Finite Group Databasebeta

← #79794

Subgroups — #79794

Order 384.

546 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#914C211yesmaximal, characteristic, complemented
2C2×D3×SD16C211yesmaximal, characteristic, complemented
3#5576C211yesmaximal, characteristic, complemented
4D4×D12C211yesmaximal, characteristic, complemented
5C3×#440C211yesmaximal, characteristic, complemented
6#5027C211yesmaximal, characteristic, complemented
7#5016C211yesmaximal, characteristic, complemented
8#4886C211yesmaximal, characteristic, complemented
9#4883C211yesmaximal, characteristic, complemented
10#4810C211yesmaximal, characteristic, complemented
11#4806C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic, complemented
13#4773C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4706C211yesmaximal, characteristic, complemented
16#3265—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×Q8C2211yescharacteristic, complemented
19C2×D3×D4C2211yescharacteristic
20C2×D3×D4C2211yescharacteristic
21C22×D12—21no—
22C2×C3×SD16C2211yescharacteristic
23C3×C4×Q8C2211yescharacteristic, complemented
24C3×C4×D4C2211yescharacteristic
25C2×#215C2211yescharacteristic, complemented
26C2×#215C2211yescharacteristic, complemented
27C2×#214C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#930—21no—
30#914—24no—
31D3×SD16—24no—
32C2×D24C2211yescharacteristic, complemented
33C2×D24C2211yescharacteristic, complemented
34C2×#204C2211yescharacteristic, complemented
35C2×#204C2211yescharacteristic, complemented
36C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
37C2×D3×C8C2211yescharacteristic, complemented
38#897—21no—
39#897C2211yescharacteristic, complemented
40#895—21no—
41#895C2211yescharacteristic
42#895C2211yescharacteristic
43#895C2211yescharacteristic, complemented
44#882—21no—
45#882—21no—
46#875—21no—
47#875C2211yescharacteristic, complemented
48#874C2211yescharacteristic
49C4×D12—21no—
50C4×D12C2211yescharacteristic
51C3×#107C2211yescharacteristic, complemented
52C3×#103C2211yescharacteristic, complemented
53C3×#102C2211yescharacteristic
54C3×#97C2211yescharacteristic, complemented
55(C2×D3)⋊C8C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#810C2211yescharacteristic, complemented
58#809C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61C4⋊(C3⋊C8)C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C22×SD16—31no—
64#548—31no—
65D42—31no—
66#496—31no—
67#468—31no—
68#466—31no—
69#463—31no—
70#460—31no—
71#450—31no—
72#449—31no—
73#448—31no—
74#447—31no—
75#440D311yescharacteristic, complemented
76#435—31no—
77C23×D3—21no—
78C23×D3—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C2×#67—21no—
83C2×#67—21no—
84#239—22no—
85#239—41no—
86D3×Q8—22no—
87D3×Q8—41no—
88D3×D4—22no—
89D3×D4—22no—
90D3×D4—41no—
91D3×D4—41no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12—21no—
96C2×D12—21no—
97C2×D12—21no—
98C2×D12—44no—
99C2×D12D411yescharacteristic
100C2×D12C2311yescharacteristic
101C2×D12C2311yescharacteristic
102C2×D12C2311yescharacteristic
103C2×C4×D3—21no—
104C2×C4×D3—21no—
105C2×C4×D3C2311yescharacteristic
106C2×C4×D3C2311yescharacteristic
107C2×Dic6—21no—
108C2×Dic6C2311yescharacteristic
109C3×SD16D414yescomplemented
110C2×C3×C8C2311yescharacteristic
111C2×C3×C8C2311yescharacteristic
112C3×C4⋊C4—21no—
113C3×C4⋊C4C2311yescharacteristic
114C3×C4⋊C4C2311yescharacteristic
115C3×C22⋊C4—21no—
116C3×C42—21no—
117C3×C42C2311yescharacteristic
118#215—22no—
119#215—22no—
120#214—22no—
121#213—22no—
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)⋊C4—21no—
128(C2×D3)⋊C4D411yescharacteristic
129(C2×D3)⋊C4D411yescharacteristic
130C4⋊Dic3—21no—
131C4⋊Dic3D411yescharacteristic
132C2×C3⋊C8C2311yescharacteristic
133C2×C3⋊C8C2311yescharacteristic
134D24—22no—
135D24—41no—
136#204—22no—
137#204—41no—
138ΓL(1,𝔽25)—22no—
139D3×C8—22no—
140C2×#41—31no—
141C22×Q8—31no—
142C22×D4—31no—
143C22×D4—31no—
144C22×D4—61no—
145AGL(1,ℤ/8)—64no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16—64no—
152C2×SD16C2×D311yescharacteristic
153C2×D8—31no—
154C2×D8—31no—
155C2×D8—31no—
156C2×M16—31no—
157C22×C8—31no—
158#127—31no—
159#124—31no—
160#124—61no—
161C22⋊Q8—31no—
162C22⋊Q8—61no—
163#121—31no—
164#121—31no—
165#121—31no—
166#121—61no—
167#121—61no—
168#120—61no—
169#120—61no—
170C4×Q8C2×D311yescharacteristic, complemented
171C4×D4—31no—
172C4×D4—61no—
173C4×D4C2×D311yescharacteristic
174#107C2×D311yescharacteristic, complemented
175C4⋊C8—31no—
176#103—31no—
177#103C2×D311yescharacteristic, complemented
178#102—31no—
179#102—31no—
180#102—31no—
181#102C2×D311yescharacteristic
182C22⋊C8—31no—
183C22⋊C8—31no—
184#97C2×D311yescharacteristic, complemented
185C23×C3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—42no—
190C22×D3—42no—
191C22×D3—42no—
192C22×D3—42no—
193C22×D3—81no—
194C22×D3—81no—
195C22×D3C2×D411yescharacteristic
196C22×D3C2×D411yescharacteristic
197C3×Q8—21no—
198C3×Q8C2×D412yescomplemented
199C3×D4—21no—
200C3×D4C2×D412yes—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—42no—
205C2×C3×C4C2×D411yescharacteristic
206C2×C3×C4C2×D411yescharacteristic
207C2×C3×C4C2411yescharacteristic
208#67—42no—
209#67—42no—
210C2×Dic3—21no—
211C2×Dic3C2×D411yescharacteristic
212C2×Dic3C2×D411yescharacteristic
213D12—22no—
214D12—21no—
215D12—21no—
216D12—21no—
217D12—21no—
218D12—21no—
219D12—21no—
220D12—42no—
221D12—42no—
222D12—42no—
223D12—42no—
224D12—42no—
225D12—81no—
226C4×D3—21no—
227C4×D3—21no—
228C4×D3—21no—
229C4×D3—21no—
230C4×D3—42no—
231C4×D3—42no—
232Dic6—21no—
233Dic6—21no—
234Dic6—42no—
235C3×C8—21no—
236C3×C8C2×D412yescomplemented
237C3⋊C8—21no—
238C3⋊C8—21no—
239C24—61no—
240C24—61no—
241#41—62no—
242#41—121no—
243C2×Q8—31no—
244C2×Q8—61no—
245C2×Q8—62no—
246C2×Q8—121no—
247C2×Q8C22×D311yescharacteristic
248C2×D4—31no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—61no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—124no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4C22×D311yescharacteristic
266C22×C4—21no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—61no—
270C22×C4—61no—
271SD16—62no—
272SD16—62no—
273SD16—62no—
274SD16—62no—
275SD16—121no—
276SD16D1214yescomplemented
277D8—62no—
278D8—62no—
279D8—121no—
280M16—62no—
281C2×C8—31no—
282C2×C8—31no—
283C2×C8—62no—
284C2×C8C22×D311yescharacteristic
285C2×C8C22×D311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—31no—
288C4⋊C4—61no—
289C4⋊C4C22×D311yescharacteristic
290C4⋊C4C22×D311yescharacteristic
291C22⋊C4—21no—
292C22⋊C4—31no—
293C22⋊C4—31no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C42—21no—
301C42C22×D311yescharacteristic
302C22×C3—22no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3C22×D411yescharacteristic
306C2×D3—21no—
307C2×D3—21no—
308C2×D3—21no—
309C2×D3—21no—
310C2×D3—21no—
311C2×D3—41no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—84no—
321C2×D3—84no—
322C2×D3AGL(1,ℤ/8)12yescomplemented
323C3×C4—22no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4C22×D411yescharacteristic
329C3×C4H(5,𝔽2)11yescharacteristic
330Dic3—21no—
331Dic3—21no—
332Dic3—41no—
333C23—21no—
334C23—31no—
335C23—31no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—122no—
340C23—122no—
341C23—122no—
342C23—122no—
343C23—241no—
344C23—241no—
345Q8—21no—
346Q8—61no—
347Q8—61no—
348Q8—122no—
349Q8C2×D1212yescomplemented
350D4—21no—
351D4—62no—
352D4—61no—
353D4—61no—
354D4—61no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—122no—
365D4—241no—
366D4C2×D1212yes—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—42no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4D3×D411yescharacteristic
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—61no—
385C8—61no—
386C8C2×D1212yescomplemented
387C2×C3—42no—
388C2×C3D4211yescharacteristic
389C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
390C2×C3#57711yescharacteristic
391D3—22no—
392D3—41no—
393D3—81no—
394D3—81no—
395D3—81no—
396C22—22no—
397C22—32no—
398C22—41no—
399C22—41no—
400C22—61no—
401C22—61no—
402C22—61no—
403C22—61no—
404C22—61no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—244no—
415C22—244no—
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—21no—
419C4—22no—
420C4—41no—
421C4—41no—
422C4—61no—
423C4—61no—
424C4—121no—
425C4C22×D1211yescharacteristic
426C4#100911yescharacteristic
427C3#326511yescharacteristic, complemented, Sylow
428C2—42no—
429C2—62no—
430C2—121no—
431C2—241no—
432C2—241no—
433C2—241no—
434C2D4×D1211yescharacteristic
435C2D3×AGL(1,ℤ/8)11yescharacteristic
436C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3265
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. C2×#914C2semidirect
2. C2×D3×SD16C2semidirect
3. #5576C2semidirect
4. D4×D12C2semidirect
5. C3×#440C2semidirect
6. #5027C2semidirect
7. #5016C2semidirect
8. #4886C2semidirect
9. #4883C2semidirect
10. #4810C2semidirect
11. #4806C2semidirect
12. #4776C2semidirect
13. #4773C2semidirect
14. #4712C2semidirect
15. #4706C2semidirect
17. C2×#239C22semidirect
18. C2×D3×Q8C22semidirect
23. C3×C4×Q8C22semidirect
25. C2×#215C22semidirect
26. C2×#215C22semidirect
32. C2×D24C22semidirect
33. C2×D24C22semidirect
34. C2×#204C22semidirect
35. C2×#204C22semidirect
36. C2×ΓL(1,𝔽25)C22semidirect
37. C2×D3×C8C22semidirect
39. #897C22semidirect
43. #895C22semidirect
47. #875C22semidirect
51. C3×#107C22semidirect
52. C3×#103C22semidirect
54. C3×#97C22semidirect
55. (C2×D3)⋊C8C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #810C22semidirect
58. #809C22semidirect
59. #809C22semidirect
60. #809C22semidirect
61. C4⋊(C3⋊C8)C22semidirect
75. #440D3semidirect
109. C3×SD16D4semidirect
170. C4×Q8C2×D3semidirect
174. #107C2×D3semidirect
177. #103C2×D3semidirect
184. #97C2×D3semidirect
198. C3×Q8C2×D4semidirect
236. C3×C8C2×D4semidirect
276. SD16D12semidirect
322. C2×D3AGL(1,ℤ/8)semidirect
349. Q8C2×D12semidirect
386. C8C2×D12semidirect
427. C3#3265semidirect

Maximal subgroups

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

subgroupindex
C2×#9142
C2×D3×SD162
#55762
D4×D122
C3×#4402
#50272
#50162
#48862
#48832
#48102
#48062
#47762
#47732
#47122
#47062
#32653