Finite Group Databasebeta

← #80142

Subgroups — #80142

Order 384.

396 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5805C211yesmaximal, characteristic
2#5625C211yesmaximal, characteristic, complemented
3C3×#450C211yesmaximal, characteristic, complemented
4#5218C211yesmaximal, characteristic, complemented
5#5175C211yesmaximal, characteristic, complemented
6#5167C211yesmaximal, characteristic, complemented
7#5166C211yesmaximal, characteristic, complemented
8#5163C211yesmaximal, characteristic, complemented
9#5047C211yesmaximal, characteristic, complemented
10#4768C211yesmaximal, characteristic, complemented
11#4767C211yesmaximal, characteristic, complemented
12#4764C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4733C211yesmaximal, characteristic, complemented
15#4720C211yesmaximal, characteristic, complemented
16#3222—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C2×#235C2211yescharacteristic
19C2×C3×SD16C2211yescharacteristic
20C2×C3×SD16C2211yescharacteristic
21C3×C22⋊Q8C2211yescharacteristic
22#953—21no—
23#953—21no—
24Q8×Dic3C2211yescharacteristic, complemented
25#944C2211yescharacteristic, complemented
26C2×#216C2211yescharacteristic, complemented
27C2×#215C2211yescharacteristic, complemented
28#940C2211yescharacteristic
29#940C2211yescharacteristic
30#938—21no—
31#938—21no—
32#936C2211yescharacteristic
33#936C2211yescharacteristic
34C2×#214C2211yescharacteristic
35C2×#214C2211yescharacteristic
36#926C2211yescharacteristic
37#924C2211yescharacteristic
38C2×Dic12C2211yescharacteristic, complemented
39C2×#204C2211yescharacteristic, complemented
40#890—21no—
41#890C2211yescharacteristic, complemented
42#889—21no—
43#887C2211yescharacteristic, complemented
44#879—21no—
45#879—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#541—31no—
61#535—31no—
62#497—31no—
63#492—31no—
64#490—31no—
65#488—31no—
66#464—31no—
67#464—31no—
68#460—31no—
69#460—31no—
70#452—31no—
71#452—31no—
72#450—31no—
73#450D311yescharacteristic, complemented
74#433—31no—
75C24×C3—21no—
76C2×C3×Q8C2311yescharacteristic
77C2×C3×D4—21no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4C2311yescharacteristic
81C2×C3×D4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C2×#67—21no—
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67—21no—
88#235—41no—
89#235—41no—
90C2×D12C2311yescharacteristic
91C2×C4×D3—21no—
92C2×Dic6C2311yescharacteristic
93C2×Dic6C2311yescharacteristic
94C3×SD16—41no—
95C3×SD16—41no—
96C2×C3×C8C2311yescharacteristic
97C2×C3×C8C2311yescharacteristic
98C3×C4⋊C4—21no—
99C3×C4⋊C4C2311yescharacteristic
100C3×C22⋊C4—21no—
101#217—21no—
102#217—21no—
103#217—21no—
104#217—21no—
105#217—21no—
106#217D411yescharacteristic
107#217D411yescharacteristic
108#216—41no—
109#215—41no—
110#214—41no—
111#214—41no—
112C4⋊Dic3—21no—
113C4⋊Dic3—21no—
114C4⋊Dic3D411yescharacteristic
115C4⋊Dic3D411yescharacteristic
116C4⋊Dic3C2311yescharacteristic
117Dic3⋊C4—21no—
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4—21no—
121Dic3⋊C4—21no—
122C4×Dic3—21no—
123C4×Dic3—21no—
124C4×Dic3C2311yescharacteristic
125C4×Dic3C2311yescharacteristic
126C2×C3⋊C8C2311yescharacteristic
127C2×C3⋊C8C2311yescharacteristic
128Dic12—41no—
129#204—41no—
130C2×#41—31no—
131C22×D4C2×D311yescharacteristic
132C2×Q16—31no—
133C2×Q16—31no—
134C2×SD16—31no—
135C2×SD16—31no—
136C2×SD16—31no—
137C2×SD16—31no—
138C2×SD16C2×D311yescharacteristic
139C2×SD16C2×D311yescharacteristic
140C4⋊Q8—31no—
141#127—31no—
142#127—31no—
143#126—61no—
144#126—61no—
145#125—31no—
146#125—61no—
147#125—61no—
148#124—31no—
149#124—31no—
150C22⋊Q8—31no—
151C22⋊Q8C2×D311yescharacteristic
152#121—61no—
153#121—61no—
154#120—61no—
155#120—61no—
156C4×Q8—31no—
157C4×Q8—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—42no—
176C23×C3—81no—
177C23×C3—81no—
178C23×C3C2×D411yescharacteristic
179C22×D3—21no—
180C3×Q8—21no—
181C3×Q8—21no—
182C3×D4—21no—
183C3×D4—21no—
184C3×D4—21no—
185C3×D4—21no—
186C3×D4—42no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4C2×D411yescharacteristic
192C2×C3×C4C2411yescharacteristic
193#67—42no—
194#67—81no—
195#67—81no—
196#67—81no—
197#67—81no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3C2×D411yescharacteristic
202C2×Dic3C2×D411yescharacteristic
203C2×Dic3C2×D411yescharacteristic
204C2×Dic3C2×D411yescharacteristic
205D12—21no—
206D12—21no—
207C4×D3—42no—
208Dic6—21no—
209Dic6—21no—
210Dic6—21no—
211Dic6—21no—
212C3×C8—21no—
213C3×C8—21no—
214C3⋊C8—21no—
215C3⋊C8—21no—
216C24—21no—
217#41—121no—
218#41—121no—
219C2×Q8—31no—
220C2×Q8—31no—
221C2×Q8C22×D311yescharacteristic
222C2×D4—21no—
223C2×D4—31no—
224C2×D4—41no—
225C2×D4—41no—
226C2×D4—61no—
227C2×D4—61no—
228C2×D4—61no—
229C2×D4—61no—
230C2×D4—61no—
231C2×D4C22×D311yescharacteristic
232C2×D4C22×D311yescharacteristic
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235Q16—121no—
236Q16—121no—
237SD16—41no—
238SD16—41no—
239SD16—121no—
240SD16—121no—
241SD16—121no—
242SD16—121no—
243C2×C8—31no—
244C2×C8—31no—
245C2×C8C22×D311yescharacteristic
246C2×C8C22×D311yescharacteristic
247C4⋊C4—21no—
248C4⋊C4—31no—
249C4⋊C4—31no—
250C4⋊C4—31no—
251C4⋊C4—61no—
252C4⋊C4—61no—
253C4⋊C4—61no—
254C4⋊C4—61no—
255C4⋊C4—61no—
256C4⋊C4—61no—
257C4⋊C4—61no—
258C4⋊C4C22×D311yescharacteristic
259C22⋊C4—21no—
260C22⋊C4—31no—
261C22⋊C4—31no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C22⋊C4—61no—
265C22⋊C4—61no—
266C22⋊C4—61no—
267C42—31no—
268C42—31no—
269C42—61no—
270C42—61no—
271C22×C3—21no—
272C22×C3—21no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—84no—
281C22×C3C22×D411yescharacteristic
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C3×C4—21no—
286C3×C4—41no—
287C3×C4—41no—
288C3×C4H(5,𝔽2)11yescharacteristic
289C3×C4H(5,𝔽2)11yescharacteristic
290Dic3—21no—
291Dic3—21no—
292Dic3—21no—
293Dic3—21no—
294Dic3—41no—
295Dic3—41no—
296Dic3—41no—
297C23—21no—
298C23—21no—
299C23—42no—
300C23—61no—
301C23—81no—
302C23—81no—
303C23D3×D411yescharacteristic
304Q8—21no—
305Q8—21no—
306Q8—61no—
307Q8—61no—
308Q8—61no—
309Q8—61no—
310D4—21no—
311D4—21no—
312D4—21no—
313D4—21no—
314D4—42no—
315D4—61no—
316D4—61no—
317D4—122no—
318D4—241no—
319D4—241no—
320D4—241no—
321D4—241no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—31no—
327C2×C4—31no—
328C2×C4—31no—
329C2×C4—31no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—122no—
334C2×C4D3×D411yescharacteristic
335C2×C4C23×D311yescharacteristic
336C8—21no—
337C8—21no—
338C8—61no—
339C8—61no—
340C2×C3—41no—
341C2×C3—81no—
342C2×C3—81no—
343C2×C3#53511yescharacteristic
344C2×C3#57711yescharacteristic
345C2×C3#57711yescharacteristic
346D3—81no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—84no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22C2×D3×D411yescharacteristic
361C4—21no—
362C4—41no—
363C4—41no—
364C4—61no—
365C4—61no—
366C4—61no—
367C4—61no—
368C4—121no—
369C4—121no—
370C4—121no—
371C4#100411yescharacteristic
372C4#100411yescharacteristic
373C3#322211yescharacteristic, complemented, Sylow
374C2—41no—
375C2—81no—
376C2—81no—
377C2—241no—
378C2#559211yescharacteristic
379C2#576211yescharacteristic
380C2#577511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#450C2
socleC22×C3C22×D4
prime radical#5805C2
hypercenter#450D3
hypocenterC3#3222
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
2. #5625C2semidirect
3. C3×#450C2semidirect
4. #5218C2semidirect
5. #5175C2semidirect
6. #5167C2semidirect
7. #5166C2semidirect
8. #5163C2semidirect
9. #5047C2semidirect
10. #4768C2semidirect
11. #4767C2semidirect
12. #4764C2semidirect
13. #4762C2semidirect
14. #4733C2semidirect
15. #4720C2semidirect
24. Q8×Dic3C22semidirect
25. #944C22semidirect
26. C2×#216C22semidirect
27. C2×#215C22semidirect
38. C2×Dic12C22semidirect
39. C2×#204C22semidirect
41. #890C22semidirect
43. #887C22semidirect
52. #821C22semidirect
53. #816C22semidirect
54. #815C22semidirect
55. Dic3⋊C8C22semidirect
56. Dic3⋊C8C22semidirect
57. C8×Dic3C22semidirect
58. #810C22semidirect
59. #809C22semidirect
73. #450D3semidirect
373. C3#3222semidirect

Maximal subgroups

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

subgroupindex
#58052
#56252
C3×#4502
#52182
#51752
#51672
#51662
#51632
#50472
#47682
#47672
#47642
#47622
#47332
#47202
#32223