Finite Group Databasebeta

← #414556

Subgroups — #414556

Order 512.

1352 conjugacy classes of subgroups.

The subgroup lattice has 565 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#446C211yesmaximal, characteristic, complemented
2C2×#1936C212yesmaximal
3C2×#1934C211yesmaximal, characteristic
4#15158C211yesmaximal, characteristic, complemented
5#15157C211yesmaximal, characteristic
6#11228C212yesmaximal, complemented
7#11227C211yesmaximal, characteristic, complemented
8#11076C212yesmaximal
9#11076C211yesmaximal, characteristic
10#10821C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C22×C4×Q8C2211yescharacteristic
13C2×#446—28no—
14C2×#446—24no—
15C22×C4⋊C8C2211yescharacteristic
16C22×C4⋊C8C2212yes—
17C22×#102C2211yescharacteristic
18C22×#102C2212yes—
19C22×#102C2211yescharacteristic
20C22×#97C2212yes—
21C22×C4×C8C2211yescharacteristic
22C2×#395—21no—
23C2×#393—21no—
24C2×#390C2211yescharacteristic
25C2×#390C2212yes—
26C2×#389C2211yescharacteristic
27C2×#389C2212yes—
28#1936—28no—
29#1934—24no—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
36C2×#340C2212yes—
37#1441—22no—
38#1441—21no—
39#1439—22no—
40#1439—21no—
41C4⋊#95C2211yescharacteristic
42C4⋊#95C2212yes—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2212yes—
45C23×Q8D411yescharacteristic
46C23×D4C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4×Q8—24no—
51C2×C4×Q8—22no—
52C2×C4×Q8—24no—
53C2×C4×Q8—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4—22no—
64C22×C22⋊C4—21no—
65C22×C22⋊C4—21no—
66C22×C22⋊C4—22no—
67C22×C42C2311yescharacteristic
68C22×C42C2312yes—
69#446—232no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—24no—
75C2×#102—22no—
76C2×#102—24no—
77C2×#102—22no—
78C2×#102—24no—
79C2×#102—22no—
80C2×#102—24no—
81C2×#102—22no—
82C2×#102—24no—
83C2×#97—24no—
84C2×#97—24no—
85C2×#97D418yes—
86C2×C4×C8—24no—
87C2×C4×C8D414yes—
88#395—44no—
89#393—44no—
90#390—22no—
91#390—24no—
92#390—22no—
93#390—24no—
94#389—22no—
95#389—24no—
96#389—22no—
97#389—24no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—21no—
107C2×#95—22no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115C25—21no—
116C22×Q8—24no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8—42no—
121C22×D4—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—22no—
125C22×D4—21no—
126C22×D4—22no—
127C22×D4—21no—
128C22×D4—22no—
129C22×D4—21no—
130C22×D4#4112yes—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4C2×D412yes—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D412yes—
139C23×C4C2×D411yescharacteristic
140C23×C4C2411yescharacteristic
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—21no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8C2×D412yes—
155C22×C8C2×D414yes—
156C22×C8C2×D414yes—
157C22×C8C2×D412yes—
158C4×Q8—28no—
159C4×Q8—216no—
160C4×Q8—28no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4#4112yes—
197C2×C4⋊C4#4114yes—
198C2×C22⋊C4—44no—
199C2×C22⋊C4—44no—
200C2×C22⋊C4—44no—
201C2×C22⋊C4—42no—
202C2×C22⋊C4—44no—
203C2×C22⋊C4—42no—
204C2×C22⋊C4—42no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4—82no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42C2×D412yes—
216C2×C42C2×D414yes—
217C2×C42#4114yes—
218C2×C42#4112yes—
219C4⋊C8—28no—
220C4⋊C8—216no—
221#102—28no—
222#102—216no—
223#102—44no—
224#97—216no—
225C4×C8—28no—
226#95—44no—
227#95—44no—
228#95—42no—
229#95—44no—
230#95—42no—
231#95—42no—
232#95—81no—
233#95—81no—
234#95—82no—
235#95—82no—
236C24—22no—
237C24—41no—
238C24—42no—
239C24—41no—
240C24—42no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—41no—
245C24—42no—
246C24—41no—
247C24C22×D411yescharacteristic
248C2×Q8—28no—
249C2×Q8—48no—
250C2×Q8—48no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×D4—22no—
254C2×D4—24no—
255C2×D4—22no—
256C2×D4—24no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—42no—
260C2×D4—41no—
261C2×D4—42no—
262C2×D4—41no—
263C2×D4—42no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—42no—
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—22no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4#12112yes—
321C22×C4#12114yes—
322C22×C4#12114yes—
323C22×C4#12112yes—
324C22×C4AGL(1,ℤ/8)12yes—
325C22×C4AGL(1,ℤ/8)12yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×D412yes—
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4C2×#4112yes—
333C22×C42−1+411yescharacteristic
334C22×C42−1+411yescharacteristic
335C22×C42−1+412yes—
336C2×C8—24no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—22no—
340C2×C8—22no—
341C2×C8—24no—
342C2×C8—22no—
343C2×C8—24no—
344C2×C8—22no—
345C2×C8—24no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8#12014yes—
353C4⋊C4—28no—
354C4⋊C4—216no—
355C4⋊C4—22no—
356C4⋊C4—24no—
357C4⋊C4—22no—
358C4⋊C4—24no—
359C4⋊C4—41no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—42no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—42no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C4⋊C4—42no—
373C4⋊C4—88no—
374C4⋊C4—84no—
375C22⋊C4—88no—
376C22⋊C4—84no—
377C22⋊C4—88no—
378C22⋊C4—84no—
379C42—22no—
380C42—24no—
381C42—22no—
382C42—24no—
383C42—24no—
384C42—24no—
385C42—24no—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—42no—
390C23—44no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—82no—
395C23—82no—
396C23—82no—
397C23—82no—
398C23—82no—
399C23—81no—
400C23—81no—
401C23—82no—
402C23—82no—
403C23—81no—
404C23—82no—
405C23—81no—
406C23—81no—
407C23—82no—
408C23—82no—
409C23C2×#12111yescharacteristic
410C23C2×#12112yes—
411C23#54811yescharacteristic
412C23#54811yescharacteristic
413C23#54812yes—
414C23C2×AGL(1,ℤ/8)11yescharacteristic
415C23C2×AGL(1,ℤ/8)11yescharacteristic
416C23#57512yes—
417C23#57611yescharacteristic
418C23#57712yes—
419C23#57811yescharacteristic
420Q8—416no—
421D4—42no—
422D4—44no—
423D4—42no—
424D4—81no—
425D4—82no—
426D4—81no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—24no—
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—42no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—44no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—82no—
460C2×C4—81no—
461C2×C4—82no—
462C2×C4—81no—
463C2×C4—82no—
464C2×C4—81no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—82no—
481C2×C4#46914yes—
482C2×C4C2×#12011yescharacteristic
483C2×C4#52511yescharacteristic
484C2×C4#52512yes—
485C2×C4#52511yescharacteristic
486C2×C4#52512yes—
487C2×C4#53211yescharacteristic
488C2×C4#53212yes—
489C2×C4#53611yescharacteristic
490C2×C4#53612yes—
491C2×C4#53611yescharacteristic
492C2×C4#53612yes—
493C2×C4#55211yescharacteristic
494C2×C4#55212yes—
495C2×C4#55211yescharacteristic
496C2×C4#55212yes—
497C2×C4#55612yes—
498C2×C4#55612yes—
499C2×C4#55612yes—
500C8—24no—
501C8—42no—
502C8—44no—
503C22—82no—
504C22—82no—
505C22—84no—
506C22—84no—
507C22—82no—
508C22—161no—
509C22—161no—
510C22—161no—
511C22—161no—
512C22—162no—
513C22—162no—
514C22#253611yescharacteristic
515C22#253612yes—
516C22#254511yescharacteristic
517C22#254512yes—
518C22#284711yescharacteristic
519C22C2×#46911yescharacteristic
520C22#305411yescharacteristic
521C22#305512yes—
522C22#305512yes—
523C22#305711yescharacteristic
524C22#306111yescharacteristic
525C22#306212yes—
526C22#306312yes—
527C22#336611yescharacteristic
528C22#336611yescharacteristic
529C22#336711yescharacteristic
530C22#336711yescharacteristic
531C22#336812yes—
532C22#337012yes—
533C22#337112yes—
534C22#337312yes—
535C22#337612yes—
536C22#337712yes—
537C4—24no—
538C4—44no—
539C4—42no—
540C4—44no—
541C4—82no—
542C4—84no—
543C4—161no—
544C4—161no—
545C4—162no—
546C4#238811yescharacteristic
547C4#259911yescharacteristic
548C4#259912yes—
549C4#260411yescharacteristic
550C4#260412yes—
551C4#261411yescharacteristic
552C2—162no—
553C2#1517611yescharacteristic
554C2#2819211yescharacteristic
555C2#2819411yescharacteristic
556C2#2819712yes—
557C2#2819912yes—
558C2#2864311yescharacteristic
559C2#2864811yescharacteristic
560C2#2864912yes—
561C2#2865012yes—
562C2#3293611yescharacteristic
563C2#3293811yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C22×#446C2semidirect
4. #15158C2semidirect
6. #11228C2semidirect
7. #11227C2semidirect
10. #10821C2semidirect
11. #10820C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4462
C2×#19362
C2×#19342
#151582
#151572
#112282
#112272
#110762
#108212
#108202