Finite Group Databasebeta

← #196423

Subgroups — #196423

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2920C211yesmaximal, characteristic
2C2×#2855C211yesmaximal, characteristic
3C2×#1511C212yesmaximal
4#14684C211yesmaximal, characteristic
5#9116C214yesmaximal, complemented
6#9112C214yesmaximal, complemented
7#8015C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C42C2211yescharacteristic
10#2920—28no—
11C2×#432—21no—
12C2×#432C2212yes—
13C2×#431—21no—
14C2×#431C2212yes—
15C2×#423—21no—
16C2×#407C2212yes—
17C2×C4×M16—21no—
18#2855—21no—
19#2855—21no—
20#2855—21no—
21#2855—21no—
22#2855C412yescomplemented
23#2855C412yescomplemented
24#2855C2212yes—
25#2855C2212yes—
26C2×#391C411yescharacteristic, complemented
27C2×#391C411yescharacteristic, complemented
28C2×#389C411yescharacteristic, complemented
29C2×#389C411yescharacteristic, complemented
30C2×#387—21no—
31C2×#387—21no—
32C2×#385—21no—
33C2×#381—21no—
34C2×#381C2212yes—
35C2×#380C2211yescharacteristic
36#1511—28no—
37C2×#328C2218yes—
38#1434—21no—
39#1434—21no—
40#1434C2212yes—
41#1434C2212yes—
42C22⋊#95—21no—
43C22⋊#95—21no—
44#1294C2218yes—
45C23×Q8C2×C411yescharacteristic
46C24×C4D411yescharacteristic
47C24×C4D411yescharacteristic
48C24×C4C2311yescharacteristic
49C22×M16—22no—
50C2×C22⋊Q8—21no—
51C2×C22⋊Q8—21no—
52C2×C22⋊Q8—24no—
53C2×C22⋊Q8—24no—
54C2×C22⋊Q8C2×C412yes—
55C2×C22⋊Q8C2312yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2×C411yescharacteristic
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C4⋊C4C2×C411yescharacteristic
62C22×C22⋊C4—21no—
63C22×C22⋊C4—21no—
64C22×C22⋊C4—22no—
65C22×C42—24no—
66C22×C42—24no—
67C22×C42C2×C411yescharacteristic
68C22×C42C2×C411yescharacteristic
69C22×C42C2×C411yescharacteristic
70C22×C42C2×C411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42D411yescharacteristic
74C22×C42D411yescharacteristic
75C22×C42D411yescharacteristic
76C22×C42D411yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#432—28no—
80#432—28no—
81#431—28no—
82#431—28no—
83#423—28no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86#407—24no—
87#407D418yes—
88C2×C22⋊C8—22no—
89C2×C22⋊C8C2314yes—
90C4×M16—28no—
91C2×#97—22no—
92C2×C4×C8—22no—
93#391—21no—
94#391—21no—
95#391—21no—
96#391—21no—
97#391C2×C412yes—
98#391C2×C412yes—
99#391C2×C412yes—
100#391C2×C412yes—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389C2×C412yes—
106#389C2×C412yes—
107#389C2×C412yes—
108#389C2×C412yes—
109#387—22no—
110#387—22no—
111#387—22no—
112#387—22no—
113#387—22no—
114#387—22no—
115#387—22no—
116#387—22no—
117#385—22no—
118#385—22no—
119#385—22no—
120#385—22no—
121#381—22no—
122#381—22no—
123#381—22no—
124#381—22no—
125#381—24no—
126#381—24no—
127#380—21no—
128#380—21no—
129#380C2×C412yes—
130#380C2312yes—
131C2×#95—21no—
132C2×#95—21no—
133C2×#95—21no—
134C2×#95—21no—
135C2×#95—21no—
136C2×#95—21no—
137C2×#95—21no—
138C2×#95—22no—
139C2×#95—22no—
140C2×#95—22no—
141C2×#95—22no—
142C2×#95—22no—
143C2×#95C2314yes—
144#328—216no—
145C25C2×D411yescharacteristic
146C22×Q8—21no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—42no—
150C22×Q8C22×C412yes—
151C23×C4—24no—
152C23×C4—24no—
153C23×C4—24no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—24no—
159C23×C4—42no—
160C23×C4—42no—
161C23×C4—42no—
162C23×C4—42no—
163C23×C4—42no—
164C23×C4C22⋊C411yescharacteristic
165C23×C4C22⋊C411yescharacteristic
166C23×C4C22⋊C411yescharacteristic
167C23×C4C22⋊C411yescharacteristic
168C23×C4C22⋊C411yescharacteristic
169C23×C4C22⋊C411yescharacteristic
170C23×C4C22⋊C411yescharacteristic
171C23×C4C22⋊C411yescharacteristic
172C23×C4C22×C411yescharacteristic
173C23×C4C22×C411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4C2×D411yescharacteristic
176C23×C4C2×D411yescharacteristic
177C23×C4C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4C2×D414yes—
182C23×C4#4114yes—
183C23×C4C2411yescharacteristic
184C2×M16—28no—
185C2×M16—44no—
186C2×M16—44no—
187C22×C8—24no—
188C22⋊Q8—28no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191C22⋊Q8—42no—
192C22⋊Q8—44no—
193C22⋊Q8—42no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4C22×C412yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C22⋊C4—44no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—44no—
222C2×C42—24no—
223C2×C42—24no—
224C2×C42—24no—
225C2×C42—24no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—22no—
240C2×C42C22⋊C411yescharacteristic
241C2×C42C22⋊C411yescharacteristic
242C2×C42C22⋊C411yescharacteristic
243C2×C42C22⋊C411yescharacteristic
244C2×C42C22⋊C411yescharacteristic
245C2×C42C22⋊C411yescharacteristic
246C2×C42C22⋊C411yescharacteristic
247C2×C42C22⋊C411yescharacteristic
248C2×C42C22×C411yescharacteristic
249C2×C42C22×C411yescharacteristic
250C2×C42C22×C411yescharacteristic
251C2×C42C22×C411yescharacteristic
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D411yescharacteristic
254C2×C42C2×D411yescharacteristic
255C2×C42C2×D411yescharacteristic
256C4⋊C8—28no—
257C4⋊C8—28no—
258C22⋊C8—28no—
259C22⋊C8—24no—
260C22⋊C8C2×D418yes—
261#97—28no—
262C4×C8—28no—
263#95—22no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268#95—22no—
269#95—22no—
270#95—22no—
271#95—22no—
272#95—22no—
273#95—22no—
274#95—22no—
275#95—22no—
276#95—22no—
277#95—24no—
278#95—24no—
279#95—44no—
280#95—44no—
281#95—44no—
282#95—44no—
283#95—44no—
284C24—22no—
285C24—22no—
286C24—22no—
287C24—22no—
288C24—41no—
289C24—41no—
290C24C2×C22⋊C411yescharacteristic
291C24C2×C22⋊C411yescharacteristic
292C24#12114yes—
293C24C22×D411yescharacteristic
294C2×Q8—24no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8—88no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—22no—
322C22×C4—22no—
323C22×C4—22no—
324C22×C4—22no—
325C22×C4—22no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—22no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—22no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
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—41no—
349C22×C4—41no—
350C22×C4—48no—
351C22×C4—48no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—416no—
355C22×C4—416no—
356C22×C4—42no—
357C22×C4—42no—
358C22×C4—42no—
359C22×C4—42no—
360C22×C4—42no—
361C22×C4—42no—
362C22×C4—42no—
363C22×C4—42no—
364C22×C4—42no—
365C22×C4—42no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—44no—
369C22×C4—44no—
370C22×C4—44no—
371C22×C4—44no—
372C22×C4—44no—
373C22×C4—44no—
374C22×C4—44no—
375C22×C4C2×C22⋊C411yescharacteristic
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C2×C22⋊C411yescharacteristic
379C22×C4C2×C22⋊C411yescharacteristic
380C22×C4C2×C22⋊C411yescharacteristic
381C22×C4C2×C22⋊C411yescharacteristic
382C22×C4C2×C22⋊C411yescharacteristic
383C22×C4C2×C22⋊C411yescharacteristic
384C22×C4C2×C22⋊C411yescharacteristic
385C22×C4C4×D414yes—
386C22×C4C4×D414yes—
387C22×C4#12012yes—
388C22×C4#12012yes—
389C22×C4#12114yes—
390C22×C4#12312yes—
391C22×C4#12312yes—
392C22×C4#13714yes—
393C22×C4C23×C411yescharacteristic
394C22×C4C22×D411yescharacteristic
395C22×C4C22×D412yes—
396C22×C4C2×#4112yes—
397M16—416no—
398C2×C8—28no—
399C2×C8—44no—
400C2×C8—44no—
401C4⋊C4—24no—
402C4⋊C4—42no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—42no—
406C4⋊C4—44no—
407C4⋊C4—44no—
408C4⋊C4—42no—
409C4⋊C4—42no—
410C4⋊C4—42no—
411C4⋊C4—44no—
412C4⋊C4—44no—
413C4⋊C4—44no—
414C22⋊C4—44no—
415C22⋊C4—44no—
416C22⋊C4—44no—
417C22⋊C4—44no—
418C22⋊C4—88no—
419C42—22no—
420C42—22no—
421C42—22no—
422C42—22no—
423C42—21no—
424C42—21no—
425C42—21no—
426C42—21no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—22no—
432C42—22no—
433C42—22no—
434C42—22no—
435C42—22no—
436C42—22no—
437C42—22no—
438C42—22no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—21no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—21no—
452C23—21no—
453C23—28no—
454C23—28no—
455C23—44no—
456C23—44no—
457C23—44no—
458C23—44no—
459C23—44no—
460C23#38612yes—
461C23#38612yes—
462C23#41012yes—
463C23#41012yes—
464C23C22×C22⋊C411yescharacteristic
465C23C2×#12112yes—
466C23C2×#13712yes—
467C23#57512yes—
468Q8—84no—
469Q8—84no—
470C2×C4—28no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—24no—
484C2×C4—28no—
485C2×C4—24no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—28no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—44no—
504C2×C4—42no—
505C2×C4—44no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—44no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
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—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—84no—
532C2×C4—84no—
533C2×C4—84no—
534C2×C4—84no—
535C2×C4—84no—
536C2×C4—84no—
537C2×C4—84no—
538C2×C4—84no—
539C2×C4—84no—
540C2×C4—84no—
541C2×C4—84no—
542C2×C4#38612yes—
543C2×C4#38612yes—
544C2×C4#41912yes—
545C2×C4#41912yes—
546C2×C4#42112yes—
547C2×C4#42112yes—
548C2×C4C2×C4×D412yes—
549C2×C4C2×#12011yescharacteristic
550C2×C4C2×#12311yescharacteristic
551C8—48no—
552C22—22no—
553C22—22no—
554C22—22no—
555C22—22no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—22no—
560C22—22no—
561C22—22no—
562C22—22no—
563C22—22no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—416no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22#189914yes—
573C22C2×#38611yescharacteristic
574C22C2×#41011yescharacteristic
575C22C2×#41011yescharacteristic
576C22C2×#41911yescharacteristic
577C22#289611yescharacteristic
578C22C2×#42111yescharacteristic
579C22C2×#42111yescharacteristic
580C22#305614yes—
581C4—22no—
582C4—22no—
583C4—22no—
584C4—22no—
585C4—21no—
586C4—22no—
587C4—21no—
588C4—22no—
589C4—22no—
590C4—41no—
591C4—41no—
592C4—41no—
593C4—41no—
594C4—82no—
595C4—82no—
596C4—82no—
597C4—82no—
598C4—164no—
599C2—24no—
600C2—24no—
601C2—42no—
602C2#1755411yescharacteristic
603C2#1756211yescharacteristic
604C2#1758011yescharacteristic
605C2#1759011yescharacteristic
606C2#2090211yescharacteristic
607C2#2097211yescharacteristic
608C2C2×#189911yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #9116C2semidirect
6. #9112C2semidirect
22. #2855C4semidirect
23. #2855C4semidirect
26. C2×#391C4semidirect
27. C2×#391C4semidirect
28. C2×#389C4semidirect
29. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29202
C2×#28552
C2×#15112
#146842
#91162
#91122
#80152