Finite Group Databasebeta

← #80263

Subgroups — #80263

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
1#5830C211yesmaximal, characteristic, complemented
2#5643C211yesmaximal, characteristic, complemented
3C3×#452C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5191C211yesmaximal, characteristic, complemented
6#5185C211yesmaximal, characteristic, complemented
7#5176C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5046C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4794C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4760C211yesmaximal, characteristic, complemented
14#4732C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3224—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C22×D12C2211yescharacteristic
19C2×C3×Q16C2211yescharacteristic, complemented
20C2×C3×SD16C2211yescharacteristic, complemented
21C3×C22⋊Q8C2211yescharacteristic
22#947C2211yescharacteristic
23#947C2211yescharacteristic
24#947C2211yescharacteristic, complemented
25Q8×Dic3C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic
27C2×#215C2211yescharacteristic
28C2×#215C2211yescharacteristic, complemented
29#940C2211yescharacteristic
30#940C2211yescharacteristic
31#937—21no—
32#937—21no—
33C2×#213C2211yescharacteristic, complemented
34#930—21no—
35#930—21no—
36#930C2211yescharacteristic
37#926C2211yescharacteristic
38C2×D24C2211yescharacteristic
39C2×D24C2211yescharacteristic
40#898—21no—
41#897—21no—
42#893C2211yescharacteristic, complemented
43#888C2211yescharacteristic, complemented
44#885—21no—
45#883—21no—
46C3×#103C2211yescharacteristic, complemented
47C3×#102C2211yescharacteristic, complemented
48C3×C22⋊C8C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic, complemented
51#830C2211yescharacteristic
52#821C2211yescharacteristic
53#821C2211yescharacteristic
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59#809C2211yescharacteristic
60#542—31no—
61#535—31no—
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#452D311yescharacteristic, complemented
71#450—31no—
72#449—31no—
73#447—31no—
74#433—31no—
75C23×D3—21no—
76C3×#41—41no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×D12—21no—
90C2×D12—41no—
91C2×D12—41no—
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×D12C2311yescharacteristic
95C2×C4×D3—21no—
96C2×Dic6—21no—
97C3×Q16—41no—
98C3×SD16—41no—
99C2×C3×C8C2311yescharacteristic
100C2×C3×C8C2311yescharacteristic
101C3×C4⋊C4—21no—
102C3×C4⋊C4C2311yescharacteristic
103C3×C22⋊C4—21no—
104#217D411yescharacteristic
105#217D411yescharacteristic
106#215—41no—
107#215—41no—
108#215—41no—
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)⋊C4—21no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118C4⋊Dic3—21no—
119C4⋊Dic3D411yescharacteristic
120C4⋊Dic3D411yescharacteristic
121C4⋊Dic3C2311yescharacteristic
122Dic3⋊C4—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129D24—41no—
130C2×#41C2×D311yescharacteristic
131C22×D4—31no—
132C2×Q16C2×D311yescharacteristic, complemented
133C2×SD16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16C2×D311yescharacteristic, complemented
137C2×D8—31no—
138C2×D8—31no—
139C2×D8—31no—
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
143#126—61no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#123—61no—
149C22⋊Q8—61no—
150C22⋊Q8C2×D311yescharacteristic
151#121—31no—
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic, complemented
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—81no—
180C22×D3—81no—
181C3×Q8—21no—
182C3×Q8—21no—
183C3×Q8—21no—
184C3×Q8—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—42no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—42no—
194C2×C3×C4C2×D411yescharacteristic
195C2×C3×C4C2411yescharacteristic
196#67—81no—
197#67—81no—
198#67—81no—
199#67—81no—
200#67—81no—
201C2×Dic3—21no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206D12—21no—
207D12—21no—
208D12—21no—
209D12—21no—
210D12—21no—
211D12—21no—
212D12—42no—
213C4×D3—81no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—61no—
220#41—41no—
221#41—41no—
222C2×Q8—61no—
223C2×Q8C22×D311yescharacteristic
224C2×Q8C22×D311yescharacteristic
225C2×D4—21no—
226C2×D4—31no—
227C2×D4—31no—
228C2×D4—31no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—61no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—121no—
236C2×D4—121no—
237C2×D4C22×D311yescharacteristic
238C22×C4—21no—
239C22×C4—61no—
240C22×C4C22×D311yescharacteristic
241Q16—41no—
242SD16—41no—
243SD16—121no—
244SD16—121no—
245SD16—121no—
246D8—121no—
247D8—121no—
248D8—121no—
249C2×C8—31no—
250C2×C8—31no—
251C2×C8C22×D311yescharacteristic
252C2×C8C22×D311yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—31no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4C22×D311yescharacteristic
260C22⋊C4—21no—
261C22⋊C4—31no—
262C22⋊C4—31no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C42—31no—
272C42—31no—
273C42—61no—
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—21no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3C22×D411yescharacteristic
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—84no—
291C3×C4—21no—
292C3×C4—41no—
293C3×C4—41no—
294C3×C4—41no—
295C3×C4H(5,𝔽2)11yescharacteristic
296C3×C4H(5,𝔽2)11yescharacteristic
297Dic3—21no—
298Dic3—21no—
299Dic3—21no—
300Dic3—21no—
301Dic3—41no—
302C23—21no—
303C23—61no—
304C23—61no—
305C23—61no—
306C23—122no—
307C23—241no—
308C23—241no—
309C23D3×D411yescharacteristic
310Q8—21no—
311Q8—21no—
312Q8—21no—
313Q8—21no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—42no—
318D4—61no—
319D4—61no—
320D4—61no—
321D4—61no—
322D4—61no—
323D4—61no—
324D4—122no—
325D4—241no—
326D4—241no—
327D4—241no—
328D4—241no—
329D4—241no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—31no—
339C2×C4—42no—
340C2×C4—61no—
341C2×C4—241no—
342C2×C4D3×D411yescharacteristic
343C2×C4C23×D311yescharacteristic
344C8—21no—
345C8—21no—
346C8—61no—
347C8—61no—
348C2×C3—41no—
349C2×C3—81no—
350C2×C3#53511yescharacteristic
351C2×C3#57611yescharacteristic
352C2×C3#57711yescharacteristic
353D3—81no—
354D3—81no—
355D3—81no—
356C22—21no—
357C22—21no—
358C22—21no—
359C22—41no—
360C22—41no—
361C22—41no—
362C22—121no—
363C22—121no—
364C22—121no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—244no—
372C22C2×D3×D411yescharacteristic
373C4—21no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—61no—
378C4—61no—
379C4—61no—
380C4—61no—
381C4—121no—
382C4#100411yescharacteristic
383C4#100411yescharacteristic
384C3#322411yescharacteristic, complemented, Sylow
385C2—41no—
386C2—81no—
387C2—241no—
388C2—241no—
389C2—241no—
390C2#559211yescharacteristic
391C2#576911yescharacteristic
392C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#452C2
socleC22×C3C22×D4
hypercenter#452D3
hypocenterC3#3224
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
1. #5830C2semidirect
2. #5643C2semidirect
3. C3×#452C2semidirect
4. #5238C2semidirect
5. #5191C2semidirect
6. #5185C2semidirect
7. #5176C2semidirect
8. #5163C2semidirect
9. #5046C2semidirect
10. #4819C2semidirect
11. #4794C2semidirect
12. #4786C2semidirect
13. #4760C2semidirect
14. #4732C2semidirect
15. #4720C2semidirect
19. C2×C3×Q16C22semidirect
20. C2×C3×SD16C22semidirect
24. #947C22semidirect
25. Q8×Dic3C22semidirect
28. C2×#215C22semidirect
33. C2×#213C22semidirect
42. #893C22semidirect
43. #888C22semidirect
46. C3×#103C22semidirect
47. C3×#102C22semidirect
49. #835C22semidirect
50. #832C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
70. #452D3semidirect
132. C2×Q16C2×D3semidirect
136. C2×SD16C2×D3semidirect
162. #103C2×D3semidirect
168. #102C2×D3semidirect
384. C3#3224semidirect

Maximal subgroups

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

subgroupindex
#58302
#56432
C3×#4522
#52382
#51912
#51852
#51762
#51632
#50462
#48192
#47942
#47862
#47602
#47322
#47202
#32243