Finite Group Databasebeta

← #406805

Subgroups — #406805

Order 512.

1316 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1852C212yesmaximal
2C2×#1825C212yesmaximal
3#14340C211yesmaximal, characteristic
4#13795C211yesmaximal, characteristic
5#11379C212yesmaximal, complemented
6#11377C212yesmaximal, complemented
7#11372C212yesmaximal, complemented
8#11324C212yesmaximal, complemented
9#10713C211yesmaximal, characteristic
10C22×#117C2211yescharacteristic
11C22×#117C2211yescharacteristic
12C23×C4⋊C4C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2212yes—
15C22×C22⋊C8C2212yes—
16C2×#384C2211yescharacteristic
17C2×#382C2211yescharacteristic
18C2×#380C2212yes—
19#1852—28no—
20#1825—28no—
21C2×#340C2212yes—
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2211yescharacteristic
29C2×#336C2211yescharacteristic
30#1441—22no—
31#1439—22no—
32C4⋊#95—22no—
33C4⋊#95C2212yes—
34C4⋊#95C2212yes—
35C4⋊#95C2212yes—
36C22⋊#95—22no—
37C24×C4—21no—
38C24×C4C2311yescharacteristic
39C23×C8C2312yes—
40C23×C8C2312yes—
41C2×#117—22no—
42C2×#117—24no—
43C2×#117—24no—
44C2×#117—24no—
45C2×#117—24no—
46C2×#117—41no—
47C2×#117—41no—
48C2×#117Q814yes—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—24no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—41no—
54C22×C4⋊C4—41no—
55C22×C4⋊C4D412yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4D412yes—
63C22×C42—21no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#107—24no—
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#107—42no—
71C2×#107—42no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—24no—
74C2×C22⋊C8D418yes—
75#384—41no—
76#384—41no—
77#382—24no—
78#380—24no—
79#380—24no—
80C2×#95—22no—
81C2×#95—22no—
82C2×#95—24no—
83C2×#95—24no—
84C2×#95—24no—
85#340—24no—
86#340—24no—
87#340—24no—
88#340—24no—
89#340—24no—
90#340—24no—
91#336—24no—
92#336—41no—
93#336—41no—
94C25C2×D411yescharacteristic
95C23×C4—22no—
96C23×C4—22no—
97C23×C4—21no—
98C23×C4—21no—
99C23×C4—21no—
100C23×C4—21no—
101C23×C4—22no—
102C23×C4—22no—
103C23×C4—24no—
104C23×C4—22no—
105C23×C4—22no—
106C23×C4—42no—
107C23×C4—42no—
108C23×C4—84no—
109C23×C4C2×D412yes—
110C23×C4C2×D412yes—
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D414yes—
113C23×C4C2×Q812yes—
114C23×C4#4114yes—
115C23×C4#4112yes—
116C23×C4C2411yescharacteristic
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—22no—
122C22×C8—22no—
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8C2×D414yes—
130C22×C8C2×D414yes—
131#117—216no—
132#117—48no—
133#117—48no—
134#117—48no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
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—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—48no—
161C2×C4⋊C4—48no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4—42no—
164C2×C4⋊C4—42no—
165C2×C4⋊C4—44no—
166C2×C4⋊C4—84no—
167C2×C4⋊C4C2×Q814yes—
168C2×C4⋊C4#4114yes—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—24no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C22⋊C4—42no—
175C2×C22⋊C4—44no—
176C2×C22⋊C4—84no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—42no—
186C2×C42—42no—
187C2×C42#4114yes—
188#107—48no—
189#107—48no—
190C22⋊C8—216no—
191#95—44no—
192#95—44no—
193#95—84no—
194#95—84no—
195#95—84no—
196C24—21no—
197C24—21no—
198C24—21no—
199C24—21no—
200C24—24no—
201C24—24no—
202C24C22⋊Q812yes—
203C24#12312yes—
204C24#12712yes—
205C24C22×D411yescharacteristic
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—24no—
223C22×C4—24no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—24no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—48no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—44no—
241C22×C4—44no—
242C22×C4—44no—
243C22×C4—44no—
244C22×C4—42no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—44no—
254C22×C4—88no—
255C22×C4—88no—
256C22×C4—88no—
257C22×C4C22⋊Q818yes—
258C22×C4C22⋊Q812yes—
259C22×C4#12314yes—
260C22×C4#12318yes—
261C22×C4#12312yes—
262C22×C4#12712yes—
263C22×C4C22×D412yes—
264C22×C4C22×Q811yescharacteristic
265C22×C4C2×#4112yes—
266C22×C4C2×#4111yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4112yes—
269C22×C4H(5,𝔽2)11yescharacteristic
270C22×C4H(5,𝔽2)11yescharacteristic
271C22×C42−1+411yescharacteristic
272C22×C42−1+411yescharacteristic
273C2×C8—24no—
274C2×C8—24no—
275C2×C8—24no—
276C2×C8—24no—
277C2×C8—24no—
278C2×C8—24no—
279C2×C8—42no—
280C2×C8—42no—
281C2×C8—42no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—42no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—48no—
292C4⋊C4—48no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—88no—
306C22⋊C4—48no—
307C22⋊C4—48no—
308C22⋊C4—88no—
309C42—24no—
310C42—24no—
311C42—42no—
312C42—42no—
313C42—48no—
314C23—28no—
315C23—28no—
316C23—22no—
317C23—22no—
318C23—24no—
319C23—24no—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—22no—
324C23—44no—
325C23—42no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—44no—
330C23#39914yes—
331C23C2×C22⋊Q811yescharacteristic
332C23C2×#12311yescharacteristic
333C23C2×#12311yescharacteristic
334C23#52511yescharacteristic
335C23C2×#12711yescharacteristic
336C23#53411yescharacteristic
337C23#53611yescharacteristic
338C23#57614yes—
339C23#57814yes—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—28no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—48no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—44no—
364C2×C4—816no—
365C2×C4—84no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—84no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4#39914yes—
382C2×C4C2×C22⋊Q812yes—
383C2×C4C2×#12312yes—
384C2×C4#52912yes—
385C2×C4#53312yes—
386C2×C4#53911yescharacteristic
387C2×C4#53911yescharacteristic
388C2×C4#54411yescharacteristic
389C2×C4#54411yescharacteristic
390C2×C4#54612yes—
391C2×C4#54612yes—
392C2×C4#54712yes—
393C2×C4#54712yes—
394C2×C4#55112yes—
395C2×C4#55112yes—
396C2×C4#55212yes—
397C2×C4#55212yes—
398C8—44no—
399C8—44no—
400C22—24no—
401C22—28no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—42no—
406C22—42no—
407C22—44no—
408C22—44no—
409C22C2×#39911yescharacteristic
410C22#248311yescharacteristic
411C22#248411yescharacteristic
412C22#250111yescharacteristic
413C22#253911yescharacteristic
414C22#262711yescharacteristic
415C22#285511yescharacteristic
416C22#308314yes—
417C22#309814yes—
418C22#309814yes—
419C22#312212yes—
420C22#312712yes—
421C22#315512yes—
422C22#315612yes—
423C22#319912yes—
424C22#320212yes—
425C22#324314yes—
426C4—24no—
427C4—44no—
428C4—84no—
429C4—84no—
430C4—82no—
431C4—162no—
432C4—162no—
433C4#241912yes—
434C4#241912yes—
435C4#243812yes—
436C4#243812yes—
437C2—44no—
438C2#1493611yescharacteristic
439C2#2373712yes—
440C2#2760812yes—
441C2#2762012yes—
442C2#2792112yes—
443C2#2831911yescharacteristic
444C2#2832211yescharacteristic
445C2#2997612yes—
446C2#3315612yes—
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
5. #11379C2semidirect
6. #11377C2semidirect
7. #11372C2semidirect
8. #11324C2semidirect

Maximal subgroups

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

subgroupindex
C2×#18522
C2×#18252
#143402
#137952
#113792
#113772
#113722
#113242
#107132