Finite Group Databasebeta

← #370090

Subgroups — #370090

Order 512.

1790 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#107C211yesmaximal, characteristic
2C2×#1947C212yesmaximal, complemented
3C2×#1852C211yesmaximal, characteristic
4C2×#1850C211yesmaximal, characteristic
5#13880C211yesmaximal, characteristic
6C22⋊C4×C4⋊C4C211yesmaximal, characteristic
7#11324C212yesmaximal, complemented
8#11232C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107—28no—
16C22×#107—24no—
17C22×#107C2212yes—
18C22×#107C2211yescharacteristic
19C22×#107C2212yes—
20C22×#107C2211yescharacteristic
21C22×C4⋊C8C2212yes—
22C22×C22⋊C8C2211yescharacteristic
23C2×#385—21no—
24C2×#385—21no—
25C2×#384C2212yes—
26C2×#380C2211yescharacteristic
27C2×#380C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C22⋊C4—21no—
30C2×C4×C22⋊C4C2211yescharacteristic
31#1947—28no—
32#1852—24no—
33#1850C418yescomplemented
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#336C2212yes—
41#1444—22no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C22⋊#95—21no—
46C24×C4—21no—
47C24×C4—21no—
48C24×C4C2311yescharacteristic
49C23×C8—22no—
50C23×C8—22no—
51C23×C8—24no—
52C23×C8—24no—
53C23×C8C2×C414yes—
54C23×C8C2311yescharacteristic
55C23×C8C2311yescharacteristic
56C23×C8C2312yes—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—24no—
59C22×C4⋊C4—24no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—42no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D414yes—
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—21no—
77C22×C22⋊C4D411yescharacteristic
78C22×C22⋊C4D411yescharacteristic
79C22×C42—22no—
80C22×C42C2312yes—
81C2×#107—232no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—22no—
88C2×#107—22no—
89C2×#107—24no—
90C2×#107—42no—
91C2×#107D418yes—
92C2×C4⋊C8—24no—
93C2×C4⋊C8—24no—
94C2×C4⋊C8C2×C418yescomplemented
95C2×C22⋊C8—22no—
96C2×C22⋊C8—22no—
97C2×C22⋊C8C2×C414yes—
98#385—44no—
99#385—44no—
100#384—24no—
101#384—42no—
102#380—22no—
103#380—22no—
104#380—22no—
105#380Q814yes—
106C4×C4⋊C4—24no—
107C4×C4⋊C4—42no—
108C4×C22⋊C4—44no—
109C4×C22⋊C4C2×C414yes—
110C4×C22⋊C4Q814yes—
111C2×#95—21no—
112C2×#95—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—21no—
117C2×#95—22no—
118C2×#95D412yes—
119#340—24no—
120#340—24no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—42no—
125#336C2×C418yescomplemented
126C25C2×D411yescharacteristic
127C23×C4—22no—
128C23×C4—22no—
129C23×C4—22no—
130C23×C4—22no—
131C23×C4—21no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—42no—
145C23×C4—82no—
146C23×C4—82no—
147C23×C4C22×C412yes—
148C23×C4C2×D412yes—
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D412yes—
151C23×C4C2×D412yes—
152C23×C4C2×Q812yes—
153C23×C4C2×Q812yes—
154C23×C4#4112yes—
155C23×C4#4112yes—
156C23×C4C2411yescharacteristic
157C22×C8—216no—
158C22×C8—28no—
159C22×C8—28no—
160C22×C8—28no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—216no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—22no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C22×C412yes—
175C22×C8C22×C412yes—
176C22×C8C22×C414yes—
177C22×C8C2×D414yes—
178C22×C8#4114yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—28no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—48no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—82no—
217C2×C4⋊C4—82no—
218C2×C4⋊C4C2×D414yes—
219C2×C4⋊C4C2×D412yes—
220C2×C4⋊C4C2×D414yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×Q814yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—22no—
225C2×C22⋊C4—24no—
226C2×C22⋊C4—42no—
227C2×C22⋊C4—42no—
228C2×C22⋊C4—42no—
229C2×C22⋊C4—42no—
230C2×C22⋊C4—42no—
231C2×C22⋊C4C4⋊C414yes—
232C2×C22⋊C4C4⋊C414yes—
233C2×C22⋊C4D814yes—
234C2×C22⋊C4Q1614yes—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—44no—
240C2×C42—44no—
241C2×C42—82no—
242C2×C42C22×C414yes—
243C2×C42C2×Q814yes—
244#107—216no—
245#107—216no—
246#107—44no—
247#107—44no—
248C4⋊C8—216no—
249C22⋊C8—28no—
250#95—24no—
251#95—44no—
252#95—44no—
253#95—42no—
254#95—44no—
255#95—44no—
256#95—82no—
257#95C4⋊C418yes—
258C24—22no—
259C24—22no—
260C24—22no—
261C24—22no—
262C24C2×C4⋊C412yes—
263C24#12112yes—
264C24C22⋊Q812yes—
265C24C2×D812yes—
266C24C2×Q1612yes—
267C24AGL(1,ℤ/8)12yes—
268C24#13712yes—
269C24C22×D411yescharacteristic
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—24no—
300C22×C4—24no—
301C22×C4—48no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—48no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—48no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—84no—
334C22×C4—88no—
335C22×C4—88no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—84no—
339C22×C4—84no—
340C22×C4C2×C4⋊C414yes—
341C22×C4C2×C4⋊C412yes—
342C22×C4C2×C4⋊C414yes—
343C22×C4C4×D414yes—
344C22×C4C4×Q814yes—
345C22×C4#12112yes—
346C22×C4C22⋊Q812yes—
347C22×C4#12514yes—
348C22×C4C4⋊Q814yes—
349C22×C4C2×D814yes—
350C22×C4C2×Q1614yes—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C22×D412yes—
354C22×C4C22×Q811yescharacteristic
355C22×C4C22×Q811yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4112yes—
358C22×C4C2×#4111yescharacteristic
359C22×C4H(5,𝔽2)12yes—
360C22×C42−1+412yes—
361C2×C8—232no—
362C2×C8—24no—
363C2×C8—24no—
364C2×C8—24no—
365C2×C8—24no—
366C2×C8—24no—
367C2×C8—24no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—24no—
371C2×C8—24no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—42no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—42no—
378C2×C8C4×D418yes—
379C4⋊C4—22no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—24no—
387C4⋊C4—48no—
388C4⋊C4—48no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—42no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—88no—
401C4⋊C4—88no—
402C4⋊C4#12014yes—
403C22⋊C4—28no—
404C22⋊C4—48no—
405C22⋊C4—84no—
406C22⋊C4#107116yescomplemented
407C42—24no—
408C42—24no—
409C42—24no—
410C42—24no—
411C42—88no—
412C23—24no—
413C23—24no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23—22no—
420C23—22no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—24no—
425C23—24no—
426C23—22no—
427C23—22no—
428C23#38914yes—
429C23C2×#10714yes—
430C23#42814yes—
431C23#45914yes—
432C23#46214yes—
433C23#47414yes—
434C23#47714yes—
435C23C22×C4⋊C411yescharacteristic
436C23C2×#12111yescharacteristic
437C23C2×C22⋊Q811yescharacteristic
438C23D4212yes—
439C23#54712yes—
440C23C22×D811yescharacteristic
441C23C22×Q1611yescharacteristic
442C23C2×AGL(1,ℤ/8)11yescharacteristic
443C23C2×#13711yescharacteristic
444C23#57612yes—
445C23#57812yes—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—24no—
458C2×C4—28no—
459C2×C4—24no—
460C2×C4—28no—
461C2×C4—24no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—44no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—44no—
475C2×C4—42no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—44no—
483C2×C4—816no—
484C2×C4—816no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—82no—
489C2×C4—82no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4—82no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—82no—
500C2×C4—82no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4#38914yes—
506C2×C4C2×#10718yescomplemented
507C2×C4C2×C4×D411yescharacteristic
508C2×C4C2×C4×D412yes—
509C2×C4C2×C4×Q811yescharacteristic
510C2×C4#52012yes—
511C2×C4C2×#12011yescharacteristic
512C2×C4#52511yescharacteristic
513C2×C4#52512yes—
514C2×C4C2×#12511yescharacteristic
515C2×C4C2×C4⋊Q811yescharacteristic
516C2×C4#53411yescharacteristic
517C2×C4#53611yescharacteristic
518C2×C4#53912yes—
519C2×C4#54612yes—
520C2×C4D4×Q812yes—
521C2×C4D4×Q812yes—
522C2×C4#55212yes—
523C2×C4#55412yes—
524C2×C4#55412yes—
525C8—28no—
526C8—42no—
527C8—42no—
528C22—24no—
529C22—24no—
530C22—24no—
531C22—24no—
532C22—28no—
533C22—28no—
534C22—24no—
535C22—24no—
536C22—28no—
537C22—28no—
538C22#194718yescomplemented
539C22C2×#38911yescharacteristic
540C22D4×C4⋊C412yes—
541C22#241011yescharacteristic
542C22#241311yescharacteristic
543C22#244912yes—
544C22C22×#10711yescharacteristic
545C22C2×#42811yescharacteristic
546C22#291312yes—
547C22C2×#45911yescharacteristic
548C22C2×#46211yescharacteristic
549C22#304111yescharacteristic
550C22#304411yescharacteristic
551C22C2×#47411yescharacteristic
552C22C2×#47711yescharacteristic
553C22#308312yes—
554C22#326412yes—
555C22#327012yes—
556C22D4×D812yes—
557C22D4×Q1612yes—
558C22#333412yes—
559C22#333812yes—
560C22#334312yes—
561C22#334612yes—
562C4—28no—
563C4—24no—
564C4—44no—
565C4—44no—
566C4—84no—
567C4—82no—
568C4—84no—
569C4—161no—
570C4—161no—
571C4C4×#12011yescharacteristic
572C4#232211yescharacteristic
573C4#236012yes—
574C4#243411yescharacteristic
575C4#247312yes—
576C4#258311yescharacteristic
577C2—28no—
578C2#1404311yescharacteristic
579C2C2×#194711yescharacteristic
580C2#2117311yescharacteristic
581C2D4×#10712yes—
582C2#2266112yes—
583C2#2435411yescharacteristic
584C2#2436011yescharacteristic
585C2#2457611yescharacteristic
586C2#2458211yescharacteristic
587C2#2633612yes—
588C2#2634212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×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×#1947C2semidirect
7. #11324C2semidirect
8. #11232C2semidirect
9. #10907C2semidirect
33. #1850C4semidirect
94. C2×C4⋊C8C2×C4semidirect
125. #336C2×C4semidirect
406. C22⋊C4#107semidirect
506. C2×C4C2×#107semidirect
538. C22#1947semidirect

Maximal subgroups

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

subgroupindex
C23×#1072
C2×#19472
C2×#18522
C2×#18502
#138802
C22⋊C4×C4⋊C42
#113242
#112322
#109072
#107302
#107212