Finite Group Databasebeta

← #78758

Subgroups — #78758

Order 384.

462 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×#67C211yesmaximal, characteristic
2#5511C211yesmaximal, characteristic, complemented
3C2×C3×#102C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5114C211yesmaximal, characteristic, complemented
6C8×#67C211yesmaximal, characteristic, complemented
7#4968C211yesmaximal, characteristic, complemented
8#4778C211yesmaximal, characteristic, complemented
9#4777C211yesmaximal, characteristic, complemented
10#4776C211yesmaximal, characteristic, complemented
11D3×#102C211yesmaximal, characteristic, complemented
12#4762C211yesmaximal, characteristic, complemented
13#4761C211yesmaximal, characteristic, complemented
14#4760C211yesmaximal, characteristic, complemented
15#4758C211yesmaximal, characteristic, complemented
16#3295—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
31C4×#67—21no—
32C4×#67C2211yescharacteristic
33#924—21no—
34#924C2211yescharacteristic, complemented
35C2×#204C2211yescharacteristic, complemented
36C2×D3×C8C2211yescharacteristic, complemented
37#897C2211yescharacteristic, complemented
38#896—21no—
39D3×C4⋊C4C2211yescharacteristic, complemented
40#889—21no—
41#888C2211yescharacteristic, complemented
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
54Dic3⋊C8C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#809C2211yescharacteristic, complemented
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#808C2211yescharacteristic, complemented
60#554—31no—
61D42—31no—
62#486—31no—
63#481—31no—
64#481—31no—
65#476—31no—
66#465—31no—
67#465—31no—
68#460—31no—
69#450—31no—
70#450—31no—
71C4×SD16—31no—
72C8×D4—31no—
73C2×#102—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×D3—21no—
96C2×C4×D3C2311yescharacteristic
97C2×Dic6—21no—
98C2×Dic6C2311yescharacteristic
99C2×C3×C8—41no—
100C2×C3×C8C2311yescharacteristic
101C2×C3×C8C2311yescharacteristic
102C3×C4⋊C4—21no—
103C3×C4⋊C4C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105#217—21no—
106#217—21no—
107#217—21no—
108#217D411yescharacteristic
109#214—22no—
110#214—41no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113(C2×D3)⋊C4D411yescharacteristic
114C4⋊Dic3—21no—
115C4⋊Dic3C2311yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4D411yescharacteristic
122C4×Dic3—21no—
123C4×Dic3C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125C2×C3⋊C8C2311yescharacteristic
126#204—41no—
127D3×C8—41no—
128C22×D4—31no—
129C22×D4—61no—
130C22×D4C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C22×C8—31no—
135C22×C8C2×D311yescharacteristic, complemented
136C4⋊Q8—31no—
137#127—31no—
138#125—61no—
139C22⋊Q8—31no—
140C22⋊Q8—31no—
141C22⋊Q8—61no—
142C22⋊Q8—61no—
143#121—31no—
144#121—31no—
145#121—61no—
146#120—61no—
147#120—61no—
148C4×Q8—31no—
149C4×D4—31no—
150C4×D4—31no—
151C4×D4—61no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4C2×D311yescharacteristic, complemented
154#106—31no—
155#106—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—
170C4×C8—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—81no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—42no—
195#67—42no—
196#67—42no—
197#67—42no—
198#67—81no—
199#67SD1614yescomplemented
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—
211C4×D3—81no—
212Dic6—21no—
213Dic6—21no—
214Dic6—81no—
215C3×C8—21no—
216C3×C8—21no—
217C3⋊C8—21no—
218C3⋊C8—21no—
219C24—21no—
220C24—61no—
221C2×Q8—31no—
222C2×Q8—61no—
223C2×D4—22no—
224C2×D4—21no—
225C2×D4—31no—
226C2×D4—31no—
227C2×D4—41no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4—62no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—121no—
235C2×D4—124no—
236C2×D4C22×D311yescharacteristic
237C2×D4C22×D311yescharacteristic
238C22×C4—21no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C22×D311yescharacteristic
243SD16—62no—
244SD16—121no—
245SD16—121no—
246C2×C8—31no—
247C2×C8—31no—
248C2×C8—41no—
249C2×C8—121no—
250C2×C8C22×D311yescharacteristic
251C2×C8C22×D311yescharacteristic
252C4⋊C4—21no—
253C4⋊C4—31no—
254C4⋊C4—31no—
255C4⋊C4—31no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4—61no—
259C4⋊C4—61no—
260C4⋊C4—61no—
261C4⋊C4C22×D311yescharacteristic
262C4⋊C4C22×D311yescharacteristic
263C22⋊C4—31no—
264C22⋊C4—31no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C22⋊C4—61no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C42—31no—
271C42—61no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—22no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—84no—
281C22×C3C2×SD1612yes—
282C22×C3C22×D411yescharacteristic
283C2×D3—21no—
284C2×D3—21no—
285C2×D3—41no—
286C2×D3—41no—
287C2×D3—41no—
288C2×D3—84no—
289C2×D3C2×SD1612yes—
290C3×C4—21no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4C2×#4111yescharacteristic
294C3×C4H(5,𝔽2)11yescharacteristic
295Dic3—21no—
296Dic3—21no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300Dic3C2×SD1612yes—
301C23—21no—
302C23—21no—
303C23—31no—
304C23—42no—
305C23—42no—
306C23—61no—
307C23—81no—
308C23—122no—
309C23—122no—
310C23—241no—
311C23D3×D411yescharacteristic
312Q8—61no—
313Q8—61no—
314Q8—241no—
315D4—21no—
316D4—21no—
317D4—21no—
318D4—34no—
319D4—42no—
320D4—61no—
321D4—61no—
322D4—122no—
323D4—122no—
324D4—122no—
325D4—122no—
326D4—241no—
327D4#23512yes—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—31no—
333C2×C4—31no—
334C2×C4—31no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—81no—
341C2×C4—122no—
342C2×C4—241no—
343C2×C4D3×D411yescharacteristic
344C2×C4C23×D311yescharacteristic
345C8—21no—
346C8—21no—
347C8—61no—
348C8—61no—
349C2×C3—22no—
350C2×C3—42no—
351C2×C3—81no—
352C2×C3#54611yescharacteristic
353C2×C3C22×SD1611yescharacteristic
354C2×C3#57611yescharacteristic
355D3—22no—
356D3—81no—
357C22—21no—
358C22—21no—
359C22—22no—
360C22—32no—
361C22—41no—
362C22—41no—
363C22—41no—
364C22—41no—
365C22—41no—
366C22—61no—
367C22—61no—
368C22—84no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22—244no—
373C22D3×SD1612yes—
374C22C2×D3×D411yescharacteristic
375C4—21no—
376C4—32no—
377C4—41no—
378C4—41no—
379C4—61no—
380C4—61no—
381C4—121no—
382C4—121no—
383C4—121no—
384C4C2×#23511yescharacteristic
385C4#100411yescharacteristic
386C3#329511yescharacteristic, complemented, Sylow
387C2—22no—
388C2—42no—
389C2—62no—
390C2—81no—
391C2—241no—
392C2#549011yescharacteristic
393C2#575711yescharacteristic
394C2C2×D3×SD1611yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC2×C3×#102C2
socleC22×C3C22×D4
prime radicalD4×#67C2
hypercenterC2×#102D3
hypocenterC3#3295
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. #5511C2semidirect
3. C2×C3×#102C2semidirect
4. #5218C2semidirect
5. #5114C2semidirect
6. C8×#67C2semidirect
7. #4968C2semidirect
8. #4778C2semidirect
9. #4777C2semidirect
10. #4776C2semidirect
11. D3×#102C2semidirect
12. #4762C2semidirect
13. #4761C2semidirect
14. #4760C2semidirect
15. #4758C2semidirect
20. C22×C3×C8C22semidirect
21. C2×C3×C4⋊C4C22semidirect
29. C2×#214C22semidirect
34. #924C22semidirect
35. C2×#204C22semidirect
36. C2×D3×C8C22semidirect
37. #897C22semidirect
39. D3×C4⋊C4C22semidirect
41. #888C22semidirect
42. #887C22semidirect
45. C3×#102C22semidirect
46. C3×#102C22semidirect
48. #832C22semidirect
49. #830C22semidirect
50. #821C22semidirect
51. (C2×D3)⋊C8C22semidirect
52. #817C22semidirect
53. #816C22semidirect
54. Dic3⋊C8C22semidirect
55. C8×Dic3C22semidirect
56. #809C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #808C22semidirect
74. C2×#102D3semidirect
135. C22×C8C2×D3semidirect
153. C2×C4⋊C4C2×D3semidirect
166. #102C2×D3semidirect
167. #102C2×D3semidirect
199. #67SD16semidirect
386. C3#3295semidirect

Maximal subgroups

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

subgroupindex
D4×#672
#55112
C2×C3×#1022
#52182
#51142
C8×#672
#49682
#47782
#47772
#47762
D3×#1022
#47622
#47612
#47602
#47582
#32953