Finite Group Databasebeta

← #399597

Subgroups — #399597

Order 512.

1466 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#427C211yesmaximal, characteristic
2C2×#2087C211yesmaximal, characteristic, complemented
3C2×#2086C211yesmaximal, characteristic, complemented
4C2×#1850C211yesmaximal, characteristic
5#14129C212yesmaximal
6#11379C211yesmaximal, characteristic, complemented
7#11377C211yesmaximal, characteristic, complemented
8#11327C212yesmaximal, complemented
9#11231C212yesmaximal, complemented
10#10731C212yesmaximal
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C22×#117C2212yes—
14C2×#427—24no—
15C2×#427—24no—
16C2×#427—24no—
17C22×#107C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C22×#106C2212yes—
20C22×C4⋊C8C2211yescharacteristic
21C22×C4⋊C8C2211yescharacteristic
22C22×C22⋊C8C2211yescharacteristic
23C2×#389C2212yes—
24C2×#389C2212yes—
25C2×#380C2212yes—
26C2×C4×C22⋊C4C2211yescharacteristic
27#2087—24no—
28#2086—24no—
29#1850—24no—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2211yescharacteristic
38C2×#336C2211yescharacteristic
39#1439—22no—
40#1439—22no—
41#1436—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C24×C4C2311yescharacteristic
46C23×C8—24no—
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—24no—
50C23×C8D414yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2311yescharacteristic
55C2×#117—24no—
56C2×#117—24no—
57C2×#117—24no—
58C2×#117—24no—
59C2×#117—42no—
60C2×#117—42no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2312yes—
66C22×C4⋊C4C2312yes—
67C22×C22⋊C4—22no—
68C22×C22⋊C4—22no—
69C22×C22⋊C4D411yescharacteristic
70C22×C22⋊C4D411yescharacteristic
71C22×C42—22no—
72C22×C42C2311yescharacteristic
73C22×C42C2311yescharacteristic
74#427—232no—
75C2×#107—22no—
76C2×#107—22no—
77C2×#107—22no—
78C2×#107—22no—
79C2×#107D414yes—
80C2×#107D414yes—
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8—22no—
87C2×C4⋊C8—22no—
88C2×C4⋊C8—24no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—24no—
91#389—24no—
92#389—24no—
93#389—42no—
94#389—42no—
95#380—24no—
96#380Q818yes—
97C4×C22⋊C4—22no—
98C4×C22⋊C4—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D411yescharacteristic
107C2×#95D411yescharacteristic
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#336—22no—
116#336—22no—
117C25C2×D411yescharacteristic
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D412yes—
134C23×C4C2×Q814yes—
135C23×C4#4112yes—
136C23×C4#4114yes—
137C23×C4C2411yescharacteristic
138C22×C8—216no—
139C22×C8—216no—
140C22×C8—28no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—28no—
147C22×C8—28no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8C2×D412yes—
159C22×C8C2×D412yes—
160C22×C8C2×D412yes—
161C22×C8C2×D412yes—
162C22×C8#4114yes—
163#117—216no—
164#117—48no—
165#117—48no—
166#117—48no—
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—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—82no—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4C2×D414yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—28no—
202C2×C22⋊C4—42no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—82no—
207C2×C22⋊C4—82no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—22no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—22no—
218C2×C42—44no—
219C2×C42—44no—
220C2×C42—44no—
221#107—28no—
222#107—28no—
223#106—216no—
224C4⋊C8—44no—
225C4⋊C8—44no—
226C22⋊C8—44no—
227#95—22no—
228#95—22no—
229#95—28no—
230#95—44no—
231#95—44no—
232#95—44no—
233#95—44no—
234#95—82no—
235#95—82no—
236C24—21no—
237C24—21no—
238C24—22no—
239C24—21no—
240C24—21no—
241C24—22no—
242C24—21no—
243C24—22no—
244C24—21no—
245C24#12112yes—
246C24C22⋊Q814yes—
247C24C22×D411yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
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—42no—
313C22×C4—42no—
314C22×C4#12112yes—
315C22×C4C22⋊Q814yes—
316C22×C4#12314yes—
317C22×C4#12518yes—
318C22×C4C4⋊Q814yes—
319C22×C4#13512yes—
320C22×C4#13512yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q812yes—
325C22×C4C2×#4112yes—
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C42−1+412yes—
331C2×C8—232no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8#12014yes—
351C2×C8#12314yes—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—24no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—88no—
369C4⋊C4—88no—
370C22⋊C4—28no—
371C22⋊C4—44no—
372C22⋊C4—416no—
373C22⋊C4—88no—
374C42—416no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—41no—
379C42—41no—
380C42—41no—
381C42—42no—
382C23—24no—
383C23—24no—
384C23—24no—
385C23—24no—
386C23—22no—
387C23—24no—
388C23—24no—
389C23—22no—
390C23—22no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#39814yes—
405C23C2×#12111yescharacteristic
406C23C2×C22⋊Q812yes—
407C23D4211yescharacteristic
408C23D4211yescharacteristic
409C23#54712yes—
410C23C2×#13511yescharacteristic
411C23C2×#13511yescharacteristic
412C23#57512yes—
413C23#57611yescharacteristic
414C23#57712yes—
415C23#57811yescharacteristic
416C2×C4—22no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—42no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—41no—
447C2×C4—44no—
448C2×C4—44no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—44no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—82no—
461C2×C4—82no—
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—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4#39814yes—
477C2×C4C2×#12011yescharacteristic
478C2×C4C2×#12311yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4#52512yes—
481C2×C4#52512yes—
482C2×C4C2×#12512yes—
483C2×C4C2×C4⋊Q811yescharacteristic
484C2×C4#53511yescharacteristic
485C2×C4#53611yescharacteristic
486C2×C4#53912yes—
487C2×C4#53912yes—
488C2×C4#54611yescharacteristic
489C2×C4#54611yescharacteristic
490C2×C4D4×Q812yes—
491C2×C4D4×Q812yes—
492C2×C4#55212yes—
493C2×C4#55412yes—
494C2×C4#55412yes—
495C8—24no—
496C8—24no—
497C8—81no—
498C8—81no—
499C22—24no—
500C22—24no—
501C22—24no—
502C22—42no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—44no—
507C22—44no—
508C22—44no—
509C22—44no—
510C22C2×#39811yescharacteristic
511C22#240811yescharacteristic
512C22#244912yes—
513C22#244912yes—
514C22#248611yescharacteristic
515C22#305011yescharacteristic
516C22#305411yescharacteristic
517C22#306011yescharacteristic
518C22#306111yescharacteristic
519C22#307612yes—
520C22#307612yes—
521C22#308112yes—
522C22#308212yes—
523C22#326712yes—
524C22#327112yes—
525C22#328011yescharacteristic
526C22#328311yescharacteristic
527C22#328511yescharacteristic
528C22#328711yescharacteristic
529C22#333712yes—
530C22#334012yes—
531C22#334712yes—
532C22#334912yes—
533C4—24no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—84no—
538C4—81no—
539C4—84no—
540C4—84no—
541C4—84no—
542C4#243411yescharacteristic
543C4#247312yes—
544C4#247312yes—
545C4#250211yescharacteristic
546C4#262311yescharacteristic
547C4#284911yescharacteristic
548C2—44no—
549C2#1478611yescharacteristic
550C2#2366611yescharacteristic
551C2#2366911yescharacteristic
552C2#2426611yescharacteristic
553C2#2427111yescharacteristic
554C2#2633912yes—
555C2#2634112yes—
556C2#2634712yes—
557C2#2634912yes—
558C2#2765911yescharacteristic
559C2#2766411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2087C2semidirect
3. C2×#2086C2semidirect
6. #11379C2semidirect
7. #11377C2semidirect
8. #11327C2semidirect
9. #11231C2semidirect

Maximal subgroups

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

subgroupindex
C22×#4272
C2×#20872
C2×#20862
C2×#18502
#141292
#113792
#113772
#113272
#112312
#107312
#107212