Finite Group Databasebeta

← #80567

Subgroups — #80567

Order 384.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#916C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5676C211yesmaximal, characteristic, complemented
4#5555C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic, complemented
7#5158C211yesmaximal, characteristic, complemented
8#5060C211yesmaximal, characteristic, complemented
9#5014C211yesmaximal, characteristic, complemented
10#4841C211yesmaximal, characteristic, complemented
11#4836C211yesmaximal, characteristic, complemented
12#4816C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4778C211yesmaximal, characteristic, complemented
15#4776C211yesmaximal, characteristic, complemented
16#3267—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#237C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic, complemented
22C2×C3×D8C2211yescharacteristic
23C3×#124C2211yescharacteristic
24C3×C4×D4C2211yescharacteristic, complemented
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic, complemented
27C2×#216C2211yescharacteristic, complemented
28#938C2211yescharacteristic
29C2×#214C2211yescharacteristic
30C2×#214C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930—21no—
33C2×C4⋊Dic3—21no—
34#916—24no—
35#911—24no—
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic, complemented
40#897C2211yescharacteristic
41#897C2211yescharacteristic, complemented
42#886—21no—
43#886—21no—
44#884—21no—
45#883—21no—
46#882—21no—
47D3×C22⋊C4—21no—
48C4×D12C2211yescharacteristic
49C4×D12C2211yescharacteristic
50#869C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic
54#832C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#809C2211yescharacteristic
60#808C2211yescharacteristic, complemented
61#803C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×#135—31no—
64#547—31no—
65#546—31no—
66#497—31no—
67#470—31no—
68#465—31no—
69#463D311yescharacteristic, complemented
70#461—31no—
71#452—31no—
72#452—31no—
73#450—31no—
74#449—31no—
75#442—31no—
76#435—31no—
77C23×D3—21no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88#239—22no—
89#239—41no—
90#237—22no—
91#237—22no—
92#237—41no—
93#237—41no—
94D3×D4—22no—
95D3×D4—41no—
96C2×D12—21no—
97C2×D12D411yescharacteristic
98C2×D12C2311yescharacteristic
99C2×C4×D3—21no—
100C2×C4×D3—21no—
101C2×C4×D3C2311yescharacteristic
102C2×C4×D3C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×SD16—22no—
106C3×D8—22no—
107C2×C3×C8C2311yescharacteristic
108C2×C3×C8C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C42C2311yescharacteristic
114#217—21no—
115#216—22no—
116#216—41no—
117#214—22no—
118#214—41no—
119#213D414yescomplemented
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4D411yescharacteristic
125(C2×D3)⋊C4D411yescharacteristic
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C4⋊Dic3C2311yescharacteristic
132Dic3⋊C4—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135#204—22no—
136#204—22no—
137ΓL(1,𝔽25)—22no—
138D3×C8—22no—
139C2×#41—31no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144#135—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16C2×D311yescharacteristic, complemented
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157#124C2×D311yescharacteristic
158#123—61no—
159#123—61no—
160#123—61no—
161C22⋊Q8—31no—
162C22⋊Q8—31no—
163C22⋊Q8—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4C2×D311yescharacteristic, complemented
171C2×C4⋊C4—61no—
172C2×C22⋊C4—61no—
173#106—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97—31no—
184C23×C3—21no—
185C23×C3—21no—
186C22×D3—21no—
187C22×D3—42no—
188C22×D3—42no—
189C22×D3—81no—
190C22×D3C2×D411yescharacteristic
191C22×D3C2×D411yescharacteristic
192C3×Q8—21no—
193C3×Q8—21no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4C2×D412yes—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—42no—
205#67—42no—
206#67—42no—
207#67—81no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215D12—21no—
216D12—41no—
217D12—42no—
218D12C2×D412yes—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—21no—
223C4×D3—42no—
224C4×D3—81no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—62no—
237#41—121no—
238#41—121no—
239#41—121no—
240C2×Q8—31no—
241C2×Q8—31no—
242C2×Q8C22×D311yescharacteristic
243C2×D4—31no—
244C2×D4—31no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—62no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—121no—
252C2×D4C22×D311yescharacteristic
253C2×D4C22×D311yescharacteristic
254C22×C4—21no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261Q16—62no—
262Q16—121no—
263SD16—22no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268D8—22no—
269D8—34no—
270M16—62no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—62no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4—61no—
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—31no—
288C22⋊C4—31no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C42C22×D311yescharacteristic
295C22×C3—22no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3C22×D411yescharacteristic
302C2×D3—21no—
303C2×D3—21no—
304C2×D3—22no—
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—84no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4C22×D411yescharacteristic
317C3×C4H(5,𝔽2)11yescharacteristic
318Dic3—21no—
319Dic3—21no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323C23—21no—
324C23—21no—
325C23—31no—
326C23—31no—
327C23—61no—
328C23—122no—
329C23—122no—
330C23—241no—
331Q8—21no—
332Q8—21no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337D4—21no—
338D4—21no—
339D4—21no—
340D4—32no—
341D4—61no—
342D4—121no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—241no—
348D4D3×D412yes—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—31no—
352C2×C4—31no—
353C2×C4—42no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—122no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4D3×D411yescharacteristic
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—21no—
370C8—32no—
371C8—61no—
372C2×C3—42no—
373C2×C3—81no—
374C2×C3D4211yescharacteristic
375C2×C3#57511yescharacteristic
376C2×C3#57711yescharacteristic
377D3—41no—
378D3—41no—
379D3—42no—
380C22—22no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—61no—
387C22—61no—
388C22—61no—
389C22—61no—
390C22—61no—
391C22—61no—
392C22—62no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—61no—
402C4—61no—
403C4—121no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100911yescharacteristic
408C3#326711yescharacteristic, complemented, Sylow
409C2—42no—
410C2—81no—
411C2—121no—
412C2—121no—
413C2—122no—
414C2#560911yescharacteristic
415C2#577011yescharacteristic
416C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3267
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. C2×#916C2semidirect
2. C2×#911C2semidirect
3. #5676C2semidirect
4. #5555C2semidirect
5. C3×#463C2semidirect
6. #5173C2semidirect
7. #5158C2semidirect
8. #5060C2semidirect
9. #5014C2semidirect
10. #4841C2semidirect
11. #4836C2semidirect
12. #4816C2semidirect
13. #4808C2semidirect
14. #4778C2semidirect
15. #4776C2semidirect
19. C2×#237C22semidirect
21. C2×C3×SD16C22semidirect
24. C3×C4×D4C22semidirect
26. C2×#216C22semidirect
27. C2×#216C22semidirect
39. C2×D3×C8C22semidirect
41. #897C22semidirect
50. #869C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
54. #832C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. #816C22semidirect
60. #808C22semidirect
61. #803C22semidirect
69. #463D3semidirect
119. #213D4semidirect
151. C2×SD16C2×D3semidirect
170. C4×D4C2×D3semidirect
174. C4⋊C8C2×D3semidirect
177. #103C2×D3semidirect
408. C3#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#9162
C2×#9112
#56762
#55552
C3×#4632
#51732
#51582
#50602
#50142
#48412
#48362
#48162
#48082
#47782
#47762
#32673