Finite Group Databasebeta

← #403042

Subgroups — #403042

Order 512.

1250 conjugacy classes of subgroups.

The subgroup lattice has 559 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1988C211yesmaximal, characteristic, complemented
2C2×#1925C211yesmaximal, characteristic
3#15144C211yesmaximal, characteristic
4#13821C211yesmaximal, characteristic, complemented
5#11384C211yesmaximal, characteristic, complemented
6#11255C211yesmaximal, characteristic, complemented
7#11219C212yesmaximal, complemented
8#11076C211yesmaximal, characteristic
9#11066C212yesmaximal
10#10931C212yesmaximal
11#10806C212yesmaximal, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#102C2211yescharacteristic
16C22×#102C2211yescharacteristic
17C22×#102C2212yes—
18C2×#393—21no—
19C2×#393—21no—
20C2×#390C2211yescharacteristic
21C2×#390C2212yes—
22C2×#389C2212yes—
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26#1988—24no—
27#1925—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2211yescharacteristic
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1439—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×D4C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C22⋊C4—22no—
63C22×C42—21no—
64C22×C42—22no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#107—22no—
68C2×#107—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×#102—22no—
72C2×#102—22no—
73C2×#102—22no—
74C2×#102—24no—
75C2×#102—24no—
76C2×#102—42no—
77C2×#102—42no—
78C2×#102Q814yescomplemented
79#393—44no—
80#393—44no—
81#390—24no—
82#390—22no—
83#390—22no—
84#390—42no—
85#389—24no—
86#389—24no—
87C4×C4⋊C4—22no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—42no—
90C4×C4⋊C4—44no—
91C4×C4⋊C4D414yes—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—21no—
99C2×#95D411yescharacteristic
100C2×#95D412yes—
101C2×#95D411yescharacteristic
102#340—22no—
103#340—24no—
104#340—24no—
105#340—24no—
106#340—24no—
107#340—42no—
108#336—22no—
109#336—42no—
110#336—42no—
111C25—21no—
112C22×Q8—44no—
113C22×Q8—42no—
114C22×Q8—42no—
115C22×Q8—42no—
116C22×Q8—82no—
117C22×D4—21no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—21no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—22no—
124C22×D4—21no—
125C22×D4—21no—
126C22×D4C2×Q812yes—
127C23×C4—21no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8C2×Q812yes—
154C4⋊Q8—28no—
155C4⋊Q8—48no—
156C4⋊Q8—44no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
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—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—28no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—42no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4C2×D412yes—
182C2×C4⋊C4C2×D412yes—
183C2×C4⋊C4C2×Q812yes—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4114yes—
186C2×C4⋊C4#4112yes—
187C2×C4⋊C4#4114yes—
188C2×C22⋊C4—44no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—42no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—42no—
196C2×C22⋊C4—42no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—44no—
211C2×C42—44no—
212C2×C42—82no—
213C2×C42C2×D412yes—
214C2×C42#4112yes—
215#107—44no—
216C4⋊C8—44no—
217#102—28no—
218#102—48no—
219#102—44no—
220#95—28no—
221#95—24no—
222#95—22no—
223#95—42no—
224#95—44no—
225#95—44no—
226#95—42no—
227#95—44no—
228#95—81no—
229#95D814yes—
230C24—22no—
231C24—41no—
232C24—42no—
233C24—42no—
234C24—41no—
235C24—41no—
236C24—41no—
237C24—42no—
238C24—42no—
239C24—41no—
240C24—41no—
241C24C22×D411yescharacteristic
242C2×Q8—48no—
243C2×Q8—88no—
244C2×Q8—88no—
245C2×Q8—84no—
246C2×Q8—84no—
247C2×D4—22no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—42no—
254C2×D4—42no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—42no—
263C2×D4—42no—
264C2×D4—42no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—24no—
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—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
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—41no—
311C22×C4—42no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4#12112yes—
316C22×C4#12112yes—
317C22×C4C2×D814yes—
318C22×C4C2×D812yes—
319C22×C4AGL(1,ℤ/8)12yes—
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q811yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4112yes—
324C22×C4C2×#4112yes—
325C22×C4C2×#4112yes—
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C42−1+411yescharacteristic
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—41no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C4⋊C4—24no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—416no—
357C4⋊C4—48no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4C22⋊Q814yes—
366C4⋊C4#12414yes—
367C22⋊C4—88no—
368C22⋊C4—84no—
369C22⋊C4—88no—
370C22⋊C4—84no—
371C42—22no—
372C42—22no—
373C42—24no—
374C42—42no—
375C42—42no—
376C42—42no—
377C42—42no—
378C42—48no—
379C42—88no—
380C23—42no—
381C23—42no—
382C23—44no—
383C23—42no—
384C23—44no—
385C23—81no—
386C23—82no—
387C23—81no—
388C23—82no—
389C23—82no—
390C23—82no—
391C23—81no—
392C23—82no—
393C23—81no—
394C23—82no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23C2×#12111yescharacteristic
404C23#52512yes—
405C23#54611yescharacteristic
406C23#54612yes—
407C23D4×Q811yescharacteristic
408C23C22×D811yescharacteristic
409C23C2×AGL(1,ℤ/8)11yescharacteristic
410C23#57512yes—
411C23#57712yes—
412C23#57711yescharacteristic
413C23#57811yescharacteristic
414Q8—816no—
415D4—42no—
416D4—44no—
417D4—42no—
418D4—81no—
419D4—81no—
420D4—82no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—24no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—24no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—82no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4#45914yes—
476C2×C4C2×C22⋊Q811yescharacteristic
477C2×C4#52511yescharacteristic
478C2×C4C2×#12411yescharacteristic
479C2×C4#52912yes—
480C2×C4#52912yes—
481C2×C4#53211yescharacteristic
482C2×C4#53312yes—
483C2×C4#53511yescharacteristic
484C2×C4#53711yescharacteristic
485C2×C4#53812yes—
486C2×C4#54212yes—
487C2×C4#55112yes—
488C2×C4#55211yescharacteristic
489C2×C4#55412yes—
490C2×C4#55411yescharacteristic
491C2×C4#55512yes—
492C2×C4#55612yes—
493C2×C4#55912yes—
494C8—42no—
495C8—81no—
496C8—82no—
497C22—82no—
498C22—84no—
499C22—84no—
500C22—82no—
501C22—82no—
502C22—161no—
503C22—161no—
504C22—162no—
505C22—162no—
506C22—161no—
507C22—161no—
508C22#241111yescharacteristic
509C22#241711yescharacteristic
510C22#245912yes—
511C22#246512yes—
512C22#285011yescharacteristic
513C22C2×#45911yescharacteristic
514C22#304011yescharacteristic
515C22#304311yescharacteristic
516C22#304611yescharacteristic
517C22#310012yes—
518C22#311512yes—
519C22#312112yes—
520C22#312512yes—
521C22#329411yescharacteristic
522C22#329412yes—
523C22#331012yes—
524C22#331312yes—
525C22#332112yes—
526C22#332612yes—
527C22#333011yescharacteristic
528C22Q8×D811yescharacteristic
529C22#338411yescharacteristic
530C22#338812yes—
531C4—24no—
532C4—44no—
533C4—44no—
534C4—42no—
535C4—44no—
536C4—84no—
537C4—82no—
538C4—84no—
539C4—161no—
540C4#239611yescharacteristic
541C4#253212yes—
542C4#256311yescharacteristic
543C4#258912yes—
544C4#259211yescharacteristic
545C4#266211yescharacteristic
546C2—162no—
547C2#1486611yescharacteristic
548C2#2447611yescharacteristic
549C2#2449711yescharacteristic
550C2#2481111yescharacteristic
551C2#2482811yescharacteristic
552C2#2672012yes—
553C2#2674812yes—
554C2#2699812yes—
555C2#2702612yes—
556C2#3302411yescharacteristic
557C2#3303711yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1988C2semidirect
4. #13821C2semidirect
5. #11384C2semidirect
6. #11255C2semidirect
7. #11219C2semidirect
11. #10806C2semidirect
78. C2×#102Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#19882
C2×#19252
#151442
#138212
#113842
#112552
#112192
#110762
#110662
#109312
#108062