Finite Group Databasebeta

← #79763

Subgroups — #79763

Order 384.

406 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5568C211yesmaximal, characteristic, complemented
2#5553C211yesmaximal, characteristic, complemented
3C3×#440C211yesmaximal, characteristic, complemented
4#5176C211yesmaximal, characteristic, complemented
5#5175C211yesmaximal, characteristic, complemented
6#5027C211yesmaximal, characteristic, complemented
7#5016C211yesmaximal, characteristic, complemented
8#4889C211yesmaximal, characteristic, complemented
9#4872C211yesmaximal, characteristic, complemented
10#4819C211yesmaximal, characteristic, complemented
11#4818C211yesmaximal, characteristic, complemented
12#4786C211yesmaximal, characteristic, complemented
13#4762C211yesmaximal, characteristic, complemented
14#4712C211yesmaximal, characteristic, complemented
15#4701C211yesmaximal, characteristic, complemented
16#3397—31nomaximal, Sylow
17C2×#235—21no—
18C2×C3×SD16C2211yescharacteristic
19C3×C4×Q8C2211yescharacteristic
20C3×C4×D4C2211yescharacteristic
21#947C2211yescharacteristic
22#944C2211yescharacteristic
23C2×#215C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#940C2211yescharacteristic
26#940C2211yescharacteristic
27C2×#214C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#930—21no—
30C2×D24C2211yescharacteristic
31C2×D24C2211yescharacteristic
32C2×#204C2211yescharacteristic
33C2×#204C2211yescharacteristic
34#898—21no—
35#896—21no—
36#893C2211yescharacteristic
37#893C2211yescharacteristic
38#893C2211yescharacteristic
39#887C2211yescharacteristic
40#884—21no—
41#884—21no—
42#876—21no—
43#875C2211yescharacteristic
44#874C2211yescharacteristic
45#872—21no—
46C4×Dic6C2211yescharacteristic
47C3×#107C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#102C2211yescharacteristic
50C3×#97C2211yescharacteristic
51#815C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54C8×Dic3C2211yescharacteristic
55#810C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#809C2211yescharacteristic
59C4⋊(C3⋊C8)C2211yescharacteristic
60#556—31no—
61#550—31no—
62#496—31no—
63#496—31no—
64#492—31no—
65#463—31no—
66#463—31no—
67#461—31no—
68#460—31no—
69#460—31no—
70#459—31no—
71#446—31no—
72#440—31no—
73#440D311yescharacteristic, complemented
74C4×D8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4C2311yescharacteristic
77C22×C3×C4—21no—
78C2×#67—21no—
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82#235—44no—
83C2×D12—21no—
84C2×D12—21no—
85C2×D12C2311yescharacteristic
86C2×D12C2311yescharacteristic
87C2×D12C2311yescharacteristic
88C2×C4×D3—21no—
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×Dic6D411yescharacteristic
92C2×Dic6C2311yescharacteristic
93C3×SD16—22no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C42—21no—
101C3×C42C2311yescharacteristic
102#215—41no—
103#215—41no—
104#214—41no—
105#213—41no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109(C2×D3)⋊C4—21no—
110(C2×D3)⋊C4—21no—
111(C2×D3)⋊C4—21no—
112(C2×D3)⋊C4—21no—
113C4⋊Dic3D411yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4—21no—
117Dic3⋊C4D411yescharacteristic
118Dic3⋊C4D411yescharacteristic
119C4×Dic3—21no—
120C4×Dic3C2311yescharacteristic
121C4×Dic3C2311yescharacteristic
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124D24—22no—
125D24—41no—
126#204—22no—
127#204—41no—
128C2×#41—61no—
129C2×SD16—31no—
130C2×SD16—31no—
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×SD16C2×D311yescharacteristic
135C2×D8—31no—
136C2×D8—31no—
137C2×D8—31no—
138C4⋊Q8—31no—
139#127—31no—
140#127—31no—
141#127—31no—
142#126—61no—
143#126—61no—
144#124—31no—
145#124—31no—
146C22⋊Q8—61no—
147#121—61no—
148#121—61no—
149#121—61no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×Q8C2×D311yescharacteristic
153C4×D4—31no—
154C4×D4—31no—
155C4×D4—31no—
156C4×D4C2×D311yescharacteristic
157#117—61no—
158#107C2×D311yescharacteristic
159C4⋊C8—31no—
160C4⋊C8—31no—
161C4⋊C8—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168#97—31no—
169#97C2×D311yescharacteristic
170C4×C8—31no—
171C23×C3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×D4—21no—
178C3×D4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—42no—
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2×D411yescharacteristic
185C2×C3×C4C2411yescharacteristic
186#67—42no—
187#67—42no—
188#67—81no—
189#67—81no—
190C2×Dic3—21no—
191C2×Dic3C2×D411yescharacteristic
192C2×Dic3C2×D411yescharacteristic
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195D12—21no—
196D12—21no—
197D12—21no—
198D12—21no—
199D12—21no—
200D12—21no—
201D12—42no—
202D12—81no—
203C4×D3—42no—
204C4×D3—42no—
205C4×D3—42no—
206Dic6—22no—
207Dic6—21no—
208Dic6—21no—
209C3×C8—21no—
210C3×C8#4112yescomplemented
211C3⋊C8—21no—
212C3⋊C8—21no—
213#41—124no—
214C2×Q8—31no—
215C2×Q8—31no—
216C2×Q8C22×D311yescharacteristic
217C2×D4—31no—
218C2×D4—31no—
219C2×D4—31no—
220C2×D4—61no—
221C2×D4—61no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—61no—
225C2×D4—61no—
226C2×D4C22×D311yescharacteristic
227C22×C4—21no—
228C22×C4—61no—
229C22×C4—61no—
230C22×C4—61no—
231SD16—22no—
232SD16—62no—
233SD16—121no—
234SD16—121no—
235SD16—121no—
236SD16—121no—
237D8—62no—
238D8—121no—
239D8—121no—
240C2×C8—31no—
241C2×C8—31no—
242C2×C8C22×D311yescharacteristic
243C2×C8C22×D311yescharacteristic
244C4⋊C4—21no—
245C4⋊C4—31no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4C22×D311yescharacteristic
252C4⋊C4C22×D311yescharacteristic
253C22⋊C4—21no—
254C22⋊C4—61no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C42—21no—
262C42—31no—
263C42—31no—
264C42—61no—
265C42C22×D311yescharacteristic
266C22×C3—41no—
267C22×C3—41no—
268C22×C3—41no—
269C22×C3C22×D411yescharacteristic
270C2×D3—41no—
271C2×D3—41no—
272C2×D3—41no—
273C2×D3—41no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C3×C4—21no—
280C3×C4—21no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4C2×#4111yescharacteristic
285C3×C4H(5,𝔽2)11yescharacteristic
286Dic3—21no—
287Dic3—21no—
288Dic3—21no—
289Dic3—41no—
290Dic3AGL(1,ℤ/8)12yes—
291C23—21no—
292C23—61no—
293C23—61no—
294C23—61no—
295Q8—21no—
296Q8—21no—
297Q8—62no—
298Q8—61no—
299Q8—61no—
300D4—21no—
301D4—21no—
302D4—61no—
303D4—61no—
304D4—61no—
305D4—61no—
306D4—61no—
307D4—61no—
308D4—122no—
309D4—122no—
310D4—122no—
311D4—241no—
312D4—241no—
313D4—241no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—31no—
318C2×C4—31no—
319C2×C4—31no—
320C2×C4—31no—
321C2×C4—42no—
322C2×C4—61no—
323C2×C4—122no—
324C2×C4—122no—
325C2×C4—122no—
326C2×C4D3×D411yescharacteristic
327C2×C4D3×D411yescharacteristic
328C2×C4C23×D311yescharacteristic
329C8—21no—
330C8—61no—
331C8—61no—
332C8#23512yescomplemented
333C2×C3—81no—
334C2×C3#55011yescharacteristic
335C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
336C2×C3#57711yescharacteristic
337D3—81no—
338D3—81no—
339D3—81no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—121no—
344C22—121no—
345C22—121no—
346C22—121no—
347C22—121no—
348C22—121no—
349C22—121no—
350C22—121no—
351C22—121no—
352C22C2×D3×D411yescharacteristic
353C4—21no—
354C4—21no—
355C4—32no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—61no—
360C4—61no—
361C4—61no—
362C4—121no—
363C4C2×#23511yescharacteristic
364C4#100911yescharacteristic
365C3#339711yescharacteristic, complemented, Sylow
366C2—81no—
367C2—241no—
368C2—241no—
369C2—241no—
370C2#555311yescharacteristic
371C2D3×AGL(1,ℤ/8)11yescharacteristic
372C2#577811yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#440C2
socleC22×C3C22×D4
hypercenter#440D3
hypocenterC3#3397
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #440

Complements

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

NH ≅ G/Ntype
1. #5568C2semidirect
2. #5553C2semidirect
3. C3×#440C2semidirect
4. #5176C2semidirect
5. #5175C2semidirect
6. #5027C2semidirect
7. #5016C2semidirect
8. #4889C2semidirect
9. #4872C2semidirect
10. #4819C2semidirect
11. #4818C2semidirect
12. #4786C2semidirect
13. #4762C2semidirect
14. #4712C2semidirect
15. #4701C2semidirect
73. #440D3semidirect
210. C3×C8#41semidirect
332. C8#235semidirect
365. C3#3397semidirect

Maximal subgroups

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

subgroupindex
#55682
#55532
C3×#4402
#51762
#51752
#50272
#50162
#48892
#48722
#48192
#48182
#47862
#47622
#47122
#47012
#33973