Finite Group Databasebeta

← #359937

Subgroups — #359937

Order 512.

1630 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34179C211yesmaximal, characteristic
2C2×#2285C211yesmaximal, characteristic
3#20701C212yesmaximal
4C2×#1822C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11210C212yesmaximal, complemented
7#10892C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10719C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C42C2211yescharacteristic
13C2×#407C2212yes—
14#2347—24no—
15#2346—24no—
16#2316C2212yes—
17#2312—22no—
18#2285—24no—
19#2285C418yescomplemented
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22#2283—22no—
23C2×#382C2211yescharacteristic
24C2×#382C2211yescharacteristic
25#2280C2212yes—
26C2×#380—22no—
27#2274C2212yes—
28C2×C4×C4⋊C4—22no—
29#1825—24no—
30#1825—24no—
31#1822—24no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C22×M16C2314yes—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117—22no—
45C2×#117—22no—
46C2×#117—22no—
47C2×#117—24no—
48C2×#117C2×C414yes—
49C2×#117C2×C414yes—
50C2×#117D414yes—
51C2×#117C2314yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4D412yes—
60C22×C42—21no—
61C22×C42—22no—
62C22×C42—24no—
63C22×C42—24no—
64C22×C42—21no—
65C22×C42C2×C414yes—
66C22×C42D412yes—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69#407—24no—
70#407—24no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73#389—28no—
74#388—216no—
75#384—22no—
76#384—22no—
77#384C2×C414yes—
78#384C2×C414yes—
79#383—24no—
80#382—24no—
81#382—22no—
82#382—22no—
83#382C2×C414yes—
84#382C2×C414yes—
85#381—22no—
86#380—24no—
87#380—216no—
88#380D414yes—
89#380D414yes—
90#379—22no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—28no—
93C4×C4⋊C4—216no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#376—24no—
98C2×#95—22no—
99C2×#95—22no—
100#343—24no—
101#343—24no—
102#340—28no—
103#340—28no—
104#339—28no—
105#339—28no—
106C25C2×D411yescharacteristic
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—24no—
111C23×C4—28no—
112C23×C4—24no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—44no—
117C23×C4—44no—
118C23×C4C22×C412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D412yes—
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4#4114yes—
126C23×C4C2411yescharacteristic
127C2×M16—24no—
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C22×C8—24no—
134#117—28no—
135#117—28no—
136#117—28no—
137#117—28no—
138#117—28no—
139#117—28no—
140#117—24no—
141#117—24no—
142#117—24no—
143#117—24no—
144#117—44no—
145#117—44no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—28no—
155C2×C4⋊C4—28no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—48no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C412yes—
164C2×C4⋊C4C22×C412yes—
165C2×C4⋊C4C2×D418yes—
166C2×C4⋊C4C2×D412yes—
167C2×C4⋊C4C2×D412yes—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4C22⋊C418yes—
177C2×C42—28no—
178C2×C42—216no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—28no—
188C2×C42—28no—
189C2×C42—24no—
190C2×C42—24no—
191C2×C42—28no—
192C2×C42—41no—
193C2×C42—41no—
194C2×C42—42no—
195C2×C42C22⋊C418yes—
196C2×C42C22×C412yes—
197C2×C42C22×C412yes—
198C2×C42#4114yes—
199C2×C42#4114yes—
200C22⋊C8—44no—
201C22⋊C8—44no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—24no—
207#95—24no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24C2×C22⋊C412yes—
213C24#12012yes—
214C24#12312yes—
215C24C22×D411yescharacteristic
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—24no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—216no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—28no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—432no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—44no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—41no—
256C22×C4C2×C22⋊C412yes—
257C22×C4C2×C22⋊C414yes—
258C22×C4C2×C22⋊C414yes—
259C22×C4C4×D414yes—
260C22×C4C4×D414yes—
261C22×C4#12011yescharacteristic
262C22×C4#12011yescharacteristic
263C22×C4#12114yes—
264C22×C4#12114yes—
265C22×C4#12312yes—
266C22×C4C23×C411yescharacteristic
267C22×C4C22×D412yes—
268C22×C4C22×D411yescharacteristic
269C22×C4C2×#4112yes—
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C42−1+414yes—
273M16—44no—
274M16—44no—
275M16—44no—
276M16—44no—
277C2×C8—44no—
278C2×C8—44no—
279C2×C8—44no—
280C4⋊C4—24no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—416no—
289C4⋊C4—416no—
290C4⋊C4—42no—
291C4⋊C4—42no—
292C4⋊C4—88no—
293C22⋊C4—216no—
294C22⋊C4—44no—
295C22⋊C4—44no—
296C22⋊C4—48no—
297C22⋊C4—84no—
298C22⋊C4—84no—
299C42—216no—
300C42—22no—
301C42—22no—
302C42—24no—
303C42—24no—
304C42—28no—
305C42—28no—
306C42—44no—
307C42—44no—
308C42—44no—
309C42—44no—
310C42—88no—
311C42#11714yes—
312C42#11714yes—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—24no—
317C23—24no—
318C23—24no—
319C23—24no—
320C23—22no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—48no—
326C23—48no—
327C23#38614yes—
328C23C22×C22⋊C411yescharacteristic
329C23C2×#12011yescharacteristic
330C23C2×#12311yescharacteristic
331C23#52512yes—
332C23#53612yes—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—216no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—24no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—432no—
352C2×C4—48no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4#38612yes—
373C2×C4#38612yes—
374C2×C4C2×#11711yescharacteristic
375C2×C4C2×#11711yescharacteristic
376C2×C4C2×C4×D412yes—
377C2×C4#51912yes—
378C2×C4C2×#12112yes—
379C2×C4#52512yes—
380C2×C4#53612yes—
381C2×C4#54714yes—
382C2×C4#54714yes—
383C2×C4#55214yes—
384C2×C4#55214yes—
385C8—84no—
386C22—24no—
387C22—24no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—24no—
393C22—24no—
394C22—24no—
395C22—44no—
396C22—44no—
397C22—48no—
398C22—48no—
399C22#228011yescharacteristic
400C22C2×#38611yescharacteristic
401C22#231011yescharacteristic
402C22#240612yes—
403C22#248412yes—
404C22#301812yes—
405C22#301812yes—
406C4—28no—
407C4—22no—
408C4—22no—
409C4—44no—
410C4—44no—
411C4—44no—
412C4—88no—
413C4—82no—
414C4—82no—
415C4#234412yes—
416C4#234412yes—
417C4#241314yes—
418C2—21no—
419C2—21no—
420C2—22no—
421C2—44no—
422C2#1387011yescharacteristic
423C2#2095211yescharacteristic
424C2#2095211yescharacteristic
425C2#2407814yes—
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #11211C2semidirect
6. #11210C2semidirect
7. #10892C2semidirect
8. #10891C2semidirect
19. #2285C4semidirect

Maximal subgroups

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

subgroupindex
#341792
C2×#22852
#207012
C2×#18222
#112112
#112102
#108922
#108912
#107192