Finite Group Databasebeta

← #214343

Subgroups — #214343

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 587 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2855C211yesmaximal, characteristic
2#20714C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#9152C214yesmaximal, complemented
5#9112C214yesmaximal, complemented
6#8713C212yesmaximal
7#8329C212yesmaximal
8C22×C22⋊Q8C2211yescharacteristic
9C23×C42C2211yescharacteristic
10C2×#431C2212yes—
11#2855—21no—
12#2855—21no—
13#2855—21no—
14#2855—21no—
15#2855C412yescomplemented
16#2855C412yescomplemented
17#2855C2212yes—
18#2855C2212yes—
19C2×#391C411yescharacteristic, complemented
20C2×#391C411yescharacteristic, complemented
21C2×#389C411yescharacteristic, complemented
22C2×#389C411yescharacteristic, complemented
23C2×#387—21no—
24C2×#387—21no—
25C2×#385—21no—
26C2×#382C2212yes—
27C2×#381—21no—
28C2×#380—21no—
29C2×#380C2211yescharacteristic
30#1817C2212yes—
31#1797—21no—
32#1797—21no—
33#1779C2212yes—
34C4×C22⋊C8—21no—
35C4×C22⋊C8—21no—
36C2×#328C2218yes—
37C4⋊#95—21no—
38#1434—21no—
39#1434C2212yes—
40C22⋊#95—21no—
41C22⋊#95C2212yes—
42#1294C2218yes—
43C23×Q8C2×C411yescharacteristic
44C24×C4D411yescharacteristic
45C24×C4D411yescharacteristic
46C24×C4C2311yescharacteristic
47C2×C22⋊Q8—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8—44no—
50C2×C22⋊Q8C2×C412yes—
51C2×C22⋊Q8C2312yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4C2×C411yescharacteristic
57C22×C4⋊C4C2×C411yescharacteristic
58C22×C4⋊C4C2×C411yescharacteristic
59C22×C22⋊C4—21no—
60C22×C22⋊C4—21no—
61C22×C22⋊C4—22no—
62C22×C42—44no—
63C22×C42C2×C411yescharacteristic
64C22×C42C2×C411yescharacteristic
65C22×C42C2×C411yescharacteristic
66C22×C42C2×C411yescharacteristic
67C22×C42D411yescharacteristic
68C22×C42D411yescharacteristic
69C22×C42D411yescharacteristic
70C22×C42D411yescharacteristic
71C22×C42D411yescharacteristic
72C22×C42D411yescharacteristic
73C22×C42C2311yescharacteristic
74C22×C42C2311yescharacteristic
75#431—44no—
76C2×C4⋊C8—22no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2314yes—
80C2×C4×C8—22no—
81#391—21no—
82#391—21no—
83#391—21no—
84#391—21no—
85#391C2×C412yes—
86#391C2×C412yes—
87#391C2×C412yes—
88#391C2×C412yes—
89#389—21no—
90#389—21no—
91#389—21no—
92#389—21no—
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#387—22no—
104#387—22no—
105#385—22no—
106#385—22no—
107#385—22no—
108#385—22no—
109#382—44no—
110#381—22no—
111#381—22no—
112#381—22no—
113#381—22no—
114#380—21no—
115#380—21no—
116#380—44no—
117#380C2×C412yes—
118#380C2312yes—
119C2×#95—21no—
120C2×#95—21no—
121C2×#95—21no—
122C2×#95—21no—
123C2×#95—21no—
124C2×#95—21no—
125C2×#95—21no—
126C2×#95—22no—
127C2×#95—22no—
128C2×#95—22no—
129C2×#95—22no—
130C2×#95C2314yes—
131#336—22no—
132#336—22no—
133#328—48no—
134C25C2×D411yescharacteristic
135C22×Q8—21no—
136C22×Q8—42no—
137C22×Q8—42no—
138C22×Q8—42no—
139C22×Q8C22×C412yes—
140C23×C4—24no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—24no—
146C23×C4—42no—
147C23×C4—42no—
148C23×C4—42no—
149C23×C4—42no—
150C23×C4—42no—
151C23×C4—42no—
152C23×C4—42no—
153C23×C4C22⋊C411yescharacteristic
154C23×C4C22⋊C411yescharacteristic
155C23×C4C22⋊C411yescharacteristic
156C23×C4C22⋊C411yescharacteristic
157C23×C4C22⋊C411yescharacteristic
158C23×C4C22⋊C411yescharacteristic
159C23×C4C22⋊C411yescharacteristic
160C23×C4C22⋊C411yescharacteristic
161C23×C4SD1614yes—
162C23×C4Q1614yes—
163C23×C4C22×C411yescharacteristic
164C23×C4C22×C411yescharacteristic
165C23×C4C2×D411yescharacteristic
166C23×C4C2×D411yescharacteristic
167C23×C4C2×D411yescharacteristic
168C23×C4C2×D411yescharacteristic
169C23×C4C2×D411yescharacteristic
170C23×C4C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2411yescharacteristic
173C22×C8—24no—
174C22⋊Q8—28no—
175C22⋊Q8—44no—
176C22⋊Q8—44no—
177C22⋊Q8—44no—
178C2×C4⋊C4—22no—
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—22no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—44no—
196C2×C4⋊C4C22×C412yes—
197C2×C4⋊C4C22×C412yes—
198C2×C4⋊C4C22×C412yes—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—44no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—44no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—44no—
208C2×C42—24no—
209C2×C42—24no—
210C2×C42—42no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42—42no—
216C2×C42—42no—
217C2×C42—42no—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42C22⋊C411yescharacteristic
221C2×C42C22⋊C411yescharacteristic
222C2×C42C22⋊C411yescharacteristic
223C2×C42C22⋊C411yescharacteristic
224C2×C42C22⋊C411yescharacteristic
225C2×C42C22⋊C411yescharacteristic
226C2×C42C22⋊C411yescharacteristic
227C2×C42C22⋊C411yescharacteristic
228C2×C42C22×C411yescharacteristic
229C2×C42C22×C411yescharacteristic
230C2×C42C22×C411yescharacteristic
231C2×C42C22×C411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42C2×D411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C4⋊C8—44no—
237C22⋊C8—28no—
238C22⋊C8—44no—
239C22⋊C8—44no—
240C4×C8—44no—
241#95—22no—
242#95—22no—
243#95—22no—
244#95—22no—
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—22no—
254#95—22no—
255#95—28no—
256#95—44no—
257#95—44no—
258#95—44no—
259#95—44no—
260C24—22no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—41no—
265C24—41no—
266C24C2×C22⋊C411yescharacteristic
267C24C2×C22⋊C411yescharacteristic
268C24C2×SD1612yes—
269C24C2×Q1612yes—
270C24C22×D411yescharacteristic
271C2×Q8—24no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—88no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
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—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
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—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—48no—
320C22×C4—48no—
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—416no—
332C22×C4—416no—
333C22×C4—42no—
334C22×C4—42no—
335C22×C4—42no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—42no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—42no—
345C22×C4—44no—
346C22×C4—44no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—44no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4#10314yes—
353C22×C4#10314yes—
354C22×C4#10314yes—
355C22×C4#10314yes—
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×SD1612yes—
367C22×C4C2×SD1612yes—
368C22×C4C2×Q1612yes—
369C22×C4C2×Q1612yes—
370C22×C4C23×C411yescharacteristic
371C22×C4C22×D411yescharacteristic
372C22×C4H(5,𝔽2)14yes—
373C2×C8—44no—
374C2×C8—44no—
375C2×C8—44no—
376C4⋊C4—24no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—44no—
381C4⋊C4—44no—
382C4⋊C4—44no—
383C4⋊C4—42no—
384C4⋊C4—44no—
385C4⋊C4—44no—
386C4⋊C4—42no—
387C4⋊C4—44no—
388C4⋊C4—88no—
389C22⋊C4—44no—
390C22⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—88no—
394C42—22no—
395C42—22no—
396C42—22no—
397C42—22no—
398C42—22no—
399C42—22no—
400C42—22no—
401C42—22no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—41no—
409C42—41no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—24no—
425C23—24no—
426C23—24no—
427C23—24no—
428C23—42no—
429C23—42no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23#41012yes—
435C23C2×#10312yes—
436C23C2×#10312yes—
437C23#42112yes—
438C23C22×C22⋊C411yescharacteristic
439C23#53412yes—
440C23C22×SD1611yescharacteristic
441C23C22×Q1611yescharacteristic
442C23#57512yes—
443Q8—84no—
444Q8—84no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—24no—
464C2×C4—24no—
465C2×C4—24no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—44no—
479C2×C4—44no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—44no—
484C2×C4—44no—
485C2×C4—44no—
486C2×C4—44no—
487C2×C4—44no—
488C2×C4—44no—
489C2×C4—44no—
490C2×C4—44no—
491C2×C4—44no—
492C2×C4—44no—
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—84no—
506C2×C4—84no—
507C2×C4—84no—
508C2×C4—84no—
509C2×C4—84no—
510C2×C4—84no—
511C2×C4—84no—
512C2×C4—84no—
513C2×C4—84no—
514C2×C4—84no—
515C2×C4—84no—
516C2×C4#41012yes—
517C2×C4C2×#10312yes—
518C2×C4C2×#10312yes—
519C2×C4C2×#10312yes—
520C2×C4C2×#10312yes—
521C2×C4#42112yes—
522C2×C4#51812yes—
523C2×C4#53412yes—
524C8—84no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—21no—
530C22—21no—
531C22—21no—
532C22—21no—
533C22—22no—
534C22—22no—
535C22—22no—
536C22—22no—
537C22—22no—
538C22—22no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—48no—
543C22—48no—
544C22—41no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22#151814yes—
549C22#230811yescharacteristic
550C22C2×#41011yescharacteristic
551C22C2×#41011yescharacteristic
552C22C22×#10311yescharacteristic
553C22#289611yescharacteristic
554C22C2×#42111yescharacteristic
555C22C2×#42111yescharacteristic
556C22#318712yes—
557C22#318912yes—
558C4—22no—
559C4—22no—
560C4—22no—
561C4—22no—
562C4—22no—
563C4—22no—
564C4—22no—
565C4—22no—
566C4—41no—
567C4—41no—
568C4—41no—
569C4—41no—
570C4—82no—
571C4—82no—
572C4—82no—
573C4—82no—
574C4—164no—
575C2—22no—
576C2—22no—
577C2—24no—
578C2—42no—
579C2#1755411yescharacteristic
580C2C2×#151811yescharacteristic
581C2#1757511yescharacteristic
582C2#1757911yescharacteristic
583C2#2174711yescharacteristic
584C2#2175911yescharacteristic
585C2#2177411yescharacteristic
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
4. #9152C2semidirect
5. #9112C2semidirect
15. #2855C4semidirect
16. #2855C4semidirect
19. C2×#391C4semidirect
20. C2×#391C4semidirect
21. C2×#389C4semidirect
22. C2×#389C4semidirect

Maximal subgroups

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

subgroupindex
C2×#28552
#207142
#146822
#91522
#91122
#87132
#83292