Finite Group Databasebeta

← #80167

Subgroups — #80167

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
1D4×#67C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3C2×#914C211yesmaximal, characteristic, complemented
4#5631C211yesmaximal, characteristic, complemented
5C3×#450C211yesmaximal, characteristic, complemented
6#5175C211yesmaximal, characteristic, complemented
7#5172C211yesmaximal, characteristic, complemented
8#5162C211yesmaximal, characteristic, complemented
9#5049C211yesmaximal, characteristic, complemented
10#4786C211yesmaximal, characteristic, complemented
11#4780C211yesmaximal, characteristic, complemented
12#4776C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3272—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×#67—21no—
19C2×#239C2211yescharacteristic, complemented
20C2×D3×D4C2211yescharacteristic
21C2×#235C2211yescharacteristic, complemented
22C2×C3×SD16C2211yescharacteristic
23C2×C3×SD16C2211yescharacteristic, complemented
24C3×C22⋊Q8C2211yescharacteristic, complemented
25#953—21no—
26#944C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28#940C2211yescharacteristic
29#939—21no—
30#938C2211yescharacteristic
31#937—21no—
32D4×Dic3C2211yescharacteristic
33C2×#214C2211yescharacteristic
34C2×#214C2211yescharacteristic, complemented
35#932—24no—
36C2×#213C2211yescharacteristic
37C2×#213C2211yescharacteristic, complemented
38#930C2211yescharacteristic
39C4×#67C2211yescharacteristic
40C2×#208C2211yescharacteristic, complemented
41#914—24no—
42C2×D24C2211yescharacteristic, complemented
43C2×#204C2211yescharacteristic, complemented
44C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
45#897C2211yescharacteristic, complemented
46#894—21no—
47#893C2211yescharacteristic, complemented
48D3×C4⋊C4—21no—
49#884—21no—
50#883—21no—
51C3×#102C2211yescharacteristic
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#835C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#809C2211yescharacteristic, complemented
61#807C2211yescharacteristic, complemented
62C2×AGL(1,ℤ/8)—31no—
63C2×AGL(1,ℤ/8)—31no—
64#547—31no—
65D42—31no—
66#492—31no—
67#468—31no—
68#468—31no—
69#461—31no—
70#459—31no—
71#450—31no—
72#450D311yescharacteristic, complemented
73#449—31no—
74#449—31no—
75C4×D8—31no—
76#436—31no—
77C24×C3—21no—
78C23×D3—21no—
79C2×C3×Q8C2311yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—21no—
82C2×C3×D4—41no—
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C2×#67—44no—
93C2×#67D411yescharacteristic
94C22×Dic3—21no—
95#239—22no—
96#239—41no—
97D3×D4—22no—
98D3×D4—41no—
99#235—22no—
100#235—41no—
101C2×D12—21no—
102C2×D12C2311yescharacteristic
103C2×D12C2311yescharacteristic
104C2×C4×D3—21no—
105C2×C4×D3—21no—
106C2×C4×D3—21no—
107C2×C4×D3C2311yescharacteristic
108C2×Dic6C2311yescharacteristic
109C3×SD16—22no—
110C3×SD16—41no—
111C2×C3×C8C2311yescharacteristic
112C2×C3×C8C2311yescharacteristic
113C3×C4⋊C4—21no—
114C3×C4⋊C4C2311yescharacteristic
115C3×C22⋊C4—21no—
116#217—21no—
117#217—21no—
118#217D411yescharacteristic
119#215—22no—
120#214—22no—
121#214—22no—
122#213—22no—
123#213D414yescomplemented
124(C2×D3)⋊C4—21no—
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4—21no—
132Dic3⋊C4D411yescharacteristic
133C4×Dic3C2311yescharacteristic
134#208—22no—
135C2×C3⋊C8C2311yescharacteristic
136C2×C3⋊C8C2311yescharacteristic
137D24—22no—
138#204—41no—
139ΓL(1,𝔽25)—22no—
140C2×#41—31no—
141C2×#41—31no—
142C22×D4—31no—
143C22×D4—61no—
144C22×D4C2×D311yescharacteristic
145AGL(1,ℤ/8)—64no—
146AGL(1,ℤ/8)—64no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×SD16—31no—
151C2×SD16C2×D311yescharacteristic
152C2×SD16C2×D311yescharacteristic, complemented
153C2×D8—31no—
154C2×D8—31no—
155C2×D8—31no—
156C2×M16—31no—
157C2×M16—31no—
158C4⋊Q8—31no—
159#127—31no—
160#123—61no—
161#123—61no—
162C22⋊Q8—31no—
163C22⋊Q8C2×D311yescharacteristic, complemented
164#121—31no—
165#121—31no—
166#121—61no—
167#121—61no—
168#120—61no—
169#120—61no—
170C4×D4—31no—
171C4×D4—31no—
172C4×D4—31no—
173C2×C4⋊C4—61no—
174#107—31no—
175C4⋊C8—31no—
176#103—31no—
177#103—31no—
178#102—31no—
179#102—31no—
180#102C2×D311yescharacteristic
181#102C2×D311yescharacteristic, complemented
182C22⋊C8—31no—
183C22⋊C8C2×D311yescharacteristic, complemented
184C4×C8—31no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—81no—
190C23×C3C2×D411yescharacteristic
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—42no—
194C22×D3—42no—
195C22×D3—81no—
196C22×D3C2×D411yescharacteristic
197C3×Q8—21no—
198C3×Q8—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—21no—
202C3×D4—42no—
203C3×D4C2×D412yes—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4C2×D411yescharacteristic
209C2×C3×C4C2411yescharacteristic
210#67—22no—
211#67—42no—
212#67—42no—
213#67—42no—
214#67—42no—
215#67—42no—
216#67—81no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—42no—
220C2×Dic3C2×D411yescharacteristic
221C2×Dic3C2×D411yescharacteristic
222C2×Dic3C2×D411yescharacteristic
223D12—21no—
224D12—21no—
225D12—21no—
226D12—42no—
227D12C2×D412yescomplemented
228C4×D3—21no—
229C4×D3—21no—
230C4×D3—42no—
231C4×D3—42no—
232C4×D3—42no—
233Dic6—21no—
234Dic6—21no—
235C3×C8—21no—
236C3×C8—21no—
237C3⋊C8—21no—
238C3⋊C8C2×D412yescomplemented
239C24—21no—
240C24—61no—
241#41—62no—
242#41—62no—
243#41—121no—
244#41—121no—
245C2×Q8—31no—
246C2×Q8C22×D311yescharacteristic
247C2×D4—22no—
248C2×D4—21no—
249C2×D4—31no—
250C2×D4—31no—
251C2×D4—31no—
252C2×D4—41no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—61no—
256C2×D4—61no—
257C2×D4—62no—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C2×D4—124no—
263C2×D4C22×D311yescharacteristic
264C2×D4C22×D311yescharacteristic
265C22×C4—31no—
266C22×C4—61no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4C22×D311yescharacteristic
271SD16—22no—
272SD16—41no—
273SD16—62no—
274SD16—62no—
275SD16—62no—
276SD16—121no—
277D8—34no—
278D8—62no—
279D8—62no—
280M16—62no—
281M16—62no—
282C2×C8—31no—
283C2×C8—31no—
284C2×C8C22×D311yescharacteristic
285C2×C8C22×D311yescharacteristic
286C4⋊C4—21no—
287C4⋊C4—31no—
288C4⋊C4—31no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4C22×D311yescharacteristic
293C22⋊C4—21no—
294C22⋊C4—31no—
295C22⋊C4—31no—
296C22⋊C4—61no—
297C22⋊C4—61no—
298C22⋊C4—61no—
299C22⋊C4—61no—
300C22⋊C4—61no—
301C42—31no—
302C22×C3—21no—
303C22×C3—22no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—84no—
312C22×C3C22×D411yescharacteristic
313C2×D3—21no—
314C2×D3—22no—
315C2×D3—21no—
316C2×D3—21no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—84no—
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4C22×D411yescharacteristic
327C3×C4H(5,𝔽2)11yescharacteristic
328Dic3—21no—
329Dic3—21no—
330Dic3—41no—
331Dic3—41no—
332Dic3AGL(1,ℤ/8)12yescomplemented
333C23—21no—
334C23—21no—
335C23—31no—
336C23—42no—
337C23—42no—
338C23—61no—
339C23—61no—
340C23—81no—
341C23—122no—
342C23—122no—
343C23—241no—
344C23D3×D411yescharacteristic
345Q8—21no—
346Q8—21no—
347Q8—61no—
348Q8—61no—
349D4—21no—
350D4—21no—
351D4—21no—
352D4—32no—
353D4—42no—
354D4—62no—
355D4—61no—
356D4—61no—
357D4—61no—
358D4—122no—
359D4—122no—
360D4—122no—
361D4—122no—
362D4—122no—
363D4—122no—
364D4—241no—
365D4D3×D412yes—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—31no—
371C2×C4—31no—
372C2×C4—31no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4D3×D411yescharacteristic
382C2×C4C23×D311yescharacteristic
383C8—21no—
384C8—21no—
385C8—32no—
386C8—61no—
387C2×C3—42no—
388C2×C3—41no—
389C2×C3—81no—
390C2×C3D4211yescharacteristic
391C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
392C2×C3#57711yescharacteristic
393D3—41no—
394D3—42no—
395D3—81no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—22no—
400C22—41no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—61no—
406C22—61no—
407C22—61no—
408C22—62no—
409C22—84no—
410C22—121no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—244no—
416C22C2×D3×D411yescharacteristic
417C4—21no—
418C4—32no—
419C4—41no—
420C4—41no—
421C4—61no—
422C4—61no—
423C4—121no—
424C4—121no—
425C4C2×D3×D411yescharacteristic
426C4#100411yescharacteristic
427C3#327211yescharacteristic, complemented, Sylow
428C2—41no—
429C2—42no—
430C2—81no—
431C2—121no—
432C2—122no—
433C2—241no—
434C2#559011yescharacteristic
435C2#576211yescharacteristic
436C2D3×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
hypercenter#450D3
hypocenterC3#3272
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. D4×#67C2semidirect
2. C2×#932C2semidirect
3. C2×#914C2semidirect
4. #5631C2semidirect
5. C3×#450C2semidirect
6. #5175C2semidirect
7. #5172C2semidirect
8. #5162C2semidirect
9. #5049C2semidirect
10. #4786C2semidirect
11. #4780C2semidirect
12. #4776C2semidirect
13. #4756C2semidirect
14. #4734C2semidirect
15. #4729C2semidirect
19. C2×#239C22semidirect
21. C2×#235C22semidirect
23. C2×C3×SD16C22semidirect
24. C3×C22⋊Q8C22semidirect
26. #944C22semidirect
27. C2×#215C22semidirect
34. C2×#214C22semidirect
37. C2×#213C22semidirect
40. C2×#208C22semidirect
42. C2×D24C22semidirect
43. C2×#204C22semidirect
44. C2×ΓL(1,𝔽25)C22semidirect
45. #897C22semidirect
47. #893C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #835C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #809C22semidirect
61. #807C22semidirect
72. #450D3semidirect
123. #213D4semidirect
152. C2×SD16C2×D3semidirect
163. C22⋊Q8C2×D3semidirect
181. #102C2×D3semidirect
183. C22⋊C8C2×D3semidirect
227. D12C2×D4semidirect
238. C3⋊C8C2×D4semidirect
332. Dic3AGL(1,ℤ/8)semidirect
427. C3#3272semidirect

Maximal subgroups

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

subgroupindex
D4×#672
C2×#9322
C2×#9142
#56312
C3×#4502
#51752
#51722
#51622
#50492
#47862
#47802
#47762
#47562
#47342
#47292
#32723