Finite Group Databasebeta

← #80555

Subgroups — #80555

Order 384.

520 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C211yesmaximal, characteristic, complemented
2C2×#911C211yesmaximal, characteristic, complemented
3#5676C211yesmaximal, characteristic, complemented
4#5554C211yesmaximal, characteristic, complemented
5C3×#463C211yesmaximal, characteristic, complemented
6#5173C211yesmaximal, characteristic, complemented
7#5156C211yesmaximal, characteristic, complemented
8#5059C211yesmaximal, characteristic, complemented
9C4×#213C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic, complemented
11#4838C211yesmaximal, characteristic, complemented
12#4812C211yesmaximal, characteristic, complemented
13#4808C211yesmaximal, characteristic, complemented
14#4780C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3273—31nomaximal, Sylow
17C2×#239C2211yescharacteristic
18C2×#237C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic, complemented
21C2×C3×SD16C2211yescharacteristic, complemented
22C2×C3×D8C2211yescharacteristic
23C3×#124C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic, complemented
25#946C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28#938C2211yescharacteristic
29C2×#214C2211yescharacteristic, complemented
30C2×#213C2211yescharacteristic
31C2×#213C2211yescharacteristic
32#930—21no—
33C2×(C2×D3)⋊C4—21no—
34#914—24no—
35#911—24no—
36C2×D24C2211yescharacteristic
37C2×#204C2211yescharacteristic, complemented
38C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
39C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
40#897C2211yescharacteristic, complemented
41#895C2211yescharacteristic
42#886—21no—
43#884—21no—
44#883—21no—
45#882—21no—
46#882—21no—
47D3×C22⋊C4—21no—
48#875C2211yescharacteristic, complemented
49C4×D12C2211yescharacteristic
50C4×D12C2211yescharacteristic, complemented
51C3×C4⋊C8C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57(C2×D3)⋊C8C2211yescharacteristic, complemented
58#816C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#807C2211yescharacteristic, complemented
61C4×C3⋊C8C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×AGL(1,ℤ/8)—31no—
64#546—31no—
65#546—31no—
66#495—31no—
67#468—31no—
68#468—31no—
69#463—31no—
70#463D311yescharacteristic, complemented
71#452—31no—
72#452—31no—
73#447—31no—
74#447—31no—
75C4×D8—31no—
76#436—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4—21no—
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C22×Dic3—21no—
88#239—22no—
89#239—41no—
90#237—22no—
91#237—41no—
92D3×D4—22no—
93D3×D4—22no—
94D3×D4—41no—
95D3×D4—41no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12D411yescharacteristic
99C2×D12C2311yescharacteristic
100C2×D12C2311yescharacteristic
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3C2311yescharacteristic
104C2×C4×D3C2311yescharacteristic
105C2×Dic6C2311yescharacteristic
106C3×SD16—22no—
107C3×D8—22no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112C3×C22⋊C4—21no—
113C3×C22⋊C4—21no—
114C3×C42C2311yescharacteristic
115#217—21no—
116#216—41no—
117#215—22no—
118#214—22no—
119#213—41no—
120#213D414yescomplemented
121(C2×D3)⋊C4—21no—
122(C2×D3)⋊C4—21no—
123(C2×D3)⋊C4—21no—
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128(C2×D3)⋊C4D411yescharacteristic
129C4⋊Dic3—21no—
130C4⋊Dic3D411yescharacteristic
131C4⋊Dic3C2311yescharacteristic
132Dic3⋊C4—21no—
133C2×C3⋊C8C2311yescharacteristic
134C2×C3⋊C8C2311yescharacteristic
135D24—22no—
136#204—22no—
137ΓL(1,𝔽25)—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×D4—31no—
142C22×D4—31no—
143AGL(1,ℤ/8)—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16C2×D311yescharacteristic, complemented
150C2×D8—31no—
151C2×D8—31no—
152C2×D8—31no—
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124C2×D311yescharacteristic, complemented
158#123—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162#121—31no—
163#121—31no—
164#121—61no—
165#121—61no—
166#120—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4C2×D311yescharacteristic, complemented
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8C2×D311yescharacteristic, complemented
175#103—31no—
176#103C2×D311yescharacteristic, complemented
177#102—31no—
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—21no—
188C22×D3—42no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—81no—
193C22×D3—81no—
194C22×D3C2×D411yescharacteristic
195C22×D3C2×D411yescharacteristic
196C3×Q8—21no—
197C3×Q8—21no—
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4C2×D412yes—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—81no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—42no—
209#67—42no—
210#67—42no—
211#67—81no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—42no—
215C2×Dic3C2×D411yescharacteristic
216C2×Dic3C2×D411yescharacteristic
217D12—21no—
218D12—21no—
219D12—21no—
220D12—41no—
221D12—42no—
222D12—81no—
223D12C2×D412yes—
224C4×D3—21no—
225C4×D3—21no—
226C4×D3—21no—
227C4×D3—21no—
228C4×D3—42no—
229C4×D3—81no—
230Dic6—21no—
231Dic6—21no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—61no—
237C24—61no—
238#41—62no—
239#41—62no—
240#41—121no—
241#41—121no—
242C2×Q8—31no—
243C2×Q8C22×D311yescharacteristic
244C2×D4—31no—
245C2×D4—31no—
246C2×D4—31no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4C22×D311yescharacteristic
258C2×D4C22×D311yescharacteristic
259C22×C4—21no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265Q16—121no—
266SD16—22no—
267SD16—62no—
268SD16—62no—
269SD16—62no—
270D8—22no—
271D8—34no—
272D8—62no—
273D8—121no—
274M16—62no—
275M16—62no—
276C2×C8—31no—
277C2×C8—31no—
278C2×C8C22×D311yescharacteristic
279C2×C8C22×D311yescharacteristic
280C4⋊C4—31no—
281C4⋊C4—31no—
282C4⋊C4—61no—
283C4⋊C4—61no—
284C4⋊C4C22×D311yescharacteristic
285C22⋊C4—21no—
286C22⋊C4—21no—
287C22⋊C4—21no—
288C22⋊C4—31no—
289C22⋊C4—31no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C42C22×D311yescharacteristic
298C22×C3—22no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3C22×D411yescharacteristic
305C2×D3—21no—
306C2×D3—21no—
307C2×D3—22no—
308C2×D3—21no—
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—21no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—84no—
318C2×D3—84no—
319C3×C4—21no—
320C3×C4—21no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4C22×D411yescharacteristic
324C3×C4H(5,𝔽2)11yescharacteristic
325Dic3—21no—
326Dic3—21no—
327Dic3—41no—
328Dic3—41no—
329C23—21no—
330C23—21no—
331C23—31no—
332C23—31no—
333C23—61no—
334C23—61no—
335C23—122no—
336C23—122no—
337C23—122no—
338C23—122no—
339C23—241no—
340C23—241no—
341Q8—21no—
342Q8—21no—
343Q8—61no—
344Q8—61no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—32no—
349D4—61no—
350D4—61no—
351D4—61no—
352D4—121no—
353D4—122no—
354D4—122no—
355D4—122no—
356D4—122no—
357D4—241no—
358D4—241no—
359D4D3×D412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—81no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4D3×D411yescharacteristic
375C2×C4D3×D411yescharacteristic
376C2×C4C23×D311yescharacteristic
377C8—21no—
378C8—21no—
379C8—32no—
380C8—61no—
381C2×C3—42no—
382C2×C3—81no—
383C2×C3D4211yescharacteristic
384C2×C3#57511yescharacteristic
385C2×C3#57611yescharacteristic
386D3—41no—
387D3—41no—
388D3—42no—
389D3—81no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—61no—
397C22—61no—
398C22—61no—
399C22—61no—
400C22—61no—
401C22—61no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—244no—
409C22—244no—
410C22C2×D3×D411yescharacteristic
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—61no—
416C4—61no—
417C4—121no—
418C4—121no—
419C4C2×D3×D411yescharacteristic
420C4#100911yescharacteristic
421C3#327311yescharacteristic, complemented, Sylow
422C2—42no—
423C2—81no—
424C2—121no—
425C2—121no—
426C2—122no—
427C2—241no—
428C2#560911yescharacteristic
429C2#576811yescharacteristic
430C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#463C2
socleC22×C3C22×D4
hypercenter#463D3
hypocenterC3#3273
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×#914C2semidirect
2. C2×#911C2semidirect
3. #5676C2semidirect
4. #5554C2semidirect
5. C3×#463C2semidirect
6. #5173C2semidirect
7. #5156C2semidirect
8. #5059C2semidirect
9. C4×#213C2semidirect
10. #4844C2semidirect
11. #4838C2semidirect
12. #4812C2semidirect
13. #4808C2semidirect
14. #4780C2semidirect
15. #4773C2semidirect
19. C2×D3×D4C22semidirect
20. C2×D3×D4C22semidirect
21. C2×C3×SD16C22semidirect
23. C3×#124C22semidirect
24. C3×C4×D4C22semidirect
25. #946C22semidirect
26. C2×#216C22semidirect
27. C2×#215C22semidirect
29. C2×#214C22semidirect
37. C2×#204C22semidirect
38. C2×ΓL(1,𝔽25)C22semidirect
39. C2×ΓL(1,𝔽25)C22semidirect
40. #897C22semidirect
48. #875C22semidirect
50. C4×D12C22semidirect
51. C3×C4⋊C8C22semidirect
52. C3×#103C22semidirect
53. C3×#102C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. (C2×D3)⋊C8C22semidirect
58. #816C22semidirect
59. #809C22semidirect
60. #807C22semidirect
61. C4×C3⋊C8C22semidirect
70. #463D3semidirect
120. #213D4semidirect
149. C2×SD16C2×D3semidirect
157. #124C2×D3semidirect
170. C4×D4C2×D3semidirect
174. C4⋊C8C2×D3semidirect
176. #103C2×D3semidirect
180. #102C2×D3semidirect
421. C3#3273semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C2×#9112
#56762
#55542
C3×#4632
#51732
#51562
#50592
C4×#2132
#48442
#48382
#48122
#48082
#47802
#47732
#32733