Finite Group Databasebeta

← #78902

Subgroups — #78902

Order 384.

414 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5829C211yesmaximal, characteristic, complemented
2#5514C211yesmaximal, characteristic, complemented
3C3×#417C211yesmaximal, characteristic, complemented
4#5238C211yesmaximal, characteristic, complemented
5#5135C211yesmaximal, characteristic, complemented
6#5128C211yesmaximal, characteristic, complemented
7#4967C211yesmaximal, characteristic, complemented
8#4819C211yesmaximal, characteristic, complemented
9#4815C211yesmaximal, characteristic, complemented
10#4812C211yesmaximal, characteristic, complemented
11#4800C211yesmaximal, characteristic, complemented
12#4773C211yesmaximal, characteristic, complemented
13#4770C211yesmaximal, characteristic, complemented
14#4763C211yesmaximal, characteristic, complemented
15#4756C211yesmaximal, characteristic, complemented
16#3322—31nomaximal, Sylow
17C2×C3×#41C2211yescharacteristic
18C2×D3×D4C2211yescharacteristic
19C2×C3×M16C2211yescharacteristic, complemented
20C2×C3×C4⋊C4C2211yescharacteristic, complemented
21#947C2211yescharacteristic
22#946C2211yescharacteristic
23C2×#215C2211yescharacteristic, complemented
24#939—21no—
25#937—21no—
26D4×Dic3C2211yescharacteristic
27C2×#213C2211yescharacteristic
28#930—21no—
29#930C2211yescharacteristic
30#930C2211yescharacteristic, complemented
31#929—21no—
32C4×#67—21no—
33C4×#67C2211yescharacteristic
34C2×(C2×D3)⋊C4—21no—
35C2×D24C2211yescharacteristic
36C2×ΓL(1,𝔽25)C2211yescharacteristic
37#898—21no—
38#895C2211yescharacteristic
39#894—21no—
40#893C2211yescharacteristic, complemented
41#892C2211yescharacteristic
42#890C2211yescharacteristic, complemented
43C3×#103C2211yescharacteristic, complemented
44C3×#103C2211yescharacteristic, complemented
45C3×#102C2211yescharacteristic
46C3×#102C2211yescharacteristic
47#835C2211yescharacteristic, complemented
48#832C2211yescharacteristic
49#830C2211yescharacteristic, complemented
50#821C2211yescharacteristic
51#821C2211yescharacteristic, complemented
52(C2×D3)⋊C8C2211yescharacteristic
53#817C2211yescharacteristic, complemented
54Dic3⋊C8C2211yescharacteristic, complemented
55C8×Dic3C2211yescharacteristic, complemented
56#809C2211yescharacteristic
57#809C2211yescharacteristic
58#807C2211yescharacteristic, complemented
59#807C2211yescharacteristic, complemented
60#553—31no—
61#546—31no—
62#488—31no—
63#485—31no—
64#480—31no—
65#478—31no—
66#469—31no—
67#468—31no—
68#463—31no—
69#449—31no—
70#447—31no—
71C4×D8—31no—
72#436—31no—
73#418—31no—
74#417D311yescharacteristic, complemented
75C23×D3—21no—
76C3×#41—22no—
77C3×#41—41no—
78C2×C3×Q8C2311yescharacteristic
79C2×C3×D4—21no—
80C2×C3×D4C2311yescharacteristic
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4C2311yescharacteristic
84C2×#67—21no—
85C2×#67—21no—
86C2×#67—21no—
87C2×#67D411yescharacteristic
88C22×Dic3—21no—
89D3×D4—22no—
90D3×D4—41no—
91C2×D12—21no—
92C2×D12C2311yescharacteristic
93C2×D12C2311yescharacteristic
94C2×C4×D3—21no—
95C2×C4×D3C2311yescharacteristic
96C3×M16—41no—
97C2×C3×C8C2311yescharacteristic
98C2×C3×C8C2311yescharacteristic
99C3×C4⋊C4—21no—
100C3×C4⋊C4C2311yescharacteristic
101C3×C4⋊C4C2311yescharacteristic
102#217—21no—
103#217D411yescharacteristic
104#215—41no—
105#213—22no—
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—
113(C2×D3)⋊C4—21no—
114(C2×D3)⋊C4D411yescharacteristic
115C4⋊Dic3—21no—
116C4⋊Dic3C2311yescharacteristic
117C4⋊Dic3C2311yescharacteristic
118Dic3⋊C4—21no—
119Dic3⋊C4—21no—
120Dic3⋊C4D411yescharacteristic
121C4×Dic3—21no—
122C4×Dic3C2311yescharacteristic
123C2×C3⋊C8C2311yescharacteristic
124C2×C3⋊C8C2311yescharacteristic
125D24—41no—
126ΓL(1,𝔽25)—41no—
127C2×#41C2×D311yescharacteristic
128C22×D4—31no—
129C2×SD16—31no—
130C2×D8—31no—
131C2×D8—31no—
132C2×M16—31no—
133C2×M16C2×D311yescharacteristic, complemented
134#126—61no—
135#125—31no—
136#124—31no—
137#123—61no—
138#123—61no—
139C22⋊Q8—31no—
140#121—31no—
141#121—31no—
142#121—31no—
143#121—61no—
144#121—61no—
145#120—61no—
146C4×D4—31no—
147C4×D4—31no—
148C4×D4—31no—
149C4×D4—61no—
150#117—31no—
151C2×C4⋊C4C2×D311yescharacteristic, complemented
152C2×C22⋊C4—61no—
153#107—31no—
154#107—31no—
155#106—31no—
156C4⋊C8—31no—
157#103—31no—
158#103C2×D311yescharacteristic, complemented
159#103C2×D311yescharacteristic, complemented
160#102—31no—
161#102—31no—
162#102—31no—
163#102—31no—
164#102—31no—
165#102C2×D311yescharacteristic
166#102C2×D311yescharacteristic
167C22⋊C8—31no—
168C22⋊C8—31no—
169C4×C8—31no—
170C23×C3—21no—
171C23×C3C2×D411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—42no—
175C22×D3—42no—
176C22×D3—81no—
177C22×D3C2×D411yescharacteristic
178C3×Q8—21no—
179C3×Q8—21no—
180C3×D4—21no—
181C3×D4—42no—
182C3×D4#4112yes—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—42no—
189C2×C3×C4—81no—
190C2×C3×C4C2×D411yescharacteristic
191C2×C3×C4C2411yescharacteristic
192#67—41no—
193#67—42no—
194#67—42no—
195#67—81no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—81no—
199C2×Dic3C2×D411yescharacteristic
200C2×Dic3C2×D411yescharacteristic
201C2×Dic3C2×D411yescharacteristic
202D12—21no—
203D12—21no—
204D12—21no—
205D12—21no—
206D12—81no—
207C4×D3—21no—
208C4×D3—21no—
209C4×D3—81no—
210C3×C8—21no—
211C3×C8—21no—
212C3⋊C8—21no—
213C3⋊C8—21no—
214C24—61no—
215#41—22no—
216#41—41no—
217C2×Q8C22×D311yescharacteristic
218C2×D4—21no—
219C2×D4—31no—
220C2×D4—31no—
221C2×D4—31no—
222C2×D4—61no—
223C2×D4—61no—
224C2×D4—62no—
225C2×D4—61no—
226C2×D4—61no—
227C2×D4—121no—
228C2×D4C22×D311yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—31no—
232C22×C4—61no—
233C22×C4—61no—
234C22×C4C22×D311yescharacteristic
235SD16—121no—
236D8—62no—
237D8—121no—
238M16—41no—
239M16—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—31no—
254C22⋊C4—31no—
255C22⋊C4—61no—
256C22⋊C4—61no—
257C22⋊C4—61no—
258C22⋊C4—61no—
259C22⋊C4—61no—
260C22⋊C4—61no—
261C22⋊C4—61no—
262C22⋊C4—61no—
263C22⋊C4—61no—
264C42—31no—
265C42—61no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3C22×D411yescharacteristic
273C2×D3—21no—
274C2×D3—21no—
275C2×D3—21no—
276C2×D3—41no—
277C2×D3—41no—
278C2×D3—41no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—84no—
283C3×C4—21no—
284C3×C4—41no—
285C3×C4—41no—
286C3×C4—41no—
287C3×C4C2×#4111yescharacteristic
288C3×C4H(5,𝔽2)11yescharacteristic
289Dic3—21no—
290Dic3—21no—
291Dic3—21no—
292Dic3—41no—
293Dic3—41no—
294C23—21no—
295C23—31no—
296C23—61no—
297C23—61no—
298C23—122no—
299C23—122no—
300C23—241no—
301C23D3×D411yescharacteristic
302Q8—21no—
303Q8—21no—
304D4—21no—
305D4—42no—
306D4—61no—
307D4—61no—
308D4—61no—
309D4—61no—
310D4—121no—
311D4—122no—
312D4—122no—
313D4—241no—
314D4—241no—
315D4#23512yes—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—31no—
322C2×C4—31no—
323C2×C4—31no—
324C2×C4—42no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—81no—
330C2×C4—241no—
331C2×C4—241no—
332C2×C4D3×D411yescharacteristic
333C2×C4C23×D311yescharacteristic
334C8—21no—
335C8—21no—
336C8—61no—
337C8—61no—
338C2×C3—42no—
339C2×C3—41no—
340C2×C3#54611yescharacteristic
341C2×C3#57511yescharacteristic
342C2×C3#57611yescharacteristic
343D3—41no—
344D3—81no—
345D3—81no—
346C22—21no—
347C22—22no—
348C22—21no—
349C22—21no—
350C22—41no—
351C22—41no—
352C22—61no—
353C22—61no—
354C22—61no—
355C22—121no—
356C22—121no—
357C22—121no—
358C22—121no—
359C22—121no—
360C22—121no—
361C22—244no—
362C22C2×D3×D411yescharacteristic
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—61no—
368C4—61no—
369C4—61no—
370C4—121no—
371C4—121no—
372C4C2×#23511yescharacteristic
373C4#100411yescharacteristic
374C3#332211yescharacteristic, complemented, Sylow
375C2—41no—
376C2—42no—
377C2—121no—
378C2—241no—
379C2—241no—
380C2#549011yescharacteristic
381C2#577411yescharacteristic
382C2#577711yescharacteristic
subgroupG/H
centerC22C2×D3×D4
commutator, cosocleC2×C3×C4C24
FrattiniC2×C4C23×D3
FittingC3×#417C2
socleC22×C3C22×D4
hypercenter#417D3
hypocenterC3#3322
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. #5829C2semidirect
2. #5514C2semidirect
3. C3×#417C2semidirect
4. #5238C2semidirect
5. #5135C2semidirect
6. #5128C2semidirect
7. #4967C2semidirect
8. #4819C2semidirect
9. #4815C2semidirect
10. #4812C2semidirect
11. #4800C2semidirect
12. #4773C2semidirect
13. #4770C2semidirect
14. #4763C2semidirect
15. #4756C2semidirect
19. C2×C3×M16C22semidirect
20. C2×C3×C4⋊C4C22semidirect
23. C2×#215C22semidirect
30. #930C22semidirect
40. #893C22semidirect
42. #890C22semidirect
43. C3×#103C22semidirect
44. C3×#103C22semidirect
47. #835C22semidirect
49. #830C22semidirect
51. #821C22semidirect
53. #817C22semidirect
54. Dic3⋊C8C22semidirect
55. C8×Dic3C22semidirect
58. #807C22semidirect
59. #807C22semidirect
74. #417D3semidirect
133. C2×M16C2×D3semidirect
151. C2×C4⋊C4C2×D3semidirect
158. #103C2×D3semidirect
159. #103C2×D3semidirect
374. C3#3322semidirect

Maximal subgroups

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

subgroupindex
#58292
#55142
C3×#4172
#52382
#51352
#51282
#49672
#48192
#48152
#48122
#48002
#47732
#47702
#47632
#47562
#33223