Finite Group Databasebeta

← #188463

Subgroups — #188463

Order 512.

1280 conjugacy classes of subgroups.

The subgroup lattice has 627 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34245C211yesmaximal, characteristic
2#21932C211yesmaximal, characteristic
3C2×#1508C211yesmaximal, characteristic
4C22×#328C211yesmaximal, characteristic
5#13782C211yesmaximal, characteristic
6#8578C212yesmaximal, complemented
7#8576C212yesmaximal, complemented
8#8574C212yesmaximal, complemented
9#8572C212yesmaximal, complemented
10#8020C211yesmaximal, characteristic
11#8012C211yesmaximal, characteristic
12C22×C22⋊Q8C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C22×C22⋊C8C2211yescharacteristic
15#2368—22no—
16#2362C412yescomplemented
17#2362C412yescomplemented
18#2344C411yescharacteristic, complemented
19#2344C411yescharacteristic, complemented
20#2342—21no—
21#2342—21no—
22Q8×C22⋊C4C411yescharacteristic, complemented
23Q8×C22⋊C4C411yescharacteristic, complemented
24#2332—22no—
25#2313—21no—
26C4×C22⋊Q8C2212yes—
27C4×C22⋊Q8C2212yes—
28C2×#385—21no—
29#2280C2211yescharacteristic
30#2274—21no—
31C2×C4×C22⋊C4C2211yescharacteristic
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1851—22no—
35#1850—21no—
36#1850—21no—
37#1797C2212yes—
38#1797C2212yes—
39#1508—24no—
40C2×#328—24no—
41C2×#328C2212yes—
42C2×#328C2212yes—
43C2×#328C2211yescharacteristic
44C2×#328C2211yescharacteristic
45C2×#328C2211yescharacteristic
46C2×#328C2211yescharacteristic
47#1436—22no—
48C4⋊#95C2212yes—
49#1434C2212yes—
50C22⋊#95—21no—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C23×Q8C2×C411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C23×C8—21no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8—22no—
64C2×C22⋊Q8C2×C411yescharacteristic
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8D414yes—
67C2×C22⋊Q8Q814yes—
68C2×C22⋊Q8C2311yescharacteristic
69C2×C22⋊Q8C2311yescharacteristic
70C2×C4×Q8C2×C412yescomplemented
71C2×C4×Q8C2×C412yescomplemented
72C2×#117—21no—
73C2×#117—21no—
74C2×#117C2×C412yes—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C22⋊C4—21no—
79C22×C22⋊C4—21no—
80C22×C22⋊C4D411yescharacteristic
81C22×C22⋊C4D411yescharacteristic
82C22×C42—21no—
83C22×C42C2312yes—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#407—22no—
87#407—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94C2×C22⋊C8C2311yescharacteristic
95#391—22no—
96#391—22no—
97#391C2×C412yescomplemented
98#391C2×C412yescomplemented
99#389C2×C412yescomplemented
100#389C2×C412yescomplemented
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#387—22no—
106#387—22no—
107#387—22no—
108#387—22no—
109#387—22no—
110#387—22no—
111#385—28no—
112#385—22no—
113#385—22no—
114#381—21no—
115#381—21no—
116#381—21no—
117#380—21no—
118#380—21no—
119#380C2×C411yescharacteristic
120#380C2×C411yescharacteristic
121#380C2312yes—
122#379—21no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yescomplemented
128C4×C4⋊C4C2×C412yescomplemented
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—22no—
132C4×C22⋊C4—21no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—21no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137C4×C22⋊C4C2×C411yescharacteristic
138C4×C22⋊C4C2×C411yescharacteristic
139C4×C22⋊C4C2×C412yes—
140C4×C22⋊C4D411yescharacteristic
141C4×C22⋊C4D412yes—
142C4×C22⋊C4D411yescharacteristic
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—21no—
147C2×#95—22no—
148C2×#95—22no—
149C2×#95D412yes—
150C2×#95C2311yescharacteristic
151C2×#95C2312yes—
152C2×#95C2311yescharacteristic
153#336—22no—
154#336—22no—
155#328—24no—
156#328—24no—
157#328—22no—
158#328—22no—
159#328—22no—
160#328—22no—
161C25C2×D411yescharacteristic
162C22×Q8—21no—
163C22×Q8—22no—
164C22×Q8—82no—
165C22×Q8C4⋊C414yescomplemented
166C22×Q8C22×C411yescharacteristic
167C22×Q8C22×C411yescharacteristic
168C23×C4—21no—
169C23×C4—21no—
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—22no—
173C23×C4—24no—
174C23×C4—42no—
175C23×C4—42no—
176C23×C4—82no—
177C23×C4—82no—
178C23×C4—82no—
179C23×C4C22×C411yescharacteristic
180C23×C4C22×C411yescharacteristic
181C23×C4C2×D412yes—
182C23×C4C2×D412yes—
183C23×C4C2×D411yescharacteristic
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D411yescharacteristic
186C23×C4C2×D412yes—
187C23×C4C2×Q812yes—
188C23×C4#4112yes—
189C23×C4#4112yes—
190C23×C4C2411yescharacteristic
191C2×M16—42no—
192C2×M16—82no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—42no—
197C22×C8—82no—
198C22⋊Q8—22no—
199C22⋊Q8—22no—
200C22⋊Q8—24no—
201C22⋊Q8—24no—
202C22⋊Q8—24no—
203C22⋊Q8—24no—
204C22⋊Q8—21no—
205C22⋊Q8—21no—
206C22⋊Q8—21no—
207C22⋊Q8—21no—
208C22⋊Q8C4⋊C414yescomplemented
209C22⋊Q8C4⋊C414yescomplemented
210C22⋊Q8C2×D412yes—
211C22⋊Q8C2×D412yes—
212C22⋊Q8C2×Q812yes—
213C22⋊Q8C2×Q812yes—
214C4×Q8—82no—
215C4×Q8—82no—
216#117—44no—
217#117—44no—
218#117—82no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—24no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—28no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—82no—
233C2×C4⋊C4—82no—
234C2×C4⋊C4C4⋊C414yescomplemented
235C2×C4⋊C4C22×C411yescharacteristic
236C2×C4⋊C4C22×C411yescharacteristic
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—24no—
242C2×C22⋊C4—24no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—24no—
246C2×C22⋊C4—24no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—21no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—42no—
252C2×C22⋊C4—42no—
253C2×C22⋊C4—42no—
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22⋊C411yescharacteristic
256C2×C22⋊C4C22⋊C411yescharacteristic
257C2×C22⋊C4C22⋊C412yes—
258C2×C22⋊C4C22⋊C412yes—
259C2×C22⋊C4C22⋊C411yescharacteristic
260C2×C42—22no—
261C2×C42—22no—
262C2×C42—22no—
263C2×C42—22no—
264C2×C42—22no—
265C2×C42—42no—
266C2×C42—82no—
267C2×C42—82no—
268C2×C42C22×C412yes—
269C2×C42C22×C412yes—
270C2×C42C2×D412yes—
271C2×C42C2×D412yes—
272C4⋊C8—82no—
273C4⋊C8—82no—
274C22⋊C8—22no—
275C22⋊C8—21no—
276C22⋊C8—21no—
277C22⋊C8—22no—
278C22⋊C8—21no—
279C22⋊C8—21no—
280C22⋊C8—42no—
281C22⋊C8—42no—
282#95—28no—
283#95—22no—
284#95—22no—
285#95—22no—
286#95—22no—
287#95—21no—
288#95—21no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—22no—
294#95—22no—
295#95—21no—
296#95—21no—
297#95—22no—
298#95—21no—
299#95—21no—
300#95—42no—
301#95—82no—
302#95—82no—
303#95C22⋊C412yes—
304#95C22⋊C412yes—
305#95C22⋊C412yes—
306#95C22⋊C412yes—
307C24—22no—
308C24—22no—
309C24—22no—
310C24—22no—
311C24—42no—
312C24C2×C22⋊C411yescharacteristic
313C24C2×C22⋊C411yescharacteristic
314C24#12112yes—
315C24C22⋊Q812yes—
316C24C22×D411yescharacteristic
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—24no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8—82no—
323C2×Q8—84no—
324C2×Q8—84no—
325C2×Q8C2×C4⋊C412yes—
326C2×Q8C2×C4⋊C412yes—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—24no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—24no—
338C22×C4—22no—
339C22×C4—24no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—42no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—42no—
349C22×C4—42no—
350C22×C4—42no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—42no—
355C22×C4—42no—
356C22×C4—42no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—48no—
362C22×C4—42no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—82no—
370C22×C4—82no—
371C22×C4—82no—
372C22×C4—82no—
373C22×C4—82no—
374C22×C4—88no—
375C22×C4—84no—
376C22×C4—84no—
377C22×C4—88no—
378C22×C4C2×C22⋊C412yes—
379C22×C4C2×C22⋊C412yes—
380C22×C4C2×C22⋊C412yes—
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C412yes—
384C22×C4C2×C4⋊C412yes—
385C22×C4C2×C4⋊C412yes—
386C22×C4#11712yes—
387C22×C4#11712yes—
388C22×C4#12112yes—
389C22×C4#12112yes—
390C22×C4#12112yes—
391C22×C4C22⋊Q812yes—
392C22×C4C22⋊Q812yes—
393C22×C4C22⋊Q812yes—
394C22×C4C23×C411yescharacteristic
395C22×C4C22×D411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C22×Q811yescharacteristic
398C22×C4C2×#4111yescharacteristic
399C22×C4C2×#4111yescharacteristic
400M16—84no—
401C2×C8—42no—
402C2×C8—42no—
403C2×C8—42no—
404C2×C8—42no—
405C2×C8—42no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—84no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—24no—
416C4⋊C4—24no—
417C4⋊C4—24no—
418C4⋊C4—42no—
419C4⋊C4—42no—
420C4⋊C4—44no—
421C4⋊C4—44no—
422C4⋊C4—82no—
423C4⋊C4—82no—
424C4⋊C4—88no—
425C4⋊C4C2×C4⋊C412yes—
426C4⋊C4C2×C4⋊C412yes—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—24no—
432C22⋊C4—24no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—24no—
437C22⋊C4—42no—
438C22⋊C4—48no—
439C22⋊C4—42no—
440C22⋊C4—42no—
441C22⋊C4—42no—
442C22⋊C4—42no—
443C22⋊C4—82no—
444C22⋊C4#10414yes—
445C42—42no—
446C42—42no—
447C42—42no—
448C42—42no—
449C42—44no—
450C42—44no—
451C42—82no—
452C42—82no—
453C42—82no—
454C42—84no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—21no—
459C23—21no—
460C23—24no—
461C23—24no—
462C23—24no—
463C23—22no—
464C23—22no—
465C23—21no—
466C23—21no—
467C23—22no—
468C23—22no—
469C23—42no—
470C23—42no—
471C23—44no—
472C23—44no—
473C23—44no—
474C23—44no—
475C23#38012yes—
476C23#38012yes—
477C23C2×#10412yes—
478C23#42112yes—
479C23C22×C22⋊C411yescharacteristic
480C23C2×#12111yescharacteristic
481C23C2×C22⋊Q811yescharacteristic
482C23#57712yes—
483C23#57812yes—
484Q8—42no—
485Q8—42no—
486Q8—42no—
487Q8—42no—
488Q8—84no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—44no—
523C2×C4—44no—
524C2×C4—48no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—44no—
534C2×C4—44no—
535C2×C4—44no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—82no—
545C2×C4—82no—
546C2×C4—84no—
547C2×C4—81no—
548C2×C4—81no—
549C2×C4—82no—
550C2×C4—82no—
551C2×C4—82no—
552C2×C4—82no—
553C2×C4—82no—
554C2×C4—816no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—81no—
558C2×C4#38012yes—
559C2×C4#38012yes—
560C2×C4C2×#10414yes—
561C2×C4C22×C4⋊C411yescharacteristic
562C2×C4C2×#11711yescharacteristic
563C2×C4C2×#12111yescharacteristic
564C2×C4C2×C22⋊Q811yescharacteristic
565C8—82no—
566C8—81no—
567C8—81no—
568C22—22no—
569C22—22no—
570C22—24no—
571C22—21no—
572C22—22no—
573C22—21no—
574C22—22no—
575C22—21no—
576C22—21no—
577C22—22no—
578C22—22no—
579C22—24no—
580C22—22no—
581C22—22no—
582C22—22no—
583C22—44no—
584C22—44no—
585C22—48no—
586C22—42no—
587C22—42no—
588C22#180614yes—
589C22C2×#38011yescharacteristic
590C22#288312yes—
591C22#289812yes—
592C22C22×#10411yescharacteristic
593C22C2×#42111yescharacteristic
594C22#304311yescharacteristic
595C22#304411yescharacteristic
596C22#308212yes—
597C4—24no—
598C4—21no—
599C4—21no—
600C4—21no—
601C4—21no—
602C4—22no—
603C4—22no—
604C4—42no—
605C4—42no—
606C4—42no—
607C4—42no—
608C4—42no—
609C4—42no—
610C4—42no—
611C4—82no—
612C4—82no—
613C4—84no—
614C4—82no—
615C4—82no—
616C2—21no—
617C2—21no—
618C2—22no—
619C2—24no—
620C2—42no—
621C2#1754812yes—
622C2#1759812yes—
623C2#2063311yescharacteristic
624C2#2065111yescharacteristic
625C2C2×#180611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8578C2semidirect
7. #8576C2semidirect
8. #8574C2semidirect
9. #8572C2semidirect
16. #2362C4semidirect
17. #2362C4semidirect
18. #2344C4semidirect
19. #2344C4semidirect
22. Q8×C22⋊C4C4semidirect
23. Q8×C22⋊C4C4semidirect
70. C2×C4×Q8C2×C4semidirect
71. C2×C4×Q8C2×C4semidirect
97. #391C2×C4semidirect
98. #391C2×C4semidirect
99. #389C2×C4semidirect
100. #389C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect
128. C4×C4⋊C4C2×C4semidirect
165. C22×Q8C4⋊C4semidirect
208. C22⋊Q8C4⋊C4semidirect
209. C22⋊Q8C4⋊C4semidirect
234. C2×C4⋊C4C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342452
#219322
C2×#15082
C22×#3282
#137822
#85782
#85762
#85742
#85722
#80202
#80122