Finite Group Databasebeta

← #481600

Subgroups — #481600

Order 512.

1190 conjugacy classes of subgroups.

The subgroup lattice has 619 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34937C211yesmaximal, characteristic
2C4×#410C211yesmaximal, characteristic, complemented
3#17361C211yesmaximal, characteristic, complemented
4#11777C212yesmaximal, complemented
5#11776C212yesmaximal, complemented
6#11775C214yesmaximal, complemented
7#11691C214yesmaximal, complemented
8C2×#532—21no—
9C2×#532C411yescharacteristic, complemented
10C2×C4×#41C2211yescharacteristic
11#2920—21no—
12C2×#410—21no—
13#2548C411yescharacteristic, complemented
14#2547C2211yescharacteristic
15#2546C411yescharacteristic, complemented
16#2545—21no—
17#2544—21no—
18#2543C412yescomplemented
19#2543C2212yes—
20#2541C2211yescharacteristic
21#2416—21no—
22#2416—21no—
23#2392C412yescomplemented
24#2392C2212yes—
25C4×#121—21no—
26C4×#121—21no—
27#2297—21no—
28C4×#117—21no—
29C2×C4×C4⋊C4C411yescharacteristic, complemented
30C4×#99—21no—
31C4×#99—21no—
32C4×#99C2212yes—
33C4×#99C2212yes—
34#1740C2212yes—
35#1467—21no—
36#1466C2212yes—
37#1465—21no—
38#1465C2212yes—
39#1458—21no—
40#1458C2212yes—
41#1344C2218yes—
42#1343C2218yes—
43C22×#41—21no—
44C22×#41D411yescharacteristic
45#532—41no—
46#532—41no—
47#532—41no—
48#532—41no—
49#532—48no—
50C2×C4⋊Q8—21no—
51C2×C4⋊Q8C2×C411yescharacteristic
52C2×#127—21no—
53C2×#127C2×C411yescharacteristic
54C2×#124—21no—
55C2×#124—21no—
56C2×#124—21no—
57C2×#121—22no—
58C2×#121—22no—
59C2×#121—21no—
60C2×#121—21no—
61C2×#121—21no—
62C2×#121C2×C412yes—
63C4×#41—41no—
64C4×#41—41no—
65C4×#41—41no—
66C4×#41—41no—
67C2×C4×Q8C2×C411yescharacteristic
68C2×C4×Q8D411yescharacteristic
69C2×C4×D4—22no—
70C2×C4×D4—22no—
71C2×C4×D4—21no—
72C2×C4×D4C2×C411yescharacteristic
73C2×C4×D4C2×C412yes—
74C2×C4×D4D411yescharacteristic
75C2×C4×D4D411yescharacteristic
76C2×C4×D4D411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C2×C4×D4C2311yescharacteristic
79C2×#117—22no—
80C2×#117C2×C411yescharacteristic
81C2×#117D411yescharacteristic
82C2×#117C2311yescharacteristic
83C22×C4⋊C4—21no—
84C22×C42—21no—
85C22×C42C2×C411yescharacteristic
86C22×C42D411yescharacteristic
87C22×C42D411yescharacteristic
88#432—22no—
89#432—41no—
90#431—22no—
91#431—41no—
92#423—41no—
93#410—28no—
94C2×#99—22no—
95C2×#99—22no—
96#407—22no—
97C4×M16—41no—
98#398C2×C412yes—
99#398C2×C412yes—
100#397C2×C412yes—
101#397C2312yes—
102#396—22no—
103#396—22no—
104#394—22no—
105#394—22no—
106#394C2×C412yes—
107#394C2×C412yes—
108#392C2×C412yes—
109#392C2312yes—
110#391—21no—
111#391—21no—
112#390—21no—
113#390—21no—
114#390—21no—
115#390—21no—
116#389—21no—
117#389—21no—
118#388—22no—
119#388—22no—
120#386—22no—
121#386—22no—
122#386—22no—
123#386—22no—
124#380—21no—
125#380—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4—21no—
128C4×C4⋊C4—41no—
129C4×C4⋊C4—41no—
130C4×C4⋊C4C2×C412yes—
131C4×C4⋊C4C2×C411yescharacteristic
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C22⋊C4—22no—
134C4×C22⋊C4—22no—
135C4×C22⋊C4—22no—
136C4×C22⋊C4—22no—
137#376—41no—
138C2×#95C2×C412yes—
139C43—41no—
140#342—22no—
141#342C2314yes—
142#324—22no—
143#324—22no—
144#323—22no—
145#323C2314yes—
146C2×#41—24no—
147C2×#41—24no—
148C2×#41—41no—
149C2×#41—41no—
150C2×#41—42no—
151C2×#41—42no—
152C2×#41—44no—
153C2×#41—44no—
154C22×Q8—21no—
155C22×Q8C22⋊C411yescharacteristic
156C22×D4—22no—
157C22×D4—22no—
158C22×D4—22no—
159C22×D4—21no—
160C22×D4—21no—
161C22×D4C22⋊C411yescharacteristic
162C22×D4C22×C412yes—
163C22×D4C2×D411yescharacteristic
164C22×D4C2×D411yescharacteristic
165C23×C4—22no—
166C23×C4—22no—
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22×C411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C2×M16—42no—
177C4⋊Q8—41no—
178C4⋊Q8—41no—
179C4⋊Q8—44no—
180#127—41no—
181#127—41no—
182#127—44no—
183#124—42no—
184#124—42no—
185#124—44no—
186#124—44no—
187#121—44no—
188#121—44no—
189#121—42no—
190#121—42no—
191#121—48no—
192#121—48no—
193#121—42no—
194#121—42no—
195C4×Q8—41no—
196C4×Q8—41no—
197C4×Q8—41no—
198C4×Q8—41no—
199C4×D4—41no—
200C4×D4—41no—
201C4×D4—41no—
202C4×D4—41no—
203C4×D4—41no—
204C4×D4—41no—
205C4×D4—41no—
206C4×D4—41no—
207C4×D4—41no—
208C4×D4—41no—
209C4×D4—41no—
210C4×D4—41no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—48no—
214C4×D4—48no—
215C4×D4—41no—
216C4×D4—41no—
217C4×D4—41no—
218C4×D4—41no—
219#117—28no—
220#117—41no—
221#117—41no—
222#117—41no—
223#117—41no—
224#117—41no—
225#117—41no—
226#117—44no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—44no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C411yescharacteristic
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C2×D411yescharacteristic
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C22⋊C4—24no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—22no—
245C2×C22⋊C4—22no—
246C2×C22⋊C4—22no—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4C22⋊C411yescharacteristic
254C2×C22⋊C4C22⋊C411yescharacteristic
255C2×C22⋊C4C22×C411yescharacteristic
256C2×C22⋊C4C2×D411yescharacteristic
257C2×C22⋊C4C2×D411yescharacteristic
258C2×C22⋊C4C2411yescharacteristic
259C2×C42—21no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—22no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—42no—
275C2×C42—44no—
276C2×C42—42no—
277C2×C42—42no—
278C2×C42C22⋊C411yescharacteristic
279C2×C42C22⋊C411yescharacteristic
280C2×C42C22⋊C411yescharacteristic
281C2×C42C22⋊C411yescharacteristic
282C2×C42C22⋊C411yescharacteristic
283C2×C42C22⋊C411yescharacteristic
284C2×C42C22×C411yescharacteristic
285C2×C42C22×C411yescharacteristic
286C2×C42C2×D411yescharacteristic
287C2×C42C2×D411yescharacteristic
288C4⋊C8—42no—
289C4⋊C8—42no—
290#99—28no—
291#99—28no—
292C22⋊C8—24no—
293C22⋊C8—42no—
294#97—42no—
295C4×C8—42no—
296#95—22no—
297#95—22no—
298#95—22no—
299#95—22no—
300#95—22no—
301#95—42no—
302#95—42no—
303#95C22×C412yes—
304#95C22×C412yes—
305C24—22no—
306C24C2×C22⋊C411yescharacteristic
307C24C2×C22⋊C411yescharacteristic
308C24C22×D411yescharacteristic
309#41—48no—
310#41—48no—
311#41—88no—
312#41—88no—
313C2×Q8—22no—
314C2×Q8—22no—
315C2×Q8—41no—
316C2×Q8—42no—
317C2×Q8—42no—
318C2×Q8—42no—
319C2×D4—22no—
320C2×D4—22no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—42no—
329C2×D4—42no—
330C2×D4—42no—
331C2×D4—44no—
332C2×D4—44no—
333C2×D4—42no—
334C2×D4—42no—
335C2×D4—44no—
336C2×D4—44no—
337C2×D4—44no—
338C2×D4—44no—
339C2×D4—44no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—42no—
343C2×D4—82no—
344C2×D4—82no—
345C2×D4—82no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—22no—
352C22×C4—22no—
353C22×C4—21no—
354C22×C4—21no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—42no—
372C22×C4—42no—
373C22×C4—42no—
374C22×C4—44no—
375C22×C4—44no—
376C22×C4—44no—
377C22×C4—42no—
378C22×C4—42no—
379C22×C4—44no—
380C22×C4—44no—
381C22×C4—81no—
382C22×C4—81no—
383C22×C4—82no—
384C22×C4—81no—
385C22×C4#10414yes—
386C22×C4C2×C22⋊C411yescharacteristic
387C22×C4C2×C22⋊C411yescharacteristic
388C22×C4C2×C22⋊C411yescharacteristic
389C22×C4C2×C22⋊C411yescharacteristic
390C22×C4C2×C22⋊C411yescharacteristic
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4AGL(1,ℤ/8)14yes—
397C22×C4C23×C411yescharacteristic
398C22×C4C22×D411yescharacteristic
399M16—84no—
400C2×C8—44no—
401C4⋊C4—24no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—48no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—44no—
416C22⋊C4—28no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—41no—
421C22⋊C4—41no—
422C22⋊C4—41no—
423C22⋊C4—44no—
424C22⋊C4—44no—
425C22⋊C4—44no—
426C22⋊C4—44no—
427C22⋊C4—44no—
428C22⋊C4—44no—
429C22⋊C4—44no—
430C22⋊C4—42no—
431C22⋊C4—42no—
432C22⋊C4—44no—
433C22⋊C4—82no—
434C22⋊C4—82no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—42no—
454C42—42no—
455C42—48no—
456C42—44no—
457C42—84no—
458C42—84no—
459C42#9911yescharacteristic
460C42#9911yescharacteristic
461C42#9911yescharacteristic
462C42#9911yescharacteristic
463C23—21no—
464C23—21no—
465C23—21no—
466C23—21no—
467C23—21no—
468C23—21no—
469C23—21no—
470C23—41no—
471C23—41no—
472C23—41no—
473C23—41no—
474C23—41no—
475C23—41no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23—42no—
480C23—42no—
481C23—42no—
482C23—42no—
483C23C2×#10412yes—
484C23C22×C22⋊C411yescharacteristic
485C23C2×AGL(1,ℤ/8)12yes—
486Q8—44no—
487Q8—84no—
488D4—44no—
489D4—44no—
490D4—44no—
491D4—82no—
492D4—84no—
493D4—88no—
494D4—88no—
495D4—84no—
496D4—88no—
497D4—82no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—21no—
516C2×C4—21no—
517C2×C4—41no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—42no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—48no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—42no—
539C2×C4—42no—
540C2×C4—42no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4—81no—
547C2×C4—88no—
548C2×C4—84no—
549C2×C4—84no—
550C2×C4—84no—
551C2×C4—84no—
552C2×C4C2×#9911yescharacteristic
553C2×C4C2×#9911yescharacteristic
554C2×C4C2×#9911yescharacteristic
555C2×C4C2×#9911yescharacteristic
556C2×C4C2×#9911yescharacteristic
557C2×C4C2×#9911yescharacteristic
558C2×C4#41812yes—
559C2×C4#41812yes—
560C2×C4C2×#10412yes—
561C2×C4C2×#10412yes—
562C8—84no—
563C22—21no—
564C22—21no—
565C22—22no—
566C22—22no—
567C22—42no—
568C22—42no—
569C22—42no—
570C22—42no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—41no—
575C22—41no—
576C22—41no—
577C22—41no—
578C22—81no—
579C22—81no—
580C22—82no—
581C22—82no—
582C22—82no—
583C22—82no—
584C22—82no—
585C22—82no—
586C22—82no—
587C22C22×#9911yescharacteristic
588C22C2×#41811yescharacteristic
589C22C22×#10411yescharacteristic
590C4—21no—
591C4—21no—
592C4—21no—
593C4—22no—
594C4—22no—
595C4—44no—
596C4—41no—
597C4—41no—
598C4—41no—
599C4—41no—
600C4—41no—
601C4—41no—
602C4—41no—
603C4—82no—
604C4—88no—
605C4—82no—
606C4#150712yes—
607C4#150712yes—
608C4#212612yes—
609C2—21no—
610C2—21no—
611C2—42no—
612C2—42no—
613C2—81no—
614C2—162no—
615C2C2×#150711yescharacteristic
616C2#3334711yescharacteristic
617C2C2×#212611yescharacteristic
subgroupG/H
centerC2×C4C2×#99
commutatorC22×C4C23×C4
Frattini, cosocleC2×C22⋊C4C24
socleC22C22×#99
prime radical#34937C2
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×#41 ◁ G
LECSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C4×#410C2semidirect
3. #17361C2semidirect
4. #11777C2semidirect
5. #11776C2semidirect
6. #11775C2semidirect
7. #11691C2semidirect
9. C2×#532C4semidirect
13. #2548C4semidirect
15. #2546C4semidirect
18. #2543C4semidirect
23. #2392C4semidirect
29. C2×C4×C4⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#349372
C4×#4102
#173612
#117772
#117762
#117752
#116912