Finite Group Databasebeta

← #83389

Subgroups — #83389

Order 384.

408 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#535C211yesmaximal, characteristic, complemented
2#5534C211yesmaximal, characteristic, complemented
3#5239C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5218C211yesmaximal, characteristic, complemented
6#5217C211yesmaximal, characteristic, complemented
7#5075C211yesmaximal, characteristic, complemented
8#5074C211yesmaximal, characteristic, complemented
9#5072C211yesmaximal, characteristic, complemented
10#5070C211yesmaximal, characteristic, complemented
11#5059C211yesmaximal, characteristic, complemented
12#5057C211yesmaximal, characteristic, complemented
13#5055C211yesmaximal, characteristic, complemented
14#5054C211yesmaximal, characteristic, complemented
15#5000C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C3×#127C2211yescharacteristic
20C3×#127C2211yescharacteristic
21C3×#124C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×#121—21no—
24C3×#121—21no—
25C3×#120—21no—
26C3×#120—21no—
27C3×#117C2211yescharacteristic
28C2×#216C2211yescharacteristic, complemented
29C2×#215C2211yescharacteristic, complemented
30C2×#214C2211yescharacteristic
31C2×#214C2211yescharacteristic
32C2×#214C2211yescharacteristic, complemented
33C2×#213C2211yescharacteristic
34C2×#213C2211yescharacteristic
35C2×#213C2211yescharacteristic, complemented
36#930C2211yescharacteristic
37#924C2211yescharacteristic
38#898—21no—
39#896—21no—
40#885—21no—
41#883—21no—
42#875C2211yescharacteristic, complemented
43C4×D12C2211yescharacteristic, complemented
44#869C2211yescharacteristic, complemented
45C4×Dic6C2211yescharacteristic, complemented
46#835C2211yescharacteristic, complemented
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#832C2211yescharacteristic
52#832C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#830C2211yescharacteristic
55#830C2211yescharacteristic
56C4⋊(C3⋊C8)C2211yescharacteristic, complemented
57C4⋊(C3⋊C8)C2211yescharacteristic, complemented
58#803C2211yescharacteristic, complemented
59C4×C3⋊C8C2211yescharacteristic, complemented
60#542—31no—
61#535D311yescharacteristic, complemented
62#496—31no—
63#495—31no—
64#489—31no—
65#486—31no—
66#464—31no—
67#463—31no—
68#460—31no—
69#459—31no—
70#452—31no—
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C24×C3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—21no—
83C2×C3×D4—21no—
84C2×C3×D4—21no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C2×C3×D4C2311yescharacteristic
88C2×C3×D4C2311yescharacteristic
89C2×C3×D4C2311yescharacteristic
90C22×C3×C4—21no—
91C22×C3×C4C2311yescharacteristic
92C2×#67—21no—
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×Dic6—21no—
96C2×Dic6C2311yescharacteristic
97C3×C4⋊C4D411yescharacteristic
98C3×C4⋊C4D411yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101C3×C22⋊C4—21no—
102C3×C22⋊C4—21no—
103C3×C22⋊C4D411yescharacteristic
104C3×C22⋊C4D411yescharacteristic
105C3×C42C2311yescharacteristic
106C3×C42C2311yescharacteristic
107#217—21no—
108#216—41no—
109#215—41no—
110#214—41no—
111#214—41no—
112#214—41no—
113#213—41no—
114#213—41no—
115#213—41no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120C4⋊Dic3—21no—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129C2×C3⋊C8C2311yescharacteristic
130C2×#41C2×D311yescharacteristic
131C22×D4C2×D311yescharacteristic
132C2×Q16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141#127C2×D311yescharacteristic
142#127C2×D311yescharacteristic
143#126—61no—
144#124—31no—
145#124—61no—
146#124C2×D311yescharacteristic
147#124C2×D311yescharacteristic
148#123—61no—
149C22⋊Q8—31no—
150C22⋊Q8—61no—
151#121—21no—
152#121—21no—
153#121—31no—
154#120—21no—
155#120—21no—
156C4×Q8—31no—
157C4×D4—31no—
158#117C2×D311yescharacteristic
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102—31no—
169C22⋊C8—31no—
170C22⋊C8—31no—
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—21no—
189C3×D4—42no—
190C3×D4—42no—
191C3×D4—81no—
192C3×D4—81no—
193C3×D4—81no—
194C3×D4—81no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4C2×D411yescharacteristic
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2×D411yescharacteristic
204C2×C3×C4C2411yescharacteristic
205#67—81no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209D12—21no—
210D12—21no—
211C4×D3—81no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—21no—
220#41—41no—
221#41—41no—
222C2×Q8—31no—
223C2×Q8—61no—
224C2×Q8C22×D311yescharacteristic
225C2×D4—21no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—21no—
229C2×D4—21no—
230C2×D4—21no—
231C2×D4—31no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—61no—
235C2×D4C22×D311yescharacteristic
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—21no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241Q16—121no—
242SD16—121no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—121no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8—31no—
252C2×C8—31no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4#6711yescharacteristic
259C4⋊C4#6711yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—21no—
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4#6711yescharacteristic
270C22⋊C4#6711yescharacteristic
271C42—61no—
272C42C22×D311yescharacteristic
273C42C22×D311yescharacteristic
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
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×C3—84no—
287C22×C3C22×D411yescharacteristic
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—21no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—41no—
297C3×C4H(5,𝔽2)11yescharacteristic
298C3×C4H(5,𝔽2)11yescharacteristic
299Dic3—41no—
300Dic3—41no—
301Dic3—41no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—61no—
307C23—81no—
308C23—81no—
309C23C2×#6711yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—42no—
322D4—42no—
323D4—61no—
324D4—61no—
325D4—81no—
326D4—81no—
327D4—81no—
328D4—81no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—42no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—241no—
338C2×C4C2×#6711yescharacteristic
339C2×C4C2×#6711yescharacteristic
340C2×C4C2×#6711yescharacteristic
341C2×C4C2×#6711yescharacteristic
342C2×C4C2×#6711yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—61no—
345C8—61no—
346C8—61no—
347C8—61no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3—81no—
351C2×C3—81no—
352C2×C3#53511yescharacteristic
353C2×C3#57611yescharacteristic
354C2×C3#57711yescharacteristic
355D3—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—84no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22C22×#6711yescharacteristic
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—41no—
379C4—121no—
380C4—121no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100411yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—41no—
386C2—81no—
387C2—81no—
388C2—81no—
389C2—241no—
390C2#580511yescharacteristic
391C2#583511yescharacteristic
392C2#583611yescharacteristic
subgroupG/H
centerC22C22×#67
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#535C2
socleC22×C3C22×D4
hypercenter#535D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ #117 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C3×#535C2semidirect
2. #5534C2semidirect
3. #5239C2semidirect
4. #5238C2semidirect
5. #5218C2semidirect
6. #5217C2semidirect
7. #5075C2semidirect
8. #5074C2semidirect
9. #5072C2semidirect
10. #5070C2semidirect
11. #5059C2semidirect
12. #5057C2semidirect
13. #5055C2semidirect
14. #5054C2semidirect
15. #5000C2semidirect
28. C2×#216C22semidirect
29. C2×#215C22semidirect
32. C2×#214C22semidirect
35. C2×#213C22semidirect
42. #875C22semidirect
43. C4×D12C22semidirect
44. #869C22semidirect
45. C4×Dic6C22semidirect
46. #835C22semidirect
47. #835C22semidirect
52. #832C22semidirect
53. #832C22semidirect
56. C4⋊(C3⋊C8)C22semidirect
57. C4⋊(C3⋊C8)C22semidirect
58. #803C22semidirect
59. C4×C3⋊C8C22semidirect
61. #535D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
C3×#5352
#55342
#52392
#52382
#52182
#52172
#50752
#50742
#50722
#50702
#50592
#50572
#50552
#50542
#50002
#32243