Finite Group Databasebeta

← #78762

Subgroups — #78762

Order 384.

450 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic
2#5513C211yesmaximal, characteristic, complemented
3C2×C3×#102C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5114C211yesmaximal, characteristic, complemented
6#5111C211yesmaximal, characteristic, complemented
7#4967C211yesmaximal, characteristic, complemented
8#4780C211yesmaximal, characteristic, complemented
9#4779C211yesmaximal, characteristic, complemented
10#4776C211yesmaximal, characteristic, complemented
11#4770C211yesmaximal, characteristic, complemented
12#4764C211yesmaximal, characteristic, complemented
13#4763C211yesmaximal, characteristic, complemented
14#4762C211yesmaximal, characteristic, complemented
15#4757C211yesmaximal, characteristic, complemented
16#3310—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×D3×D4C2211yescharacteristic
20C22×C3×C8C2211yescharacteristic, complemented
21C2×C3×C4⋊C4C2211yescharacteristic, complemented
22#953—21no—
23#940C2211yescharacteristic
24#939—21no—
25#938C2211yescharacteristic
26#937—21no—
27D4×Dic3C2211yescharacteristic
28C2×#214C2211yescharacteristic
29C2×#214C2211yescharacteristic, complemented
30#930C2211yescharacteristic
31#929—21no—
32C4×#67C2211yescharacteristic
33#926—21no—
34#924C2211yescharacteristic, complemented
35C2×#204C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37#898—21no—
38#897C2211yescharacteristic, complemented
39#892C2211yescharacteristic, complemented
40#890C2211yescharacteristic, complemented
41#889—21no—
42#887C2211yescharacteristic, complemented
43C3×#102C2211yescharacteristic
44C3×#102C2211yescharacteristic
45C3×#102C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic, complemented
47#832C2211yescharacteristic
48#832C2211yescharacteristic, complemented
49#830C2211yescharacteristic, complemented
50#821C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic, complemented
52#817C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#552—31no—
61D42—31no—
62#490—31no—
63#482—31no—
64#480—31no—
65#478—31no—
66#468—31no—
67#465—31no—
68#460—31no—
69#450—31no—
70#450—31no—
71#440—31no—
72#435—31no—
73#418—31no—
74C2×#102D311yescharacteristic, complemented
75C24×C3—21no—
76C23×D3—21no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—41no—
80C2×C3×D4C2311yescharacteristic
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×#67—44no—
90C2×#67D411yescharacteristic
91C22×Dic3—21no—
92D3×D4—22no—
93D3×D4—41no—
94C2×D12C2311yescharacteristic
95C2×C4×D3C2311yescharacteristic
96C2×Dic6C2311yescharacteristic
97C2×C3×C8—41no—
98C2×C3×C8C2311yescharacteristic
99C2×C3×C8C2311yescharacteristic
100C3×C4⋊C4—21no—
101C3×C4⋊C4C2311yescharacteristic
102C3×C4⋊C4C2311yescharacteristic
103#217—21no—
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#214—22no—
108#214—41no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4D411yescharacteristic
113C4⋊Dic3—21no—
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127ΓL(1,𝔽25)—41no—
128C22×D4—31no—
129C22×D4—61no—
130C22×D4C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×M16—31no—
135C22×C8C2×D311yescharacteristic, complemented
136#127—31no—
137#126—61no—
138#125—31no—
139#125—61no—
140#123—61no—
141C22⋊Q8—31no—
142C22⋊Q8—31no—
143#121—31no—
144#121—31no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151#117—31no—
152#117—61no—
153C2×C4⋊C4C2×D311yescharacteristic, complemented
154#107—31no—
155#107—31no—
156#106—31no—
157C4⋊C8—31no—
158#103—31no—
159#102—31no—
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102C2×D311yescharacteristic
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic, complemented
167#102C2×D311yescharacteristic, complemented
168C22⋊C8—31no—
169C22⋊C8—31no—
170#97—31no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—81no—
176C23×C3C2×D411yescharacteristic
177C22×D3—21no—
178C22×D3—42no—
179C22×D3—42no—
180C22×D3—81no—
181C22×D3C2×D411yescharacteristic
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—42no—
186C3×D4#4112yes—
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—42no—
198#67—42no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—42no—
204C2×Dic3C2×D411yescharacteristic
205C2×Dic3C2×D411yescharacteristic
206C2×Dic3C2×D411yescharacteristic
207D12—21no—
208D12—21no—
209C4×D3—21no—
210C4×D3—21no—
211Dic6—21no—
212Dic6—21no—
213C3×C8—21no—
214C3×C8—21no—
215C3⋊C8—21no—
216C3⋊C8—21no—
217C24—21no—
218C24—61no—
219C2×Q8—31no—
220C2×D4—22no—
221C2×D4—21no—
222C2×D4—31no—
223C2×D4—31no—
224C2×D4—41no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—62no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—121no—
232C2×D4—124no—
233C2×D4C22×D311yescharacteristic
234C2×D4C22×D311yescharacteristic
235C22×C4—21no—
236C22×C4—31no—
237C22×C4—61no—
238C22×C4C22×D311yescharacteristic
239SD16—62no—
240SD16—121no—
241SD16—121no—
242M16—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8—41no—
246C2×C8C22×D311yescharacteristic
247C2×C8C22×D311yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—31no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4C22×D311yescharacteristic
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—31no—
260C22⋊C4—31no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—31no—
268C42—61no—
269C42—61no—
270C22×C3—21no—
271C22×C3—21no—
272C22×C3—22no—
273C22×C3—41no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—84no—
279C22×C3AGL(1,ℤ/8)12yes—
280C22×C3C22×D411yescharacteristic
281C2×D3—21no—
282C2×D3—21no—
283C2×D3—21no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—84no—
288C3×C4—21no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4C2×#4111yescharacteristic
292C3×C4H(5,𝔽2)11yescharacteristic
293Dic3—21no—
294Dic3—21no—
295Dic3—21no—
296Dic3—41no—
297Dic3—41no—
298Dic3—41no—
299C23—21no—
300C23—21no—
301C23—31no—
302C23—42no—
303C23—42no—
304C23—61no—
305C23—81no—
306C23—122no—
307C23—122no—
308C23—241no—
309C23D3×D411yescharacteristic
310Q8—61no—
311Q8—61no—
312D4—21no—
313D4—21no—
314D4—21no—
315D4—42no—
316D4—62no—
317D4—61no—
318D4—61no—
319D4—122no—
320D4—122no—
321D4—122no—
322D4—122no—
323D4—241no—
324D4#23512yes—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—31no—
330C2×C4—31no—
331C2×C4—31no—
332C2×C4—42no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—122no—
339C2×C4D3×D411yescharacteristic
340C2×C4C23×D311yescharacteristic
341C8—21no—
342C8—21no—
343C8—61no—
344C8—61no—
345C2×C3—22no—
346C2×C3—42no—
347C2×C3—81no—
348C2×C3#54611yescharacteristic
349C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
350C2×C3#57711yescharacteristic
351D3—41no—
352D3—81no—
353C22—21no—
354C22—21no—
355C22—22no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—41no—
361C22—61no—
362C22—61no—
363C22—61no—
364C22—84no—
365C22—121no—
366C22—121no—
367C22—121no—
368C22—244no—
369C22#91112yes—
370C22C2×D3×D411yescharacteristic
371C4—21no—
372C4—41no—
373C4—41no—
374C4—61no—
375C4—61no—
376C4—61no—
377C4—121no—
378C4—121no—
379C4—121no—
380C4C2×#23511yescharacteristic
381C4#100411yescharacteristic
382C3#331011yescharacteristic, complemented, Sylow
383C2—22no—
384C2—42no—
385C2—81no—
386C2—121no—
387C2—241no—
388C2#549011yescharacteristic
389C2C2×#91111yescharacteristic
390C2#576211yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#102C2
socleC22×C3C22×D4
prime radicalD4×#67C2
hypercenterC2×#102D3
hypocenterC3#3310
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#102

Complements

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

NH ≅ G/Ntype
2. #5513C2semidirect
3. C2×C3×#102C2semidirect
4. #5218C2semidirect
5. #5114C2semidirect
6. #5111C2semidirect
7. #4967C2semidirect
8. #4780C2semidirect
9. #4779C2semidirect
10. #4776C2semidirect
11. #4770C2semidirect
12. #4764C2semidirect
13. #4763C2semidirect
14. #4762C2semidirect
15. #4757C2semidirect
20. C22×C3×C8C22semidirect
21. C2×C3×C4⋊C4C22semidirect
29. C2×#214C22semidirect
34. #924C22semidirect
35. C2×#204C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
38. #897C22semidirect
39. #892C22semidirect
40. #890C22semidirect
42. #887C22semidirect
45. C3×#102C22semidirect
46. C3×#102C22semidirect
48. #832C22semidirect
49. #830C22semidirect
50. #821C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #817C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #807C22semidirect
59. #807C22semidirect
74. C2×#102D3semidirect
135. C22×C8C2×D3semidirect
153. C2×C4⋊C4C2×D3semidirect
166. #102C2×D3semidirect
167. #102C2×D3semidirect
382. C3#3310semidirect

Maximal subgroups

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

subgroupindex
D4×#672
#55132
C2×C3×#1022
#52182
#51142
#51112
#49672
#47802
#47792
#47762
#47702
#47642
#47632
#47622
#47572
#33103