Finite Group Databasebeta

← #80556

Subgroups — #80556

Order 384.

498 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×#915C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5677C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5169C211yesmaximal, characteristic, complemented
7#5158C211yesmaximal, characteristic, complemented
8#5059C211yesmaximal, characteristic, complemented
9C4×#214C211yesmaximal, characteristic, complemented
10#4845C211yesmaximal, characteristic, complemented
11#4838C211yesmaximal, characteristic, complemented
12#4816C211yesmaximal, characteristic, complemented
13#4806C211yesmaximal, characteristic, complemented
14#4783C211yesmaximal, characteristic, complemented
15#4782C211yesmaximal, characteristic, complemented
16#3275—31nomaximal, Sylow
17C2×D3×Q8C2211yescharacteristic, complemented
18C2×#237C2211yescharacteristic
19C2×#237C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C2×C3×SD16C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C3×#124C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic
25#946C2211yescharacteristic
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic
28#938C2211yescharacteristic, complemented
29C2×#214C2211yescharacteristic
30C2×#214C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic
32#930—21no—
33C2×(C2×D3)⋊C4—21no—
34#915—24no—
35#911—24no—
36C2×Dic12C2211yescharacteristic, complemented
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
40#897C2211yescharacteristic
41#895C2211yescharacteristic
42#886—21no—
43#885—21no—
44#884—21no—
45#882—21no—
46#67⋊C4—21no—
47#878—21no—
48#875C2211yescharacteristic
49C4×D12C2211yescharacteristic
50C4×Dic6C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic
55#821C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61C4×C3⋊C8C2211yescharacteristic, complemented
62C2×#137—31no—
63C2×AGL(1,ℤ/8)—31no—
64#548—31no—
65#546—31no—
66#495—31no—
67#470—31no—
68#469—31no—
69#464—31no—
70#463D311yescharacteristic, complemented
71#452—31no—
72#450—31no—
73#449—31no—
74#448—31no—
75C4×SD16—31no—
76#436—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—
88D3×Q8—22no—
89D3×Q8—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×D3C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×Dic6—21no—
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#215—41no—
117#214—41no—
118#214D414yescomplemented
119#213—22no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4D411yescharacteristic
126(C2×D3)⋊C4D411yescharacteristic
127C4⋊Dic3—21no—
128C4⋊Dic3D411yescharacteristic
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132C4×Dic3—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135Dic12—22no—
136#204—22no—
137ΓL(1,𝔽25)—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143#137—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×Q16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16C2×D311yescharacteristic
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124—61no—
158#124C2×D311yescharacteristic, complemented
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—61no—
166#121—61no—
167#120—61no—
168C4×Q8—31no—
169C4×D4—31no—
170C4×D4—61no—
171C4×D4C2×D311yescharacteristic
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, complemented
181C22⋊C8—31no—
182C22⋊C8—31no—
183C4×C8—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—81no—
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2×D411yescharacteristic
203C2×C3×C4C2411yescharacteristic
204#67—42no—
205#67—42no—
206#67—42no—
207#67—42no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—42no—
212C2×Dic3—42no—
213C2×Dic3C2×D411yescharacteristic
214C2×Dic3C2×D411yescharacteristic
215D12—21no—
216D12—21no—
217D12—41no—
218D12—81no—
219C4×D3—21no—
220C4×D3—21no—
221C4×D3—21no—
222C4×D3—21no—
223C4×D3—42no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—42no—
228Dic6C2×D412yes—
229C3×C8—21no—
230C3×C8—21no—
231C3⋊C8—21no—
232C3⋊C8C2×D412yes—
233C24—61no—
234#41—62no—
235#41—62no—
236#41—121no—
237#41—121no—
238C2×Q8—31no—
239C2×Q8—31no—
240C2×Q8—61no—
241C2×Q8—62no—
242C2×Q8—121no—
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—62no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C2×D4—121no—
253C2×D4C22×D311yescharacteristic
254C2×D4C22×D311yescharacteristic
255C22×C4—21no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261Q16—62no—
262Q16—62no—
263SD16—22no—
264SD16—34no—
265SD16—62no—
266SD16—121no—
267SD16—121no—
268D8—22no—
269D8—62no—
270M16—62no—
271M16—62no—
272C2×C8—31no—
273C2×C8—31no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—31no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4C22×D311yescharacteristic
282C22⋊C4—21no—
283C22⋊C4—21no—
284C22⋊C4—21no—
285C22⋊C4—31no—
286C22⋊C4—31no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—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—21no—
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—41no—
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—22no—
319Dic3—21no—
320Dic3—21no—
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—32no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337Q8—122no—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—61no—
342D4—61no—
343D4—121no—
344D4—122no—
345D4—122no—
346D4—122no—
347D4—122no—
348D4—241no—
349D4D3×D412yes—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—31no—
353C2×C4—31no—
354C2×C4—61no—
355C2×C4—61no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—81no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
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—81no—
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—121no—
393C22—121no—
394C22—121no—
395C22—244no—
396C22C2×D3×D411yescharacteristic
397C4—21no—
398C4—21no—
399C4—41no—
400C4—41no—
401C4—61no—
402C4—61no—
403C4—62no—
404C4—121no—
405C4—121no—
406C4C2×D3×D411yescharacteristic
407C4#100911yescharacteristic
408C3#327511yescharacteristic, complemented, Sylow
409C2—42no—
410C2—81no—
411C2—121no—
412C2—121no—
413C2—241no—
414C2#560911yescharacteristic
415C2#576811yescharacteristic
416C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3275
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×#915C2semidirect
2. C2×#911C2semidirect
3. #5677C2semidirect
4. #5554C2semidirect
5. C3×#463C2semidirect
6. #5169C2semidirect
7. #5158C2semidirect
8. #5059C2semidirect
9. C4×#214C2semidirect
10. #4845C2semidirect
11. #4838C2semidirect
12. #4816C2semidirect
13. #4806C2semidirect
14. #4783C2semidirect
15. #4782C2semidirect
17. C2×D3×Q8C22semidirect
23. C3×#124C22semidirect
26. C2×#216C22semidirect
28. #938C22semidirect
30. C2×#214C22semidirect
36. C2×Dic12C22semidirect
39. C2×ΓL(1,𝔽25)C22semidirect
50. C4×Dic6C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
59. #810C22semidirect
60. #808C22semidirect
61. C4×C3⋊C8C22semidirect
70. #463D3semidirect
118. #214D4semidirect
158. #124C2×D3semidirect
174. C4⋊C8C2×D3semidirect
177. #103C2×D3semidirect
180. #102C2×D3semidirect
408. C3#3275semidirect

Maximal subgroups

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

subgroupindex
C2×#9152
C2×#9112
#56772
#55542
C3×#4632
#51692
#51582
#50592
C4×#2142
#48452
#48382
#48162
#48062
#47832
#47822
#32753