Finite Group Databasebeta

← #440879

Subgroups — #440879

Order 512.

1514 conjugacy classes of subgroups.

The subgroup lattice has 527 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2073C211yesmaximal, characteristic
3C2×#1836C211yesmaximal, characteristic, complemented
4#15235C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11371C211yesmaximal, characteristic, complemented
7#11267C212yesmaximal, complemented
8#11189C212yesmaximal
9#10927C211yesmaximal, characteristic
10#10875C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—22no—
14C22×C4×D4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×#97C2211yescharacteristic
18#2542—24no—
19#2542—24no—
20C2×#399—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27#2073—24no—
28#1836—24no—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1443—22no—
39#1442—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C24×C4—21no—
46C23×C8C2312yes—
47C23×C8C2312yes—
48C2×C4⋊Q8—22no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—42no—
52C2×C22⋊Q8—216no—
53C2×C22⋊Q8—48no—
54C2×C22⋊Q8—48no—
55C2×C4×D4—22no—
56C2×C4×D4—22no—
57C2×C4×D4—24no—
58C2×C4×D4—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4D412yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C22⋊C4—21no—
69C22×C22⋊C4—22no—
70C22×C22⋊C4—22no—
71C22×C42C2311yescharacteristic
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—24no—
79C2×#102—42no—
80C2×#97—24no—
81C2×#97—22no—
82#399—216no—
83#390—22no—
84#390—22no—
85#390—22no—
86#390D414yes—
87#384—22no—
88#384—22no—
89#384—22no—
90#384D414yes—
91#382—22no—
92#382—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102#340—24no—
103#340—24no—
104#340—24no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336—24no—
109#336—22no—
110#336—24no—
111C25—21no—
112C22×Q8—24no—
113C22×Q8—44no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—44no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4C2×D412yes—
124C22×D4#4114yes—
125C23×C4—24no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—42no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2411yescharacteristic
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8#4114yes—
154C4⋊Q8—48no—
155C4⋊Q8—48no—
156C22⋊Q8—464no—
157C4×D4—48no—
158C4×D4—48no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—28no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—42no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—44no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42C2×D412yes—
214C2×C42C2×D412yes—
215C2×C42#4114yes—
216#102—48no—
217#102—48no—
218#97—28no—
219#95—24no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—82no—
224#95—82no—
225#95—82no—
226C24—22no—
227C24—24no—
228C24—41no—
229C24—41no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24C22×D411yescharacteristic
236C2×Q8—416no—
237C2×Q8—48no—
238C2×Q8—84no—
239C2×Q8—88no—
240C2×Q8—84no—
241C2×D4—24no—
242C2×D4—22no—
243C2×D4—24no—
244C2×D4—24no—
245C2×D4—24no—
246C2×D4—24no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—41no—
250C2×D4—41no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—42no—
256C2×D4#12414yes—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—48no—
280C22×C4—416no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—84no—
308C22×C4—88no—
309C22×C4—84no—
310C22×C4#12014yes—
311C22×C4#12412yes—
312C22×C4#12412yes—
313C22×C4AGL(1,ℤ/8)14yes—
314C22×C4AGL(1,ℤ/8)14yes—
315C22×C4C22×D411yescharacteristic
316C22×C4C22×D411yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4C2×#4112yes—
319C22×C4C2×#4112yes—
320C22×C4H(5,𝔽2)12yes—
321C22×C42−1+412yes—
322C22×C42−1+412yes—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C4⋊C4—216no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—416no—
342C4⋊C4—416no—
343C4⋊C4—48no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C22⋊C4—48no—
356C22⋊C4—416no—
357C22⋊C4—416no—
358C42—22no—
359C42—22no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—41no—
364C42—41no—
365C42—42no—
366C23—24no—
367C23—44no—
368C23—44no—
369C23—44no—
370C23—44no—
371C23—42no—
372C23—42no—
373C23—44no—
374C23—44no—
375C23—44no—
376C23—44no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—81no—
382C23—82no—
383C23—81no—
384C23—82no—
385C23—81no—
386C23—81no—
387C23C2×#12011yescharacteristic
388C23C2×#12411yescharacteristic
389C23#53511yescharacteristic
390C23#54712yes—
391C23#54812yes—
392C23C2×AGL(1,ℤ/8)12yes—
393C23C2×AGL(1,ℤ/8)12yes—
394C23#57712yes—
395C23#57812yes—
396Q8—816no—
397D4—28no—
398D4—44no—
399D4—44no—
400D4—44no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—816no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#49014yes—
455C2×C4#52511yescharacteristic
456C2×C4C2×#12411yescharacteristic
457C2×C4#52912yes—
458C2×C4#53611yescharacteristic
459C2×C4#53611yescharacteristic
460C2×C4#53912yes—
461C2×C4#54112yes—
462C2×C4#54212yes—
463C2×C4#54612yes—
464C2×C4#54612yes—
465C2×C4#54712yes—
466C2×C4#54812yes—
467C2×C4#55112yes—
468C2×C4#55212yes—
469C2×C4#55212yes—
470C2×C4#56312yes—
471C8—44no—
472C8—82no—
473C22—44no—
474C22—48no—
475C22—82no—
476C22—82no—
477C22—84no—
478C22—84no—
479C22—84no—
480C22—84no—
481C22—84no—
482C22#240611yescharacteristic
483C22#243511yescharacteristic
484C22#243611yescharacteristic
485C22#248912yes—
486C22#272312yes—
487C22#300312yes—
488C22#301312yes—
489C22C2×#49011yescharacteristic
490C22#313712yes—
491C22#313811yescharacteristic
492C22#321712yes—
493C22#322111yescharacteristic
494C22#322311yescharacteristic
495C22#334312yes—
496C22#334512yes—
497C22#335312yes—
498C22#335412yes—
499C22#336712yes—
500C22#336712yes—
501C22#337312yes—
502C22#337312yes—
503C4—24no—
504C4—42no—
505C4—44no—
506C4—44no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—162no—
511C4—162no—
512C4#247712yes—
513C4#248812yes—
514C4#269012yes—
515C4#269812yes—
516C2—84no—
517C2#1567011yescharacteristic
518C2#2410012yes—
519C2#2586011yescharacteristic
520C2#2586811yescharacteristic
521C2#2593912yes—
522C2#2773612yes—
523C2#2774412yes—
524C2#3244612yes—
525C2#3244812yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1836C2semidirect
4. #15235C2semidirect
5. #11384C2semidirect
6. #11371C2semidirect
7. #11267C2semidirect
10. #10875C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20732
C2×#18362
#152352
#113842
#113712
#112672
#111892
#109272
#108752
#107882