Finite Group Databasebeta

← #223087

Subgroups — #223087

Order 512.

1024 conjugacy classes of subgroups.

The subgroup lattice has 595 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36186C211yesmaximal, characteristic
2#22599C211yesmaximal, characteristic, complemented
3#13918C211yesmaximal, characteristic, complemented
4#9909C211yesmaximal, characteristic, complemented
5#9885C211yesmaximal, characteristic, complemented
6#9156C212yesmaximal
7#9114C212yesmaximal
8#8716C212yesmaximal, complemented
9#8571C211yesmaximal, characteristic
10C4×#327C211yesmaximal, characteristic
11#8026C212yesmaximal, complemented
12C2×#552—21no—
13C2×#431C2212yes—
14#2857C2211yescharacteristic
15#2854C2212yes—
16#2552C411yescharacteristic, complemented
17#2542C411yescharacteristic, complemented
18#2519—22no—
19#2512C412yescomplemented
20#2506C412yescomplemented
21#2498—22no—
22#2489—21no—
23#2417—21no—
24#2365C411yescharacteristic, complemented
25#2362—21no—
26#2354C411yescharacteristic, complemented
27C4×#121C2211yescharacteristic
28C2×#389—21no—
29#2285C2212yes—
30C2×#382C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#1939—21no—
33#1916—22no—
34#1817C2212yes—
35#1797C2211yescharacteristic
36C8×C4⋊C4—21no—
37C4×C22⋊C8C2211yescharacteristic
38C2×#327C2211yescharacteristic
39C2×#327C2211yescharacteristic
40C2×#327C2212yes—
41#1442—22no—
42C4⋊#95—21no—
43C4⋊#95C2211yescharacteristic
44#1434C2212yes—
45C4×#95C2211yescharacteristic
46#1294C2211yescharacteristic
47#1294C2212yes—
48#1294C2212yes—
49#1294C2212yes—
50#1294C2211yescharacteristic
51#1294C2212yes—
52#1294C2212yes—
53#552—44no—
54#552—44no—
55C2×C4⋊Q8C2×C411yescharacteristic
56C2×#126—22no—
57C2×#125—21no—
58C2×#125—22no—
59C2×#123—22no—
60C2×C22⋊Q8—21no—
61C2×C22⋊Q8—21no—
62C2×C22⋊Q8—21no—
63C2×#121C2311yescharacteristic
64C2×C4×Q8—21no—
65C2×C4×D4—21no—
66C2×C4×D4C2×C411yescharacteristic
67C2×#117—22no—
68C2×#117—21no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76#431—24no—
77#431—42no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—22no—
80C2×C22⋊C8C2312yes—
81C2×C22⋊C8C2311yescharacteristic
82C2×C22⋊C8C2311yescharacteristic
83C2×C4×C8—22no—
84C2×C4×C8—21no—
85#400—22no—
86#399—22no—
87#398—22no—
88#398—22no—
89#397—22no—
90#396—22no—
91#391—21no—
92#391—21no—
93#390C2×C412yes—
94#390C2×C412yes—
95#390C2×C411yescharacteristic
96#389—21no—
97#389—21no—
98#389—44no—
99#389C2×C412yes—
100#389C2×C412yes—
101#389C2×C412yes—
102#389C2×C411yescharacteristic
103#388—22no—
104#388—22no—
105#388—21no—
106#388—22no—
107#388—22no—
108#387—22no—
109#387—22no—
110#386—22no—
111#386—21no—
112#384C2×C411yescharacteristic
113#384C2×C412yes—
114#384C2×C412yes—
115#382—22no—
116#382—24no—
117#382—42no—
118#380—21no—
119#380—22no—
120#380—21no—
121#380C2312yes—
122#380C2311yescharacteristic
123C4×C4⋊C4—22no—
124C4×C4⋊C4C2×C412yes—
125C4×C4⋊C4C2×C412yes—
126C4×C4⋊C4C2×C411yescharacteristic
127C4×C4⋊C4C2×C411yescharacteristic
128C4×C4⋊C4C2×C412yes—
129C4×C4⋊C4C2×C411yescharacteristic
130C4×C4⋊C4D411yescharacteristic
131C4×C4⋊C4D412yes—
132C4×C4⋊C4D411yescharacteristic
133C4×C22⋊C4—21no—
134C2×#95—21no—
135C2×#95—22no—
136C2×#95—22no—
137C2×#95—22no—
138C2×#95—21no—
139C2×#95D412yes—
140C2×#95C2311yescharacteristic
141C2×#95C2312yes—
142C2×#95C2311yescharacteristic
143#336—21no—
144#336—22no—
145#336—22no—
146#336—21no—
147#327—22no—
148#327—22no—
149#327—42no—
150C22×Q8—21no—
151C22×D4—21no—
152C22×D4C22×C411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—22no—
165C4⋊Q8—42no—
166#126—48no—
167#125—48no—
168#125—44no—
169#123—44no—
170#123—44no—
171C22⋊Q8—44no—
172C22⋊Q8—42no—
173C22⋊Q8—42no—
174C22⋊Q8—44no—
175#121—21no—
176#121—21no—
177#121—21no—
178#121—21no—
179C4×Q8—44no—
180C4×D4—42no—
181C4×D4—42no—
182C4×D4—42no—
183#117—44no—
184#117—44no—
185#117—44no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—42no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—82no—
204C2×C4⋊C4C22⋊C411yescharacteristic
205C2×C4⋊C4C22⋊C411yescharacteristic
206C2×C4⋊C4C22⋊C412yes—
207C2×C4⋊C4C22⋊C411yescharacteristic
208C2×C4⋊C4C22⋊C411yescharacteristic
209C2×C4⋊C4C22⋊C412yes—
210C2×C4⋊C4D814yes—
211C2×C4⋊C4SD1614yes—
212C2×C4⋊C4C22×C411yescharacteristic
213C2×C4⋊C4C22×C411yescharacteristic
214C2×C4⋊C4C22×C412yes—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C412yes—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—22no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—21no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42—42no—
237C2×C42—42no—
238C2×C42C22×C411yescharacteristic
239C2×C42C22×C412yes—
240C2×C42C22×C412yes—
241C2×C42C22×C411yescharacteristic
242C2×C42C2×D412yes—
243C2×C42C2×D412yes—
244C2×C42C2×D411yescharacteristic
245C2×C42C2×D411yescharacteristic
246C4⋊C8—44no—
247C4⋊C8—42no—
248C22⋊C8—21no—
249C22⋊C8—21no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—21no—
253C22⋊C8#4112yes—
254C4×C8—44no—
255C4×C8—42no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—21no—
260#95—22no—
261#95—22no—
262#95—22no—
263#95—21no—
264#95—22no—
265#95—21no—
266#95—21no—
267#95—22no—
268#95—22no—
269#95—21no—
270#95—44no—
271#95—42no—
272#95—44no—
273#95—44no—
274#95—81no—
275#95C22⋊C412yes—
276#95C22⋊C412yes—
277#95C22⋊C412yes—
278#95C22⋊C412yes—
279#95#4112yes—
280C24—21no—
281C24C22×D411yescharacteristic
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—41no—
288C2×D4—41no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—22no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—22no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—42no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
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—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4C2×C22⋊C411yescharacteristic
341C22×C4C2×C22⋊C412yes—
342C22×C4C2×C22⋊C411yescharacteristic
343C22×C4C2×C22⋊C412yes—
344C22×C4C2×C22⋊C412yes—
345C22×C4C2×C22⋊C412yes—
346C22×C4C2×C22⋊C411yescharacteristic
347C22×C4C2×C22⋊C411yescharacteristic
348C22×C4C2×D814yes—
349C22×C4C2×D812yes—
350C22×C4C2×SD1614yes—
351C22×C4C2×SD1612yes—
352C22×C4C23×C411yescharacteristic
353C22×C4C22×D411yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4H(5,𝔽2)11yescharacteristic
357C22×C42−1+411yescharacteristic
358C2×C8—22no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—42no—
372C4⋊C4—41no—
373C4⋊C4—42no—
374C4⋊C4—41no—
375C4⋊C4—42no—
376C4⋊C4—44no—
377C4⋊C4—42no—
378C4⋊C4—44no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—42no—
382C4⋊C4—44no—
383C4⋊C4—44no—
384C4⋊C4—44no—
385C4⋊C4—82no—
386C4⋊C4—81no—
387C4⋊C4—88no—
388C4⋊C4—84no—
389C4⋊C4#10214yes—
390C4⋊C4#10214yes—
391C4⋊C4#10218yes—
392C22⋊C4—42no—
393C22⋊C4—44no—
394C22⋊C4—41no—
395C22⋊C4—42no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—44no—
401C22⋊C4—42no—
402C22⋊C4—44no—
403C22⋊C4—81no—
404C22⋊C4—82no—
405C42—22no—
406C42—22no—
407C42—22no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42—22no—
412C42—22no—
413C42—22no—
414C42—22no—
415C42—41no—
416C42—41no—
417C42—44no—
418C42—44no—
419C42—42no—
420C42—84no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—41no—
426C23—41no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23C22×C22⋊C411yescharacteristic
433C23#54611yescharacteristic
434C23#54711yescharacteristic
435C23C22×D811yescharacteristic
436C23C22×SD1611yescharacteristic
437C23#57711yescharacteristic
438C23#57811yescharacteristic
439Q8—84no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444D4—81no—
445D4—81no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—24no—
459C2×C4—24no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—21no—
473C2×C4—22no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—42no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—42no—
491C2×C4—42no—
492C2×C4—41no—
493C2×C4—41no—
494C2×C4—42no—
495C2×C4—41no—
496C2×C4—41no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4—82no—
503C2×C4—81no—
504C2×C4—81no—
505C2×C4—81no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—84no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#41012yes—
520C2×C4C2×#10214yes—
521C2×C4C2×#10212yes—
522C2×C4C2×#10214yes—
523C2×C4C2×#10212yes—
524C2×C4#42112yes—
525C2×C4C4×#4111yescharacteristic
526C2×C4#52011yescharacteristic
527C2×C4#54611yescharacteristic
528C2×C4#54711yescharacteristic
529C8—42no—
530C8—82no—
531C8—81no—
532C22—21no—
533C22—21no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—81no—
548C22—81no—
549C22—81no—
550C22—81no—
551C22#233911yescharacteristic
552C22C2×#41011yescharacteristic
553C22#288411yescharacteristic
554C22C22×#10211yescharacteristic
555C22#289811yescharacteristic
556C22C2×#42111yescharacteristic
557C22#290111yescharacteristic
558C22#329411yescharacteristic
559C22#329811yescharacteristic
560C22#333411yescharacteristic
561C22#333511yescharacteristic
562C4—22no—
563C4—21no—
564C4—21no—
565C4—21no—
566C4—21no—
567C4—24no—
568C4—24no—
569C4—41no—
570C4—41no—
571C4—42no—
572C4—42no—
573C4—82no—
574C4—82no—
575C4—81no—
576C4—82no—
577C4—81no—
578C4—82no—
579C4—81no—
580C4—162no—
581C4#151714yes—
582C2—21no—
583C2—21no—
584C2—21no—
585C2—21no—
586C2—161no—
587C2C2×#151711yescharacteristic
588C2#1760111yescharacteristic
589C2#1760511yescharacteristic
590C2#1761811yescharacteristic
591C2#2247811yescharacteristic
592C2#2248511yescharacteristic
593C2#2250711yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#121C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22599C2semidirect
3. #13918C2semidirect
4. #9909C2semidirect
5. #9885C2semidirect
8. #8716C2semidirect
11. #8026C2semidirect
16. #2552C4semidirect
17. #2542C4semidirect
19. #2512C4semidirect
20. #2506C4semidirect
24. #2365C4semidirect
26. #2354C4semidirect

Maximal subgroups

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

subgroupindex
#361862
#225992
#139182
#99092
#98852
#91562
#91142
#87162
#85712
C4×#3272
#80262