Finite Group Databasebeta

← #72225890

Subgroups — #72225890

Order 1536.

1612 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10803808C212yesmaximal, complemented
2#10782888C211yesmaximal, characteristic
3#10765754C214yesmaximal, complemented
4#10695159C212yesmaximal, complemented
5#10695092C214yesmaximal, complemented
6#10688988C211yesmaximal, characteristic, complemented
7C3×#7846C211yesmaximal, characteristic, complemented
8#213274—31nomaximal, Sylow
9C2×#4717—24no—
10#78470C2214yes—
11#78467C2212yes—
12#78458C2214yes—
13C3×C4×#97C2212yes—
14C3×C4×#97C2211yescharacteristic
15C2×C3×#322C2211yescharacteristic
16C2×C3×#322C2212yes—
17C2×C3×C82C2211yescharacteristic
18#69845—24no—
19#69792C2212yes—
20#69735C2218yes—
21#69730C2214yes—
22#69395C2214yes—
23#68083—28no—
24#68077—28no—
25#68065C411yescharacteristic, complemented
26#68062C411yescharacteristic, complemented
27#68060C412yescomplemented
28#68058C414yescomplemented
29#21706—32no—
30#19603—31no—
31#17962—34no—
32#9862—32no—
33#9849—34no—
34#8822—31no—
35#7846D311yescharacteristic, complemented
36C2×#904—24no—
37C2×#901—24no—
38#5477C2314yes—
39C2×C3×#97—24no—
40C2×C3×#97D414yes—
41C2×C3×C4×C8C2312yes—
42C2×C3×C4×C8C2312yes—
43C2×C3×C4×C8C2311yescharacteristic
44C2×C3×C4×C8C2311yescharacteristic
45C3×C43C2311yescharacteristic
46#5115—28no—
47#5110—24no—
48C2×#819—24no—
49C2×#805—28no—
50#4924C2314yes—
51#4717—416no—
52#4699C2×C412yes—
53#4696—24no—
54#4695C2×C412yes—
55#4695C2×C414yes—
56#4694C2×C418yes—
57#4693C2×C414yes—
58C4×D24—28no—
59#4688—28no—
60C8×D12—28no—
61#4682C2×C414yes—
62C3×#322—22no—
63C3×#322D418yes—
64C3×C82C2×C412yes—
65C3×C82C2×C411yescharacteristic
66C3×C82C2×C411yescharacteristic
67#4459—216no—
68#4457—28no—
69#3169—34no—
70#3147—32no—
71C2×#444—64no—
72#2931—34no—
73#1975—64no—
74#1962—34no—
75#1848—32no—
76C4×#97C2×D312yes—
77C4×#97C2×D311yescharacteristic
78#1713—31no—
79#1712—31no—
80#1709—32no—
81#1706—34no—
82#1594—68no—
83#1580—68no—
84#1566—34no—
85#1492—38no—
86C2×#322C2×D311yescharacteristic
87C2×#322C2×D312yes—
88C2×C82C2×D311yescharacteristic
89C2×#235—24no—
90C22×C3×C8—22no—
91C22×C3×C8C2×D412yes—
92C22×C3×C8C2×D412yes—
93C22×C3×C8#4114yes—
94C2×C3×C42—22no—
95C2×C3×C42C2×D412yes—
96C2×C3×C42C2411yescharacteristic
97#930—28no—
98C2×#208—24no—
99C2×Dic12—24no—
100#904—48no—
101#904—48no—
102C2×D24—28no—
103C2×D24—24no—
104C2×#204—28no—
105#901—416no—
106C2×ΓL(1,𝔽25)—44no—
107C2×D3×C8—44no—
108#875—24no—
109#874C22×C414yes—
110C4×D12—28no—
111#869C22×C414yes—
112C3×#97—28no—
113C3×#97—48no—
114C3×C4×C8—22no—
115C3×C4×C8C22×C412yes—
116C3×C4×C8C22×C411yescharacteristic
117C3×C4×C8C22×C412yes—
118C3×C4×C8C22×C411yescharacteristic
119C3×C4×C8C2×D414yes—
120C3×C4×C8C2×D414yes—
121C3×C4×C8C2×D414yes—
122C3×C4×C8#4112yes—
123#821—216no—
124(C2×D3)⋊C8—48no—
125#819—48no—
126#818—28no—
127#805—416no—
128C4⋊(C3⋊C8)—28no—
129C2×#135—64no—
130C2×#131—64no—
131#532—34no—
132#500—34no—
133#495—64no—
134#494—32no—
135#493—32no—
136#493—34no—
137#492—34no—
138#486—38no—
139#466—68no—
140#444—1216no—
141C4×D8—68no—
142#436—68no—
143C8×D4—68no—
144C2×#108—64no—
145#423—34no—
146C2×#104—68no—
147#408—64no—
148C2×#97—24no—
149C2×#97D1214yes—
150C2×C4×C8C22×D312yes—
151C2×C4×C8C22×D312yes—
152C2×C4×C8C22×D311yescharacteristic
153C2×C4×C8C22×D311yescharacteristic
154C43C22×D311yescharacteristic
155#334—68no—
156D4⋊C8—616no—
157#322—22no—
158#322D1218yes—
159C82C4×D312yes—
160C82C4×D311yescharacteristic
161C82C4×D311yescharacteristic
162C22×C3×C4—22no—
163C22×C3×C4C22×D411yescharacteristic
164C22×C3×C4C2×#4112yes—
165C2×#67—44no—
166#235—48no—
167#235—48no—
168C2×D12—28no—
169C2×D12—24no—
170C2×C4×D3—44no—
171C2×Dic6—24no—
172C2×C3×C8—24no—
173C2×C3×C8—28no—
174C2×C3×C8—24no—
175C2×C3×C8—44no—
176C2×C3×C8C4×D414yes—
177C2×C3×C8C4×D412yes—
178C2×C3×C8C4×D412yes—
179C2×C3×C8C4×D414yes—
180C2×C3×C8#12714yes—
181C2×C3×C8AGL(1,ℤ/8)18yes—
182C3×C42—24no—
183C3×C42—44no—
184C3×C42C23×C411yescharacteristic
185C3×C42C22×D412yes—
186C3×C42C2×#4111yescharacteristic
187(C2×D3)⋊C4—48no—
188C4⋊Dic3—28no—
189#208—48no—
190C2×C3⋊C8—44no—
191Dic12—48no—
192D24—48no—
193D24—416no—
194#204—416no—
195ΓL(1,𝔽25)—88no—
196D3×C8—88no—
197C2×#41—64no—
198#135—128no—
199#135—128no—
200C2×Q16—64no—
201C2×SD16—68no—
202C2×D8—64no—
203C2×D8—68no—
204#131—1216no—
205C2×M16—64no—
206C2×M16—124no—
207C22×C8—22no—
208C22×C8—124no—
209C22×C8C2×D1212yes—
210C22×C8C2×D1212yes—
211C22×C8#23514yes—
212C4⋊Q8—34no—
213#127—34no—
214#124—64no—
215#121—68no—
216C4×D4—68no—
217C2×C42—22no—
218C2×C42C2×D1212yes—
219C2×C42C23×D311yescharacteristic
220#108—128no—
221#107—68no—
222C4⋊C8—68no—
223#104—1216no—
224#102—616no—
225C22⋊C8—128no—
226#97—28no—
227#97—48no—
228C4×C8—22no—
229C4×C8C2×C4×D312yes—
230C4×C8C2×C4×D311yescharacteristic
231C4×C8C2×C4×D311yescharacteristic
232C4×C8C2×C4×D312yes—
233C4×C8C2×D1214yes—
234C4×C8C2×D1214yes—
235C4×C8C2×D1214yes—
236C4×C8#23512yes—
237C23×C3#53211yescharacteristic
238C22×D3—44no—
239C2×C3×C4—24no—
240C2×C3×C4—44no—
241C2×C3×C4—42no—
242C2×C3×C4C2×C4×D411yescharacteristic
243C2×C3×C4C2×C4×D412yes—
244C2×C3×C4C2×#12711yescharacteristic
245C2×C3×C4#53212yes—
246C2×C3×C4C2×AGL(1,ℤ/8)14yes—
247#67—88no—
248C2×Dic3—44no—
249D12—48no—
250D12—416no—
251C4×D3—84no—
252C4×D3—84no—
253Dic6—48no—
254C3×C8—24no—
255C3×C8—24no—
256C3×C8—44no—
257C3×C8#44214yes—
258C3×C8#44214yes—
259C3×C8#44418yescomplemented
260C3⋊C8—84no—
261#41—128no—
262#41—128no—
263C2×Q8—64no—
264C2×D4—68no—
265C2×D4—64no—
266C2×D4—124no—
267C22×C4—22no—
268C22×C4—124no—
269C22×C4C22×D1211yescharacteristic
270C22×C4C2×#23512yes—
271Q16—128no—
272SD16—1216no—
273D8—128no—
274D8—1216no—
275M16—128no—
276M16—248no—
277C2×C8—24no—
278C2×C8—28no—
279C2×C8—24no—
280C2×C8—44no—
281C2×C8—124no—
282C2×C8—248no—
283C2×C8C4×D1214yes—
284C2×C8C4×D1212yes—
285C2×C8C4×D1212yes—
286C2×C8C4×D1214yes—
287C2×C8#87414yes—
288C2×C8#90818yes—
289C4⋊C4—68no—
290C22⋊C4—128no—
291C42—24no—
292C42—44no—
293C42C22×C4×D311yescharacteristic
294C42C22×D1212yes—
295C42C2×#23511yescharacteristic
296C22×C3—22no—
297C22×C3C4×#12711yescharacteristic
298C22×C3#318412yes—
299C2×D3—84no—
300C2×D3—88no—
301C3×C4—22no—
302C3×C4—82no—
303C3×C4C2×#44212yes—
304C3×C4C2×#44414yes—
305C3×C4#315112yes—
306Dic3—84no—
307C23—124no—
308C23#547711yescharacteristic
309Q8—128no—
310D4—128no—
311D4—1216no—
312D4—248no—
313C2×C4—24no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—124no—
317C2×C4—244no—
318C2×C4—244no—
319C2×C4C2×C4×D1211yescharacteristic
320C2×C4C2×C4×D1212yes—
321C2×C4C2×#87411yescharacteristic
322C2×C4#547712yes—
323C2×C4C2×#90814yes—
324C8—24no—
325C8—24no—
326C8—44no—
327C8—244no—
328C8#471114yes—
329C8#471114yes—
330C8#471718yescomplemented
331C2×C3—22no—
332C2×C3#2169011yescharacteristic
333C2×C3#2170612yes—
334D3—164no—
335C22—22no—
336C22—248no—
337C22—244no—
338C22C4×#87411yescharacteristic
339C22#7853512yes—
340C4—22no—
341C4—82no—
342C4—244no—
343C4C2×#471112yes—
344C4C2×#471714yes—
345C4#7853012yes—
346C3#21327411yescharacteristic, complemented, Sylow
347C2—22no—
348C2—484no—
349C2#1080364211yescharacteristic
350C2#1080380812yes—
subgroupG/H
centerC2×C4C2×#874
commutatorC3×C42C23×C4
FrattiniC2×C42C23×D3
FittingC3×#7846C2
socleC22×C3C4×#127
prime radical#10782888C2
cosocleC2×C3×C42C24
hypercenter#7846D3
hypocenterC3#213274
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #7846

Complements

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

NH ≅ G/Ntype
1. #10803808C2semidirect
3. #10765754C2semidirect
4. #10695159C2semidirect
5. #10695092C2semidirect
6. #10688988C2semidirect
7. C3×#7846C2semidirect
25. #68065C4semidirect
26. #68062C4semidirect
27. #68060C4semidirect
28. #68058C4semidirect
35. #7846D3semidirect
259. C3×C8#444semidirect
330. C8#4717semidirect
346. C3#213274semidirect

Maximal subgroups

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

subgroupindex
#108038082
#107828882
#107657542
#106951592
#106950922
#106889882
C3×#78462
#2132743