C2#583511yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#546C2
socleC22×C3C22×D4
hypercenter#546D3
hypocenterC3#3297
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C4×D4 ◁ #546

Complements

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

NH ≅ G/Ntype
1. C3×#546C2semidirect
2. #5565C2semidirect
3. #5238C2semidirect
4. #5233C2semidirect
5. #5217C2semidirect
6. C2×#832C2semidirect
7. #5057C2semidirect
8. #5056C2semidirect
9. #5048C2semidirect
10. #5044C2semidirect
11. #5038C2semidirect
12. #5035C2semidirect
13. C4×#213C2semidirect
14. D4×C3⋊C8C2semidirect
15. #5009C2semidirect
28. C2×#215C22semidirect
33. #930C22semidirect
34. #926C22semidirect
37. C22×C3⋊C8C22semidirect
41. C4×D12C22semidirect
42. #870C22semidirect
43. #835C22semidirect
44. #835C22semidirect
45. #835C22semidirect
53. #830C22semidirect
54. #809C22semidirect
55. #808C22semidirect
56. #807C22semidirect
57. #807C22semidirect
58. C4⋊(C3⋊C8)C22semidirect
59. C4×C3⋊C8C22semidirect
61. #546D3semidirect
374. C3#3297semidirect

Maximal subgroups

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

subgroupindex
C3×#5462
#55652
#52382
#52332
#52172
C2×#8322
#50572
#50562
#50482
#50442
#50382
#50352
C4×#2132
D4×C3⋊C82
#50092
#32973