Finite Group Databasebeta

← #79998

Subgroups — #79998

Order 384.

420 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5805C211yesmaximal, characteristic, complemented
2#5619C211yesmaximal, characteristic, complemented
3C3×#447C211yesmaximal, characteristic, complemented
4#5217C211yesmaximal, characteristic, complemented
5#5154C211yesmaximal, characteristic, complemented
6#5153C211yesmaximal, characteristic, complemented
7#5151C211yesmaximal, characteristic, complemented
8#5150C211yesmaximal, characteristic, complemented
9#5037C211yesmaximal, characteristic, complemented
10#4787C211yesmaximal, characteristic, complemented
11#4786C211yesmaximal, characteristic, complemented
12#4768C211yesmaximal, characteristic, complemented
13#4764C211yesmaximal, characteristic, complemented
14#4734C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3220—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×D8C2211yescharacteristic
20C2×C3×D8C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#953—21no—
23#953—21no—
24#940C2211yescharacteristic
25#940C2211yescharacteristic
26#940C2211yescharacteristic, complemented
27#939—21no—
28#938—21no—
29#938—21no—
30#938—21no—
31#936C2211yescharacteristic
32#936C2211yescharacteristic
33D4×Dic3C2211yescharacteristic, complemented
34C2×#214C2211yescharacteristic, complemented
35C2×#213C2211yescharacteristic
36C2×#213C2211yescharacteristic
37C2×#213C2211yescharacteristic, complemented
38#930C2211yescharacteristic
39#926C2211yescharacteristic
40C2×D24C2211yescharacteristic, complemented
41C2×#204C2211yescharacteristic, complemented
42#893C2211yescharacteristic, complemented
43#890C2211yescharacteristic, complemented
44#886—21no—
45#883—21no—
46C3×#102C2211yescharacteristic
47C3×#102C2211yescharacteristic
48C3×C22⋊C8C2211yescharacteristic
49#832C2211yescharacteristic
50#832C2211yescharacteristic
51#830C2211yescharacteristic
52#821C2211yescharacteristic, complemented
53#816C2211yescharacteristic, complemented
54#815C2211yescharacteristic, complemented
55Dic3⋊C8C2211yescharacteristic, complemented
56Dic3⋊C8C2211yescharacteristic, complemented
57C8×Dic3C2211yescharacteristic, complemented
58#810C2211yescharacteristic, complemented
59#809C2211yescharacteristic, complemented
60#540—31no—
61#535—31no—
62#496—31no—
63#493—31no—
64#490—31no—
65#488—31no—
66#463—31no—
67#463—31no—
68#459—31no—
69#459—31no—
70#449—31no—
71#449—31no—
72#447—31no—
73#447D311yescharacteristic, complemented
74#433—31no—
75C24×C3—21no—
76C2×C3×D4—21no—
77C2×C3×D4—21no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C2×C3×D4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88C2×#67—21no—
89C2×#67—21no—
90C2×#67—21no—
91C2×#67—21no—
92C22×Dic3—21no—
93#235—41no—
94#235—41no—
95C2×D12C2311yescharacteristic
96C2×D12C2311yescharacteristic
97C2×C4×D3—21no—
98C2×C4×D3—21no—
99C2×Dic6C2311yescharacteristic
100C3×D8—41no—
101C3×D8—41no—
102C2×C3×C8C2311yescharacteristic
103C2×C3×C8C2311yescharacteristic
104C3×C4⋊C4C2311yescharacteristic
105C3×C22⋊C4—21no—
106#217—21no—
107#217—21no—
108#217—21no—
109#217—21no—
110#217—21no—
111#217D411yescharacteristic
112#217D411yescharacteristic
113#214—41no—
114#213—41no—
115#213—41no—
116#213—41no—
117(C2×D3)⋊C4—21no—
118(C2×D3)⋊C4—21no—
119C4⋊Dic3—21no—
120C4⋊Dic3D411yescharacteristic
121C4⋊Dic3D411yescharacteristic
122C4⋊Dic3C2311yescharacteristic
123Dic3⋊C4—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128D24—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4C2×D311yescharacteristic
132C2×SD16—31no—
133C2×SD16—31no—
134C2×D8—31no—
135C2×D8—31no—
136C2×D8—31no—
137C2×D8—31no—
138C2×D8C2×D311yescharacteristic
139C2×D8C2×D311yescharacteristic
140#127—31no—
141#127—31no—
142#127—31no—
143#125—31no—
144#124—31no—
145#124—31no—
146#123—61no—
147#123—61no—
148#121—31no—
149#121—61no—
150#121—61no—
151#121—61no—
152#121—61no—
153#121C2×D311yescharacteristic
154#120—61no—
155#120—61no—
156C4×D4—31no—
157C4×D4—31no—
158#117—31no—
159C4⋊C8—31no—
160C4⋊C8—31no—
161#103—31no—
162#103—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#102C2×D311yescharacteristic
169C22⋊C8—31no—
170C22⋊C8C2×D311yescharacteristic
171#97—31no—
172C4×C8—31no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—42no—
177C23×C3—81no—
178C23×C3—81no—
179C23×C3C2×D411yescharacteristic
180C22×D3—21no—
181C22×D3—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—21no—
187C3×D4—21no—
188C3×D4—42no—
189C3×D4—81no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4C2×D411yescharacteristic
194C2×C3×C4C2411yescharacteristic
195#67—42no—
196#67—81no—
197#67—81no—
198#67—81no—
199#67—81no—
200#67—81no—
201#67—81no—
202#67—81no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—81no—
206C2×Dic3C2×D411yescharacteristic
207C2×Dic3C2×D411yescharacteristic
208C2×Dic3C2×D411yescharacteristic
209C2×Dic3C2×D411yescharacteristic
210D12—21no—
211D12—21no—
212D12—21no—
213D12—21no—
214C4×D3—42no—
215C4×D3—81no—
216Dic6—21no—
217Dic6—21no—
218C3×C8—21no—
219C3×C8—21no—
220C3⋊C8—21no—
221C3⋊C8—21no—
222C24—21no—
223#41—121no—
224#41—121no—
225C2×Q8—31no—
226C2×D4—21no—
227C2×D4—21no—
228C2×D4—31no—
229C2×D4—31no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—61no—
233C2×D4—61no—
234C2×D4—61no—
235C2×D4—61no—
236C2×D4—61no—
237C2×D4—61no—
238C2×D4—61no—
239C2×D4—61no—
240C2×D4C22×D311yescharacteristic
241C2×D4C22×D311yescharacteristic
242C2×D4C22×D311yescharacteristic
243C22×C4—61no—
244C22×C4—61no—
245C22×C4—61no—
246C22×C4C22×D311yescharacteristic
247SD16—121no—
248SD16—121no—
249D8—41no—
250D8—41no—
251D8—121no—
252D8—121no—
253D8—121no—
254D8—121no—
255C2×C8—31no—
256C2×C8—31no—
257C2×C8C22×D311yescharacteristic
258C2×C8C22×D311yescharacteristic
259C4⋊C4—31no—
260C4⋊C4—31no—
261C4⋊C4—31no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4C22×D311yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—31no—
267C22⋊C4—31no—
268C22⋊C4—61no—
269C22⋊C4—61no—
270C22⋊C4—61no—
271C22⋊C4—61no—
272C22⋊C4—61no—
273C22⋊C4—61no—
274C22⋊C4—61no—
275C42—31no—
276C42—31no—
277C22×C3—21no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—84no—
290C22×C3C22×D411yescharacteristic
291C2×D3—41no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C3×C4—21no—
298C3×C4—41no—
299C3×C4H(5,𝔽2)11yescharacteristic
300C3×C4H(5,𝔽2)11yescharacteristic
301Dic3—21no—
302Dic3—21no—
303Dic3—21no—
304Dic3—21no—
305Dic3—41no—
306Dic3—41no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—42no—
311C23—61no—
312C23—61no—
313C23—81no—
314C23—81no—
315C23D3×D411yescharacteristic
316Q8—61no—
317Q8—61no—
318D4—21no—
319D4—21no—
320D4—21no—
321D4—21no—
322D4—21no—
323D4—21no—
324D4—42no—
325D4—61no—
326D4—61no—
327D4—61no—
328D4—61no—
329D4—81no—
330D4—122no—
331D4—241no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336D4—241no—
337D4—241no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—31no—
342C2×C4—31no—
343C2×C4—31no—
344C2×C4—31no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—122no—
348C2×C4—241no—
349C2×C4—241no—
350C2×C4D3×D411yescharacteristic
351C2×C4C23×D311yescharacteristic
352C8—21no—
353C8—21no—
354C8—61no—
355C8—61no—
356C2×C3—41no—
357C2×C3—81no—
358C2×C3—81no—
359C2×C3—81no—
360C2×C3#53511yescharacteristic
361C2×C3#57611yescharacteristic
362C2×C3#57611yescharacteristic
363D3—81no—
364D3—81no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—84no—
378C22—121no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22—121no—
384C22C2×D3×D411yescharacteristic
385C4—21no—
386C4—41no—
387C4—61no—
388C4—61no—
389C4—61no—
390C4—61no—
391C4—121no—
392C4—121no—
393C4#100411yescharacteristic
394C4#100411yescharacteristic
395C3#322011yescharacteristic, complemented, Sylow
396C2—41no—
397C2—81no—
398C2—81no—
399C2—81no—
400C2—241no—
401C2—241no—
402C2#559211yescharacteristic
403C2#575711yescharacteristic
404C2#577411yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#447C2
socleC22×C3C22×D4
hypercenter#447D3
hypocenterC3#3220
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. #5805C2semidirect
2. #5619C2semidirect
3. C3×#447C2semidirect
4. #5217C2semidirect
5. #5154C2semidirect
6. #5153C2semidirect
7. #5151C2semidirect
8. #5150C2semidirect
9. #5037C2semidirect
10. #4787C2semidirect
11. #4786C2semidirect
12. #4768C2semidirect
13. #4764C2semidirect
14. #4734C2semidirect
15. #4720C2semidirect
26. #940C22semidirect
33. D4×Dic3C22semidirect
34. C2×#214C22semidirect
37. C2×#213C22semidirect
40. C2×D24C22semidirect
41. C2×#204C22semidirect
42. #893C22semidirect
43. #890C22semidirect
52. #821C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
59. #809C22semidirect
73. #447D3semidirect
395. C3#3220semidirect

Maximal subgroups

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

subgroupindex
#58052
#56192
C3×#4472
#52172
#51542
#51532
#51512
#51502
#50372
#47872
#47862
#47682
#47642
#47342
#47202
#32203