Finite Group Databasebeta

← #80471

Subgroups — #80471

Order 384.

356 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5732C211yesmaximal, characteristic, complemented
2#5568C211yesmaximal, characteristic, complemented
3C3×#462C211yesmaximal, characteristic, complemented
4#5191C211yesmaximal, characteristic
5#5190C211yesmaximal, characteristic, complemented
6#5185C211yesmaximal, characteristic, complemented
7#5182C211yesmaximal, characteristic, complemented
8#5086C211yesmaximal, characteristic, complemented
9#5026C211yesmaximal, characteristic, complemented
10#4844C211yesmaximal, characteristic
11#4828C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic
13#4817C211yesmaximal, characteristic, complemented
14#4798C211yesmaximal, characteristic, complemented
15#4790C211yesmaximal, characteristic, complemented
16#3402—31nomaximal, Sylow
17C2×C3×Q16C2211yescharacteristic
18C2×C3×Q16C2211yescharacteristic
19C3×C4⋊Q8C2211yescharacteristic
20C3×C4×Q8C2211yescharacteristic
21#947C2211yescharacteristic
22#947C2211yescharacteristic
23#947C2211yescharacteristic
24Q8×Dic3C2211yescharacteristic
25Q8×Dic3C2211yescharacteristic
26#944C2211yescharacteristic
27C2×#216C2211yescharacteristic
28C2×#216C2211yescharacteristic
29C2×#215C2211yescharacteristic
30C2×#215C2211yescharacteristic
31C2×#215C2211yescharacteristic
32C2×D24C2211yescharacteristic
33C2×#204C2211yescharacteristic
34#898—21no—
35#898—21no—
36#898—21no—
37#897—21no—
38#896—21no—
39#893C2211yescharacteristic
40#892—21no—
41#887C2211yescharacteristic
42#876—21no—
43#875C2211yescharacteristic
44C4×D12C2211yescharacteristic
45#872—21no—
46C4×Dic6C2211yescharacteristic
47C3×C4⋊C8C2211yescharacteristic
48C3×#103C2211yescharacteristic
49C3×#103C2211yescharacteristic
50#835C2211yescharacteristic
51#821C2211yescharacteristic
52#816C2211yescharacteristic
53#815C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56C8×Dic3C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic
59#803C2211yescharacteristic
60#556—31no—
61#556—31no—
62#497—31no—
63#497—31no—
64#495—31no—
65#464—31no—
66#464—31no—
67#463—31no—
68#463—31no—
69#462—31no—
70#462D311yescharacteristic, complemented
71#446—31no—
72#440—31no—
73#440—31no—
74C4×Q16—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×Q8C2311yescharacteristic
77C2×C3×Q8C2311yescharacteristic
78C2×D12C2311yescharacteristic
79C2×D12C2311yescharacteristic
80C2×C4×D3—21no—
81C2×C4×D3—21no—
82C2×Dic6D411yescharacteristic
83C2×Dic6C2311yescharacteristic
84C3×Q16—41no—
85C3×Q16—41no—
86C2×C3×C8C2311yescharacteristic
87C2×C3×C8C2311yescharacteristic
88C3×C4⋊C4—21no—
89C3×C4⋊C4—21no—
90C3×C4⋊C4—21no—
91C3×C4⋊C4C2311yescharacteristic
92C3×C42—21no—
93C3×C42C2311yescharacteristic
94#216—22no—
95#216—41no—
96#215—22no—
97#215—22no—
98#215—41no—
99(C2×D3)⋊C4—21no—
100(C2×D3)⋊C4—21no—
101(C2×D3)⋊C4—21no—
102(C2×D3)⋊C4—21no—
103(C2×D3)⋊C4—21no—
104(C2×D3)⋊C4—21no—
105(C2×D3)⋊C4—21no—
106(C2×D3)⋊C4—21no—
107(C2×D3)⋊C4—21no—
108(C2×D3)⋊C4—21no—
109C4⋊Dic3—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3D411yescharacteristic
112C4⋊Dic3C2311yescharacteristic
113Dic3⋊C4—21no—
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116Dic3⋊C4D411yescharacteristic
117Dic3⋊C4D411yescharacteristic
118C4×Dic3—21no—
119C4×Dic3—21no—
120C4×Dic3—21no—
121C4×Dic3C2311yescharacteristic
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125D24—41no—
126#204—41no—
127C2×Q16—31no—
128C2×Q16—31no—
129C2×Q16C2×D311yescharacteristic
130C2×Q16C2×D311yescharacteristic
131C2×SD16—31no—
132C2×SD16—31no—
133C2×SD16—31no—
134C2×SD16—31no—
135C2×D8—31no—
136C4⋊Q8—31no—
137C4⋊Q8C2×D311yescharacteristic
138#126—61no—
139#126—61no—
140#126—61no—
141#126—61no—
142#124—31no—
143#124—31no—
144#124—31no—
145#124—31no—
146C22⋊Q8—61no—
147C22⋊Q8—61no—
148C4×Q8—31no—
149C4×Q8—31no—
150C4×Q8—31no—
151C4×Q8—31no—
152C4×Q8C2×D311yescharacteristic
153C4×D4—31no—
154C4×D4—31no—
155#117—61no—
156#117—61no—
157#107—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160C4⋊C8C2×D311yescharacteristic
161#103—31no—
162#103—31no—
163#103C2×D311yescharacteristic
164#103C2×D311yescharacteristic
165#102—31no—
166#102—31no—
167#97—31no—
168#97—31no—
169C4×C8—31no—
170C22×D3—21no—
171C22×D3—21no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—21no—
177C3×Q8—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2×D411yescharacteristic
184C2×C3×C4C2411yescharacteristic
185C2×Dic3—21no—
186C2×Dic3—21no—
187C2×Dic3C2×D411yescharacteristic
188C2×Dic3C2×D411yescharacteristic
189C2×Dic3C2×D411yescharacteristic
190C2×Dic3C2×D411yescharacteristic
191D12—21no—
192D12—21no—
193D12—21no—
194D12—21no—
195C4×D3—42no—
196C4×D3—42no—
197Dic6—21no—
198Dic6—21no—
199Dic6—41no—
200C3×C8—21no—
201C3×C8—21no—
202C3⋊C8—21no—
203C3⋊C8#4112yes—
204C2×Q8—31no—
205C2×Q8—31no—
206C2×Q8C22×D311yescharacteristic
207C2×Q8C22×D311yescharacteristic
208C2×Q8C22×D311yescharacteristic
209C2×D4—31no—
210C2×D4—31no—
211C22×C4—61no—
212C22×C4—61no—
213Q16—41no—
214Q16—41no—
215Q16—62no—
216Q16—121no—
217SD16—62no—
218SD16—62no—
219SD16—121no—
220SD16—121no—
221D8—121no—
222C2×C8—31no—
223C2×C8—31no—
224C2×C8C22×D311yescharacteristic
225C2×C8C22×D311yescharacteristic
226C4⋊C4—21no—
227C4⋊C4—21no—
228C4⋊C4—21no—
229C4⋊C4—31no—
230C4⋊C4—31no—
231C4⋊C4—31no—
232C4⋊C4—31no—
233C4⋊C4—61no—
234C4⋊C4—61no—
235C4⋊C4—61no—
236C4⋊C4—61no—
237C4⋊C4—61no—
238C4⋊C4C22×D311yescharacteristic
239C22⋊C4—61no—
240C22⋊C4—61no—
241C22⋊C4—61no—
242C22⋊C4—61no—
243C22⋊C4—61no—
244C22⋊C4—61no—
245C22⋊C4—61no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C42—21no—
250C42—31no—
251C42—31no—
252C42—61no—
253C42—61no—
254C42—61no—
255C42C22×D311yescharacteristic
256C22×C3C22×D411yescharacteristic
257C2×D3—41no—
258C2×D3—41no—
259C2×D3—41no—
260C2×D3—41no—
261C2×D3—41no—
262C2×D3—41no—
263C3×C4—21no—
264C3×C4—21no—
265C3×C4—41no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—41no—
269C3×C4C2×#4111yescharacteristic
270C3×C4H(5,𝔽2)11yescharacteristic
271Dic3—21no—
272Dic3—21no—
273Dic3—21no—
274Dic3—21no—
275Dic3—41no—
276Dic3—41no—
277C23—61no—
278C23—61no—
279Q8—21no—
280Q8—21no—
281Q8—21no—
282Q8—21no—
283Q8—21no—
284Q8—21no—
285Q8—61no—
286Q8—61no—
287Q8—121no—
288D4—61no—
289D4—61no—
290D4—61no—
291D4—61no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—31no—
297C2×C4—31no—
298C2×C4—31no—
299C2×C4—31no—
300C2×C4—61no—
301C2×C4—61no—
302C2×C4—122no—
303C2×C4—122no—
304C2×C4D3×D411yescharacteristic
305C2×C4D3×D411yescharacteristic
306C2×C4C23×D311yescharacteristic
307C8—21no—
308C8—21no—
309C8—32no—
310C8—61no—
311C2×C3#55011yescharacteristic
312C2×C3#57511yescharacteristic
313C2×C3#57811yescharacteristic
314D3—81no—
315D3—81no—
316C22—121no—
317C22—121no—
318C22—121no—
319C22—121no—
320C22—121no—
321C22—121no—
322C22C2×D3×D411yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—61no—
330C4—61no—
331C4—61no—
332C4—61no—
333C4—121no—
334C4—121no—
335C4#100411yescharacteristic
336C4D3×#4111yescharacteristic
337C3#340211yescharacteristic, complemented, Sylow
338C2—241no—
339C2—241no—
340C2#559911yescharacteristic
341C2#576611yescharacteristic
342C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#462C2
socleC22×C3C22×D4
prime radicalC2×D24C22
hypercenter#462D3
hypocenterC3#3402
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. #5732C2semidirect
2. #5568C2semidirect
3. C3×#462C2semidirect
5. #5190C2semidirect
6. #5185C2semidirect
7. #5182C2semidirect
8. #5086C2semidirect
9. #5026C2semidirect
11. #4828C2semidirect
13. #4817C2semidirect
14. #4798C2semidirect
15. #4790C2semidirect
70. #462D3semidirect
337. C3#3402semidirect

Maximal subgroups

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

subgroupindex
#57322
#55682
C3×#4622
#51912
#51902
#51852
#51822
#50862
#50262
#48442
#48282
#48192
#48172
#47982
#47902
#34023