Finite Group Databasebeta

← #410000

Subgroups — #410000

Order 512.

1896 conjugacy classes of subgroups.

The subgroup lattice has 459 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#484C211yesmaximal, characteristic
2C23×#107C211yesmaximal, characteristic
3C2×#2502C211yesmaximal, characteristic
4C2×#1892C211yesmaximal, characteristic
5C2×#1824C211yesmaximal, characteristic
6#13924C211yesmaximal, characteristic
7#11324C214yesmaximal, complemented
8#11124C214yesmaximal, complemented
9#10714C211yesmaximal, characteristic
10C24×C8C2211yescharacteristic
11C22×C22⋊Q8C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C2×#484—24no—
15C2×#484—24no—
16C2×#484—24no—
17C22×#107—24no—
18C22×#107—24no—
19C22×#107—24no—
20C22×#107C2212yes—
21C22×#107C2214yes—
22C22×#103C2212yes—
23C22×#103C2212yes—
24C22×C22⋊C8C2211yescharacteristic
25#2502—24no—
26#2502—24no—
27C2×#398—22no—
28C2×#391C2212yes—
29C2×#385—22no—
30C2×#384C2212yes—
31C2×#380C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33#1892—24no—
34#1824—24no—
35C2×#340C2212yes—
36C2×#340C2214yes—
37C2×#336C2214yes—
38#1445—22no—
39C4⋊#95C2214yes—
40C22⋊#95—22no—
41C23×Q8C2311yescharacteristic
42C24×C4—21no—
43C24×C4—21no—
44C24×C4C2311yescharacteristic
45C23×C8—22no—
46C23×C8—22no—
47C23×C8—28no—
48C23×C8D414yes—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×C22⋊Q8—22no—
52C2×C22⋊Q8—22no—
53C2×C22⋊Q8—22no—
54C2×C22⋊Q8—22no—
55C2×C22⋊Q8—22no—
56C2×C22⋊Q8D414yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—22no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—24no—
71C22×C4⋊C4D414yes—
72C22×C4⋊C4D414yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C22⋊C4—21no—
78C22×C22⋊C4—21no—
79C22×C22⋊C4D412yes—
80C22×C42C2312yes—
81#484—232no—
82C2×#107—232no—
83C2×#107—24no—
84C2×#107—28no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—28no—
88C2×#107—28no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—28no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8D414yes—
97#398—216no—
98#391—24no—
99#391—24no—
100#385—216no—
101#384—24no—
102#384—24no—
103#380—22no—
104#380—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—24no—
108C2×#95—21no—
109C2×#95—28no—
110C2×#95D412yes—
111#340—28no—
112#340—24no—
113#336—28no—
114C25C2×D411yescharacteristic
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—24no—
121C22×Q8—24no—
122C22×Q8C2×D412yes—
123C23×C4—24no—
124C23×C4—24no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—42no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—84no—
137C23×C4—84no—
138C23×C4C2×D414yes—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4#4112yes—
146C23×C4C2411yescharacteristic
147C22×C8—216no—
148C22×C8—216no—
149C22×C8—28no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—24no—
153C22×C8—216no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—24no—
157C22×C8—24no—
158C22×C8C2×D414yes—
159C22×C8C2×D414yes—
160C22×C8#4118yes—
161C22⋊Q8—28no—
162C22⋊Q8—28no—
163C22⋊Q8—44no—
164C22⋊Q8—44no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—28no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—216no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4—44no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—88no—
203C2×C4⋊C4C2×D414yes—
204C2×C4⋊C4C2×D414yes—
205C2×C4⋊C4C2×D412yes—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—24no—
208C2×C22⋊C4—24no—
209C2×C22⋊C4—42no—
210C2×C22⋊C4—42no—
211C2×C22⋊C4—42no—
212C2×C22⋊C4—44no—
213C2×C22⋊C4Q1618yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—24no—
217C2×C42—24no—
218#107—232no—
219#107—216no—
220#103—216no—
221#103—48no—
222C22⋊C8—28no—
223#95—216no—
224#95—24no—
225#95—42no—
226#95—88no—
227C24—24no—
228C24—24no—
229C24#12012yes—
230C24#12112yes—
231C24#12712yes—
232C24C2×Q1614yes—
233C24#13714yes—
234C24C22×D411yescharacteristic
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—28no—
238C2×Q8—41no—
239C2×Q8—41no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—44no—
247C2×Q8—44no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—28no—
256C22×C4—28no—
257C22×C4—28no—
258C22×C4—24no—
259C22×C4—28no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—22no—
263C22×C4—24no—
264C22×C4—22no—
265C22×C4—24no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—44no—
275C22×C4—48no—
276C22×C4—44no—
277C22×C4—48no—
278C22×C4—48no—
279C22×C4—48no—
280C22×C4—42no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—88no—
285C22×C4—88no—
286C22×C4—88no—
287C22×C4—88no—
288C22×C4—88no—
289C22×C4—88no—
290C22×C4#12014yes—
291C22×C4#12014yes—
292C22×C4#12012yes—
293C22×C4#12114yes—
294C22×C4#12114yes—
295C22×C4#12112yes—
296C22×C4#12712yes—
297C22×C4C2×Q1618yes—
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4114yes—
305C22×C42−1+414yes—
306C2×C8—232no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—28no—
310C2×C8—28no—
311C2×C8—24no—
312C2×C8—28no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—44no—
316C2×C8#12318yes—
317C4⋊C4—24no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—22no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—42no—
324C4⋊C4—41no—
325C4⋊C4—42no—
326C4⋊C4—42no—
327C4⋊C4—42no—
328C4⋊C4—41no—
329C4⋊C4—48no—
330C4⋊C4—48no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—44no—
335C4⋊C4—816no—
336C22⋊C4—216no—
337C22⋊C4—48no—
338C22⋊C4—84no—
339C42—42no—
340C42—42no—
341C42—44no—
342C23—216no—
343C23—216no—
344C23—24no—
345C23—24no—
346C23—24no—
347C23—24no—
348C23—24no—
349C23—24no—
350C23#39214yes—
351C23#45118yes—
352C23#46218yes—
353C23#49414yes—
354C23#49714yes—
355C23C2×#12011yescharacteristic
356C23C2×#12111yescharacteristic
357C23C2×#12711yescharacteristic
358C23#54714yes—
359C23C22×Q1612yes—
360C23C2×#13712yes—
361C23#57614yes—
362Q8—44no—
363Q8—44no—
364Q8—81no—
365Q8—81no—
366Q8—82no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—28no—
373C2×C4—28no—
374C2×C4—28no—
375C2×C4—216no—
376C2×C4—28no—
377C2×C4—48no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—44no—
383C2×C4—44no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—816no—
389C2×C4—816no—
390C2×C4—82no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—82no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4—84no—
398C2×C4—88no—
399C2×C4—82no—
400C2×C4—82no—
401C2×C4—82no—
402C2×C4#39214yes—
403C2×C4C2×#12011yescharacteristic
404C2×C4C2×#12011yescharacteristic
405C2×C4C2×#12111yescharacteristic
406C2×C4C2×#12111yescharacteristic
407C2×C4C2×#12312yes—
408C2×C4#52512yes—
409C2×C4#52512yes—
410C2×C4#53612yes—
411C2×C4#53612yes—
412C2×C4#53612yes—
413C2×C4#54714yes—
414C2×C4#54714yes—
415C2×C4#55214yes—
416C8—28no—
417C8—44no—
418C22—216no—
419C22—28no—
420C22—216no—
421C22—28no—
422C22—24no—
423C22—24no—
424C22#201618yes—
425C22C2×#39211yescharacteristic
426C22#240612yes—
427C22#241312yes—
428C22#250112yes—
429C22C2×#45112yes—
430C22#300812yes—
431C22C2×#46212yes—
432C22#304112yes—
433C22C2×#49411yescharacteristic
434C22C2×#49711yescharacteristic
435C22#315512yes—
436C22#333818yes—
437C22#334618yes—
438C4—24no—
439C4—48no—
440C4—82no—
441C4—84no—
442C4—84no—
443C4—82no—
444C4—162no—
445C4#240612yes—
446C4#241312yes—
447C4#248612yes—
448C4#261411yescharacteristic
449C4#284911yescharacteristic
450C2—28no—
451C2#1506911yescharacteristic
452C2C2×#201611yescharacteristic
453C2#2342211yescharacteristic
454C2#2409814yes—
455C2#2458214yes—
456C2#2793012yes—
457C2#2793812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #11324C2semidirect
8. #11124C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4842
C23×#1072
C2×#25022
C2×#18922
C2×#18242
#139242
#113242
#111242
#107142