Finite Group Databasebeta

← #80542

Subgroups — #80542

Order 384.

444 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#124C211yesmaximal, characteristic, complemented
2#5548C211yesmaximal, characteristic, complemented
3C3×#463C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5170C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5154C211yesmaximal, characteristic, complemented
8#5057C211yesmaximal, characteristic, complemented
9#5016C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4835C211yesmaximal, characteristic, complemented
12#4806C211yesmaximal, characteristic, complemented
13#4794C211yesmaximal, characteristic, complemented
14#4773C211yesmaximal, characteristic, complemented
15#4765C211yesmaximal, characteristic, complemented
16#3213—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×#235—21no—
20C2×C3×SD16C2211yescharacteristic, complemented
21C2×C3×D8C2211yescharacteristic
22C3×#124C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24#947C2211yescharacteristic
25C2×#215C2211yescharacteristic, complemented
26#940C2211yescharacteristic, complemented
27#938C2211yescharacteristic
28#936C2211yescharacteristic
29C2×#214C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32C4×#67—21no—
33C2×D24C2211yescharacteristic
34C2×#204C2211yescharacteristic, complemented
35#895C2211yescharacteristic
36#888C2211yescharacteristic, complemented
37#885—21no—
38#885—21no—
39#885—21no—
40#884—21no—
41D3×C22⋊C4—21no—
42D3×C22⋊C4—21no—
43#875C2211yescharacteristic
44C4×D12C2211yescharacteristic, complemented
45C42×D3C2211yescharacteristic
46C3×C4⋊C8C2211yescharacteristic, complemented
47C3×#103C2211yescharacteristic, complemented
48C3×#102C2211yescharacteristic
49#835C2211yescharacteristic, complemented
50#832C2211yescharacteristic
51#821C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic
54#816C2211yescharacteristic, complemented
55#815C2211yescharacteristic, complemented
56#815C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic, complemented
60#532—31no—
61C2×#124—31no—
62#496—31no—
63#496—31no—
64#489—31no—
65#489—31no—
66#463—31no—
67#463—31no—
68#463—31no—
69#463D311yescharacteristic, complemented
70#448—31no—
71#448—31no—
72#447—31no—
73#447—31no—
74#432—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4C2311yescharacteristic
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84D3×Q8—41no—
85D3×Q8—41no—
86D3×D4—41no—
87D3×D4—41no—
88#235—44no—
89C2×D12—21no—
90C2×D12C2311yescharacteristic
91C2×D12C2311yescharacteristic
92C2×C4×D3—21no—
93C2×C4×D3—21no—
94C2×C4×D3D411yescharacteristic
95C2×C4×D3D411yescharacteristic
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6—21no—
99C2×Dic6C2311yescharacteristic
100C3×SD16—41no—
101C3×D8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106C3×C22⋊C4—21no—
107C3×C22⋊C4—21no—
108C3×C42C2311yescharacteristic
109#217—21no—
110#217—21no—
111#217—21no—
112#215—41no—
113#214—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—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124C4×Dic3D411yescharacteristic
125C4×Dic3D411yescharacteristic
126C4×Dic3C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129D24—41no—
130#204—41no—
131C2×#41—61no—
132C22×Q8—31no—
133C22×D4—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16C2×D311yescharacteristic, complemented
138C2×D8—31no—
139C2×D8—31no—
140C2×D8—31no—
141C2×D8C2×D311yescharacteristic
142C4⋊Q8—31no—
143#127—31no—
144#124—31no—
145#124—31no—
146#124—31no—
147#124—61no—
148#124—61no—
149#124—61no—
150#124C2×D311yescharacteristic
151#121—31no—
152#121—31no—
153#121—61no—
154C4×D4—31no—
155C4×D4—61no—
156C4×D4C2×D311yescharacteristic, complemented
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C42—31no—
160C4⋊C8—31no—
161C4⋊C8C2×D311yescharacteristic, complemented
162#103—31no—
163#103—31no—
164#103—31no—
165#103C2×D311yescharacteristic, complemented
166#102—31no—
167#102—31no—
168#102—31no—
169#102C2×D311yescharacteristic
170C22⋊C8—31no—
171C22⋊C8—31no—
172#97—31no—
173#97—31no—
174C23×C3—21no—
175C23×C3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—42no—
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—42no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—42no—
197#67—42no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203D12—21no—
204D12—21no—
205D12—21no—
206D12—21no—
207D12—42no—
208C4×D3—22no—
209C4×D3—21no—
210C4×D3—21no—
211C4×D3—41no—
212C4×D3—42no—
213C4×D3—42no—
214Dic6—21no—
215Dic6—21no—
216Dic6—42no—
217Dic6—42no—
218C3×C8—21no—
219C3×C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C24—61no—
223#41—124no—
224C2×Q8—31no—
225C2×Q8—61no—
226C2×Q8—61no—
227C2×Q8—121no—
228C2×Q8—121no—
229C2×Q8C22×D311yescharacteristic
230C2×D4—31no—
231C2×D4—31no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4C22×D311yescharacteristic
240C2×D4C22×D311yescharacteristic
241C22×C4—21no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—31no—
245C22×C4—61no—
246C22×C4—61no—
247SD16—41no—
248SD16—121no—
249SD16—121no—
250SD16—121no—
251D8—41no—
252D8—121no—
253D8—121no—
254D8—121no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8C22×D311yescharacteristic
258C2×C8C22×D311yescharacteristic
259C4⋊C4—31no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—21no—
264C22⋊C4—21no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C42—31no—
275C42—31no—
276C42—31no—
277C42C22×D311yescharacteristic
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—41no—
288C2×D3—41no—
289C2×D3—41no—
290C2×D3—41no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—84no—
294C2×D3AGL(1,ℤ/8)12yes—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4H(5,𝔽2)11yescharacteristic
300C3×C4H(5,𝔽2)11yescharacteristic
301Dic3—21no—
302Dic3—21no—
303Dic3—21no—
304Dic3—41no—
305Dic3—41no—
306C23—21no—
307C23—21no—
308C23—31no—
309C23—61no—
310C23—61no—
311C23—122no—
312C23—122no—
313C23—122no—
314Q8—21no—
315Q8—21no—
316Q8—61no—
317Q8—61no—
318Q8—122no—
319Q8—122no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—61no—
325D4—61no—
326D4—61no—
327D4—61no—
328D4—122no—
329D4—122no—
330D4—122no—
331D4—122no—
332D4—122no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—31no—
338C2×C4—42no—
339C2×C4—62no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—121no—
345C2×C4—122no—
346C2×C4—122no—
347C2×C4D3×D411yescharacteristic
348C2×C4D3×D411yescharacteristic
349C2×C4C23×D311yescharacteristic
350C8—21no—
351C8—21no—
352C8—61no—
353C8—61no—
354C2×C3—81no—
355C2×C3—81no—
356C2×C3#53511yescharacteristic
357C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
358C2×C3#57511yescharacteristic
359D3—22no—
360D3—81no—
361D3—81no—
362C22—32no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—41no—
367C22—41no—
368C22—41no—
369C22—61no—
370C22—61no—
371C22—121no—
372C22—121no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22—121no—
377C22—244no—
378C22C2×D3×D411yescharacteristic
379C4—21no—
380C4—21no—
381C4—41no—
382C4—41no—
383C4—61no—
384C4—61no—
385C4—61no—
386C4—121no—
387C4—121no—
388C4#100411yescharacteristic
389C4#100911yescharacteristic
390C3#321311yescharacteristic, complemented, Sylow
391C2—62no—
392C2—81no—
393C2—81no—
394C2—241no—
395C2—241no—
396C2#560711yescharacteristic
397C2#576811yescharacteristic
398C2D3×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#3213
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. D3×#124C2semidirect
2. #5548C2semidirect
3. C3×#463C2semidirect
4. #5176C2semidirect
5. #5170C2semidirect
6. #5156C2semidirect
7. #5154C2semidirect
8. #5057C2semidirect
9. #5016C2semidirect
10. #4844C2semidirect
11. #4835C2semidirect
12. #4806C2semidirect
13. #4794C2semidirect
14. #4773C2semidirect
15. #4765C2semidirect
20. C2×C3×SD16C22semidirect
23. C3×C4×D4C22semidirect
25. C2×#215C22semidirect
26. #940C22semidirect
29. C2×#214C22semidirect
34. C2×#204C22semidirect
36. #888C22semidirect
44. C4×D12C22semidirect
46. C3×C4⋊C8C22semidirect
47. C3×#103C22semidirect
49. #835C22semidirect
54. #816C22semidirect
55. #815C22semidirect
56. #815C22semidirect
57. #810C22semidirect
59. C4⋊(C3⋊C8)C22semidirect
69. #463D3semidirect
137. C2×SD16C2×D3semidirect
156. C4×D4C2×D3semidirect
161. C4⋊C8C2×D3semidirect
165. #103C2×D3semidirect
390. C3#3213semidirect

Maximal subgroups

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

subgroupindex
D3×#1242
#55482
C3×#4632
#51762
#51702
#51562
#51542
#50572
#50162
#48442
#48352
#48062
#47942
#47732
#47652
#32133