Finite Group Databasebeta

← #418188

Subgroups — #418188

Order 512.

1726 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1852C212yesmaximal, complemented
5#11200C212yesmaximal, complemented
6#11096C214yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10885C212yesmaximal, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C23×C4⋊C4C2211yescharacteristic
12C22×#107C2214yes—
13C22×C22⋊C8C2212yes—
14C22×C22⋊C8C2212yes—
15#2563—28no—
16#2487—28no—
17C2×#400—21no—
18C2×#399—21no—
19C2×#393—21no—
20C2×#393—21no—
21C2×#390—21no—
22C2×#389C2212yes—
23C2×#386—21no—
24C2×#385—21no—
25C2×#382—21no—
26C2×#380C2212yes—
27C2×#380C2212yes—
28C2×#380C2211yescharacteristic
29C2×#380C2212yes—
30C2×C4×C4⋊C4C2211yescharacteristic
31#2057—28no—
32#1852—28no—
33C2×#340C2214yes—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C23×Q8—21no—
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4×D4—24no—
47C2×C4×D4—24no—
48C2×C4×D4—42no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—22no—
52C22×C4⋊C4—24no—
53C22×C4⋊C4—24no—
54C22×C4⋊C4—42no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C22⋊C4—22no—
64C22×C22⋊C4D411yescharacteristic
65C22×C22⋊C4D411yescharacteristic
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C2×#107—28no—
69C2×#107—28no—
70C2×#107—44no—
71C2×C22⋊C8—24no—
72C2×C22⋊C8—28no—
73C2×C22⋊C8—28no—
74C2×C22⋊C8—42no—
75#400—28no—
76#399—28no—
77#393—28no—
78#393—28no—
79#390—28no—
80#389—24no—
81#389D418yes—
82#386—28no—
83#385—28no—
84#382—28no—
85#380—24no—
86#380—24no—
87#380—24no—
88#380—24no—
89#380—24no—
90#380—24no—
91#380D418yes—
92C4×C4⋊C4—24no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99#340—28no—
100#340—28no—
101#340—28no—
102#336—24no—
103C25C2×D411yescharacteristic
104C25C2×D411yescharacteristic
105C22×Q8—24no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8—84no—
109C22×D4—22no—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—22no—
113C22×D4—41no—
114C22×D4—41no—
115C22×D4Q1614yes—
116C23×C4—22no—
117C23×C4—24no—
118C23×C4—22no—
119C23×C4—24no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—24no—
128C23×C4—22no—
129C23×C4—24no—
130C23×C4—42no—
131C23×C4—42no—
132C23×C4—84no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D414yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—24no—
140C22×C8—24no—
141C22×C8—24no—
142C22×C8—24no—
143C22×C8—24no—
144C22×C8—24no—
145C22×C8#4118yes—
146C4⋊Q8—416no—
147C4×D4—416no—
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—24no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—28no—
165C2×C4⋊C4—416no—
166C2×C4⋊C4—41no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4C2×D414yes—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C22⋊C4—24no—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4—22no—
180C2×C22⋊C4—22no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—41no—
185C2×C22⋊C4—41no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—84no—
189C2×C22⋊C4—84no—
190C2×C22⋊C4—84no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—24no—
194C2×C42—42no—
195C2×C42—42no—
196C2×C42C2×D414yes—
197#107—416no—
198C22⋊C8—216no—
199C22⋊C8—48no—
200#95—24no—
201#95—28no—
202#95—28no—
203#95—24no—
204#95—28no—
205#95—28no—
206C24—21no—
207C24—21no—
208C24—24no—
209C24—21no—
210C24—21no—
211C24—24no—
212C24—21no—
213C24—21no—
214C24—21no—
215C24—24no—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—24no—
220C24#12012yes—
221C24C2×Q1612yes—
222C24C2×Q1612yes—
223C24#13712yes—
224C24C22×D411yescharacteristic
225C2×Q8—48no—
226C2×Q8—88no—
227C2×Q8—88no—
228C2×Q8—88no—
229C2×D4—28no—
230C2×D4—28no—
231C2×D4—44no—
232C2×D4—44no—
233C2×D4—44no—
234C2×D4—44no—
235C2×D4—44no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—44no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—48no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—44no—
270C22×C4—44no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—44no—
276C22×C4—44no—
277C22×C4—44no—
278C22×C4—44no—
279C22×C4—44no—
280C22×C4—44no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—44no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—88no—
292C22×C4—88no—
293C22×C4—88no—
294C22×C4#12012yes—
295C22×C4#12012yes—
296C22×C4#12012yes—
297C22×C4C2×Q1612yes—
298C22×C4#13712yes—
299C22×C4C22×D412yes—
300C22×C4C22×D412yes—
301C22×C4C2×#4114yes—
302C22×C4H(5,𝔽2)12yes—
303C22×C42−1+414yes—
304C2×C8—28no—
305C2×C8—28no—
306C2×C8—28no—
307C2×C8—44no—
308C2×C8—44no—
309C2×C8—44no—
310C2×C8—44no—
311C4⋊C4—24no—
312C4⋊C4—24no—
313C4⋊C4—28no—
314C4⋊C4—28no—
315C4⋊C4—28no—
316C4⋊C4—44no—
317C4⋊C4—48no—
318C4⋊C4—48no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—88no—
325C22⋊C4—44no—
326C22⋊C4—44no—
327C22⋊C4—48no—
328C22⋊C4—88no—
329C22⋊C4—88no—
330C22⋊C4—88no—
331C42—28no—
332C42—48no—
333C23—28no—
334C23—28no—
335C23—22no—
336C23—24no—
337C23—22no—
338C23—22no—
339C23—22no—
340C23—22no—
341C23—22no—
342C23—24no—
343C23—28no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—44no—
348C23—44no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—44no—
354C23—44no—
355C23—44no—
356C23—41no—
357C23—42no—
358C23—41no—
359C23—42no—
360C23—42no—
361C23—42no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—42no—
366C23#45114yes—
367C23C2×#12011yescharacteristic
368C23C2×#12011yescharacteristic
369C23#53411yescharacteristic
370C23#54714yes—
371C23C22×Q1611yescharacteristic
372C23C2×#13711yescharacteristic
373C23C2×#13711yescharacteristic
374C23#57511yescharacteristic
375C23#57614yes—
376Q8—816no—
377D4—416no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—41no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—44no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—816no—
411C2×C4—84no—
412C2×C4—84no—
413C2×C4—84no—
414C2×C4—84no—
415C2×C4—84no—
416C2×C4—84no—
417C2×C4—84no—
418C2×C4—84no—
419C2×C4—84no—
420C2×C4—84no—
421C2×C4#45114yes—
422C2×C4#52512yes—
423C2×C4#53212yes—
424C2×C4#53612yes—
425C2×C4#53612yes—
426C2×C4#54214yes—
427C2×C4D4214yes—
428C2×C4#54714yes—
429C2×C4#54814yes—
430C2×C4#55112yes—
431C2×C4#56312yes—
432C8—48no—
433C22—28no—
434C22—24no—
435C22—44no—
436C22—44no—
437C22—44no—
438C22—44no—
439C22—44no—
440C22—42no—
441C22—42no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—42no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22—81no—
450C22—81no—
451C22#240311yescharacteristic
452C22#240612yes—
453C22#243212yes—
454C22#271712yes—
455C22C2×#45111yescharacteristic
456C22C2×#45111yescharacteristic
457C22#300711yescharacteristic
458C22#300711yescharacteristic
459C22#300812yes—
460C22#300812yes—
461C22#318911yescharacteristic
462C22#319511yescharacteristic
463C22#319912yes—
464C22#333814yes—
465C22#334614yes—
466C22#334714yes—
467C22#335714yes—
468C4—24no—
469C4—42no—
470C4—42no—
471C4—84no—
472C4—84no—
473C4—84no—
474C4—84no—
475C4—81no—
476C4#246114yes—
477C4#267414yes—
478C2—44no—
479C2—82no—
480C2#1526211yescharacteristic
481C2#2392111yescharacteristic
482C2#2392511yescharacteristic
483C2#2409812yes—
484C2#2410512yes—
485C2#2570612yes—
486C2#2571312yes—
487C2#3226112yes—
488C2#3226812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2057C2semidirect
4. C2×#1852C2semidirect
5. #11200C2semidirect
6. #11096C2semidirect
8. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#24872
C2×#20572
C2×#18522
#112002
#110962
#109642
#108852