Finite Group Databasebeta

← #222223

Subgroups — #222223

Order 512.

968 conjugacy classes of subgroups.

The subgroup lattice has 718 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3102C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic, complemented
3#22531C211yesmaximal, characteristic, complemented
4C2×#1591C211yesmaximal, characteristic, complemented
5C2×#1584C211yesmaximal, characteristic, complemented
6#9663C211yesmaximal, characteristic, complemented
7#9657C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8952C211yesmaximal, characteristic, complemented
10#8952C211yesmaximal, characteristic, complemented
11#8948C211yesmaximal, characteristic, complemented
12#8599C211yesmaximal, characteristic, complemented
13#8315C211yesmaximal, characteristic, complemented
14#8312C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic, complemented
16C2×#529C2211yescharacteristic
17C2×C4×#41C2211yescharacteristic
18#3102—24no—
19#3102—24no—
20C2×#485C411yescharacteristic
21C2×#482C411yescharacteristic
22C2×#479C411yescharacteristic
23C2×#478C411yescharacteristic
24C2×#464C411yescharacteristic
25C2×#463C411yescharacteristic
26C2×#452C411yescharacteristic
27C2×#449C411yescharacteristic
28C2×#446C2211yescharacteristic
29C2×#446C2211yescharacteristic
30C2×#446C2211yescharacteristic
31C2×C8×Q8C2211yescharacteristic
32C2×C4×Q16C2211yescharacteristic
33C2×C4×SD16C2211yescharacteristic
34C2×C4×SD16C2211yescharacteristic
35C2×C4×D8C2211yescharacteristic
36C2×#431C2211yescharacteristic
37#1955—21no—
38#1955—21no—
39#1955—21no—
40#1954—21no—
41#1954C2211yescharacteristic
42#1954C2211yescharacteristic
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1798—21no—
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1750—21no—
50#1749—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1591—24no—
54#1584—24no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×D4⋊C8C2211yescharacteristic
62C2×D4⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×C82C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×SD16C2×C411yescharacteristic
74C22×SD16C2×C411yescharacteristic
75C22×D8C2×C411yescharacteristic
76#529—22no—
77#529—22no—
78#529—41no—
79#529—41no—
80C2×#126—21no—
81C2×#125C2×C411yescharacteristic
82C2×#124C2×C411yescharacteristic
83C2×#123—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×#121C2×C411yescharacteristic
86C4×#41—22no—
87C4×#41—22no—
88C4×#41—41no—
89C4×#41—41no—
90C2×C4×Q8C2×C411yescharacteristic
91C2×C4×Q8C2311yescharacteristic
92C2×C4×Q8C2311yescharacteristic
93C2×C4×D4—21no—
94C2×C4×D4—21no—
95C2×C4×D4—21no—
96C2×C4×D4C2×C411yescharacteristic
97C2×C4×D4C2311yescharacteristic
98C2×C4×D4C2311yescharacteristic
99C2×#117—21no—
100C2×#117—21no—
101C2×#117C2×C411yescharacteristic
102C22×C42—21no—
103C22×C42C2311yescharacteristic
104#485—24no—
105#482—24no—
106#479—24no—
107#478—24no—
108#464—24no—
109#463—24no—
110#452—24no—
111#449—24no—
112#446—22no—
113#446—22no—
114#446—22no—
115#446—41no—
116#446—41no—
117#446—41no—
118C8×Q8—22no—
119C8×Q8—41no—
120C4×Q16D414yes—
121C4×Q16D414yes—
122C4×SD16—22no—
123C4×SD16—22no—
124C4×SD16—22no—
125C4×SD16—22no—
126C4×D8—22no—
127C4×D8—22no—
128#431—22no—
129#431—22no—
130C2×#107C2×C411yescharacteristic
131C2×#107C2×C411yescharacteristic
132C2×#106C2×C411yescharacteristic
133C2×#106C2×C411yescharacteristic
134C2×C4⋊C8—21no—
135C2×C4⋊C8—21no—
136C2×C4⋊C8—21no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8C2×C411yescharacteristic
139C2×C4⋊C8C2×C411yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142C2×C4⋊C8C2311yescharacteristic
143C2×C4⋊C8C2311yescharacteristic
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#102C2×C411yescharacteristic
149C2×#102C2×C411yescharacteristic
150C2×#102C2×C411yescharacteristic
151C2×#102C2×C411yescharacteristic
152C2×C22⋊C8—21no—
153C2×C22⋊C8—21no—
154C2×C22⋊C8—21no—
155C2×C22⋊C8—21no—
156C2×C22⋊C8—21no—
157C2×C22⋊C8—21no—
158C2×C22⋊C8—21no—
159C2×C22⋊C8—21no—
160C2×C22⋊C8—21no—
161C2×C22⋊C8—21no—
162C2×C22⋊C8—21no—
163C2×C22⋊C8C2×C411yescharacteristic
164C2×C22⋊C8C2×C411yescharacteristic
165C2×C22⋊C8D411yescharacteristic
166C2×C22⋊C8D411yescharacteristic
167C2×#97—21no—
168C2×#97—21no—
169C2×#97—21no—
170C2×C4×C8—21no—
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175C2×C4×C8C2311yescharacteristic
176C2×C4×C8C2311yescharacteristic
177#336—21no—
178#336—21no—
179#336—21no—
180#336D411yescharacteristic
181#336D411yescharacteristic
182#335—22no—
183#335—22no—
184Q8⋊C8—22no—
185Q8⋊C8—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188D4⋊C8—41no—
189D4⋊C8—41no—
190D4⋊C8—41no—
191D4⋊C8—41no—
192#322—22no—
193C82—22no—
194C2×#41—22no—
195C2×#41—22no—
196C2×#41—41no—
197C2×#41—41no—
198C2×#41—41no—
199C2×#41—41no—
200C22×Q8C22×C411yescharacteristic
201C22×Q8C22×C411yescharacteristic
202C22×D4—21no—
203C22×D4—21no—
204C22×D4C22×C411yescharacteristic
205C22×D4C22×C411yescharacteristic
206C23×C4—21no—
207C23×C4—21no—
208C23×C4—21no—
209C23×C4—21no—
210C23×C4C22×C411yescharacteristic
211C23×C4C2×D411yescharacteristic
212C23×C4#4111yescharacteristic
213C2×Q16—41no—
214C2×Q16—41no—
215C2×Q16C22⋊C414yes—
216C2×SD16—22no—
217C2×SD16—22no—
218C2×SD16—22no—
219C2×SD16—41no—
220C2×SD16—41no—
221C2×SD16—41no—
222C2×D8—22no—
223C2×D8—22no—
224C2×D8—41no—
225C22×C8—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8—21no—
231C22×C8C22×C411yescharacteristic
232C22×C8C22×C411yescharacteristic
233C22×C8C22×C411yescharacteristic
234C22×C8C22×C411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C22×C8C2×D411yescharacteristic
237C22×C8C2×D411yescharacteristic
238C22×C8C2×D411yescharacteristic
239#126—42no—
240#126—42no—
241#125—22no—
242#125—41no—
243#124—22no—
244#124—41no—
245#123—42no—
246#123—42no—
247C22⋊Q8—22no—
248C22⋊Q8—41no—
249#121—22no—
250#121—41no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8—21no—
255C4×Q8—22no—
256C4×Q8—41no—
257C4×Q8C2×D412yes—
258C4×Q8C2×D412yes—
259C4×Q8C2×D412yes—
260C4×Q8C2×D412yes—
261C4×D4—22no—
262C4×D4—21no—
263C4×D4—21no—
264C4×D4—21no—
265C4×D4—21no—
266C4×D4—21no—
267C4×D4—21no—
268C4×D4—21no—
269C4×D4—21no—
270C4×D4—41no—
271C4×D4—42no—
272C4×D4—42no—
273C4×D4—42no—
274C4×D4—42no—
275C4×D4—42no—
276C4×D4—42no—
277#117—22no—
278#117—42no—
279#117—42no—
280#117—42no—
281#117—42no—
282#117—41no—
283C2×C4⋊C4—21no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C22×C411yescharacteristic
290C2×C4⋊C4C22×C411yescharacteristic
291C2×C4⋊C4C22×C411yescharacteristic
292C2×C4⋊C4C22×C411yescharacteristic
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4—21no—
296C2×C22⋊C4—21no—
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—21no—
304C2×C42—21no—
305C2×C42—21no—
306C2×C42—21no—
307C2×C42—21no—
308C2×C42—21no—
309C2×C42—42no—
310C2×C42—42no—
311C2×C42C22×C411yescharacteristic
312C2×C42C2×D411yescharacteristic
313C2×C42#4111yescharacteristic
314C2×C42C2411yescharacteristic
315#107—22no—
316#107C22⋊C414yes—
317#106—22no—
318#106—22no—
319C4⋊C8—22no—
320C4⋊C8—22no—
321C4⋊C8—21no—
322C4⋊C8—21no—
323C4⋊C8—21no—
324C4⋊C8—21no—
325C4⋊C8—21no—
326C4⋊C8—21no—
327C4⋊C8—21no—
328C4⋊C8—21no—
329C4⋊C8—42no—
330C4⋊C8—42no—
331C4⋊C8—42no—
332C4⋊C8—42no—
333#103—22no—
334#103—22no—
335#103C22⋊C414yes—
336#103C22⋊C414yes—
337#102—22no—
338#102—22no—
339#102—22no—
340#102—22no—
341C22⋊C8—22no—
342C22⋊C8—22no—
343C22⋊C8—22no—
344C22⋊C8—22no—
345C22⋊C8—42no—
346C22⋊C8—42no—
347C22⋊C8—42no—
348C22⋊C8—42no—
349C22⋊C8—42no—
350C22⋊C8—81no—
351C22⋊C8—81no—
352C22⋊C8—81no—
353C22⋊C8—81no—
354C22⋊C8—81no—
355C22⋊C8—81no—
356#97—42no—
357#97—42no—
358#97—42no—
359C4×C8—21no—
360C4×C8—21no—
361C4×C8—21no—
362C4×C8—21no—
363C4×C8—21no—
364C4×C8—21no—
365C4×C8—21no—
366C4×C8—21no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—42no—
370C4×C8C2×D412yes—
371C4×C8C2×D412yes—
372C24—21no—
373C24—21no—
374C24C4×D411yescharacteristic
375#41—44no—
376#41—44no—
377#41—44no—
378#41—44no—
379C2×Q8—21no—
380C2×Q8—21no—
381C2×Q8—21no—
382C2×Q8—21no—
383C2×Q8—21no—
384C2×Q8—21no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—21no—
389C2×Q8C2×C22⋊C412yes—
390C2×Q8C2×C22⋊C412yes—
391C2×D4—21no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—21no—
395C2×D4—21no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4—21no—
402C2×D4—42no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—81no—
408C2×D4#13112yes—
409C22×C4—21no—
410C22×C4—21no—
411C22×C4—21no—
412C22×C4—21no—
413C22×C4—21no—
414C22×C4—21no—
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—42no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—42no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—41no—
444C22×C4—41no—
445C22×C4—41no—
446C22×C4—41no—
447C22×C4—41no—
448C22×C4—81no—
449C22×C4C4×D411yescharacteristic
450C22×C4C4×D411yescharacteristic
451C22×C4C4×D411yescharacteristic
452C22×C4C23×C411yescharacteristic
453C22×C4C22×D411yescharacteristic
454C22×C4C2×#4111yescharacteristic
455Q16—44no—
456SD16—44no—
457SD16—44no—
458D8—44no—
459C2×C8—21no—
460C2×C8—21no—
461C2×C8—21no—
462C2×C8—21no—
463C2×C8—21no—
464C2×C8—21no—
465C2×C8—21no—
466C2×C8—21no—
467C2×C8—22no—
468C2×C8—21no—
469C2×C8—21no—
470C2×C8—21no—
471C2×C8—21no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—21no—
479C2×C8—21no—
480C2×C8—41no—
481C2×C8—41no—
482C2×C8—41no—
483C2×C8—41no—
484C2×C8—41no—
485C2×C8—41no—
486C2×C8—41no—
487C2×C8—41no—
488C2×C8—41no—
489C2×C8—41no—
490C2×C8—41no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8C2×C22⋊C412yes—
498C2×C8C2×C22⋊C412yes—
499C2×C8#13512yes—
500C2×C8#13512yes—
501C4⋊C4—21no—
502C4⋊C4—21no—
503C4⋊C4—22no—
504C4⋊C4—22no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—21no—
508C4⋊C4—21no—
509C4⋊C4—21no—
510C4⋊C4—21no—
511C4⋊C4—42no—
512C4⋊C4—42no—
513C4⋊C4—42no—
514C4⋊C4—42no—
515C4⋊C4C2×C22⋊C412yes—
516C4⋊C4C2×C22⋊C412yes—
517C22⋊C4—42no—
518C22⋊C4—42no—
519C22⋊C4—42no—
520C22⋊C4—42no—
521C22⋊C4—42no—
522C22⋊C4—42no—
523C22⋊C4—41no—
524C22⋊C4—41no—
525C22⋊C4—41no—
526C22⋊C4—41no—
527C42—21no—
528C42—21no—
529C42—22no—
530C42—22no—
531C42—22no—
532C42—22no—
533C42—21no—
534C42—21no—
535C42—21no—
536C42—21no—
537C42—42no—
538C42—42no—
539C42C22×D411yescharacteristic
540C42C22×D411yescharacteristic
541C42H(5,𝔽2)11yescharacteristic
542C42H(5,𝔽2)11yescharacteristic
543C23—21no—
544C23—21no—
545C23—21no—
546C23—22no—
547C23—21no—
548C23—21no—
549C23—21no—
550C23—21no—
551C23—41no—
552C23—41no—
553C23—41no—
554C23—41no—
555C23—41no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23—41no—
563C23—41no—
564C23C2×C4×D411yescharacteristic
565Q8—22no—
566Q8—22no—
567Q8—22no—
568Q8—22no—
569Q8—41no—
570Q8—41no—
571Q8—41no—
572Q8—41no—
573D4—22no—
574D4—22no—
575D4—48no—
576D4—41no—
577D4—41no—
578D4—41no—
579D4—41no—
580D4—41no—
581D4—41no—
582D4—84no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—22no—
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—22no—
592C2×C4—21no—
593C2×C4—21no—
594C2×C4—21no—
595C2×C4—21no—
596C2×C4—21no—
597C2×C4—21no—
598C2×C4—48no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—81no—
622C2×C4—81no—
623C2×C4—84no—
624C2×C4—81no—
625C2×C4—81no—
626C2×C4—81no—
627C2×C4—81no—
628C2×C4—81no—
629C2×C4—81no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4C22×C22⋊C411yescharacteristic
633C2×C4#51811yescharacteristic
634C2×C4D4211yescharacteristic
635C2×C4D4211yescharacteristic
636C2×C4#54611yescharacteristic
637C2×C4#54611yescharacteristic
638C2×C4C2×#13111yescharacteristic
639C2×C4#56811yescharacteristic
640C2×C4#56811yescharacteristic
641C2×C4#56811yescharacteristic
642C2×C4C2×#13511yescharacteristic
643C2×C4C2×#13511yescharacteristic
644C2×C4#57611yescharacteristic
645C2×C4#57811yescharacteristic
646C8—22no—
647C8—22no—
648C8—22no—
649C8—22no—
650C8—42no—
651C8—41no—
652C8—41no—
653C8—41no—
654C8—41no—
655C8—81no—
656C8—81no—
657C8—81no—
658C8—81no—
659C8—81no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—42no—
664C22—41no—
665C22—42no—
666C22—42no—
667C22—41no—
668C22—41no—
669C22—41no—
670C22—81no—
671C22—81no—
672C22—81no—
673C22—81no—
674C22—81no—
675C22—81no—
676C22—81no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22D4×C22⋊C411yescharacteristic
682C22#293011yescharacteristic
683C22#293311yescharacteristic
684C22C4×#13511yescharacteristic
685C22#295111yescharacteristic
686C22#295511yescharacteristic
687C22#295611yescharacteristic
688C4—21no—
689C4—21no—
690C4—41no—
691C4—42no—
692C4—42no—
693C4—41no—
694C4—41no—
695C4—42no—
696C4—42no—
697C4—81no—
698C4—81no—
699C4#287911yescharacteristic
700C4#288011yescharacteristic
701C4#297611yescharacteristic
702C4#297811yescharacteristic
703C4#328511yescharacteristic
704C4#328511yescharacteristic
705C4#328711yescharacteristic
706C4#328711yescharacteristic
707C2—81no—
708C2—82no—
709C2—161no—
710C2#1793211yescharacteristic
711C2#1795411yescharacteristic
712C2#1799611yescharacteristic
713C2#1802711yescharacteristic
714C2#2240011yescharacteristic
715C2#2245111yescharacteristic
716C2#2245311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC2×#449C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22532C2semidirect
3. #22531C2semidirect
4. C2×#1591C2semidirect
5. C2×#1584C2semidirect
6. #9663C2semidirect
7. #9657C2semidirect
8. #8952C2semidirect
9. #8952C2semidirect
10. #8952C2semidirect
11. #8948C2semidirect
12. #8599C2semidirect
13. #8315C2semidirect
14. #8312C2semidirect
15. #8148C2semidirect

Maximal subgroups

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

subgroupindex
C2×#31022
#225322
#225312
C2×#15912
C2×#15842
#96632
#96572
#89522
#89482
#85992
#83152
#83122
#81482