Finite Group Databasebeta

← #207274

Subgroups — #207274

Order 512.

1088 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#21934C211yesmaximal, characteristic
3#13802C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9912C211yesmaximal, characteristic, complemented
6#9911C211yesmaximal, characteristic, complemented
7#9910C212yesmaximal, complemented
8#9909C211yesmaximal, characteristic, complemented
9#9908C211yesmaximal, characteristic, complemented
10#8020C212yesmaximal
11#8016C212yesmaximal
12C2×#407C2212yes—
13#2857C2211yescharacteristic
14#2857C2211yescharacteristic
15#2851C2212yes—
16#2626—21no—
17#2626—21no—
18#2608—22no—
19#2607—22no—
20#2441C412yescomplemented
21#2441C412yescomplemented
22#2439C412yescomplemented
23#2439C412yescomplemented
24#2329—21no—
25#2329—21no—
26#2280C2212yes—
27C2×#380C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29C2×C4×C22⋊C4C2211yescharacteristic
30C22×#95C2211yescharacteristic
31#1851—22no—
32#1851—22no—
33#1797C2211yescharacteristic
34#1797C2211yescharacteristic
35#1797C2212yes—
36#1436—21no—
37#1436—21no—
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40C22⋊#95—21no—
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4—21no—
52C24×C4C2311yescharacteristic
53C22×M16—22no—
54C2×C4⋊Q8C2×C411yescharacteristic, complemented
55C2×C4⋊Q8C2×C411yescharacteristic, complemented
56C2×#125C2×C412yescomplemented
57C2×C22⋊Q8—21no—
58C2×C22⋊Q8—21no—
59C2×#117C2×C412yes—
60C2×#117C2×C412yes—
61C22×C4⋊C4—21no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D411yescharacteristic
64C22×C22⋊C4D411yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72#407—24no—
73#407—24no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2312yes—
76C2×C22⋊C8C2312yes—
77#399—22no—
78#399—22no—
79#396—22no—
80#396—22no—
81#396—22no—
82#396—22no—
83#393—22no—
84#393—22no—
85#391—21no—
86#391—21no—
87#391—22no—
88#391—22no—
89#389—21no—
90#389—21no—
91#389—22no—
92#389—22no—
93#389C2×C412yescomplemented
94#389C2×C412yescomplemented
95#389C2×C412yescomplemented
96#389C2×C412yescomplemented
97#387—22no—
98#387—22no—
99#387—22no—
100#387—22no—
101#387—22no—
102#387—22no—
103#385—22no—
104#385—22no—
105#384C2×C411yescharacteristic, complemented
106#384C2×C411yescharacteristic, complemented
107#382C2×C412yescomplemented
108#381—22no—
109#380—21no—
110#380—21no—
111#380—22no—
112#380C2×C412yes—
113#380C2×C412yes—
114#380Q814yescomplemented
115#380C2312yes—
116#380C2312yes—
117#379—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—44no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95Q814yescomplemented
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139C2×#95C2311yescharacteristic
140#336—22no—
141#336—21no—
142#336—21no—
143C25C2×D411yescharacteristic
144C22×Q8—21no—
145C22×Q8—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4—24no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4—42no—
154C23×C4—42no—
155C23×C4—81no—
156C23×C4—81no—
157C23×C4—81no—
158C23×C4—81no—
159C23×C4C22×C412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D411yescharacteristic
164C23×C4C2×D411yescharacteristic
165C23×C4C2×D412yes—
166C23×C4C2×Q812yes—
167C23×C4C2×Q812yes—
168C23×C4#4112yes—
169C23×C4#4112yes—
170C23×C4C2411yescharacteristic
171C2×M16—44no—
172C2×M16—82no—
173C2×M16—82no—
174C22×C8—22no—
175C22×C8—22no—
176C4⋊Q8—81no—
177C4⋊Q8—81no—
178#125—82no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8—42no—
183#117—82no—
184#117—82no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4C22×C412yes—
201C2×C4⋊C4C22×C412yes—
202C2×C4⋊C4C22×C412yes—
203C2×C4⋊C4C22×C412yes—
204C2×C22⋊C4—22no—
205C2×C22⋊C4—22no—
206C2×C22⋊C4—24no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C22⋊C4—22no—
213C2×C22⋊C4—41no—
214C2×C22⋊C4—41no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—42no—
217C2×C22⋊C4C22⋊C412yes—
218C2×C22⋊C4C22⋊C412yes—
219C2×C22⋊C4C22⋊C412yes—
220C2×C22⋊C4C22⋊C412yes—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—42no—
224C2×C42—42no—
225C2×C42—42no—
226C2×C42—42no—
227C2×C42—81no—
228C2×C42—81no—
229C2×C42—81no—
230C2×C42—81no—
231C2×C42C22×C412yes—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D412yes—
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—82no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—22no—
243C22⋊C8—22no—
244C22⋊C8—44no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—22no—
249#95—22no—
250#95—22no—
251#95—22no—
252#95—22no—
253#95—24no—
254#95—22no—
255#95—22no—
256#95—21no—
257#95—21no—
258#95—21no—
259#95—21no—
260#95—42no—
261#95—42no—
262#95—42no—
263#95—42no—
264#95—81no—
265#95—81no—
266#95C22⋊C412yes—
267#95C22⋊C412yes—
268#95C22⋊C412yes—
269#95C22⋊C412yes—
270#95C2×Q812yes—
271#95C2×Q812yes—
272#95C2×Q812yes—
273#95C2×Q812yes—
274C24—21no—
275C24—21no—
276C24—21no—
277C24—21no—
278C24—41no—
279C24—41no—
280C24—41no—
281C24—41no—
282C24C2×C22⋊C412yes—
283C24C22⋊Q812yes—
284C24C22⋊Q812yes—
285C24C22×D411yescharacteristic
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—81no—
289C2×Q8—81no—
290C2×Q8—81no—
291C2×Q8—81no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—48no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—48no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—82no—
349C22×C4—82no—
350C22×C4—82no—
351C22×C4—82no—
352C22×C4—84no—
353C22×C4—84no—
354C22×C4—84no—
355C22×C4—84no—
356C22×C4—84no—
357C22×C4C2×C22⋊C412yes—
358C22×C4C2×C22⋊C412yes—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C412yes—
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C4×Q814yes—
365C22×C4C4×Q814yes—
366C22×C4C22⋊Q812yes—
367C22×C4C22⋊Q812yes—
368C22×C4#12414yes—
369C22×C4C4⋊Q814yes—
370C22×C4AGL(1,ℤ/8)12yes—
371C22×C4#13712yes—
372C22×C4C23×C411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C22×C4C22×Q811yescharacteristic
375C22×C4C22×Q811yescharacteristic
376C22×C4C2×#4111yescharacteristic
377C22×C4C2×#4111yescharacteristic
378M16—88no—
379C2×C8—42no—
380C2×C8—42no—
381C2×C8—42no—
382C2×C8—42no—
383C2×C8—42no—
384C2×C8—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—44no—
393C4⋊C4—81no—
394C4⋊C4—81no—
395C4⋊C4—82no—
396C4⋊C4—82no—
397C4⋊C4—82no—
398C4⋊C4—82no—
399C4⋊C4—81no—
400C4⋊C4—81no—
401C4⋊C4—84no—
402C22⋊C4—24no—
403C22⋊C4—24no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—48no—
407C22⋊C4—42no—
408C22⋊C4—42no—
409C22⋊C4—42no—
410C22⋊C4—42no—
411C22⋊C4—82no—
412C22⋊C4—82no—
413C22⋊C4—82no—
414C22⋊C4—82no—
415C42—41no—
416C42—41no—
417C42—42no—
418C42—42no—
419C42—41no—
420C42—41no—
421C42—82no—
422C42—82no—
423C42—84no—
424C42—84no—
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—22no—
430C23—22no—
431C23—22no—
432C23—22no—
433C23—22no—
434C23—22no—
435C23—22no—
436C23—22no—
437C23—22no—
438C23—22no—
439C23—42no—
440C23—42no—
441C23—42no—
442C23—42no—
443C23—42no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—42no—
451C23#39114yes—
452C23C22×C22⋊C411yescharacteristic
453C23C2×C22⋊Q811yescharacteristic
454C23C2×C22⋊Q811yescharacteristic
455C23C2×AGL(1,ℤ/8)11yescharacteristic
456C23C2×#13711yescharacteristic
457C23#57511yescharacteristic
458C23#57511yescharacteristic
459Q8—162no—
460Q8—162no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—24no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—21no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—42no—
490C2×C4—44no—
491C2×C4—41no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—41no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—42no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
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—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—816no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—81no—
534C2×C4—81no—
535C2×C4—81no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—81no—
541C2×C4—81no—
542C2×C4#39114yes—
543C2×C4#41712yes—
544C2×C4#41712yes—
545C2×C4C2×C4×Q811yescharacteristic
546C2×C4C2×C4×Q811yescharacteristic
547C2×C4C2×#12411yescharacteristic
548C2×C4C2×C4⋊Q811yescharacteristic
549C8—82no—
550C8—82no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—24no—
564C22—24no—
565C22—24no—
566C22—44no—
567C22—44no—
568C22—44no—
569C22—44no—
570C22—42no—
571C22—42no—
572C22—42no—
573C22—42no—
574C22C2×#39111yescharacteristic
575C22#288311yescharacteristic
576C22#288311yescharacteristic
577C22C2×#41711yescharacteristic
578C22#289611yescharacteristic
579C22#290111yescharacteristic
580C22#290111yescharacteristic
581C22#307711yescharacteristic
582C22#307711yescharacteristic
583C22#307811yescharacteristic
584C22#307811yescharacteristic
585C4—21no—
586C4—21no—
587C4—21no—
588C4—21no—
589C4—24no—
590C4—22no—
591C4—22no—
592C4—44no—
593C4—41no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—82no—
598C4—82no—
599C4—82no—
600C4—82no—
601C4—82no—
602C4—82no—
603C4—82no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—44no—
609C2#1762211yescharacteristic
610C2#1762311yescharacteristic
611C2#1762911yescharacteristic
612C2#1763011yescharacteristic
613C2#2131211yescharacteristic
614C2#2133011yescharacteristic
615C2#2133311yescharacteristic
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
4. #9913C2semidirect
5. #9912C2semidirect
6. #9911C2semidirect
7. #9910C2semidirect
8. #9909C2semidirect
9. #9908C2semidirect
20. #2441C4semidirect
21. #2441C4semidirect
22. #2439C4semidirect
23. #2439C4semidirect
54. C2×C4⋊Q8C2×C4semidirect
55. C2×C4⋊Q8C2×C4semidirect
56. C2×#125C2×C4semidirect
93. #389C2×C4semidirect
94. #389C2×C4semidirect
95. #389C2×C4semidirect
96. #389C2×C4semidirect
105. #384C2×C4semidirect
106. #384C2×C4semidirect
107. #382C2×C4semidirect
114. #380Q8semidirect
135. C2×#95Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
#219342
#138022
#99132
#99122
#99112
#99102
#99092
#99082
#80202
#80162