Finite Group Databasebeta

← #403517

Subgroups — #403517

Order 512.

1350 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2331C211yesmaximal, characteristic, complemented
2C2×#2067C212yesmaximal
3C2×#2057C211yesmaximal, characteristic, complemented
4C2×#2023C211yesmaximal, characteristic
5#14260C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11208C211yesmaximal, characteristic, complemented
8#11051C211yesmaximal, characteristic
9#10931C212yesmaximal
10#10907C212yesmaximal, complemented
11#10833C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×C4⋊C8C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17#2331—28no—
18C2×#391—22no—
19C2×#390C2211yescharacteristic
20C2×#390C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#389C2211yescharacteristic
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4C2212yes—
27C2×#376C2211yescharacteristic
28#2067—28no—
29#2057—24no—
30#2023—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#336C2211yescharacteristic
37C2×#336C2211yescharacteristic
38C2×#336C2212yes—
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×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—42no—
73C2×#102—24no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—24no—
77C2×#102—24no—
78C2×#102—42no—
79#391—216no—
80#390—24no—
81#390—22no—
82#390—22no—
83#390—22no—
84#390Q814yescomplemented
85#389—24no—
86#389—22no—
87#389—22no—
88#389D418yes—
89#387—216no—
90#382—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—24no—
93#376—24no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D412yes—
102C2×#95D412yes—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#336—22no—
109#336—22no—
110#336—24no—
111C25—21no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—82no—
116C22×Q8—82no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—21no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4—22no—
126C22×D4C2×Q812yes—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D412yes—
136C23×C4C2411yescharacteristic
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—22no—
149C22×C8—22no—
150C22×C8#4114yes—
151C4⋊Q8—216no—
152C4⋊Q8—216no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—28no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4—82no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4C2×D414yes—
183C2×C4⋊C4C2×D414yes—
184C2×C4⋊C4C2×D414yes—
185C2×C4⋊C4C2×Q812yes—
186C2×C4⋊C4#4114yes—
187C2×C22⋊C4—28no—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—82no—
192C2×C22⋊C4—82no—
193C2×C22⋊C4—82no—
194C2×C22⋊C4—82no—
195C2×C22⋊C4—82no—
196C2×C22⋊C4—82no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—28no—
205C2×C42—82no—
206C2×C42—82no—
207C2×C42C2×D414yes—
208C2×C42C2×D412yes—
209C2×C42C2×Q812yes—
210C2×C42#4112yes—
211C2×C42#4112yes—
212#107—48no—
213C4⋊C8—48no—
214#102—48no—
215#102—48no—
216#95—28no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—42no—
223#95—82no—
224#95—82no—
225C24—22no—
226C24—41no—
227C24—41no—
228C24—41no—
229C24—41no—
230C24—41no—
231C24—41no—
232C24—42no—
233C24—42no—
234C24—42no—
235C24—42no—
236C24C22×D411yescharacteristic
237C2×Q8—216no—
238C2×Q8—84no—
239C2×Q8—84no—
240C2×Q8—88no—
241C2×Q8—88no—
242C2×D4—24no—
243C2×D4—22no—
244C2×D4—24no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—42no—
251C2×D4—42no—
252C2×D4—42no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—42no—
257C2×D4—41no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
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—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4#12014yes—
308C22×C4#12112yes—
309C22×C4#12112yes—
310C22×C4C22⋊Q814yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C22×Q811yescharacteristic
314C22×C4C2×#4112yes—
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4112yes—
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C42−1+411yescharacteristic
320C22×C42−1+412yes—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—24no—
324C2×C8—42no—
325C2×C8—42no—
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—
335C4⋊C4—216no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—24no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—88no—
352C4⋊C4—84no—
353C4⋊C4#12118yes—
354C22⋊C4—88no—
355C22⋊C4—88no—
356C22⋊C4—84no—
357C22⋊C4—84no—
358C42—24no—
359C42—24no—
360C42—24no—
361C42—24no—
362C42—24no—
363C42—24no—
364C42—88no—
365C23—44no—
366C23—44no—
367C23—42no—
368C23—42no—
369C23—42no—
370C23—81no—
371C23—82no—
372C23—82no—
373C23—81no—
374C23—82no—
375C23—82no—
376C23—81no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23—82no—
381C23—82no—
382C23—82no—
383C23—81no—
384C23—81no—
385C23—81no—
386C23—81no—
387C23—81no—
388C23C2×#12011yescharacteristic
389C23C2×#12111yescharacteristic
390C23C2×C22⋊Q811yescharacteristic
391C23#52511yescharacteristic
392C23#53211yescharacteristic
393C23#53611yescharacteristic
394C23#53711yescharacteristic
395C23#57612yes—
396C23#57612yes—
397C23#57712yes—
398C23#57812yes—
399Q8—816no—
400D4—44no—
401D4—44no—
402D4—82no—
403D4—82no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—28no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—44no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4C2×#12112yes—
458C2×C4#52512yes—
459C2×C4#52912yes—
460C2×C4#53312yes—
461C2×C4#54112yes—
462C2×C4#54212yes—
463C2×C4#54612yes—
464C2×C4#54712yes—
465C2×C4#54812yes—
466C2×C4#54812yes—
467C2×C4D4×Q811yescharacteristic
468C2×C4D4×Q812yes—
469C2×C4#55011yescharacteristic
470C2×C4#55411yescharacteristic
471C2×C4#55612yes—
472C2×C4#55611yescharacteristic
473C8—44no—
474C8—82no—
475C22—82no—
476C22—82no—
477C22—82no—
478C22—84no—
479C22—84no—
480C22—161no—
481C22—161no—
482C22—162no—
483C22—162no—
484C22—161no—
485C22—161no—
486C22#239611yescharacteristic
487C22#240911yescharacteristic
488C22#243011yescharacteristic
489C22#253811yescharacteristic
490C22#254511yescharacteristic
491C22#263711yescharacteristic
492C22#284711yescharacteristic
493C22#300812yes—
494C22#301312yes—
495C22#304111yescharacteristic
496C22#304111yescharacteristic
497C22#304311yescharacteristic
498C22#304411yescharacteristic
499C22#308112yes—
500C22#308312yes—
501C22#312112yes—
502C22#312212yes—
503C22#317512yes—
504C22#317912yes—
505C22#324112yes—
506C22#324312yes—
507C22#325712yes—
508C22#326212yes—
509C4—42no—
510C4—48no—
511C4—42no—
512C4—44no—
513C4—44no—
514C4—82no—
515C4—84no—
516C4—161no—
517C4—161no—
518C4#242612yes—
519C4#242912yes—
520C4#246512yes—
521C4#246912yes—
522C2—162no—
523C2#1487711yescharacteristic
524C2#2378712yes—
525C2#2430812yes—
526C2#2551012yes—
527C2#2828811yescharacteristic
528C2#2829011yescharacteristic
529C2#2866311yescharacteristic
530C2#2866711yescharacteristic
531C2#3015812yes—
532C2#3291111yescharacteristic
533C2#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
3. C2×#2057C2semidirect
6. #11228C2semidirect
7. #11208C2semidirect
10. #10907C2semidirect
11. #10833C2semidirect
84. #390Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#23312
C2×#20672
C2×#20572
C2×#20232
#142602
#112282
#112082
#110512
#109312
#109072
#108332