Finite Group Databasebeta

← #207251

Subgroups — #207251

Order 512.

1012 conjugacy classes of subgroups.

The subgroup lattice has 605 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35033C211yesmaximal, characteristic
2#21930C211yesmaximal, characteristic
3#13807C211yesmaximal, characteristic
4#9913C212yesmaximal, complemented
5#9912C211yesmaximal, characteristic, complemented
6#9911C211yesmaximal, characteristic, complemented
7#9910C212yesmaximal, complemented
8#9909C211yesmaximal, characteristic, complemented
9#9908C211yesmaximal, characteristic, complemented
10#8024C212yesmaximal
11#8012C212yesmaximal
12C22×C22⋊C8C2212yes—
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×#381C2211yescharacteristic
28C2×C4×C22⋊C4C2211yescharacteristic
29C2×C4×C22⋊C4C2211yescharacteristic
30#1850—22no—
31#1850—22no—
32#1797C2211yescharacteristic
33#1797C2211yescharacteristic
34#1797C2212yes—
35#1437—21no—
36#1437—21no—
37#1434—21no—
38#1434C2211yescharacteristic
39#1434C2211yescharacteristic
40C4×#95—21no—
41C4×#95C2211yescharacteristic
42C4×#95C2211yescharacteristic
43#1294C2212yes—
44#1294C2212yes—
45#1294C2212yes—
46#1294C2212yes—
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2212yes—
51C24×C4C2311yescharacteristic
52C23×C8—22no—
53C2×C4⋊Q8C2×C411yescharacteristic
54C2×C4⋊Q8C2×C411yescharacteristic
55C2×#125C2×C412yes—
56C2×C22⋊Q8—21no—
57C2×C22⋊Q8—21no—
58C2×#117C2×C412yes—
59C2×#117C2×C412yes—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—21no—
62C22×C22⋊C4D411yescharacteristic
63C22×C22⋊C4D411yescharacteristic
64C22×C42—21no—
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2311yescharacteristic
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C22⋊C8—24no—
73C2×C22⋊C8—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×C412yes—
94#389C2×C412yes—
95#389C2×C412yes—
96#389C2×C412yes—
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
106#384C2×C411yescharacteristic
107#382C2×C412yes—
108#381—22no—
109#381—22no—
110#381Q814yescomplemented
111#380—21no—
112#380—21no—
113#380C2×C412yes—
114#380C2×C412yes—
115#380C2312yes—
116#380C2312yes—
117#379—22no—
118C4×C22⋊C4—22no—
119C4×C22⋊C4—22no—
120C4×C22⋊C4—22no—
121C4×C22⋊C4C2×C412yes—
122C4×C22⋊C4C2×C412yes—
123C4×C22⋊C4D412yes—
124C4×C22⋊C4D412yes—
125C4×C22⋊C4Q814yescomplemented
126C2×#95—21no—
127C2×#95—21no—
128C2×#95—21no—
129C2×#95—21no—
130C2×#95—21no—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95D411yescharacteristic
134C2×#95D411yescharacteristic
135C2×#95C2311yescharacteristic
136C2×#95C2311yescharacteristic
137C2×#95C2311yescharacteristic
138C2×#95C2311yescharacteristic
139#336—22no—
140#336—21no—
141#336—21no—
142C25C2×D411yescharacteristic
143C22×Q8—21no—
144C22×Q8—21no—
145C23×C4—21no—
146C23×C4—21no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4—81no—
150C23×C4—81no—
151C23×C4C22×C412yes—
152C23×C4C2×D411yescharacteristic
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D411yescharacteristic
155C23×C4C2×D411yescharacteristic
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×Q812yes—
159C23×C4C2×Q812yes—
160C23×C4#4112yes—
161C23×C4#4112yes—
162C23×C4C2411yescharacteristic
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—44no—
166C22×C8—82no—
167C22×C8—82no—
168C4⋊Q8—81no—
169C4⋊Q8—81no—
170#125—82no—
171C22⋊Q8—42no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—42no—
175#117—82no—
176#117—82no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C22×C412yes—
190C2×C4⋊C4C22×C412yes—
191C2×C4⋊C4C22×C412yes—
192C2×C4⋊C4C22×C412yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—24no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—42no—
207C2×C22⋊C4—81no—
208C2×C22⋊C4—81no—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22⋊C412yes—
211C2×C22⋊C4C22⋊C412yes—
212C2×C22⋊C4C22⋊C412yes—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—24no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—81no—
221C2×C42—81no—
222C2×C42—81no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42C22×C412yes—
227C2×C42C22×C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C2×D412yes—
230C2×C42C2×D411yescharacteristic
231C2×C42C2×D411yescharacteristic
232C4⋊C8—81no—
233C4⋊C8—81no—
234C4⋊C8—82no—
235C22⋊C8—22no—
236C22⋊C8—22no—
237C22⋊C8—22no—
238C22⋊C8—22no—
239C22⋊C8—44no—
240#95—22no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
245#95—22no—
246#95—22no—
247#95—22no—
248#95—24no—
249#95—22no—
250#95—22no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—21no—
255#95—42no—
256#95—42no—
257#95—42no—
258#95—42no—
259#95—81no—
260#95—81no—
261#95C22⋊C412yes—
262#95C22⋊C412yes—
263#95C22⋊C412yes—
264#95C22⋊C412yes—
265#95C2×Q812yes—
266#95C2×Q812yes—
267#95C2×Q812yes—
268#95C2×Q812yes—
269C24—21no—
270C24—21no—
271C24—21no—
272C24—21no—
273C24—41no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24C2×C22⋊C412yes—
278C24C22⋊Q812yes—
279C24C22⋊Q812yes—
280C24C22×D411yescharacteristic
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×Q8—81no—
286C2×Q8—81no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
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—41no—
340C22×C4—41no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C412yes—
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×C22⋊C411yescharacteristic
349C22×C4C2×C22⋊C412yes—
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C4×Q814yes—
353C22×C4C4×Q814yes—
354C22×C4C22⋊Q812yes—
355C22×C4C22⋊Q812yes—
356C22×C4#12414yes—
357C22×C4C4⋊Q814yes—
358C22×C4#13512yes—
359C22×C4#13512yes—
360C22×C4C23×C411yescharacteristic
361C22×C4C22×D411yescharacteristic
362C22×C4C22×Q811yescharacteristic
363C22×C4C22×Q811yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4C2×#4111yescharacteristic
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—88no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—44no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—81no—
382C4⋊C4—81no—
383C4⋊C4—82no—
384C4⋊C4—82no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4—81no—
388C4⋊C4—81no—
389C22⋊C4—24no—
390C22⋊C4—24no—
391C22⋊C4—42no—
392C22⋊C4—42no—
393C22⋊C4—48no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—42no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C22⋊C4—82no—
401C22⋊C4—82no—
402C22⋊C4—84no—
403C42—41no—
404C42—41no—
405C42—42no—
406C42—42no—
407C42—41no—
408C42—41no—
409C42—82no—
410C42—82no—
411C42—84no—
412C42—84no—
413C42—84no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—22no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—42no—
431C23—42no—
432C23—81no—
433C23—81no—
434C23—81no—
435C23—81no—
436C23—81no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23#39114yes—
441C23C22×C22⋊C411yescharacteristic
442C23C2×C22⋊Q811yescharacteristic
443C23C2×C22⋊Q811yescharacteristic
444C23C2×#13511yescharacteristic
445C23C2×#13511yescharacteristic
446C23#57511yescharacteristic
447C23#57511yescharacteristic
448Q8—162no—
449Q8—162no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—22no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—44no—
476C2×C4—44no—
477C2×C4—44no—
478C2×C4—42no—
479C2×C4—44no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—42no—
485C2×C4—42no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
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—81no—
515C2×C4—81no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4#39114yes—
531C2×C4#41612yes—
532C2×C4#41612yes—
533C2×C4C2×C4×Q811yescharacteristic
534C2×C4C2×C4×Q811yescharacteristic
535C2×C4C2×#12411yescharacteristic
536C2×C4C2×C4⋊Q811yescharacteristic
537C8—82no—
538C8—82no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—24no—
552C22—24no—
553C22—24no—
554C22—42no—
555C22—42no—
556C22—42no—
557C22—42no—
558C22—82no—
559C22—82no—
560C22—82no—
561C22—82no—
562C22C2×#39111yescharacteristic
563C22#288311yescharacteristic
564C22#288311yescharacteristic
565C22C2×#41611yescharacteristic
566C22#289611yescharacteristic
567C22#290111yescharacteristic
568C22#290111yescharacteristic
569C22#307611yescharacteristic
570C22#307611yescharacteristic
571C22#307611yescharacteristic
572C22#307611yescharacteristic
573C4—21no—
574C4—21no—
575C4—21no—
576C4—21no—
577C4—24no—
578C4—22no—
579C4—22no—
580C4—44no—
581C4—42no—
582C4—41no—
583C4—41no—
584C4—42no—
585C4—82no—
586C4—82no—
587C4—82no—
588C4—82no—
589C4—82no—
590C4—82no—
591C4—82no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—44no—
597C2#1761511yescharacteristic
598C2#1761711yescharacteristic
599C2#1762511yescharacteristic
600C2#1762911yescharacteristic
601C2#2131111yescharacteristic
602C2#2133011yescharacteristic
603C2#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
110. #381Q8semidirect
125. C4×C22⋊C4Q8semidirect

Maximal subgroups

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

subgroupindex
#350332
#219302
#138072
#99132
#99122
#99112
#99102
#99092
#99082
#80242
#80122