Finite Group Databasebeta

← #80864

Subgroups — #80864

Order 384.

548 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#911C211yesmaximal, characteristic, complemented
2C2×D3×M16C211yesmaximal, characteristic, complemented
3#5606C211yesmaximal, characteristic, complemented
4D3×#121C211yesmaximal, characteristic, complemented
5C3×#469C211yesmaximal, characteristic, complemented
6#5157C211yesmaximal, characteristic, complemented
7#5134C211yesmaximal, characteristic, complemented
8#5042C211yesmaximal, characteristic, complemented
9#5039C211yesmaximal, characteristic, complemented
10#4864C211yesmaximal, characteristic, complemented
11#4861C211yesmaximal, characteristic, complemented
12#4778C211yesmaximal, characteristic, complemented
13#4775C211yesmaximal, characteristic, complemented
14#4771C211yesmaximal, characteristic, complemented
15#4770C211yesmaximal, characteristic, complemented
16#3055—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×D3×D4—21no—
19C2×D3×D4C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic, complemented
22C2×C3×M16C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24C3×#121C2211yescharacteristic, complemented
25#939—21no—
26#938C2211yescharacteristic
27#938C2211yescharacteristic, complemented
28#937—21no—
29D4×Dic3—21no—
30C2×#214C2211yescharacteristic, complemented
31C2×#213C2211yescharacteristic, complemented
32#930C2211yescharacteristic
33#924C2211yescharacteristic, complemented
34C2×#208C2211yescharacteristic, complemented
35#911—24no—
36D3×M16—24no—
37C2×#204C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×ΓL(1,𝔽25)C2211yescharacteristic
40C2×D3×C8C2211yescharacteristic, complemented
41C2×D3×C8C2211yescharacteristic, complemented
42#897C2211yescharacteristic, complemented
43#895C2211yescharacteristic
44#892C2211yescharacteristic
45D3×C4⋊C4C2211yescharacteristic
46#884—21no—
47#883—21no—
48#67⋊C4—21no—
49D3×C22⋊C4—21no—
50C3×#106C2211yescharacteristic, complemented
51C3×#102C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53#832C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56#818C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#808C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×M16—31no—
64#525—31no—
65C2×#121—31no—
66#469—31no—
67#469D311yescharacteristic, complemented
68#468—31no—
69#468—31no—
70#466—31no—
71#466—31no—
72#465—31no—
73#465—31no—
74#428—31no—
75#418—31no—
76#417—31no—
77C23×D3—21no—
78C23×D3D411yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C22×Dic3—21no—
90C22×Dic3D411yescharacteristic
91#237—22no—
92#237—22no—
93D3×D4—22no—
94D3×D4—22no—
95D3×D4—44no—
96C2×D12—21no—
97C2×D12C2311yescharacteristic
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—21no—
101C2×C4×D3D411yescharacteristic
102C2×C4×D3D411yescharacteristic
103C2×C4×D3C2311yescharacteristic
104C2×C4×D3C2311yescharacteristic
105C2×Dic6C2311yescharacteristic
106C3×D8—22no—
107C3×M16—22no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C4⋊C4C2311yescharacteristic
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#214—22no—
118#213—22no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123C4⋊Dic3—21no—
124C4⋊Dic3C2311yescharacteristic
125C4⋊Dic3C2311yescharacteristic
126Dic3⋊C4—21no—
127Dic3⋊C4—21no—
128C4×Dic3—21no—
129#208—22no—
130C2×C3⋊C8C2311yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132#204—22no—
133ΓL(1,𝔽25)—22no—
134ΓL(1,𝔽25)D414yescomplemented
135D3×C8—22no—
136D3×C8—22no—
137C2×#41—31no—
138C22×D4—31no—
139C22×D4—61no—
140C23×C4—31no—
141AGL(1,ℤ/8)—64no—
142C2×SD16—31no—
143C2×SD16—31no—
144C2×D8—31no—
145C2×D8C2×D311yescharacteristic, complemented
146C2×M16—31no—
147C2×M16—31no—
148C2×M16—31no—
149C2×M16—64no—
150C2×M16C2×D311yescharacteristic, complemented
151C22×C8—31no—
152C22×C8—31no—
153#123—61no—
154C22⋊Q8—31no—
155C22⋊Q8—31no—
156#121—31no—
157#121—31no—
158#121—31no—
159#121—31no—
160#121—61no—
161#121—61no—
162#121C2×D311yescharacteristic
163#121C2×D311yescharacteristic, complemented
164#120—61no—
165C4×D4—61no—
166C4×D4—61no—
167#117—31no—
168C2×C4⋊C4—31no—
169C2×C22⋊C4—61no—
170#107—31no—
171#107—31no—
172#106—31no—
173#106C2×D311yescharacteristic, complemented
174#103—31no—
175#103—31no—
176#102—31no—
177#102—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic, complemented
181#102C2×D311yescharacteristic, complemented
182C23×C3—21no—
183C23×C3—21no—
184C23×C3C2×D411yescharacteristic
185C22×D3—22no—
186C22×D3—22no—
187C22×D3—21no—
188C22×D3—41no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3C2×D411yescharacteristic
193C22×D3C2×D411yescharacteristic
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—42no—
199C3×D4—42no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4C2×D411yescharacteristic
205C2×C3×C4C2411yescharacteristic
206#67—42no—
207#67—42no—
208#67—42no—
209#67—42no—
210#67—42no—
211C2×Dic3—22no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—42no—
216C2×Dic3C2×D411yescharacteristic
217C2×Dic3C2×D411yescharacteristic
218D12—21no—
219D12—21no—
220D12—42no—
221C4×D3—22no—
222C4×D3—21no—
223C4×D3—21no—
224C4×D3—22no—
225C4×D3—42no—
226C4×D3C2×D412yes—
227C4×D3#4112yes—
228Dic6—21no—
229Dic6—21no—
230C3×C8—21no—
231C3×C8C2×D412yescomplemented
232C3⋊C8—21no—
233C3⋊C8C2×D412yescomplemented
234C24—31no—
235C24—61no—
236#41—62no—
237#41—62no—
238C2×Q8—31no—
239C2×D4—21no—
240C2×D4—21no—
241C2×D4—31no—
242C2×D4—61no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—61no—
246C2×D4—61no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—124no—
251C2×D4C22×D311yescharacteristic
252C2×D4C22×D311yescharacteristic
253C22×C4—31no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—62no—
259C22×C4—62no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4C22×D311yescharacteristic
263SD16—62no—
264SD16—62no—
265D8—22no—
266D8—62no—
267M16—22no—
268M16—34no—
269M16—62no—
270M16—62no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—62no—
274C2×C8—62no—
275C2×C8C22×D311yescharacteristic
276C2×C8C22×D311yescharacteristic
277C4⋊C4—31no—
278C4⋊C4—31no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—61no—
287C22⋊C4—61no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C42—61no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3C22×D411yescharacteristic
304C2×D3—21no—
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—44no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—84no—
313C2×D3#12112yes—
314C2×D3AGL(1,ℤ/8)12yescomplemented
315C3×C4—21no—
316C3×C4—41no—
317C3×C4—41no—
318C3×C4C22×D411yescharacteristic
319C3×C4C2×#4111yescharacteristic
320Dic3—21no—
321Dic3—41no—
322Dic3—41no—
323Dic3—41no—
324Dic3#12112yes—
325C23—21no—
326C23—21no—
327C23—31no—
328C23—31no—
329C23—62no—
330C23—62no—
331C23—61no—
332C23—121no—
333C23—122no—
334C23—122no—
335C23—122no—
336C23D3×D411yescharacteristic
337Q8—61no—
338Q8—61no—
339D4—21no—
340D4—21no—
341D4—21no—
342D4—21no—
343D4—42no—
344D4—42no—
345D4—61no—
346D4—61no—
347D4—122no—
348D4—122no—
349D4—122no—
350D4—122no—
351D4—122no—
352D4—122no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—31no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—31no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—62no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—122no—
370C2×C4—122no—
371C2×C4D3×D411yescharacteristic
372C2×C4C23×D311yescharacteristic
373C8—21no—
374C8—32no—
375C8—61no—
376C8D3×D412yescomplemented
377C2×C3—41no—
378C2×C3—81no—
379C2×C3—81no—
380C2×C3C2×#12111yescharacteristic
381C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
382C2×C3#57511yescharacteristic
383D3—22no—
384D3—22no—
385D3—81no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—32no—
390C22—32no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22—124no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—244no—
406C22C2×D3×D411yescharacteristic
407C4—21no—
408C4—32no—
409C4—41no—
410C4—41no—
411C4—61no—
412C4—121no—
413C4—121no—
414C4—121no—
415C4C2×D3×D411yescharacteristic
416C4D3×#4111yescharacteristic
417C3#305511yescharacteristic, complemented, Sylow
418C2—41no—
419C2—62no—
420C2—62no—
421C2—81no—
422C2—81no—
423C2—241no—
424C2D3×#12111yescharacteristic
425C2D3×AGL(1,ℤ/8)11yescharacteristic
426C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#469C2
socleC22×C3C22×D4
hypercenter#469D3
hypocenterC3#3055
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×#911C2semidirect
2. C2×D3×M16C2semidirect
3. #5606C2semidirect
4. D3×#121C2semidirect
5. C3×#469C2semidirect
6. #5157C2semidirect
7. #5134C2semidirect
8. #5042C2semidirect
9. #5039C2semidirect
10. #4864C2semidirect
11. #4861C2semidirect
12. #4778C2semidirect
13. #4775C2semidirect
14. #4771C2semidirect
15. #4770C2semidirect
21. C2×C3×D8C22semidirect
22. C2×C3×M16C22semidirect
24. C3×#121C22semidirect
27. #938C22semidirect
30. C2×#214C22semidirect
31. C2×#213C22semidirect
33. #924C22semidirect
34. C2×#208C22semidirect
37. C2×#204C22semidirect
40. C2×D3×C8C22semidirect
41. C2×D3×C8C22semidirect
42. #897C22semidirect
50. C3×#106C22semidirect
51. C3×#102C22semidirect
52. C3×#102C22semidirect
53. #832C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. #818C22semidirect
57. #816C22semidirect
58. #810C22semidirect
59. #809C22semidirect
60. #808C22semidirect
61. #807C22semidirect
67. #469D3semidirect
134. ΓL(1,𝔽25)D4semidirect
145. C2×D8C2×D3semidirect
150. C2×M16C2×D3semidirect
163. #121C2×D3semidirect
173. #106C2×D3semidirect
180. #102C2×D3semidirect
181. #102C2×D3semidirect
231. C3×C8C2×D4semidirect
233. C3⋊C8C2×D4semidirect
314. C2×D3AGL(1,ℤ/8)semidirect
376. C8D3×D4semidirect
417. C3#3055semidirect

Maximal subgroups

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

subgroupindex
C2×#9112
C2×D3×M162
#56062
D3×#1212
C3×#4692
#51572
#51342
#50422
#50392
#48642
#48612
#47782
#47752
#47712
#47702
#30553