Finite Group Databasebeta

← #80815

Subgroups — #80815

Order 384.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5629C211yesmaximal, characteristic, complemented
2#5607C211yesmaximal, characteristic, complemented
3C3×#468C211yesmaximal, characteristic, complemented
4#5172C211yesmaximal, characteristic, complemented
5#5169C211yesmaximal, characteristic, complemented
6#5135C211yesmaximal, characteristic, complemented
7#5129C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4885C211yesmaximal, characteristic, complemented
11#4882C211yesmaximal, characteristic, complemented
12#4815C211yesmaximal, characteristic, complemented
13#4810C211yesmaximal, characteristic, complemented
14#4777C211yesmaximal, characteristic, complemented
15#4773C211yesmaximal, characteristic, complemented
16#3198—31nomaximal, Sylow
17C2×#239C2211yescharacteristic, complemented
18C2×D3×D4C2211yescharacteristic
19C2×#235C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C2×C3×M16C2211yescharacteristic, complemented
22C3×C22⋊Q8C2211yescharacteristic, complemented
23C3×#121C2211yescharacteristic
24#946C2211yescharacteristic
25C2×#215C2211yescharacteristic
26#940—21no—
27#938C2211yescharacteristic
28#937—21no—
29C2×#213C2211yescharacteristic
30#930C2211yescharacteristic
31#930C2211yescharacteristic, complemented
32C22×C3⋊C8C2211yescharacteristic, complemented
33C2×D24C2211yescharacteristic
34#896—21no—
35#895—21no—
36#895C2211yescharacteristic
37#895C2211yescharacteristic
38#892C2211yescharacteristic
39D3×C4⋊C4C2211yescharacteristic, complemented
40#885—21no—
41#884—21no—
42#884—21no—
43#882—21no—
44C3×#107C2211yescharacteristic, complemented
45C3×#103C2211yescharacteristic, complemented
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic, complemented
51(C2×D3)⋊C8C2211yescharacteristic
52(C2×D3)⋊C8C2211yescharacteristic
53(C2×D3)⋊C8C2211yescharacteristic, complemented
54(C2×D3)⋊C8C2211yescharacteristic, complemented
55#817C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic, complemented
58#808C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#537—31no—
61#535—31no—
62#485—31no—
63#482—31no—
64#481—31no—
65#480—31no—
66#469—31no—
67#468D311yescharacteristic, complemented
68#466—31no—
69#465—31no—
70#450—31no—
71#449—31no—
72#449—31no—
73#447—31no—
74#408—31no—
75C23×D3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C2×#67—21no—
84C2×#67D411yescharacteristic
85C2×#67D411yescharacteristic
86#239—41no—
87#239—41no—
88D3×D4—41no—
89D3×D4—41no—
90#235—41no—
91#235—41no—
92C2×D12—21no—
93C2×D12—21no—
94C2×D12—21no—
95C2×D12D411yescharacteristic
96C2×D12C2311yescharacteristic
97C2×D12C2311yescharacteristic
98C2×C4×D3—21no—
99C2×C4×D3—21no—
100C2×C4×D3C2311yescharacteristic
101C2×C4×D3C2311yescharacteristic
102C2×Dic6D411yescharacteristic
103C3×SD16—41no—
104C3×M16—41no—
105C2×C3×C8C2311yescharacteristic
106C2×C3×C8C2311yescharacteristic
107C3×C4⋊C4—21no—
108C3×C4⋊C4C2311yescharacteristic
109C3×C4⋊C4C2311yescharacteristic
110C3×C22⋊C4—21no—
111C3×C22⋊C4—21no—
112#217—21no—
113#215—41no—
114#213—41no—
115(C2×D3)⋊C4—21no—
116(C2×D3)⋊C4—21no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119(C2×D3)⋊C4—21no—
120(C2×D3)⋊C4—21no—
121C4⋊Dic3C2311yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124Dic3⋊C4—21no—
125C4×Dic3—21no—
126C2×C3⋊C8—41no—
127C2×C3⋊C8C2311yescharacteristic
128C2×C3⋊C8C2311yescharacteristic
129D24—41no—
130C2×#41—31no—
131C2×#41—31no—
132C22×D4—31no—
133C2×SD16—31no—
134C2×SD16C2×D311yescharacteristic
135C2×D8—31no—
136C2×D8—31no—
137C2×M16C2×D311yescharacteristic, complemented
138C22×C8—31no—
139#127—61no—
140#124—61no—
141C22⋊Q8—31no—
142C22⋊Q8—61no—
143C22⋊Q8C2×D311yescharacteristic, complemented
144#121—31no—
145#121—31no—
146#121—31no—
147#121—31no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121C2×D311yescharacteristic
153#120—61no—
154#120—61no—
155#117—31no—
156C2×C4⋊C4—31no—
157#107—31no—
158#107C2×D311yescharacteristic, complemented
159#106—31no—
160#106—31no—
161#103—31no—
162#103C2×D311yescharacteristic, complemented
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102—31no—
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8—31no—
171C22⋊C8—31no—
172C22⋊C8—31no—
173C23×C3—21no—
174C23×C3C2×D411yescharacteristic
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—42no—
178C22×D3—81no—
179C22×D3—81no—
180C22×D3C2×D411yescharacteristic
181C22×D3C2×D411yescharacteristic
182C3×Q8—21no—
183C3×Q8—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—81no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4C2×D411yescharacteristic
193C2×C3×C4C2411yescharacteristic
194#67—41no—
195#67—41no—
196#67—42no—
197#67—81no—
198#67—81no—
199#67—81no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204D12—21no—
205D12—21no—
206D12—21no—
207D12—21no—
208D12—41no—
209D12—42no—
210D12—81no—
211D12—81no—
212C4×D3—21no—
213C4×D3—21no—
214C4×D3—21no—
215C4×D3—21no—
216C4×D3—42no—
217C4×D3—81no—
218Dic6—41no—
219C3×C8—21no—
220C3×C8—21no—
221C3⋊C8—21no—
222C3⋊C8—21no—
223C24—61no—
224#41—121no—
225#41—121no—
226#41—121no—
227#41—121no—
228C2×Q8—31no—
229C2×Q8C22×D311yescharacteristic
230C2×D4—21no—
231C2×D4—31no—
232C2×D4—31no—
233C2×D4—31no—
234C2×D4—31no—
235C2×D4—31no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—61no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4C22×D311yescharacteristic
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—61no—
249C22×C4—61no—
250C22×C4C22×D311yescharacteristic
251SD16—41no—
252SD16—121no—
253D8—121no—
254D8—121no—
255M16—41no—
256C2×C8—31no—
257C2×C8—31no—
258C2×C8—121no—
259C2×C8C22×D311yescharacteristic
260C2×C8C22×D311yescharacteristic
261C4⋊C4—21no—
262C4⋊C4—31no—
263C4⋊C4—31no—
264C4⋊C4—61no—
265C4⋊C4—61no—
266C4⋊C4C22×D311yescharacteristic
267C4⋊C4C22×D311yescharacteristic
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C22⋊C4—61no—
276C22⋊C4—61no—
277C42—61no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3C22×D411yescharacteristic
285C2×D3—21no—
286C2×D3—21no—
287C2×D3—21no—
288C2×D3—21no—
289C2×D3—21no—
290C2×D3—21no—
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—84no—
298C3×C4—21no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4H(5,𝔽2)11yescharacteristic
303C3×C4H(5,𝔽2)11yescharacteristic
304Dic3—21no—
305Dic3—21no—
306Dic3—41no—
307Dic3—41no—
308C23—21no—
309C23—31no—
310C23—31no—
311C23—61no—
312C23—61no—
313C23—122no—
314C23—241no—
315C23—241no—
316C23D3×D411yescharacteristic
317Q8—21no—
318Q8—21no—
319Q8—121no—
320D4—21no—
321D4—21no—
322D4—61no—
323D4—61no—
324D4—61no—
325D4—61no—
326D4—81no—
327D4—121no—
328D4—122no—
329D4—121no—
330D4—121no—
331D4—122no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—122no—
351C2×C4—241no—
352C2×C4D3×D411yescharacteristic
353C2×C4C23×D311yescharacteristic
354C8—21no—
355C8—21no—
356C8—61no—
357C8—61no—
358C2×C3—41no—
359C2×C3—81no—
360C2×C3#53411yescharacteristic
361C2×C3#57611yescharacteristic
362C2×C3#57711yescharacteristic
363D3—41no—
364D3—41no—
365D3—81no—
366D3—81no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—61no—
374C22—61no—
375C22—61no—
376C22—61no—
377C22—61no—
378C22—61no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22—121no—
385C22—244no—
386C22C2×D3×D411yescharacteristic
387C4—21no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—61no—
392C4—61no—
393C4—121no—
394C4—121no—
395C4#100411yescharacteristic
396C4#100911yescharacteristic
397C3#319811yescharacteristic, complemented, Sylow
398C2—41no—
399C2—81no—
400C2—121no—
401C2—121no—
402C2—241no—
403C2—241no—
404C2#560511yescharacteristic
405C2#577411yescharacteristic
406C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#468C2
socleC22×C3C22×D4
hypercenter#468D3
hypocenterC3#3198
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #468

Complements

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

NH ≅ G/Ntype
1. #5629C2semidirect
2. #5607C2semidirect
3. C3×#468C2semidirect
4. #5172C2semidirect
5. #5169C2semidirect
6. #5135C2semidirect
7. #5129C2semidirect
8. #5048C2semidirect
9. #5038C2semidirect
10. #4885C2semidirect
11. #4882C2semidirect
12. #4815C2semidirect
13. #4810C2semidirect
14. #4777C2semidirect
15. #4773C2semidirect
17. C2×#239C22semidirect
21. C2×C3×M16C22semidirect
22. C3×C22⋊Q8C22semidirect
31. #930C22semidirect
32. C22×C3⋊C8C22semidirect
39. D3×C4⋊C4C22semidirect
44. C3×#107C22semidirect
45. C3×#103C22semidirect
47. #835C22semidirect
50. #821C22semidirect
53. (C2×D3)⋊C8C22semidirect
54. (C2×D3)⋊C8C22semidirect
55. #817C22semidirect
57. #809C22semidirect
58. #808C22semidirect
59. #807C22semidirect
67. #468D3semidirect
137. C2×M16C2×D3semidirect
143. C22⋊Q8C2×D3semidirect
158. #107C2×D3semidirect
162. #103C2×D3semidirect
397. C3#3198semidirect

Maximal subgroups

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

subgroupindex
#56292
#56072
C3×#4682
#51722
#51692
#51352
#51292
#50482
#50382
#48852
#48822
#48152
#48102
#47772
#47732
#31983