Finite Group Databasebeta

← #79879

Subgroups — #79879

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×#912C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5555C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#442C211yesmaximal, characteristic, complemented
6#5019C211yesmaximal, characteristic, complemented
7#5016C211yesmaximal, characteristic, complemented
8#4867C211yesmaximal, characteristic, complemented
9#4864C211yesmaximal, characteristic, complemented
10#4783C211yesmaximal, characteristic, complemented
11#4783C211yesmaximal, characteristic, complemented
12#4781C211yesmaximal, characteristic, complemented
13#4776C211yesmaximal, characteristic, complemented
14#4715C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3267—31nomaximal, Sylow
17C2×#237C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×#237C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C3×C4×D4C2211yescharacteristic
23C3×C4×D4C2211yescharacteristic, complemented
24C2×#214C2211yescharacteristic
25C2×#214C2211yescharacteristic
26C2×#214C2211yescharacteristic, complemented
27C2×#213C2211yescharacteristic
28#930—21no—
29#930—21no—
30C2×(C2×D3)⋊C4—21no—
31C2×C4⋊Dic3—21no—
32#912—24no—
33#911—24no—
34C2×Dic12C2211yescharacteristic, complemented
35C2×Dic12C2211yescharacteristic, complemented
36C2×#204C2211yescharacteristic
37C2×#204C2211yescharacteristic
38C2×ΓL(1,𝔽25)C2211yescharacteristic
39C2×D3×C8C2211yescharacteristic, complemented
40#897C2211yescharacteristic
41#897C2211yescharacteristic
42#897C2211yescharacteristic, complemented
43#895C2211yescharacteristic
44#886—21no—
45#886—21no—
46#886—21no—
47#882—21no—
48#875C2211yescharacteristic
49C4×D12C2211yescharacteristic
50#869C2211yescharacteristic, complemented
51C3×#106C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic
53C3×#102C2211yescharacteristic, complemented
54C3×#97C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#810C2211yescharacteristic, complemented
60#809C2211yescharacteristic
61C4⋊(C3⋊C8)C2211yescharacteristic, 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#463—31no—
70#461—31no—
71#452—31no—
72#452—31no—
73#450—31no—
74#449—31no—
75#442D311yescharacteristic, complemented
76#435—31no—
77C23×D3—21no—
78C2×C3×D4C2311yescharacteristic
79C2×C3×D4C2311yescharacteristic
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C22×Dic3—21no—
87C22×Dic3—21no—
88C22×Dic3—21no—
89#237—22no—
90#237—22no—
91#237—22no—
92#237—41no—
93#237—41no—
94#237—41no—
95D3×D4—22no—
96D3×D4—41no—
97C2×D12—21no—
98C2×D12D411yescharacteristic
99C2×D12C2311yescharacteristic
100C2×C4×D3C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×Dic6C2311yescharacteristic
103C2×Dic6C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C3×D8D414yescomplemented
106C2×C3×C8C2311yescharacteristic
107C2×C3×C8C2311yescharacteristic
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112C3×C42C2311yescharacteristic
113#214—22no—
114#214—22no—
115#214—22no—
116#213—22no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4D411yescharacteristic
124(C2×D3)⋊C4D411yescharacteristic
125C4⋊Dic3—21no—
126C4⋊Dic3—21no—
127C4⋊Dic3—21no—
128C4⋊Dic3—21no—
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C2×C3⋊C8C2311yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133Dic12—22no—
134Dic12—41no—
135#204—22no—
136#204—41no—
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×SD16—31no—
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C22×C8—31no—
156C4⋊Q8—31no—
157#124—31no—
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×D4C2×D311yescharacteristic
170C4×D4C2×D311yescharacteristic, complemented
171C2×C4⋊C4—61no—
172C2×C22⋊C4—61no—
173#106C2×D311yescharacteristic, complemented
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102C2×D311yescharacteristic
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8—31no—
183#97C2×D311yescharacteristic, complemented
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×D4—21no—
193C3×D4—21no—
194C3×D4C2×D412yes—
195C3×D4C2×D412yes—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—42no—
199C2×C3×C4—81no—
200C2×C3×C4C2×D411yescharacteristic
201C2×C3×C4C2×D411yescharacteristic
202C2×C3×C4C2411yescharacteristic
203#67—42no—
204#67—42no—
205#67—42no—
206#67—42no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—42no—
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—
223Dic6—21no—
224Dic6—21no—
225Dic6—21no—
226Dic6—21no—
227Dic6—21no—
228Dic6—21no—
229C3×C8—21no—
230C3×C8C2×D412yes—
231C3⋊C8—21no—
232C3⋊C8—21no—
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×Q8—31no—
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—21no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261Q16—62no—
262Q16—121no—
263SD16—62no—
264SD16—62no—
265SD16—62no—
266SD16—62no—
267SD16—121no—
268D8—62no—
269D8D1214yescomplemented
270M16—62no—
271C2×C8—31no—
272C2×C8—31no—
273C2×C8—62no—
274C2×C8C22×D311yescharacteristic
275C2×C8C22×D311yescharacteristic
276C4⋊C4—31no—
277C4⋊C4—61no—
278C4⋊C4—61no—
279C4⋊C4—61no—
280C4⋊C4—61no—
281C4⋊C4—61no—
282C4⋊C4C22×D311yescharacteristic
283C4⋊C4C22×D311yescharacteristic
284C22⋊C4—21no—
285C22⋊C4—21no—
286C22⋊C4—31no—
287C22⋊C4—31no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C42C22×D311yescharacteristic
295C22×C3—22no—
296C22×C3—22no—
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—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—61no—
332Q8—61no—
333Q8—61no—
334Q8—61no—
335Q8—61no—
336Q8—61no—
337D4—21no—
338D4—21no—
339D4—61no—
340D4—61no—
341D4—121no—
342D4—122no—
343D4—122no—
344D4—122no—
345D4—122no—
346D4—241no—
347D4C2×D1212yes—
348D4C2×D1212yes—
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—81no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4D3×D411yescharacteristic
366C2×C4D3×D411yescharacteristic
367C2×C4C23×D311yescharacteristic
368C8—21no—
369C8—61no—
370C8—61no—
371C8C2×D1212yes—
372C2×C3—42no—
373C2×C3—42no—
374C2×C3D4211yescharacteristic
375C2×C3#57511yescharacteristic
376C2×C3#57711yescharacteristic
377D3—41no—
378D3—41no—
379D3—81no—
380C22—22no—
381C22—22no—
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—121no—
404C4—121no—
405C4—121no—
406C4C22×D1211yescharacteristic
407C4#100911yescharacteristic
408C3#326711yescharacteristic, complemented, Sylow
409C2—42no—
410C2—42no—
411C2—121no—
412C2—121no—
413C2—241no—
414C2D4×D1211yescharacteristic
415C2#577311yescharacteristic
416C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#442C2
socleC22×C3C22×D4
hypercenter#442D3
hypocenterC3#3267
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #442

Complements

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

NH ≅ G/Ntype
1. C2×#912C2semidirect
2. C2×#911C2semidirect
3. #5555C2semidirect
4. #5554C2semidirect
5. C3×#442C2semidirect
6. #5019C2semidirect
7. #5016C2semidirect
8. #4867C2semidirect
9. #4864C2semidirect
10. #4783C2semidirect
11. #4783C2semidirect
12. #4781C2semidirect
13. #4776C2semidirect
14. #4715C2semidirect
15. #4706C2semidirect
19. C2×#237C22semidirect
23. C3×C4×D4C22semidirect
26. C2×#214C22semidirect
34. C2×Dic12C22semidirect
35. C2×Dic12C22semidirect
39. C2×D3×C8C22semidirect
42. #897C22semidirect
50. #869C22semidirect
51. C3×#106C22semidirect
53. C3×#102C22semidirect
54. C3×#97C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #810C22semidirect
58. #810C22semidirect
59. #810C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
75. #442D3semidirect
105. C3×D8D4semidirect
170. C4×D4C2×D3semidirect
173. #106C2×D3semidirect
180. #102C2×D3semidirect
183. #97C2×D3semidirect
269. D8D12semidirect
408. C3#3267semidirect

Maximal subgroups

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

subgroupindex
C2×#9122
C2×#9112
#55552
#55542
C3×#4422
#50192
#50162
#48672
#48642
#47832
#47812
#47762
#47152
#47062
#32673