Finite Group Databasebeta

← #188534

Subgroups — #188534

Order 512.

1302 conjugacy classes of subgroups.

The subgroup lattice has 644 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34223C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic, complemented
3C2×#1513C211yesmaximal, characteristic
4C2×#1504C211yesmaximal, characteristic
5#13965C211yesmaximal, characteristic, complemented
6#8578C212yesmaximal, complemented
7#8577C212yesmaximal, complemented
8#8572C212yesmaximal, complemented
9#8571C212yesmaximal, complemented
10#8020C211yesmaximal, characteristic, complemented
11#8016C211yesmaximal, characteristic, complemented
12C2×#525C2211yescharacteristic
13C2×#407C2211yescharacteristic
14C2×#407C2211yescharacteristic
15#2374—22no—
16#2363C412yescomplemented
17#2362C412yescomplemented
18#2357—21no—
19#2353—21no—
20#2347C411yescharacteristic, complemented
21#2346C411yescharacteristic, complemented
22#2341C411yescharacteristic, complemented
23#2339C411yescharacteristic, complemented
24#2321—22no—
25#2318—21no—
26C4×C22⋊Q8C2212yes—
27C4×#121C2212yes—
28C2×#385—22no—
29#2280C2211yescharacteristic
30C2×#380C2211yescharacteristic
31#2275—21no—
32C2×C4×C22⋊C4C2211yescharacteristic
33C22×#95C2211yescharacteristic
34#1851—21no—
35#1851—21no—
36#1851—22no—
37#1797C2212yes—
38#1797C2212yes—
39#1513—24no—
40#1504—24no—
41C2×#328C2211yescharacteristic
42C2×#328C2212yes—
43C2×#328C2211yescharacteristic
44C2×#327C2211yescharacteristic
45C2×#327C2212yes—
46C2×#327C2211yescharacteristic
47#1444—21no—
48#1444—21no—
49C4⋊#95C2212yes—
50C4⋊#95C2212yes—
51#1294C2212yes—
52#1294C2212yes—
53#1294C2212yes—
54#1294C2212yes—
55C22×#41C2×C411yescharacteristic
56C24×C4—21no—
57C24×C4C2311yescharacteristic
58C22×M16—21no—
59C22×M16—21no—
60#525—22no—
61#525—22no—
62#525D414yes—
63#525Q814yes—
64C2×#123—21no—
65C2×C22⋊Q8C2×C411yescharacteristic
66C2×C22⋊Q8C2311yescharacteristic
67C2×#121C2×C411yescharacteristic
68C2×#121C2311yescharacteristic
69C2×#120—21no—
70C2×C4×Q8C2×C412yescomplemented
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×C4×D4—22no—
74C2×C4×D4C2×C412yescomplemented
75C2×#117C2×C411yescharacteristic
76C2×#117C2×C412yes—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—22no—
79C22×C4⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4D412yes—
82C22×C42C2312yes—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85#407—22no—
86#407—22no—
87#407—22no—
88#407—22no—
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8C2311yescharacteristic
92C2×C22⋊C8C2312yes—
93C2×C22⋊C8C2311yescharacteristic
94#391C2×C412yescomplemented
95#390—22no—
96#390—22no—
97#390C2×C412yescomplemented
98#389C2×C412yescomplemented
99#389C2×C412yescomplemented
100#388—22no—
101#388—22no—
102#388—22no—
103#388—22no—
104#388—22no—
105#388—22no—
106#387—22no—
107#387—22no—
108#386—22no—
109#386—22no—
110#386—22no—
111#385—22no—
112#385—48no—
113#383—21no—
114#382—21no—
115#381—21no—
116#381—21no—
117#380—22no—
118#380—22no—
119#380—21no—
120#380C2×C411yescharacteristic
121#380C2×C411yescharacteristic
122#380C2312yes—
123#379—21no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4—22no—
127C4×C4⋊C4C2×C412yescomplemented
128C4×C4⋊C4C2×C412yescomplemented
129C4×C22⋊C4—21no—
130C4×C22⋊C4—22no—
131C4×C22⋊C4—21no—
132C4×C22⋊C4—22no—
133C4×C22⋊C4—21no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4C2×C411yescharacteristic
136C4×C22⋊C4C2×C411yescharacteristic
137C4×C22⋊C4C2×C412yes—
138C4×C22⋊C4D411yescharacteristic
139C4×C22⋊C4D411yescharacteristic
140C4×C22⋊C4D412yes—
141#376—21no—
142C2×#95—21no—
143C2×#95—22no—
144C2×#95—22no—
145C2×#95—22no—
146C2×#95—22no—
147C2×#95—21no—
148C2×#95D411yescharacteristic
149C2×#95D411yescharacteristic
150C2×#95C2311yescharacteristic
151C2×#95C2311yescharacteristic
152C2×#95C2312yes—
153#336—22no—
154#336—22no—
155#328—22no—
156#328—22no—
157#328—24no—
158#327—22no—
159#327—22no—
160#327—24no—
161C25C2×D411yescharacteristic
162C2×#41—22no—
163C2×#41—82no—
164C2×#41C4⋊C414yescomplemented
165C22×Q8C22×C411yescharacteristic
166C22×D4—21no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—22no—
174C23×C4—22no—
175C23×C4—42no—
176C23×C4—42no—
177C23×C4—42no—
178C23×C4—44no—
179C23×C4—82no—
180C23×C4—82no—
181C23×C4C22×C411yescharacteristic
182C23×C4C22×C411yescharacteristic
183C23×C4C2×D412yes—
184C23×C4C2×D411yescharacteristic
185C23×C4C2×D412yes—
186C23×C4C2×D411yescharacteristic
187C23×C4C2×D411yescharacteristic
188C23×C4C2×D412yes—
189C23×C4C2×Q812yes—
190C23×C4#4112yes—
191C23×C4#4112yes—
192C23×C4C2411yescharacteristic
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—82no—
196C2×M16—82no—
197C22×C8—22no—
198C22×C8—21no—
199C22×C8—21no—
200#123—24no—
201#123—24no—
202C22⋊Q8—22no—
203C22⋊Q8—21no—
204C22⋊Q8—21no—
205C22⋊Q8C4⋊C414yescomplemented
206C22⋊Q8C2×D412yes—
207C22⋊Q8C2×Q812yes—
208#121—22no—
209#121—21no—
210#121—21no—
211#121C4⋊C414yescomplemented
212#121C2×D412yes—
213#121C2×Q812yes—
214#120—24no—
215#120—24no—
216C4×Q8—82no—
217C4×D4—24no—
218C4×D4—24no—
219C4×D4—24no—
220C4×D4—24no—
221C4×D4—44no—
222C4×D4—44no—
223C4×D4—82no—
224#117—22no—
225#117—82no—
226#117C4⋊C414yescomplemented
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—44no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—44no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4—44no—
237C2×C4⋊C4—82no—
238C2×C4⋊C4—82no—
239C2×C4⋊C4—82no—
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C4⋊C4C22×C411yescharacteristic
242C2×C4⋊C4C22×C412yes—
243C2×C4⋊C4C22×C412yes—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4—24no—
249C2×C22⋊C4—24no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—44no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4C22⋊C412yes—
261C2×C22⋊C4C22⋊C412yes—
262C2×C22⋊C4C22⋊C412yes—
263C2×C22⋊C4C22⋊C412yes—
264C2×C42—21no—
265C2×C42—22no—
266C2×C42—22no—
267C2×C42—22no—
268C2×C42—22no—
269C2×C42—82no—
270C2×C42C22×C412yes—
271C2×C42C22×C412yes—
272C2×C42C2×D412yes—
273C2×C42C2×D412yes—
274C4⋊C8—82no—
275C4⋊C8—82no—
276C22⋊C8—21no—
277C22⋊C8—21no—
278C22⋊C8—22no—
279C22⋊C8—21no—
280C22⋊C8—22no—
281C22⋊C8—21no—
282C22⋊C8—42no—
283C22⋊C8—42no—
284#95—21no—
285#95—22no—
286#95—21no—
287#95—22no—
288#95—22no—
289#95—22no—
290#95—22no—
291#95—22no—
292#95—22no—
293#95—21no—
294#95—21no—
295#95—22no—
296#95—21no—
297#95—21no—
298#95—22no—
299#95—44no—
300#95—42no—
301#95—44no—
302#95C22⋊C411yescharacteristic
303#95C22⋊C411yescharacteristic
304#95C22⋊C412yes—
305#95C22⋊C411yescharacteristic
306#95C22⋊C411yescharacteristic
307#95C22⋊C412yes—
308C24—22no—
309C24—22no—
310C24—21no—
311C24—41no—
312C24—41no—
313C24—42no—
314C24C2×C22⋊C411yescharacteristic
315C24C2×C22⋊C411yescharacteristic
316C24#12112yes—
317C24C22⋊Q812yes—
318C24C22×D411yescharacteristic
319#41—84no—
320#41—84no—
321C2×Q8—21no—
322C2×Q8—42no—
323C2×Q8C2×C4⋊C412yes—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—24no—
327C2×D4—24no—
328C2×D4—21no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—42no—
332C2×D4—81no—
333C2×D4—81no—
334C2×D4—82no—
335C2×D4C2×C4⋊C412yes—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—24no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
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—42no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—48no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—44no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—42no—
369C22×C4—42no—
370C22×C4—42no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—82no—
374C22×C4—82no—
375C22×C4—82no—
376C22×C4—82no—
377C22×C4—82no—
378C22×C4—82no—
379C22×C4—82no—
380C22×C4—88no—
381C22×C4—88no—
382C22×C4—84no—
383C22×C4—84no—
384C22×C4C2×C22⋊C412yes—
385C22×C4C2×C22⋊C412yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C412yes—
389C22×C4C2×C22⋊C412yes—
390C22×C4C2×C4⋊C412yes—
391C22×C4C2×C4⋊C412yes—
392C22×C4#11712yes—
393C22×C4#11712yes—
394C22×C4#12112yes—
395C22×C4#12112yes—
396C22×C4#12112yes—
397C22×C4C22⋊Q812yes—
398C22×C4C22⋊Q812yes—
399C22×C4C22⋊Q812yes—
400C22×C4AGL(1,ℤ/8)14yes—
401C22×C4#13714yes—
402C22×C4C23×C411yescharacteristic
403C22×C4C22×D411yescharacteristic
404C22×C4C22×D411yescharacteristic
405C22×C4C22×Q811yescharacteristic
406C22×C4C2×#4111yescharacteristic
407C22×C4C2×#4111yescharacteristic
408M16—84no—
409M16—84no—
410C2×C8—42no—
411C2×C8—42no—
412C2×C8—42no—
413C2×C8—42no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—42no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—24no—
422C4⋊C4—42no—
423C4⋊C4—42no—
424C4⋊C4—44no—
425C4⋊C4—82no—
426C4⋊C4—82no—
427C4⋊C4—88no—
428C4⋊C4—88no—
429C4⋊C4—84no—
430C4⋊C4C2×C4⋊C412yes—
431C4⋊C4C2×C4⋊C412yes—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—24no—
435C22⋊C4—24no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—24no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—24no—
442C22⋊C4—24no—
443C22⋊C4—42no—
444C22⋊C4—42no—
445C22⋊C4—44no—
446C22⋊C4—82no—
447C22⋊C4—82no—
448C22⋊C4—81no—
449C22⋊C4—81no—
450C22⋊C4—82no—
451C22⋊C4—84no—
452C42—24no—
453C42—42no—
454C42—42no—
455C42—42no—
456C42—42no—
457C42—44no—
458C42—82no—
459C42—82no—
460C42—82no—
461C23—22no—
462C23—21no—
463C23—21no—
464C23—24no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—21no—
470C23—21no—
471C23—21no—
472C23—21no—
473C23—22no—
474C23—22no—
475C23—22no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—44no—
481C23—44no—
482C23—44no—
483C23—44no—
484C23—42no—
485C23—42no—
486C23—41no—
487C23—41no—
488C23#38012yes—
489C23#38012yes—
490C23C22×C22⋊C411yescharacteristic
491C23C2×#12111yescharacteristic
492C23C2×C22⋊Q811yescharacteristic
493C23C2×AGL(1,ℤ/8)12yes—
494C23C2×#13712yes—
495Q8—42no—
496Q8—42no—
497D4—42no—
498D4—42no—
499D4—44no—
500D4—42no—
501D4—42no—
502D4—84no—
503D4—84no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—22no—
512C2×C4—22no—
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—22no—
520C2×C4—22no—
521C2×C4—42no—
522C2×C4—44no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—44no—
537C2×C4—44no—
538C2×C4—44no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—41no—
542C2×C4—41no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—42no—
548C2×C4—42no—
549C2×C4—42no—
550C2×C4—42no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—84no—
554C2×C4—81no—
555C2×C4—81no—
556C2×C4—82no—
557C2×C4—82no—
558C2×C4—816no—
559C2×C4—81no—
560C2×C4—81no—
561C2×C4—82no—
562C2×C4—82no—
563C2×C4—82no—
564C2×C4—82no—
565C2×C4—82no—
566C2×C4—82no—
567C2×C4—82no—
568C2×C4—164no—
569C2×C4#38012yes—
570C2×C4#38012yes—
571C2×C4#41714yes—
572C2×C4#41714yes—
573C2×C4C22×C4⋊C411yescharacteristic
574C2×C4C2×#11711yescharacteristic
575C2×C4C2×#12111yescharacteristic
576C2×C4C2×C22⋊Q811yescharacteristic
577C8—81no—
578C8—81no—
579C8—82no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—22no—
585C22—22no—
586C22—22no—
587C22—22no—
588C22—22no—
589C22—22no—
590C22—22no—
591C22—22no—
592C22—22no—
593C22—22no—
594C22—44no—
595C22—48no—
596C22—44no—
597C22—42no—
598C22—42no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—42no—
604C22—82no—
605C22—82no—
606C22C2×#38011yescharacteristic
607C22#288312yes—
608C22C2×#41712yes—
609C22#290111yescharacteristic
610C22#290111yescharacteristic
611C22#303811yescharacteristic
612C22#303911yescharacteristic
613C22#307912yes—
614C4—21no—
615C4—22no—
616C4—21no—
617C4—21no—
618C4—21no—
619C4—22no—
620C4—24no—
621C4—24no—
622C4—42no—
623C4—42no—
624C4—42no—
625C4—42no—
626C4—42no—
627C4—82no—
628C4—82no—
629C4—82no—
630C4—82no—
631C4—162no—
632C2—21no—
633C2—22no—
634C2—21no—
635C2—42no—
636C2—42no—
637C2—82no—
638C2#1762212yes—
639C2#1763012yes—
640C2#2063311yescharacteristic
641C2#2064811yescharacteristic
642C2#2065511yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#525C22
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
2. #21934C2semidirect
5. #13965C2semidirect
6. #8578C2semidirect
7. #8577C2semidirect
8. #8572C2semidirect
9. #8571C2semidirect
10. #8020C2semidirect
11. #8016C2semidirect
16. #2363C4semidirect
17. #2362C4semidirect
20. #2347C4semidirect
21. #2346C4semidirect
22. #2341C4semidirect
23. #2339C4semidirect
70. C2×C4×Q8C2×C4semidirect
74. C2×C4×D4C2×C4semidirect
94. #391C2×C4semidirect
97. #390C2×C4semidirect
98. #389C2×C4semidirect
99. #389C2×C4semidirect
127. C4×C4⋊C4C2×C4semidirect
128. C4×C4⋊C4C2×C4semidirect
164. C2×#41C4⋊C4semidirect
205. C22⋊Q8C4⋊C4semidirect
211. #121C4⋊C4semidirect
226. #117C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
#342232
#219342
C2×#15132
C2×#15042
#139652
#85782
#85772
#85722
#85712
#80202
#80162