Finite Group Databasebeta

← #440930

Subgroups — #440930

Order 512.

1426 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2C2×#2025C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic, complemented
4#15218C211yesmaximal, characteristic, complemented
5#11384C212yesmaximal, complemented
6#11372C211yesmaximal, characteristic, complemented
7#11208C212yesmaximal, complemented
8#11066C212yesmaximal
9#10925C211yesmaximal, characteristic
10#10874C211yesmaximal, characteristic, complemented
11#10788C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18#2563—24no—
19#2563—24no—
20C2×#393—22no—
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#380—22no—
27C2×C4×C4⋊C4C2211yescharacteristic
28#2025—24no—
29#1991—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
39#1444—22no—
40#1441—22no—
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×#125—24no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—42no—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—24no—
59C2×C4×D4—42no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4D412yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4—21no—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—42no—
78C2×#102—42no—
79C2×#102—42no—
80C2×#102Q818yescomplemented
81C2×C4×C8—22no—
82C2×C4×C8—42no—
83#393—216no—
84#390—22no—
85#390—22no—
86#390—22no—
87#390D414yes—
88#389—24no—
89#389—42no—
90#382—22no—
91#382—22no—
92#382—22no—
93#382D414yes—
94#380—216no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95D412yes—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#340—42no—
109#340—42no—
110#336—22no—
111#336—24no—
112#336—42no—
113C25—21no—
114C22×Q8—24no—
115C22×Q8—42no—
116C22×Q8—42no—
117C22×Q8—82no—
118C22×Q8—82no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—22no—
125C22×D4C2×D412yes—
126C22×D4C2×Q814yes—
127C23×C4—24no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—42no—
134C23×C4—44no—
135C23×C4—42no—
136C23×C4—44no—
137C23×C4C2×D412yes—
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2411yescharacteristic
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—22no—
154C22×C8—22no—
155C22×C8C2×Q814yes—
156C4⋊Q8—48no—
157C4⋊Q8—48no—
158#125—28no—
159#125—48no—
160#125—44no—
161C4×D4—48no—
162C4×D4—48no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
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—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4C2×Q814yes—
198C2×C4⋊C4#4114yes—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—28no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—82no—
205C2×C22⋊C4—82no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42C2×D412yes—
218C2×C42C2×D412yes—
219C2×C42#4114yes—
220#102—48no—
221#102—48no—
222C4×C8—44no—
223#95—28no—
224#95—28no—
225#95—24no—
226#95—44no—
227#95—44no—
228#95—82no—
229#95—82no—
230C24—22no—
231C24—24no—
232C24—41no—
233C24—41no—
234C24—42no—
235C24—42no—
236C24—42no—
237C24—42no—
238C24—42no—
239C24C22×D411yescharacteristic
240C2×Q8—48no—
241C2×Q8—88no—
242C2×Q8—88no—
243C2×Q8—84no—
244C2×Q8—84no—
245C2×D4—24no—
246C2×D4—22no—
247C2×D4—24no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—24no—
251C2×D4—24no—
252C2×D4—24no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—42no—
256C2×D4—42no—
257C2×D4—42no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4C4⋊Q814yes—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—21no—
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—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—48no—
287C22×C4—416no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
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—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—84no—
313C22×C4—88no—
314C22×C4—84no—
315C22×C4#12014yes—
316C22×C4#12412yes—
317C22×C4#12412yes—
318C22×C4#13514yes—
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q812yes—
322C22×C4C2×#4112yes—
323C22×C4C2×#4112yes—
324C22×C4H(5,𝔽2)12yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+412yes—
327C2×C8—24no—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C4⋊C4—24no—
342C4⋊C4—24no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—48no—
349C4⋊C4—48no—
350C4⋊C4—416no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—88no—
359C22⋊C4—48no—
360C22⋊C4—88no—
361C22⋊C4—88no—
362C42—22no—
363C42—22no—
364C42—24no—
365C42—24no—
366C42—24no—
367C42—41no—
368C42—41no—
369C42—42no—
370C23—24no—
371C23—44no—
372C23—44no—
373C23—44no—
374C23—44no—
375C23—42no—
376C23—42no—
377C23—44no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—81no—
382C23—82no—
383C23—81no—
384C23—82no—
385C23—81no—
386C23—81no—
387C23—82no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23C2×#12011yescharacteristic
392C23C2×#12411yescharacteristic
393C23#53511yescharacteristic
394C23#54612yes—
395C23D4×Q812yes—
396C23C2×#13512yes—
397C23#57512yes—
398C23#57612yes—
399C23#57712yes—
400Q8—816no—
401D4—28no—
402D4—44no—
403D4—44no—
404D4—44no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—44no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—42no—
430C2×C4—44no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—816no—
441C2×C4—81no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4#48814yes—
459C2×C4#52511yescharacteristic
460C2×C4C2×C4⋊Q811yescharacteristic
461C2×C4#53312yes—
462C2×C4#53511yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#54212yes—
465C2×C4#54212yes—
466C2×C4#54312yes—
467C2×C4#54612yes—
468C2×C4#54612yes—
469C2×C4#54612yes—
470C2×C4D4×Q812yes—
471C2×C4#55412yes—
472C2×C4#55412yes—
473C2×C4#55612yes—
474C2×C4#56212yes—
475C8—44no—
476C8—82no—
477C22—44no—
478C22—48no—
479C22—82no—
480C22—82no—
481C22—84no—
482C22—84no—
483C22—84no—
484C22—84no—
485C22—84no—
486C22#240411yescharacteristic
487C22#243411yescharacteristic
488C22#243511yescharacteristic
489C22#249212yes—
490C22#269212yes—
491C22#300212yes—
492C22#301212yes—
493C22C2×#48811yescharacteristic
494C22#313612yes—
495C22#314011yescharacteristic
496C22#321112yes—
497C22#322011yescharacteristic
498C22#322211yescharacteristic
499C22#330312yes—
500C22#330512yes—
501C22#331912yes—
502C22#332112yes—
503C22#338012yes—
504C22#338112yes—
505C22#338812yes—
506C22#338912yes—
507C4—24no—
508C4—42no—
509C4—44no—
510C4—82no—
511C4—82no—
512C4—82no—
513C4—84no—
514C4—82no—
515C4—84no—
516C4#246512yes—
517C4#254912yes—
518C4#269212yes—
519C4#270012yes—
520C2—84no—
521C2#1567111yescharacteristic
522C2#2396512yes—
523C2#2584312yes—
524C2#2585911yescharacteristic
525C2#2586711yescharacteristic
526C2#2781412yes—
527C2#2782612yes—
528C2#3152712yes—
529C2#3152912yes—
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×#1991C2semidirect
4. #15218C2semidirect
5. #11384C2semidirect
6. #11372C2semidirect
7. #11208C2semidirect
10. #10874C2semidirect
80. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
C2×#20252
C2×#19912
#152182
#113842
#113722
#112082
#110662
#109252
#108742
#107882