Finite Group Databasebeta

← #414565

Subgroups — #414565

Order 512.

1330 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic, complemented
2C2×#2057C211yesmaximal, characteristic, complemented
3C2×#2023C211yesmaximal, characteristic
4C2×#1925C212yesmaximal
5#14260C211yesmaximal, characteristic
6#11219C212yesmaximal, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11066C212yesmaximal
9#11051C211yesmaximal, characteristic
10#10807C212yesmaximal, complemented
11#10796C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#102C2212yes—
15C22×#102C2212yes—
16#2331—28no—
17C2×#391—22no—
18C2×#390C2211yescharacteristic
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#389C2212yes—
22C2×#389C2211yescharacteristic
23C2×#387—22no—
24C2×#382C2211yescharacteristic
25C2×C4×C4⋊C4C2212yes—
26C2×#376C2211yescharacteristic
27#2057—24no—
28#2023—24no—
29#1925—28no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1440—22no—
40#1439—22no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2312yes—
47C2×C4⋊Q8—24no—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8D414yes—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C22×C42C2311yescharacteristic
67C2×#107—24no—
68C2×#107—24no—
69C2×#107—24no—
70C2×#102—24no—
71C2×#102—24no—
72C2×#102—24no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76#391—216no—
77#390—22no—
78#390—22no—
79#390—24no—
80#390—22no—
81#390—22no—
82#389—24no—
83#389—24no—
84#389—22no—
85#389—22no—
86#387—216no—
87#382—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4D418yes—
90#376—24no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95D412yes—
99C2×#95D412yes—
100#340—24no—
101#340—24no—
102#340—24no—
103#340—24no—
104#340—24no—
105#336—24no—
106#336—22no—
107#336—24no—
108#336—22no—
109C25—21no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—82no—
114C22×Q8—82no—
115C22×D4—21no—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—22no—
120C22×D4—21no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4#4112yes—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
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#4114yes—
149C4⋊Q8—216no—
150C4⋊Q8—216no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4—82no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4C2×D414yes—
181C2×C4⋊C4C2×D414yes—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4#4112yes—
184C2×C4⋊C4#4114yes—
185C2×C22⋊C4—28no—
186C2×C22⋊C4—44no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C22⋊C4—82no—
190C2×C22⋊C4—82no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—28no—
205C2×C42—82no—
206C2×C42—82no—
207C2×C42C2×D414yes—
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210#107—48no—
211#102—216no—
212#102—48no—
213#95—24no—
214#95—24no—
215#95—28no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—42no—
220#95—82no—
221#95—82no—
222C24—22no—
223C24—41no—
224C24—41no—
225C24—41no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—42no—
230C24—42no—
231C24—42no—
232C24—42no—
233C24C22×D411yescharacteristic
234C2×Q8—216no—
235C2×Q8—84no—
236C2×Q8—84no—
237C2×Q8—88no—
238C2×Q8—88no—
239C2×D4—24no—
240C2×D4—22no—
241C2×D4—22no—
242C2×D4—24no—
243C2×D4—42no—
244C2×D4—42no—
245C2×D4—42no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—41no—
250C2×D4—42no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—22no—
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—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4#12014yes—
306C22×C4#12112yes—
307C22×C4#12112yes—
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D412yes—
310C22×C4C2×#4112yes—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4112yes—
314C22×C4C2×#4111yescharacteristic
315C22×C42−1+411yescharacteristic
316C22×C42−1+411yescharacteristic
317C22×C42−1+412yes—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—216no—
333C4⋊C4—24no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—84no—
349C4⋊C4—88no—
350C4⋊C4#12118yes—
351C22⋊C4—88no—
352C22⋊C4—84no—
353C22⋊C4—84no—
354C22⋊C4—88no—
355C42—24no—
356C42—24no—
357C42—24no—
358C42—24no—
359C42—42no—
360C42—42no—
361C42—88no—
362C23—42no—
363C23—44no—
364C23—44no—
365C23—42no—
366C23—42no—
367C23—81no—
368C23—81no—
369C23—82no—
370C23—82no—
371C23—81no—
372C23—82no—
373C23—82no—
374C23—82no—
375C23—82no—
376C23—82no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—81no—
381C23—81no—
382C23—81no—
383C23—81no—
384C23—81no—
385C23C2×#12011yescharacteristic
386C23C2×#12111yescharacteristic
387C23#52511yescharacteristic
388C23#52511yescharacteristic
389C23#53211yescharacteristic
390C23#53611yescharacteristic
391C23#53611yescharacteristic
392C23#57612yes—
393C23#57612yes—
394C23#57712yes—
395C23#57812yes—
396Q8—816no—
397D4—44no—
398D4—44no—
399D4—82no—
400D4—82no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—28no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—42no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—44no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4C2×#12112yes—
455C2×C4#52512yes—
456C2×C4#52512yes—
457C2×C4#53212yes—
458C2×C4#53612yes—
459C2×C4#53612yes—
460C2×C4#54811yescharacteristic
461C2×C4#54812yes—
462C2×C4#54811yescharacteristic
463C2×C4#55211yescharacteristic
464C2×C4#55212yes—
465C2×C4#55211yescharacteristic
466C2×C4#55212yes—
467C2×C4#55612yes—
468C2×C4#55612yes—
469C2×C4#55612yes—
470C8—44no—
471C8—82no—
472C22—84no—
473C22—82no—
474C22—82no—
475C22—82no—
476C22—84no—
477C22—162no—
478C22—161no—
479C22—161no—
480C22—162no—
481C22—161no—
482C22—161no—
483C22#238811yescharacteristic
484C22#253611yescharacteristic
485C22#254511yescharacteristic
486C22#259911yescharacteristic
487C22#260411yescharacteristic
488C22#261411yescharacteristic
489C22#284711yescharacteristic
490C22#300812yes—
491C22#301312yes—
492C22#304111yescharacteristic
493C22#304111yescharacteristic
494C22#304311yescharacteristic
495C22#304411yescharacteristic
496C22#311812yes—
497C22#312112yes—
498C22#312212yes—
499C22#312412yes—
500C22#317512yes—
501C22#317912yes—
502C22#324112yes—
503C22#324112yes—
504C22#324312yes—
505C22#324312yes—
506C4—42no—
507C4—48no—
508C4—42no—
509C4—44no—
510C4—82no—
511C4—84no—
512C4—82no—
513C4—161no—
514C4—161no—
515C4#253612yes—
516C4#254512yes—
517C4#259912yes—
518C4#260412yes—
519C2—162no—
520C2#1517611yescharacteristic
521C2#2357912yes—
522C2#2821011yescharacteristic
523C2#2821111yescharacteristic
524C2#2866311yescharacteristic
525C2#2866711yescharacteristic
526C2#2940112yes—
527C2#2952712yes—
528C2#2964612yes—
529C2#3291111yescharacteristic
530C2#3291811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2331C2semidirect
2. C2×#2057C2semidirect
6. #11219C2semidirect
7. #11208C2semidirect
10. #10807C2semidirect
11. #10796C2semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20572
C2×#20232
C2×#19252
#142602
#112192
#112082
#110662
#110512
#108072
#107962