Finite Group Databasebeta

← #80023

Subgroups — #80023

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
1#5806C211yesmaximal, characteristic, complemented
2C2×#932C211yesmaximal, characteristic, complemented
3C2×#911C211yesmaximal, characteristic, complemented
4#5612C211yesmaximal, characteristic, complemented
5C3×#447C211yesmaximal, characteristic, complemented
6#5156C211yesmaximal, characteristic, complemented
7#5155C211yesmaximal, characteristic, complemented
8#5153C211yesmaximal, characteristic, complemented
9#5042C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4783C211yesmaximal, characteristic, complemented
12#4780C211yesmaximal, characteristic, complemented
13#4756C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4729C211yesmaximal, characteristic, complemented
16#3273—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic, complemented
18C2×#237C2211yescharacteristic
19C2×D3×D4C2211yescharacteristic, complemented
20C2×#235C2211yescharacteristic
21C2×C3×D8C2211yescharacteristic
22C2×C3×D8C2211yescharacteristic
23C3×#121C2211yescharacteristic
24#953—21no—
25C2×#217—21no—
26#939—21no—
27#938—21no—
28#938C2211yescharacteristic
29#937—21no—
30#936C2211yescharacteristic, complemented
31#936C2211yescharacteristic, complemented
32#935—21no—
33D4×Dic3C2211yescharacteristic, complemented
34C2×#214C2211yescharacteristic, complemented
35C2×#214C2211yescharacteristic, complemented
36C2×#214C2211yescharacteristic, complemented
37#932—24no—
38C2×#213C2211yescharacteristic
39C2×#213C2211yescharacteristic
40C4×#67C2211yescharacteristic
41#924C2211yescharacteristic, complemented
42C2×#208C2211yescharacteristic, complemented
43#911—24no—
44C2×Dic12C2211yescharacteristic, complemented
45C2×#204C2211yescharacteristic, complemented
46C2×ΓL(1,𝔽25)C2211yescharacteristic, complemented
47#897C2211yescharacteristic, complemented
48#893C2211yescharacteristic, complemented
49#883—21no—
50D3×C22⋊C4—21no—
51C3×#102C2211yescharacteristic, complemented
52C3×#102C2211yescharacteristic, complemented
53C3×C22⋊C8C2211yescharacteristic, complemented
54#832C2211yescharacteristic, complemented
55#821C2211yescharacteristic, complemented
56(C2×D3)⋊C8C2211yescharacteristic, complemented
57#816C2211yescharacteristic, complemented
58Dic3⋊C8C2211yescharacteristic, complemented
59C8×Dic3C2211yescharacteristic, complemented
60#810C2211yescharacteristic, complemented
61#807C2211yescharacteristic, 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#463—31no—
71#452—31no—
72#452—31no—
73#447—31no—
74#447D311yescharacteristic, complemented
75C4×D8—31no—
76#436—31no—
77C24×C3—21no—
78C23×D3—21no—
79C2×C3×D4—22no—
80C2×C3×D4—21no—
81C2×C3×D4—21no—
82C2×C3×D4—41no—
83C2×C3×D4C2311yescharacteristic
84C2×C3×D4C2311yescharacteristic
85C2×C3×D4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67D411yescharacteristic
92C22×Dic3—21no—
93C22×Dic3—21no—
94#237—22no—
95#237—41no—
96D3×D4—22no—
97D3×D4—41no—
98#235—22no—
99#235—41no—
100C2×D12C2311yescharacteristic
101C2×C4×D3—21no—
102C2×C4×D3—21no—
103C2×C4×D3C2311yescharacteristic
104C2×Dic6C2311yescharacteristic
105C2×Dic6C2311yescharacteristic
106C3×D8—22no—
107C3×D8—41no—
108C2×C3×C8C2311yescharacteristic
109C2×C3×C8C2311yescharacteristic
110C3×C4⋊C4C2311yescharacteristic
111C3×C22⋊C4—21no—
112#217—21no—
113#217—21no—
114#217—21no—
115#217—21no—
116#217—21no—
117#217—21no—
118#217—21no—
119#217D411yescharacteristic
120#214—22no—
121#214—22no—
122#214—22no—
123#213—22no—
124#213D414yescomplemented
125(C2×D3)⋊C4—21no—
126(C2×D3)⋊C4—21no—
127(C2×D3)⋊C4D411yescharacteristic
128C4⋊Dic3—21no—
129C4⋊Dic3C2311yescharacteristic
130Dic3⋊C4—21no—
131Dic3⋊C4D411yescharacteristic
132C4×Dic3C2311yescharacteristic
133#208—22no—
134C2×C3⋊C8C2311yescharacteristic
135C2×C3⋊C8C2311yescharacteristic
136Dic12—41no—
137#204—22no—
138ΓL(1,𝔽25)—22no—
139C2×#41—31no—
140C2×#41—31no—
141C22×D4—31no—
142C22×D4C2×D311yescharacteristic, complemented
143AGL(1,ℤ/8)—64no—
144AGL(1,ℤ/8)—64no—
145C2×Q16—31no—
146C2×SD16—31no—
147C2×SD16—31no—
148C2×SD16—31no—
149C2×SD16—31no—
150C2×D8—31no—
151C2×D8—31no—
152C2×D8C2×D311yescharacteristic
153C2×D8C2×D311yescharacteristic
154C2×M16—31no—
155C2×M16—31no—
156#124—31no—
157#124—31no—
158#123—61no—
159#123—61no—
160C22⋊Q8—31no—
161C22⋊Q8—31no—
162#121—31no—
163#121—61no—
164#121—61no—
165#121C2×D311yescharacteristic
166#120—61no—
167#120—61no—
168C4×D4—31no—
169C4×D4—31no—
170C4×D4—31no—
171C2×C22⋊C4—61no—
172C2×C22⋊C4—61no—
173#107—31no—
174C4⋊C8—31no—
175#103—31no—
176#103—31no—
177#102—31no—
178#102—31no—
179#102C2×D311yescharacteristic, complemented
180#102C2×D311yescharacteristic, complemented
181C22⋊C8—31no—
182C22⋊C8C2×D311yescharacteristic, complemented
183C4×C8—31no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—42no—
188C23×C3—42no—
189C23×C3—81no—
190C23×C3C2×D411yescharacteristic
191C22×D3—21no—
192C22×D3—42no—
193C22×D3—42no—
194C22×D3—81no—
195C22×D3C2×D411yescharacteristic
196C3×D4—21no—
197C3×D4—21no—
198C3×D4—21no—
199C3×D4—21no—
200C3×D4—21no—
201C3×D4—42no—
202C3×D4—81no—
203C3×D4C2×D412yes—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4C2×D411yescharacteristic
208C2×C3×C4C2411yescharacteristic
209#67—41no—
210#67—42no—
211#67—42no—
212#67—42no—
213#67—81no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—42no—
218C2×Dic3—81no—
219C2×Dic3C2×D411yescharacteristic
220C2×Dic3C2×D411yescharacteristic
221C2×Dic3C2×D411yescharacteristic
222D12—21no—
223D12C2×D412yes—
224C4×D3—21no—
225C4×D3—21no—
226C4×D3—42no—
227C4×D3—81no—
228Dic6—21no—
229Dic6—21no—
230Dic6—21no—
231Dic6—21no—
232C3×C8—21no—
233C3×C8—21no—
234C3⋊C8—21no—
235C3⋊C8C2×D412yes—
236C24—21no—
237C24—61no—
238#41—62no—
239#41—62no—
240#41—121no—
241#41—121no—
242C2×Q8—31no—
243C2×Q8—31no—
244C2×D4—22no—
245C2×D4—21no—
246C2×D4—21no—
247C2×D4—31no—
248C2×D4—31no—
249C2×D4—41no—
250C2×D4—61no—
251C2×D4—62no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—61no—
255C2×D4—121no—
256C2×D4C22×D311yescharacteristic
257C2×D4C22×D311yescharacteristic
258C2×D4C22×D311yescharacteristic
259C22×C4—31no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4C22×D311yescharacteristic
265Q16—121no—
266SD16—62no—
267SD16—62no—
268SD16—62no—
269SD16—62no—
270D8—22no—
271D8—34no—
272D8—41no—
273D8—62no—
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—31no—
287C22⋊C4—31no—
288C22⋊C4—61no—
289C22⋊C4—61no—
290C22⋊C4—61no—
291C22⋊C4—61no—
292C22⋊C4—61no—
293C22⋊C4—61no—
294C22⋊C4—61no—
295C22⋊C4—61no—
296C22⋊C4—61no—
297C42—31no—
298C22×C3—22no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—84no—
311C22×C3C22×D411yescharacteristic
312C2×D3—21no—
313C2×D3—22no—
314C2×D3—21no—
315C2×D3—21no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—84no—
319C3×C4—21no—
320C3×C4—41no—
321C3×C4C22×D411yescharacteristic
322C3×C4H(5,𝔽2)11yescharacteristic
323Dic3—21no—
324Dic3—21no—
325Dic3—21no—
326Dic3—41no—
327Dic3—41no—
328Dic3—41no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—31no—
333C23—42no—
334C23—42no—
335C23—61no—
336C23—81no—
337C23—122no—
338C23—122no—
339C23—241no—
340C23D3×D411yescharacteristic
341Q8—61no—
342Q8—61no—
343Q8—61no—
344Q8—61no—
345D4—21no—
346D4—21no—
347D4—21no—
348D4—21no—
349D4—21no—
350D4—32no—
351D4—42no—
352D4—61no—
353D4—81no—
354D4—121no—
355D4—122no—
356D4—122no—
357D4—122no—
358D4—241no—
359D4D3×D412yes—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—31no—
364C2×C4—31no—
365C2×C4—31no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—122no—
372C2×C4—122no—
373C2×C4—241no—
374C2×C4—241no—
375C2×C4D3×D411yescharacteristic
376C2×C4C23×D311yescharacteristic
377C8—21no—
378C8—21no—
379C8—32no—
380C8—61no—
381C2×C3—41no—
382C2×C3—42no—
383C2×C3—81no—
384C2×C3—81no—
385C2×C3D4211yescharacteristic
386C2×C3#57511yescharacteristic
387C2×C3#57611yescharacteristic
388D3—41no—
389D3—42no—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—41no—
402C22—61no—
403C22—61no—
404C22—61no—
405C22—62no—
406C22—84no—
407C22—121no—
408C22—121no—
409C22—244no—
410C22C2×D3×D411yescharacteristic
411C4—21no—
412C4—41no—
413C4—61no—
414C4—61no—
415C4—61no—
416C4—121no—
417C4—121no—
418C4—121no—
419C4C2×D3×D411yescharacteristic
420C4#100411yescharacteristic
421C3#327311yescharacteristic, complemented, Sylow
422C2—41no—
423C2—42no—
424C2—81no—
425C2—81no—
426C2—121no—
427C2—122no—
428C2#559011yescharacteristic
429C2#575711yescharacteristic
430C2#577311yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3273
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #5806C2semidirect
2. C2×#932C2semidirect
3. C2×#911C2semidirect
4. #5612C2semidirect
5. C3×#447C2semidirect
6. #5156C2semidirect
7. #5155C2semidirect
8. #5153C2semidirect
9. #5042C2semidirect
10. #4787C2semidirect
11. #4783C2semidirect
12. #4780C2semidirect
13. #4756C2semidirect
14. #4733C2semidirect
15. #4729C2semidirect
17. C22×C3×D4C22semidirect
19. C2×D3×D4C22semidirect
30. #936C22semidirect
31. #936C22semidirect
33. D4×Dic3C22semidirect
34. C2×#214C22semidirect
35. C2×#214C22semidirect
36. C2×#214C22semidirect
41. #924C22semidirect
42. C2×#208C22semidirect
44. C2×Dic12C22semidirect
45. C2×#204C22semidirect
46. C2×ΓL(1,𝔽25)C22semidirect
47. #897C22semidirect
48. #893C22semidirect
51. C3×#102C22semidirect
52. C3×#102C22semidirect
53. C3×C22⋊C8C22semidirect
54. #832C22semidirect
55. #821C22semidirect
56. (C2×D3)⋊C8C22semidirect
57. #816C22semidirect
58. Dic3⋊C8C22semidirect
59. C8×Dic3C22semidirect
60. #810C22semidirect
61. #807C22semidirect
74. #447D3semidirect
124. #213D4semidirect
142. C22×D4C2×D3semidirect
179. #102C2×D3semidirect
180. #102C2×D3semidirect
182. C22⋊C8C2×D3semidirect
421. C3#3273semidirect

Maximal subgroups

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

subgroupindex
#58062
C2×#9322
C2×#9112
#56122
C3×#4472
#51562
#51552
#51532
#50422
#47872
#47832
#47802
#47562
#47332
#47292
#32733