Finite Group Databasebeta

← #80641

Subgroups — #80641

Order 384.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5624C211yesmaximal, characteristic, complemented
2#5596C211yesmaximal, characteristic, complemented
3C3×#465C211yesmaximal, characteristic, complemented
4#5165C211yesmaximal, characteristic, complemented
5#5163C211yesmaximal, characteristic, complemented
6#5115C211yesmaximal, characteristic, complemented
7C2×Dic3⋊C8C211yesmaximal, characteristic, complemented
8#5048C211yesmaximal, characteristic, complemented
9#5038C211yesmaximal, characteristic, complemented
10#4871C211yesmaximal, characteristic, complemented
11#4870C211yesmaximal, characteristic, complemented
12#4819C211yesmaximal, characteristic, complemented
13#4793C211yesmaximal, characteristic, complemented
14#4786C211yesmaximal, characteristic, complemented
15#4766C211yesmaximal, characteristic, complemented
16#3247—31nomaximal, Sylow
17C2×#237—21no—
18C2×C3×SD16C2211yescharacteristic
19C22×C3×C8C2211yescharacteristic
20C3×C22⋊Q8C2211yescharacteristic
21C3×#121C2211yescharacteristic
22#947C2211yescharacteristic
23Q8×Dic3C2211yescharacteristic
24C2×#215C2211yescharacteristic
25#940C2211yescharacteristic
26#939—21no—
27D4×Dic3C2211yescharacteristic
28C2×#213C2211yescharacteristic
29#930C2211yescharacteristic
30#930C2211yescharacteristic
31C2×C4×Dic3C2211yescharacteristic
32C22×C3⋊C8C2211yescharacteristic
33C2×D24C2211yescharacteristic
34#898—21no—
35#893C2211yescharacteristic
36#893C2211yescharacteristic
37#892—21no—
38#890C2211yescharacteristic
39#888C2211yescharacteristic
40#886—21no—
41#885—21no—
42#67⋊C4—21no—
43#67⋊C4—21no—
44C3×#106C2211yescharacteristic
45C3×#103C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic
48#832C2211yescharacteristic
49#821C2211yescharacteristic
50#821C2211yescharacteristic
51#818C2211yescharacteristic
52Dic3⋊C8C2211yescharacteristic
53Dic3⋊C8C2211yescharacteristic
54Dic3⋊C8C2211yescharacteristic
55Dic3⋊C8C2211yescharacteristic
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#808C2211yescharacteristic
59#807C2211yescharacteristic
60#532—31no—
61#529—31no—
62#478—31no—
63#478—31no—
64#477—31no—
65#476—31no—
66#466—31no—
67#466—31no—
68#465—31no—
69#465D311yescharacteristic, complemented
70#463—31no—
71#463—31no—
72#460—31no—
73#459—31no—
74C2×C4⋊C8—31no—
75C2×C3×Q8C2311yescharacteristic
76C2×C3×D4—21no—
77C2×C3×D4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—21no—
80C2×#67—21no—
81C2×#67—21no—
82C2×#67—21no—
83C22×Dic3—21no—
84C22×Dic3D411yescharacteristic
85C22×Dic3D411yescharacteristic
86#237—44no—
87C2×D12C2311yescharacteristic
88C2×D12C2311yescharacteristic
89C2×C4×D3—21no—
90C2×C4×D3—21no—
91C2×Dic6—21no—
92C3×SD16—41no—
93C2×C3×C8—41no—
94C2×C3×C8C2311yescharacteristic
95C2×C3×C8C2311yescharacteristic
96C3×C4⋊C4—21no—
97C3×C4⋊C4C2311yescharacteristic
98C3×C4⋊C4C2311yescharacteristic
99C3×C22⋊C4—21no—
100C3×C22⋊C4—21no—
101#217—21no—
102#215—41no—
103#213—41no—
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—
109(C2×D3)⋊C4—21no—
110C4⋊Dic3—21no—
111C4⋊Dic3—21no—
112C4⋊Dic3C2311yescharacteristic
113C4⋊Dic3C2311yescharacteristic
114Dic3⋊C4—21no—
115Dic3⋊C4—21no—
116C4×Dic3—21no—
117C4×Dic3D411yescharacteristic
118C4×Dic3D411yescharacteristic
119C4×Dic3C2311yescharacteristic
120C4×Dic3C2311yescharacteristic
121C2×C3⋊C8—41no—
122C2×C3⋊C8C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124D24—41no—
125C2×#41—61no—
126C2×SD16—31no—
127C2×SD16C2×D311yescharacteristic
128C2×D8—31no—
129C2×D8—31no—
130C22×C8—31no—
131C22×C8C2×D311yescharacteristic
132C4⋊Q8—31no—
133#127—31no—
134#126—61no—
135#125—31no—
136#124—31no—
137#124—61no—
138#123—61no—
139C22⋊Q8C2×D311yescharacteristic
140#121—31no—
141#121—31no—
142#121—61no—
143#121C2×D311yescharacteristic
144C4×Q8—31no—
145C4×D4—31no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—61no—
149C4×D4—61no—
150#117—61no—
151C2×C42—31no—
152#107—31no—
153#107—31no—
154#106—31no—
155#106C2×D311yescharacteristic
156C4⋊C8—31no—
157C4⋊C8—31no—
158C4⋊C8—31no—
159C4⋊C8—31no—
160#103—31no—
161#103C2×D311yescharacteristic
162#102—31no—
163#102—31no—
164#102—31no—
165#102—31no—
166#102—31no—
167#102C2×D311yescharacteristic
168C23×C3—21no—
169C23×C3C2×D411yescharacteristic
170C22×D3—21no—
171C22×D3—21no—
172C3×Q8—21no—
173C3×Q8—21no—
174C3×D4—21no—
175C3×D4—21no—
176C3×D4—42no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4C2×D411yescharacteristic
183C2×C3×C4C2411yescharacteristic
184#67—42no—
185#67—42no—
186#67—42no—
187#67—42no—
188C2×Dic3—22no—
189C2×Dic3—22no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—42no—
193C2×Dic3C2×D411yescharacteristic
194C2×Dic3C2×D411yescharacteristic
195C2×Dic3C2×D411yescharacteristic
196C2×Dic3C2×D411yescharacteristic
197D12—21no—
198D12—21no—
199D12—21no—
200D12—21no—
201C4×D3—42no—
202C4×D3—42no—
203Dic6—42no—
204C3×C8—21no—
205C3×C8—21no—
206C3⋊C8—21no—
207C3⋊C8—21no—
208#41—124no—
209C2×Q8—61no—
210C2×Q8C22×D311yescharacteristic
211C2×D4—21no—
212C2×D4—31no—
213C2×D4—31no—
214C2×D4—61no—
215C2×D4—61no—
216C2×D4—61no—
217C2×D4—61no—
218C2×D4C22×D311yescharacteristic
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—61no—
222C22×C4—61no—
223C22×C4—61no—
224C22×C4C22×D311yescharacteristic
225SD16—41no—
226SD16—121no—
227D8—121no—
228D8—121no—
229C2×C8—31no—
230C2×C8—31no—
231C2×C8—41no—
232C2×C8—121no—
233C2×C8C22×D311yescharacteristic
234C2×C8C22×D311yescharacteristic
235C4⋊C4—21no—
236C4⋊C4—31no—
237C4⋊C4—31no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—61no—
242C4⋊C4C22×D311yescharacteristic
243C4⋊C4C22×D311yescharacteristic
244C22⋊C4—21no—
245C22⋊C4—21no—
246C22⋊C4—61no—
247C22⋊C4—61no—
248C22⋊C4—61no—
249C22⋊C4—61no—
250C22⋊C4—61no—
251C22⋊C4—61no—
252C22⋊C4—61no—
253C42—31no—
254C42—31no—
255C42—31no—
256C42—31no—
257C42—61no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—41no—
263C22×C3AGL(1,ℤ/8)12yes—
264C22×C3C22×D411yescharacteristic
265C2×D3—41no—
266C2×D3—41no—
267C2×D3—41no—
268C2×D3—41no—
269C2×D3—41no—
270C2×D3—41no—
271C3×C4—21no—
272C3×C4—41no—
273C3×C4—41no—
274C3×C4—41no—
275C3×C4H(5,𝔽2)11yescharacteristic
276C3×C42−1+411yescharacteristic
277Dic3—21no—
278Dic3—21no—
279Dic3—21no—
280Dic3—41no—
281Dic3—41no—
282Dic3#13512yes—
283C23—21no—
284C23—61no—
285C23—61no—
286C23D3×D411yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—122no—
290D4—21no—
291D4—21no—
292D4—42no—
293D4—61no—
294D4—61no—
295D4—61no—
296D4—61no—
297D4—122no—
298D4—122no—
299D4—122no—
300D4—122no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—31no—
307C2×C4—31no—
308C2×C4—31no—
309C2×C4—31no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—61no—
313C2×C4—61no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4—122no—
317C2×C4D3×D411yescharacteristic
318C2×C4C23×D311yescharacteristic
319C8—21no—
320C8—21no—
321C8—61no—
322C8—61no—
323C2×C3—22no—
324C2×C3—81no—
325C2×C3#53711yescharacteristic
326C2×C3C2×#13511yescharacteristic
327C2×C3C2×AGL(1,ℤ/8)11yescharacteristic
328D3—81no—
329D3—81no—
330C22—21no—
331C22—21no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—121no—
336C22—121no—
337C22—121no—
338C22—121no—
339C22—121no—
340C22—121no—
341C22#91412yes—
342C22C2×D3×D411yescharacteristic
343C4—21no—
344C4—32no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—61no—
349C4—61no—
350C4—61no—
351C4—121no—
352C4—121no—
353C4#100411yescharacteristic
354C4#101011yescharacteristic
355C3#324711yescharacteristic, complemented, Sylow
356C2—22no—
357C2—81no—
358C2—241no—
359C2—241no—
360C2#559711yescharacteristic
361C2C2×#91411yescharacteristic
362C2D3×#13511yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#465C2
socleC22×C3C22×D4
hypercenter#465D3
hypocenterC3#3247
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #465

Complements

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

NH ≅ G/Ntype
1. #5624C2semidirect
2. #5596C2semidirect
3. C3×#465C2semidirect
4. #5165C2semidirect
5. #5163C2semidirect
6. #5115C2semidirect
7. C2×Dic3⋊C8C2semidirect
8. #5048C2semidirect
9. #5038C2semidirect
10. #4871C2semidirect
11. #4870C2semidirect
12. #4819C2semidirect
13. #4793C2semidirect
14. #4786C2semidirect
15. #4766C2semidirect
69. #465D3semidirect
355. C3#3247semidirect

Maximal subgroups

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

subgroupindex
#56242
#55962
C3×#4652
#51652
#51632
#51152
C2×Dic3⋊C82
#50482
#50382
#48712
#48702
#48192
#47932
#47862
#47662
#32473