Finite Group Databasebeta

← #394830

Subgroups — #394830

Order 512.

1692 conjugacy classes of subgroups.

The subgroup lattice has 451 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2628C211yesmaximal, characteristic
2C2×#2317C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1852C212yesmaximal
5#11328C214yesmaximal, complemented
6#11208C212yesmaximal, complemented
7#10731C212yesmaximal
8#10713C211yesmaximal, characteristic
9C22×C4⋊Q8C2211yescharacteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2214yes—
14C22×C22⋊C8C2212yes—
15#2628—28no—
16C2×#400—21no—
17C2×#399—21no—
18C2×#395—21no—
19C2×#393—21no—
20#2317—28no—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#385—21no—
25C2×#385—21no—
26C2×#384—21no—
27C2×#382—21no—
28C2×#382C2211yescharacteristic
29C2×#380C2212yes—
30#2057—28no—
31#1852—28no—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C23×Q8—21no—
38C24×C4—21no—
39C24×C4C2311yescharacteristic
40C23×C8C2314yes—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—42no—
44C2×#117—22no—
45C2×#117—24no—
46C2×#117—24no—
47C2×#117—24no—
48C2×#117—24no—
49C2×#117—42no—
50C2×#117Q814yescomplemented
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—42no—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C22⋊C4D411yescharacteristic
68C22×C22⋊C4D411yescharacteristic
69C22×C42—21no—
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—28no—
73C2×#107—28no—
74C2×#107—44no—
75C2×C22⋊C8—24no—
76C2×C22⋊C8—24no—
77C2×C22⋊C8Q818yescomplemented
78#400—28no—
79#399—28no—
80#395—28no—
81#393—28no—
82#389—24no—
83#389—24no—
84#389—24no—
85#389—24no—
86#389—24no—
87#389D418yes—
88#385—28no—
89#385—28no—
90#384—28no—
91#382—28no—
92#382—24no—
93#380—24no—
94#380D418yes—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101#340—28no—
102#340—28no—
103#340—28no—
104#336—24no—
105#336—42no—
106C25C2×D411yescharacteristic
107C22×Q8—24no—
108C22×Q8—42no—
109C22×Q8—42no—
110C22×Q8—84no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—84no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D414yes—
129C23×C4C2×Q814yes—
130C23×C4C2×Q812yes—
131C23×C4#4112yes—
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8C2×Q818yes—
140C4⋊Q8—416no—
141#117—216no—
142#117—216no—
143#117—416no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
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—28no—
163C2×C4⋊C4—28no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—416no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—41no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4—84no—
174C2×C4⋊C4—84no—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4C2×D414yes—
177C2×C4⋊C4C2×D414yes—
178C2×C4⋊C4C2×D414yes—
179C2×C4⋊C4C2×Q814yes—
180C2×C4⋊C4#4114yes—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—24no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—22no—
185C2×C22⋊C4—22no—
186C2×C22⋊C4—22no—
187C2×C22⋊C4—22no—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—41no—
190C2×C22⋊C4—41no—
191C2×C22⋊C4—84no—
192C2×C42—24no—
193C2×C42—24no—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42C2×D414yes—
201#107—416no—
202C22⋊C8—216no—
203#95—28no—
204#95—24no—
205#95—24no—
206#95—28no—
207#95—28no—
208#95—28no—
209C24—21no—
210C24—21no—
211C24—21no—
212C24—21no—
213C24—24no—
214C24—24no—
215C24#12012yes—
216C24C22⋊Q812yes—
217C24C4⋊Q812yes—
218C24C22×D411yescharacteristic
219C2×Q8—48no—
220C2×Q8—88no—
221C2×Q8—88no—
222C2×Q8—88no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—28no—
232C22×C4—28no—
233C22×C4—28no—
234C22×C4—28no—
235C22×C4—21no—
236C22×C4—24no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—24no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—48no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—44no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4—88no—
271C22×C4—88no—
272C22×C4—88no—
273C22×C4#12012yes—
274C22×C4#12012yes—
275C22×C4#12012yes—
276C22×C4C22⋊Q818yes—
277C22×C4C22⋊Q812yes—
278C22×C4C4⋊Q818yes—
279C22×C4C4⋊Q812yes—
280C22×C4C22×D412yes—
281C22×C4C22×D412yes—
282C22×C4C22×Q811yescharacteristic
283C22×C4C22×Q812yes—
284C22×C4C2×#4111yescharacteristic
285C22×C4C2×#4112yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+412yes—
288C2×C8—28no—
289C2×C8—28no—
290C2×C8—28no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—24no—
297C4⋊C4—24no—
298C4⋊C4—28no—
299C4⋊C4—28no—
300C4⋊C4—28no—
301C4⋊C4—44no—
302C4⋊C4—44no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—44no—
306C4⋊C4—44no—
307C4⋊C4—44no—
308C4⋊C4—88no—
309C4⋊C4—88no—
310C4⋊C4—88no—
311C4⋊C4C22⋊Q818yes—
312C22⋊C4—216no—
313C22⋊C4—44no—
314C22⋊C4—44no—
315C22⋊C4—88no—
316C42—216no—
317C42—28no—
318C42—44no—
319C23—28no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—28no—
324C23—22no—
325C23—22no—
326C23—22no—
327C23—24no—
328C23—24no—
329C23—44no—
330C23—42no—
331C23—42no—
332C23—44no—
333C23—42no—
334C23—42no—
335C23#39514yes—
336C23C2×#12011yescharacteristic
337C23C2×#12011yescharacteristic
338C23C2×C22⋊Q811yescharacteristic
339C23#52511yescharacteristic
340C23C2×C4⋊Q811yescharacteristic
341C23#53411yescharacteristic
342C23#53611yescharacteristic
343C23#57614yes—
344C23#57814yes—
345Q8—816no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—216no—
353C2×C4—216no—
354C2×C4—28no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—48no—
360C2×C4—48no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—44no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—816no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—84no—
391C2×C4#39514yes—
392C2×C4C2×C22⋊Q812yes—
393C2×C4C2×C22⋊Q812yes—
394C2×C4#52512yes—
395C2×C4C2×C4⋊Q812yes—
396C2×C4#53912yes—
397C2×C4#54412yes—
398C2×C4D4214yes—
399C2×C4#54714yes—
400C2×C4D4×Q814yes—
401C2×C4#55414yes—
402C8—48no—
403C22—28no—
404C22—24no—
405C22—44no—
406C22—44no—
407C22—42no—
408C22—42no—
409C22—44no—
410C22—44no—
411C22—44no—
412C22C2×#39511yescharacteristic
413C22#240311yescharacteristic
414C22#240611yescharacteristic
415C22#243411yescharacteristic
416C22#258311yescharacteristic
417C22#262711yescharacteristic
418C22#284811yescharacteristic
419C22#300812yes—
420C22#300812yes—
421C22#301112yes—
422C22#301112yes—
423C22#308314yes—
424C22#312414yes—
425C22#316414yes—
426C22#319912yes—
427C22#320212yes—
428C22#324314yes—
429C4—24no—
430C4—42no—
431C4—42no—
432C4—48no—
433C4—84no—
434C4—81no—
435C4—84no—
436C4—84no—
437C4—81no—
438C4#244014yes—
439C4#244914yes—
440C2—44no—
441C2#1472011yescharacteristic
442C2#2355512yes—
443C2#2392511yescharacteristic
444C2#2392811yescharacteristic
445C2#2407312yes—
446C2#2582112yes—
447C2#2931012yes—
448C2#2997612yes—
449C2#3296312yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2057C2semidirect
5. #11328C2semidirect
6. #11208C2semidirect
50. C2×#117Q8semidirect
77. C2×C22⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#26282
C2×#23172
C2×#20572
C2×#18522
#113282
#112082
#107312
#107132