Finite Group Databasebeta

← #359938

Subgroups — #359938

Order 512.

1570 conjugacy classes of subgroups.

The subgroup lattice has 435 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34178C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3#20701C212yesmaximal
4C2×#1821C211yesmaximal, characteristic
5#11211C212yesmaximal, complemented
6#11210C212yesmaximal, complemented
7#10892C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10720C212yesmaximal
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C2×#407C2212yes—
14#2345—24no—
15#2344—24no—
16#2317—24no—
17#2317C418yescomplemented
18#2316C2212yes—
19#2313—22no—
20C2×#389—22no—
21C2×#385—22no—
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26#2280—22no—
27#2280C2212yes—
28#2274C2212yes—
29#1825—24no—
30#1825—24no—
31#1821—24no—
32C2×#343C2212yes—
33C2×#343C2212yes—
34#1736C2214yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C24×C4D412yes—
39C24×C4C2311yescharacteristic
40C22×M16C2314yes—
41C2×#117—22no—
42C2×#117—22no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—24no—
49C2×#117C2×C414yes—
50C2×#117C2×C414yes—
51C2×#117C2314yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—24no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4C2×C414yes—
59C22×C4⋊C4D412yes—
60C22×C4⋊C4D414yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2311yescharacteristic
65C22×C22⋊C4—21no—
66C22×C22⋊C4—21no—
67C22×C42—21no—
68C22×C42C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#407—24no—
71#407—24no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74#389—216no—
75#389—28no—
76#387—216no—
77#385—216no—
78#384—22no—
79#384—22no—
80#384C2×C414yes—
81#384C2×C414yes—
82#382—22no—
83#382—22no—
84#382C2×C414yes—
85#382C2×C414yes—
86#381—24no—
87#381—22no—
88#380—24no—
89#380—24no—
90#380D414yes—
91#380D414yes—
92#379—22no—
93C4×C4⋊C4—28no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4—22no—
97C4×C22⋊C4—22no—
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—22no—
108C23×C4—22no—
109C23×C4—24no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—84no—
118C23×C4C22⋊C418yes—
119C23×C4C22×C412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2×D412yes—
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4#4114yes—
127C23×C4C2411yescharacteristic
128C2×M16—24no—
129C2×M16—24no—
130C2×M16—24no—
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C22×C8—24no—
135#117—216no—
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—
146#117—88no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—28no—
153C2×C4⋊C4—28no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—84no—
169C2×C4⋊C4—84no—
170C2×C4⋊C4C22⋊C418yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C412yes—
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C22×C412yes—
175C2×C4⋊C4C2×D418yes—
176C2×C4⋊C4C2×D412yes—
177C2×C4⋊C4C2×D412yes—
178C2×C22⋊C4—28no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—42no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—28no—
194C2×C42—42no—
195C2×C42—42no—
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#12011yescharacteristic
214C24#12011yescharacteristic
215C24#12312yes—
216C24C22×D411yescharacteristic
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—24no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—28no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—28no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—28no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—48no—
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×C4—88no—
257C22×C4—84no—
258C22×C4—88no—
259C22×C4C2×C22⋊C412yes—
260C22×C4C2×C22⋊C414yes—
261C22×C4C2×C22⋊C414yes—
262C22×C4C4×D414yes—
263C22×C4C4×D414yes—
264C22×C4#12012yes—
265C22×C4#12114yes—
266C22×C4#12114yes—
267C22×C4#12312yes—
268C22×C4C23×C411yescharacteristic
269C22×C4C22×D412yes—
270C22×C4C22×D411yescharacteristic
271C22×C4C2×#4112yes—
272C22×C4C2×#4112yes—
273C22×C4C2×#4112yes—
274C22×C42−1+414yes—
275M16—44no—
276M16—44no—
277M16—44no—
278M16—44no—
279C2×C8—44no—
280C2×C8—44no—
281C2×C8—44no—
282C4⋊C4—216no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—24no—
287C4⋊C4—216no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—28no—
291C4⋊C4—28no—
292C4⋊C4—42no—
293C4⋊C4—42no—
294C4⋊C4—88no—
295C4⋊C4—88no—
296C4⋊C4—88no—
297C22⋊C4—216no—
298C22⋊C4—48no—
299C22⋊C4—48no—
300C22⋊C4—44no—
301C22⋊C4—44no—
302C22⋊C4—44no—
303C22⋊C4—44no—
304C42—22no—
305C42—22no—
306C42—24no—
307C42—24no—
308C42—44no—
309C42—44no—
310C42—48no—
311C42—88no—
312C42#11714yescomplemented
313C42#11714yescomplemented
314C23—22no—
315C23—22no—
316C23—22no—
317C23—22no—
318C23—22no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—22no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—48no—
330C23—48no—
331C23#38612yes—
332C23#38612yes—
333C23C22×C22⋊C411yescharacteristic
334C23C2×#12011yescharacteristic
335C23C2×#12311yescharacteristic
336C23#52512yes—
337C23#53612yes—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—22no—
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—24no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—416no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4—48no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—82no—
371C2×C4—82no—
372C2×C4—88no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4#38614yes—
378C2×C4C2×#11711yescharacteristic
379C2×C4C2×#11711yescharacteristic
380C2×C4C2×C4×D412yes—
381C2×C4#51912yes—
382C2×C4C2×#12112yes—
383C2×C4#52512yes—
384C2×C4#53612yes—
385C2×C4#54714yes—
386C2×C4#54714yes—
387C2×C4#55214yes—
388C2×C4#55214yes—
389C8—84no—
390C22—24no—
391C22—24no—
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—44no—
403C22—44no—
404C22—48no—
405C22—44no—
406C22—44no—
407C22#228011yescharacteristic
408C22C2×#38611yescharacteristic
409C22#231011yescharacteristic
410C22#240612yes—
411C22#248412yes—
412C22#301912yes—
413C22#302012yes—
414C4—24no—
415C4—44no—
416C4—44no—
417C4—44no—
418C4—44no—
419C4—88no—
420C4—82no—
421C4—82no—
422C4#234412yes—
423C4#234412yes—
424C4#241314yes—
425C2—21no—
426C2—21no—
427C2—22no—
428C2—42no—
429C2—42no—
430C2#1387011yescharacteristic
431C2#2095311yescharacteristic
432C2#2095411yescharacteristic
433C2#2407914yes—
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
17. #2317C4semidirect
312. C42#117semidirect
313. C42#117semidirect

Maximal subgroups

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

subgroupindex
#341782
C2×#23172
#207012
C2×#18212
#112112
#112102
#108922
#108912
#107202