Finite Group Databasebeta

← #393499

Subgroups — #393499

Order 512.

1704 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
1C2×#2499C211yesmaximal, characteristic
2#33996C211yesmaximal, characteristic
3#20911C212yesmaximal
4C2×#1822C211yesmaximal, characteristic
5#10957C212yesmaximal, complemented
6#10957C212yesmaximal, complemented
7#10892C212yesmaximal, complemented
8#10891C212yesmaximal, complemented
9#10733C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C22×#117C2212yes—
12C22×#117C2211yescharacteristic
13C2×#419C2214yes—
14C2×#407C2212yes—
15#2499—24no—
16#2499C418yescomplemented
17C2×#397—22no—
18C2×#391C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2211yescharacteristic
23#2274—24no—
24#2274C2212yes—
25#2274C2212yes—
26#2273—22no—
27C4×#117—22no—
28C4×#117—24no—
29C2×C4×C22⋊C4C2211yescharacteristic
30#1873—24no—
31#1872—24no—
32#1822—24no—
33C2×#343C2212yes—
34C2×#343C2212yes—
35#1736C2214yes—
36C2×#339C2214yes—
37C2×#339C2214yes—
38C23×Q8C2311yescharacteristic
39C24×C4C2311yescharacteristic
40C22×M16C2314yes—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8C2×C414yes—
46C2×#117—22no—
47C2×#117—22no—
48C2×#117—22no—
49C2×#117—22no—
50C2×#117—24no—
51C2×#117—28no—
52C2×#117—28no—
53C2×#117C2×C414yes—
54C2×#117D414yes—
55C2×#117D414yes—
56C2×#117C2314yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4D414yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4D412yes—
67C22×C42—22no—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71#419—28no—
72#419—28no—
73C2×#103—28no—
74#407—24no—
75#407—24no—
76C2×C22⋊C8—22no—
77C2×C22⋊C8—22no—
78#397—216no—
79#391—22no—
80#391—22no—
81#391C2×C414yescomplemented
82#391C2×C414yescomplemented
83#387—216no—
84#384—22no—
85#384C2×C414yescomplemented
86#382—22no—
87#382C2×C414yescomplemented
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—28no—
91C4×C4⋊C4—28no—
92C4×C22⋊C4—216no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—24no—
95C4×C22⋊C4—24no—
96C4×C22⋊C4C2×C414yes—
97#376—24no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C43—24no—
103#343—24no—
104#343—24no—
105#340—28no—
106#339—28no—
107#339—28no—
108C25C2×D411yescharacteristic
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8C22×C412yes—
115C23×C4—24no—
116C23×C4—24no—
117C23×C4—21no—
118C23×C4—24no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4C22×C412yes—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D412yes—
127C23×C4C2×D412yes—
128C23×C4C2×D412yes—
129C23×C4#4114yes—
130C23×C4C2411yescharacteristic
131C2×M16—24no—
132C2×M16—24no—
133C2×M16—24no—
134C2×M16—24no—
135C2×M16—24no—
136C2×M16—24no—
137C22×C8—24no—
138C22⋊Q8—28no—
139C22⋊Q8—28no—
140C22⋊Q8—44no—
141C22⋊Q8—44no—
142#117—28no—
143#117—28no—
144#117—216no—
145#117—28no—
146#117—28no—
147#117—28no—
148#117—28no—
149#117—24no—
150#117—24no—
151#117—216no—
152#117#4118yes—
153#117#4118yes—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—216no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—48no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4C22×C412yes—
170C2×C4⋊C4C22×C412yes—
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C2×D412yes—
173C2×C4⋊C4C2×D412yes—
174C2×C4⋊C4C2×D418yes—
175C2×C22⋊C4—28no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—44no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4C22⋊C418yes—
187C2×C42—216no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—24no—
191C2×C42—24no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42C22⋊C418yes—
200C2×C42C22×C412yes—
201C2×C42C22×C412yes—
202#103—416no—
203C22⋊C8—44no—
204C22⋊C8—44no—
205#95—28no—
206#95—28no—
207#95—24no—
208#95—28no—
209#95—28no—
210#95—28no—
211C24—22no—
212C24—22no—
213C24—42no—
214C24—42no—
215C24C2×C22⋊C412yes—
216C24#12012yes—
217C24#12312yes—
218C24C22×D411yescharacteristic
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—44no—
228C2×Q8—44no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—28no—
238C22×C4—28no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—28no—
246C22×C4—216no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—41no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4C2×C22⋊C412yes—
269C22×C4C2×C22⋊C414yes—
270C22×C4C2×C22⋊C414yes—
271C22×C4C4×D414yes—
272C22×C4C4×D414yes—
273C22×C4#12011yescharacteristic
274C22×C4#12011yescharacteristic
275C22×C4#12114yes—
276C22×C4#12114yes—
277C22×C4#12312yes—
278C22×C4C23×C411yescharacteristic
279C22×C4C22×D412yes—
280C22×C4C22×D411yescharacteristic
281C22×C4C2×#4112yes—
282C22×C4C2×#4114yes—
283C22×C4C2×#4114yes—
284M16—44no—
285M16—44no—
286M16—44no—
287M16—44no—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C4⋊C4—216no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—416no—
299C4⋊C4—42no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4#123116yes—
303C22⋊C4—216no—
304C22⋊C4—28no—
305C22⋊C4—28no—
306C22⋊C4—216no—
307C22⋊C4—48no—
308C22⋊C4—48no—
309C42—216no—
310C42—28no—
311C42—28no—
312C42—28no—
313C42—28no—
314C42—24no—
315C42—24no—
316C42—42no—
317C42—42no—
318C42—88no—
319C42—88no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—24no—
325C23—24no—
326C23—24no—
327C23—22no—
328C23—22no—
329C23—42no—
330C23—42no—
331C23—42no—
332C23—42no—
333C23—84no—
334C23—84no—
335C23#38614yes—
336C23C22×C22⋊C411yescharacteristic
337C23C2×#12011yescharacteristic
338C23C2×#12311yescharacteristic
339C23#52512yes—
340C23#52512yes—
341Q8—48no—
342Q8—84no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—24no—
353C2×C4—216no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—42no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—48no—
360C2×C4—44no—
361C2×C4—48no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—816no—
376C2×C4—84no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#38612yes—
382C2×C4#38612yes—
383C2×C4C2×C4×D412yes—
384C2×C4C4×#4112yes—
385C2×C4C4×#4112yes—
386C2×C4C2×#12112yes—
387C2×C4C2×#12314yes—
388C2×C4#52514yes—
389C2×C4#52914yes—
390C2×C4#52914yes—
391C2×C4#53212yes—
392C2×C4#53212yes—
393C8—84no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—24no—
399C22—24no—
400C22—24no—
401C22—24no—
402C22—24no—
403C22—44no—
404C22—44no—
405C22—82no—
406C22—82no—
407C22—84no—
408C22C2×#38611yescharacteristic
409C22#229711yescharacteristic
410C22#229711yescharacteristic
411C22#284912yes—
412C22#285312yes—
413C22#301812yes—
414C22#302212yes—
415C4—22no—
416C4—22no—
417C4—44no—
418C4—44no—
419C4—44no—
420C4—416no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4C4×#12314yes—
425C4#285414yes—
426C2—21no—
427C2—21no—
428C2—22no—
429C2—44no—
430C2#1470611yescharacteristic
431C2#2095211yescharacteristic
432C2#2095511yescharacteristic
433C2#3300114yes—
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. #10957C2semidirect
6. #10957C2semidirect
7. #10892C2semidirect
8. #10891C2semidirect
16. #2499C4semidirect
81. #391C2×C4semidirect
82. #391C2×C4semidirect
85. #384C2×C4semidirect
87. #382C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#24992
#339962
#209112
C2×#18222
#109572
#108922
#108912
#107332