Finite Group Databasebeta

← #189541

Subgroups — #189541

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 658 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2929C211yesmaximal, characteristic
2C2×#1810C211yesmaximal, characteristic, complemented
3C2×#1810C211yesmaximal, characteristic, complemented
4#17400C212yesmaximal
5#8613C214yesmaximal, complemented
6#8612C214yesmaximal, complemented
7#8009C212yesmaximal, complemented
8C22×#131C411yescharacteristic
9C24×C8C2211yescharacteristic
10C2×#525C2211yescharacteristic
11#2929—21no—
12#2929—21no—
13#2929—21no—
14#2929—21no—
15#2929—21no—
16#2929—21no—
17#2929—21no—
18#2929—21no—
19C2×#435C412yes—
20C2×#435C2212yes—
21C2×C8×D4C412yes—
22C2×C8×D4C2212yes—
23C22×#108—21no—
24C22×#108—21no—
25C2×#424C411yescharacteristic
26C2×#408C411yescharacteristic
27C2×#407C2212yes—
28C2×#407C2212yes—
29C2×#407C2211yescharacteristic
30C2×#405C411yescharacteristic
31#1810—21no—
32#1810—21no—
33#1810—21no—
34#1810—21no—
35#1810C2212yes—
36#1810C2212yes—
37#1810C2212yes—
38#1810C2212yes—
39C2×#341—21no—
40C2×#341—21no—
41#1472—22no—
42#1471—22no—
43#1468—24no—
44#1465C2218yes—
45#1293C2218yes—
46C22×#41C2×C411yescharacteristic
47C24×C4C2311yescharacteristic
48C2×#131—21no—
49C2×#131—21no—
50C2×#131—21no—
51C2×#131—21no—
52C2×#131—44no—
53C22×M16—22no—
54C22×M16—22no—
55C22×M16C2×C411yescharacteristic
56C22×M16C2×C411yescharacteristic
57C22×M16C2×C412yes—
58C23×C8—44no—
59C23×C8C2×C411yescharacteristic
60C23×C8C2×C412yes—
61C23×C8C2×C411yescharacteristic
62C23×C8C2×C411yescharacteristic
63C23×C8C2×C411yescharacteristic
64C23×C8D411yescharacteristic
65C23×C8D411yescharacteristic
66C23×C8D411yescharacteristic
67C23×C8D411yescharacteristic
68C23×C8D411yescharacteristic
69C23×C8D411yescharacteristic
70C23×C8D411yescharacteristic
71C23×C8D411yescharacteristic
72C23×C8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74#525—21no—
75#525—21no—
76#525—21no—
77#525—21no—
78#525—21no—
79#525—21no—
80#525—21no—
81#525—21no—
82C2×#123D412yes—
83C2×C22⋊Q8D411yescharacteristic
84C2×C22⋊Q8Q811yescharacteristic
85C2×#121D411yescharacteristic
86C2×#121Q811yescharacteristic
87C2×#120Q812yes—
88C2×C4×D4C2×C412yes—
89C2×C4×D4C2312yes—
90C2×#117C2×C411yescharacteristic
91#435—22no—
92#435—22no—
93#435—22no—
94#435—22no—
95#435—22no—
96#435—22no—
97#435—22no—
98#435—22no—
99C8×D4—22no—
100C8×D4—22no—
101C8×D4—22no—
102C8×D4—22no—
103C8×D4—22no—
104C8×D4—22no—
105C8×D4—22no—
106C8×D4—22no—
107C2×#108—22no—
108C2×#108—22no—
109C2×#108—22no—
110C2×#108—22no—
111C2×#108—44no—
112C2×#108—44no—
113#424—21no—
114#424—21no—
115#424—21no—
116#424—21no—
117C2×C4⋊C8C2×C412yes—
118C2×C4⋊C8C2×C412yes—
119#408—21no—
120#408—21no—
121#408—21no—
122#408—21no—
123#407—21no—
124#407—21no—
125#407—21no—
126#407—21no—
127#407—22no—
128#407—22no—
129#407—22no—
130#407—22no—
131#407—22no—
132#407—22no—
133#407C2314yes—
134#407C2314yes—
135C2×C22⋊C8—22no—
136C2×C22⋊C8—22no—
137C2×C22⋊C8—22no—
138C2×C22⋊C8—22no—
139C2×C22⋊C8C2×C412yes—
140C2×C22⋊C8C2×C412yes—
141C2×C22⋊C8C2×C412yes—
142C2×C22⋊C8C2312yes—
143#405—21no—
144#405—21no—
145#405—21no—
146#405—21no—
147C2×#97C2×C412yes—
148C2×C4×C8C2×C412yes—
149#341—22no—
150#341—22no—
151#341—22no—
152#341—22no—
153#324—28no—
154#323—28no—
155C25C2×Q811yescharacteristic
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—42no—
161C2×#41—42no—
162C22×Q8C4⋊C411yescharacteristic
163C22×D4C4⋊C412yes—
164C22×D4C4⋊C412yes—
165C22×D4C4⋊C411yescharacteristic
166C22×D4C2×Q812yes—
167C23×C4—42no—
168C23×C4—42no—
169C23×C4C22×C411yescharacteristic
170C23×C4C22×C412yes—
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×Q811yescharacteristic
179C23×C4#4111yescharacteristic
180C23×C4#4111yescharacteristic
181C23×C4#4111yescharacteristic
182C23×C4#4111yescharacteristic
183C23×C4C2411yescharacteristic
184#131—44no—
185#131—44no—
186#131—44no—
187#131—44no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—22no—
193C2×M16—22no—
194C2×M16—24no—
195C2×M16—24no—
196C2×M16—44no—
197C2×M16—42no—
198C2×M16—42no—
199C2×M16—42no—
200C2×M16—42no—
201C2×M16—44no—
202C2×M16—44no—
203C2×M16—44no—
204C2×M16—44no—
205C22×C8—24no—
206C22×C8—24no—
207C22×C8—24no—
208C22×C8—22no—
209C22×C8—22no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8—42no—
213C22×C8—42no—
214C22×C8—42no—
215C22×C8—42no—
216C22×C8—42no—
217C22×C8—42no—
218C22×C8—42no—
219C22×C8—42no—
220C22×C8—42no—
221C22×C8—44no—
222C22×C8—42no—
223C22×C8—42no—
224C22×C8C22⋊C411yescharacteristic
225C22×C8C22⋊C411yescharacteristic
226C22×C8C22⋊C411yescharacteristic
227C22×C8C22⋊C411yescharacteristic
228C22×C8C22⋊C411yescharacteristic
229C22×C8C22⋊C411yescharacteristic
230C22×C8C22⋊C411yescharacteristic
231C22×C8C22⋊C411yescharacteristic
232C22×C8C22⋊C411yescharacteristic
233C22×C8C22⋊C411yescharacteristic
234C22×C8C22⋊C411yescharacteristic
235C22×C8C22⋊C411yescharacteristic
236C22×C8C22⋊C411yescharacteristic
237C22×C8C22⋊C411yescharacteristic
238C22×C8C22⋊C411yescharacteristic
239C22×C8C22⋊C411yescharacteristic
240C22×C8C22×C411yescharacteristic
241C22×C8C22×C411yescharacteristic
242C22×C8C22×C412yes—
243C22×C8C22×C412yes—
244C22×C8C22×C411yescharacteristic
245C22×C8C22×C411yescharacteristic
246C22×C8C2×D411yescharacteristic
247C22×C8C2×D411yescharacteristic
248C22×C8C2×D411yescharacteristic
249C22×C8C2×D411yescharacteristic
250C22×C8C2×D411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254#123—22no—
255#123—22no—
256#123—22no—
257#123—22no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260C22⋊Q8—21no—
261C22⋊Q8—21no—
262C22⋊Q8—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266#121—21no—
267#121—21no—
268#121—21no—
269#121—21no—
270#121—21no—
271#121—21no—
272#121—21no—
273#121—21no—
274#120—22no—
275#120—22no—
276#120—22no—
277#120—22no—
278C4×D4—22no—
279C4×D4—22no—
280C4×D4—22no—
281C4×D4—22no—
282C4×D4—22no—
283C4×D4—22no—
284C4×D4—22no—
285C4×D4—22no—
286#117—21no—
287#117—21no—
288#117—21no—
289#117—21no—
290C2×C4⋊C4C4⋊C411yescharacteristic
291C2×C4⋊C4C4⋊C411yescharacteristic
292C2×C4⋊C4C4⋊C412yes—
293C2×C4⋊C4C2×D412yes—
294C2×C22⋊C4C4⋊C412yes—
295C2×C22⋊C4C4⋊C412yes—
296C2×C22⋊C4C4⋊C412yes—
297C2×C22⋊C4C2×D412yes—
298C2×C22⋊C4C2×Q812yes—
299C2×C42C22×C412yes—
300#108—44no—
301#108—44no—
302#108—44no—
303#108—44no—
304C4⋊C8—22no—
305C4⋊C8—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C22⋊C8—22no—
309C22⋊C8—22no—
310C22⋊C8—22no—
311C22⋊C8—22no—
312C22⋊C8—22no—
313C22⋊C8—22no—
314C22⋊C8—22no—
315C22⋊C8—22no—
316C22⋊C8—24no—
317C22⋊C8—24no—
318C22⋊C8—24no—
319C22⋊C8—24no—
320C22⋊C8—42no—
321C22⋊C8—42no—
322C22⋊C8—42no—
323C22⋊C8—42no—
324#97—22no—
325#97—22no—
326C4×C8—22no—
327C4×C8—22no—
328C24—42no—
329C24—42no—
330C24—42no—
331C24C2×C4⋊C411yescharacteristic
332C24C2×C4⋊C411yescharacteristic
333C24C2×C4⋊C412yes—
334C24C22⋊Q811yescharacteristic
335C24C22⋊Q811yescharacteristic
336C24C22⋊Q811yescharacteristic
337C24C22⋊Q811yescharacteristic
338C24C22×Q811yescharacteristic
339#41—42no—
340#41—44no—
341#41—44no—
342#41—42no—
343#41—42no—
344#41—42no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—42no—
348C2×D4—22no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—22no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—44no—
357C2×D4—42no—
358C2×D4—44no—
359C2×D4—44no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—42no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—42no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—41no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4C2×C22⋊C411yescharacteristic
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C4⋊C412yes—
390C22×C4C2×C4⋊C411yescharacteristic
391C22×C4C2×C4⋊C412yes—
392C22×C4C2×C4⋊C412yes—
393C22×C4C2×C4⋊C411yescharacteristic
394C22×C4#11711yescharacteristic
395C22×C4#11711yescharacteristic
396C22×C4#11711yescharacteristic
397C22×C4#11711yescharacteristic
398C22×C4#12111yescharacteristic
399C22×C4#12111yescharacteristic
400C22×C4#12111yescharacteristic
401C22×C4#12111yescharacteristic
402C22×C4#12111yescharacteristic
403C22×C4#12111yescharacteristic
404C22×C4#12111yescharacteristic
405C22×C4#12111yescharacteristic
406C22×C4C22⋊Q811yescharacteristic
407C22×C4C22⋊Q811yescharacteristic
408C22×C4C22⋊Q811yescharacteristic
409C22×C4C22⋊Q811yescharacteristic
410C22×C4C23×C411yescharacteristic
411C22×C4C22×D411yescharacteristic
412C22×C4C22×D411yescharacteristic
413C22×C4C22×D411yescharacteristic
414C22×C4C2×#4111yescharacteristic
415C22×C4C2×#4111yescharacteristic
416M16—44no—
417M16—42no—
418M16—42no—
419M16—42no—
420M16—42no—
421M16—44no—
422M16—44no—
423M16—44no—
424M16—44no—
425M16—44no—
426M16—84no—
427M16—84no—
428C2×C8—22no—
429C2×C8—22no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—22no—
433C2×C8—22no—
434C2×C8—22no—
435C2×C8—22no—
436C2×C8—22no—
437C2×C8—22no—
438C2×C8—22no—
439C2×C8—22no—
440C2×C8—22no—
441C2×C8—22no—
442C2×C8—22no—
443C2×C8—22no—
444C2×C8—22no—
445C2×C8—41no—
446C2×C8—41no—
447C2×C8—41no—
448C2×C8—41no—
449C2×C8—41no—
450C2×C8—41no—
451C2×C8—41no—
452C2×C8—41no—
453C2×C8—41no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—44no—
458C2×C8—44no—
459C2×C8—42no—
460C2×C8—42no—
461C2×C8—44no—
462C2×C8—44no—
463C2×C8—84no—
464C2×C8—84no—
465C2×C8#9914yes—
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C2×C22⋊C411yescharacteristic
473C2×C8C2×C22⋊C411yescharacteristic
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—22no—
481C4⋊C4—22no—
482C22⋊C4—22no—
483C22⋊C4—22no—
484C22⋊C4—22no—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—22no—
488C22⋊C4—22no—
489C22⋊C4—22no—
490C22⋊C4—22no—
491C22⋊C4—22no—
492C42—22no—
493C42—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—22no—
500C23—22no—
501C23—22no—
502C23—22no—
503C23—22no—
504C23—24no—
505C23—41no—
506C23—41no—
507C23—41no—
508C23—41no—
509C23—41no—
510C23—41no—
511C23—41no—
512C23—41no—
513C23—41no—
514C23—41no—
515C23—41no—
516C23—41no—
517C23—42no—
518C23—42no—
519C23—416no—
520C23#38011yescharacteristic
521C23#38011yescharacteristic
522C23#38011yescharacteristic
523C23#38011yescharacteristic
524C23C22×C4⋊C411yescharacteristic
525C23C2×C22⋊Q811yescharacteristic
526C23C2×C22⋊Q811yescharacteristic
527Q8—42no—
528Q8—42no—
529D4—42no—
530D4—44no—
531D4—44no—
532D4—44no—
533D4—44no—
534D4—44no—
535D4—44no—
536D4—42no—
537C2×C4—22no—
538C2×C4—22no—
539C2×C4—22no—
540C2×C4—22no—
541C2×C4—22no—
542C2×C4—22no—
543C2×C4—22no—
544C2×C4—22no—
545C2×C4—22no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—21no—
550C2×C4—21no—
551C2×C4—21no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—24no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—22no—
562C2×C4—41no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—41no—
571C2×C4—41no—
572C2×C4—41no—
573C2×C4—41no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4#38011yescharacteristic
580C2×C4#38011yescharacteristic
581C2×C4#38011yescharacteristic
582C2×C4#38011yescharacteristic
583C2×C4C2×#9912yes—
584C2×C4#41012yes—
585C2×C4C22×C22⋊C411yescharacteristic
586C2×C4C2×#11711yescharacteristic
587C2×C4C2×#12111yescharacteristic
588C2×C4C2×#12111yescharacteristic
589C8—21no—
590C8—21no—
591C8—21no—
592C8—21no—
593C8—21no—
594C8—21no—
595C8—42no—
596C8—42no—
597C8—84no—
598C8—84no—
599C8C2×#9912yes—
600C8C2×#9912yes—
601C22—21no—
602C22—21no—
603C22—21no—
604C22—21no—
605C22—21no—
606C22—21no—
607C22—21no—
608C22—21no—
609C22—21no—
610C22—21no—
611C22—21no—
612C22—21no—
613C22—21no—
614C22—21no—
615C22—21no—
616C22—21no—
617C22—21no—
618C22—21no—
619C22—44no—
620C22—44no—
621C22—44no—
622C22—44no—
623C22—44no—
624C22—44no—
625C22—44no—
626C22—42no—
627C22—42no—
628C22—42no—
629C22—84no—
630C22#179912yes—
631C22#181312yes—
632C22C2×#38011yescharacteristic
633C22#291612yes—
634C4—21no—
635C4—21no—
636C4—21no—
637C4—21no—
638C4—21no—
639C4—21no—
640C4—42no—
641C4—42no—
642C4—42no—
643C4#179912yes—
644C4C22×#9911yescharacteristic
645C4C2×#41011yescharacteristic
646C2—22no—
647C2—22no—
648C2—22no—
649C2—41no—
650C2—41no—
651C2—41no—
652C2—41no—
653C2—82no—
654C2C2×#179911yescharacteristic
655C2#2066811yescharacteristic
656C2C2×#181311yescharacteristic
subgroupG/H
center, socleC22C2×#380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1810C2semidirect
3. C2×#1810C2semidirect
5. #8613C2semidirect
6. #8612C2semidirect
7. #8009C2semidirect

Maximal subgroups

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

subgroupindex
C2×#29292
C2×#18102
#174002
#86132
#86122
#80092