Finite Group Databasebeta

← #481253

Subgroups — #481253

Order 512.

700 conjugacy classes of subgroups.

The subgroup lattice has 670 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60331C211yesmaximal, characteristic
2#33504C211yesmaximal, characteristic, complemented
3#33479C211yesmaximal, characteristic, complemented
4#11640C211yesmaximal, characteristic, complemented
5#11639C211yesmaximal, characteristic, complemented
6#11637C211yesmaximal, characteristic, complemented
7#11624C211yesmaximal, characteristic, complemented
8#11622C211yesmaximal, characteristic, complemented
9#11621C211yesmaximal, characteristic, complemented
10#11616C211yesmaximal, characteristic, complemented
11#11615C211yesmaximal, characteristic, complemented
12#11614C211yesmaximal, characteristic, complemented
13#11607C211yesmaximal, characteristic, complemented
14#11606C211yesmaximal, characteristic, complemented
15C2×Q8⋊C16C211yesmaximal, characteristic, complemented
16#3482C411yescharacteristic
17#2994—21no—
18#2994—21no—
19#2994C411yescharacteristic
20#2994C411yescharacteristic
21#2991C411yescharacteristic
22#2990—21no—
23#2990C411yescharacteristic
24#2986—21no—
25#2986C411yescharacteristic
26#2984C411yescharacteristic
27#2984C411yescharacteristic
28#2966—21no—
29#2966—21no—
30#2966C2211yescharacteristic
31#2966C2211yescharacteristic
32C8×#41—21no—
33C8×#41C2211yescharacteristic
34C8×#41C2211yescharacteristic
35#2960—21no—
36#2960C2211yescharacteristic
37C2×C8×Q8C2211yescharacteristic
38C2×C8×Q8C2211yescharacteristic
39#2183—21no—
40#2183—21no—
41#2169—21no—
42#2168—21no—
43#2168—21no—
44#2168—21no—
45#2168—21no—
46D4×C16—21no—
47#2163C2211yescharacteristic
48#2163C2211yescharacteristic
49#2162C2211yescharacteristic
50#2162C2211yescharacteristic
51#2162C2211yescharacteristic
52#2162C2211yescharacteristic
53#2150C2211yescharacteristic
54C2×C4⋊C16C2211yescharacteristic
55C2×C4⋊C16C2211yescharacteristic
56C4×M32C2211yescharacteristic
57C4×M32C2211yescharacteristic
58C2×C4×C16C2211yescharacteristic
59Q8⋊C16C2211yescharacteristic
60Q8⋊C16C2211yescharacteristic
61Q8⋊C16C2211yescharacteristic
62Q8⋊C16C2211yescharacteristic
63Q8⋊C16C2211yescharacteristic
64Q8⋊C16C2211yescharacteristic
65Q8⋊C16C2211yescharacteristic
66Q8⋊C16C2211yescharacteristic
67Q8⋊C16C2211yescharacteristic
68Q8⋊C16C2211yescharacteristic
69Q8⋊C16C2211yescharacteristic
70Q8⋊C16C2211yescharacteristic
71D4⋊C16C2211yescharacteristic
72D4⋊C16C2211yescharacteristic
73D4⋊C16C2211yescharacteristic
74D4⋊C16C2211yescharacteristic
75#557C811yescharacteristic
76#556—21no—
77#556C811yescharacteristic
78#552—21no—
79#552—21no—
80#552C811yescharacteristic
81#552C811yescharacteristic
82D4×Q8C811yescharacteristic
83#548C811yescharacteristic
84#548C811yescharacteristic
85#547—21no—
86#547C811yescharacteristic
87#533—21no—
88#533C2×C411yescharacteristic
89#529—21no—
90#529—21no—
91#529C2×C411yescharacteristic
92#529C2×C411yescharacteristic
93C4×#41—21no—
94C4×#41C2×C411yescharacteristic
95C4×#41C2×C411yescharacteristic
96C2×C4×Q8C2×C411yescharacteristic
97C2×C4×Q8C2×C411yescharacteristic
98C2×M32—21no—
99C2×M32—21no—
100C2×M32—21no—
101C22×C16—21no—
102C22×C16—21no—
103C22×C16—21no—
104#446—21no—
105#446—21no—
106#446—21no—
107#446—21no—
108#446—21no—
109#446—21no—
110#446—21no—
111#446C2×C411yescharacteristic
112#446C2×C411yescharacteristic
113#446C2×C411yescharacteristic
114#446C2×C411yescharacteristic
115C8×Q8—21no—
116C8×Q8—21no—
117C8×Q8—21no—
118C8×Q8—21no—
119C8×Q8—21no—
120C8×Q8—21no—
121C8×Q8C2×C411yescharacteristic
122C8×Q8C2×C411yescharacteristic
123C8×Q8C2×C411yescharacteristic
124C8×Q8C2×C411yescharacteristic
125C8×Q8C2×C411yescharacteristic
126C8×Q8D411yescharacteristic
127C8×Q8D411yescharacteristic
128C8×Q8C2311yescharacteristic
129C8×Q8C2311yescharacteristic
130C8×Q8C2311yescharacteristic
131#436—21no—
132#436—21no—
133#436C2×C411yescharacteristic
134#436C2×C411yescharacteristic
135#435—21no—
136#435—21no—
137#435—21no—
138#435—21no—
139#435—21no—
140#435—21no—
141C8×D4—21no—
142C8×D4—21no—
143C8×D4—21no—
144C8×D4—21no—
145C8×D4—21no—
146C8×D4—21no—
147C8×D4—21no—
148C8×D4C2×C411yescharacteristic
149C8×D4D411yescharacteristic
150C8×D4D411yescharacteristic
151C8×D4D411yescharacteristic
152C8×D4D411yescharacteristic
153C8×D4D411yescharacteristic
154C8×D4D411yescharacteristic
155C8×D4C2311yescharacteristic
156#433—21no—
157#433—21no—
158#433—21no—
159#433—21no—
160#433—21no—
161#433—21no—
162#432—21no—
163#432—21no—
164#432C2×C411yescharacteristic
165#432C2×C411yescharacteristic
166#431—21no—
167#431—21no—
168#431—21no—
169#431—21no—
170#431C2×C411yescharacteristic
171#431C2×C411yescharacteristic
172#431C2×C411yescharacteristic
173#431C2×C411yescharacteristic
174#424—21no—
175#424—21no—
176#424—21no—
177#423—21no—
178#423C2×C411yescharacteristic
179C2×C4⋊C8—21no—
180C2×C4⋊C8—21no—
181C2×C4⋊C8C2×C411yescharacteristic
182C2×C4⋊C8C2×C411yescharacteristic
183#405—21no—
184#405—21no—
185#405—21no—
186C2×C4×C8—21no—
187C2×C4×C8—21no—
188C2×C4×C8—21no—
189C2×C4×C8C2311yescharacteristic
190C2×C4×C8C2311yescharacteristic
191C2×C4×C8C2311yescharacteristic
192C4⋊C16—21no—
193C4⋊C16—21no—
194C4⋊C16—21no—
195C4⋊C16—21no—
196C4⋊C16—21no—
197C4⋊C16—21no—
198C4⋊C16C2311yescharacteristic
199C4⋊C16C2311yescharacteristic
200C4⋊C16C2311yescharacteristic
201C4⋊C16C2311yescharacteristic
202C22⋊C16—21no—
203C22⋊C16—21no—
204C22⋊C16—21no—
205C22⋊C16—21no—
206C22⋊C16—21no—
207C22⋊C16—21no—
208#346—21no—
209#346—21no—
210#346—21no—
211#346—21no—
212C4×C16—21no—
213C4×C16—21no—
214C4×C16C2311yescharacteristic
215C4×C16C2311yescharacteristic
216C4×C16C2311yescharacteristic
217C4×C16C2311yescharacteristic
218C2×#41—21no—
219C2×#41C2×C811yescharacteristic
220C2×#41M1611yescharacteristic
221C22×Q8C2×C811yescharacteristic
222C22×Q8C2×C811yescharacteristic
223C2×M16—21no—
224C2×M16—21no—
225C2×M16—21no—
226C22×C8—21no—
227C22×C8—21no—
228C22×C8—21no—
229C22×C8—21no—
230C22×C8C2×D411yescharacteristic
231C22×C8C2×D411yescharacteristic
232C22×C8C2×D411yescharacteristic
233C22×C8C2×D411yescharacteristic
234C22×C8C2×D411yescharacteristic
235C22×C8C2×D411yescharacteristic
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8C2×C811yescharacteristic
239C4⋊Q8C2×C811yescharacteristic
240#126—21no—
241#126—21no—
242#126—21no—
243#126—21no—
244#126—21no—
245#126—21no—
246#125—21no—
247#125—21no—
248#125—21no—
249#125—21no—
250#125—21no—
251#125C2×C811yescharacteristic
252#125C2×C811yescharacteristic
253#124—21no—
254#124—21no—
255#124C2×C811yescharacteristic
256#124C2×C811yescharacteristic
257#123—21no—
258#123—21no—
259#123—21no—
260#123—21no—
261#123—21no—
262#123—21no—
263C22⋊Q8—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C22⋊Q8—21no—
269C22⋊Q8—21no—
270C22⋊Q8C2×C811yescharacteristic
271C22⋊Q8C2×C811yescharacteristic
272C22⋊Q8C2×C811yescharacteristic
273C22⋊Q8C2×C811yescharacteristic
274#121—21no—
275#121—21no—
276#121C2×C811yescharacteristic
277#121C2×C811yescharacteristic
278C4×Q8—21no—
279C4×Q8—21no—
280C4×Q8—21no—
281C4×Q8—21no—
282C4×Q8—21no—
283C4×Q8—21no—
284C4×Q8C22⋊C411yescharacteristic
285C4×Q8C2×C811yescharacteristic
286C4×Q8C2×C811yescharacteristic
287C4×Q8C2×C811yescharacteristic
288C4×Q8C2×C811yescharacteristic
289C4×Q8C2×C811yescharacteristic
290C4×Q8C2×C811yescharacteristic
291C4×Q8C22×C411yescharacteristic
292C4×Q8C22×C411yescharacteristic
293C4×Q8C22×C411yescharacteristic
294C4×D4—21no—
295C4×D4—21no—
296C4×D4—21no—
297C4×D4—21no—
298C4×D4—21no—
299C4×D4—21no—
300C4×D4—21no—
301C4×D4C22⋊C411yescharacteristic
302C4×D4C22⋊C411yescharacteristic
303C4×D4C22⋊C411yescharacteristic
304C4×D4C2×C811yescharacteristic
305C4×D4C2×C811yescharacteristic
306C4×D4C2×C811yescharacteristic
307C4×D4C2×C811yescharacteristic
308C4×D4C22×C411yescharacteristic
309#117—21no—
310#117—21no—
311#117—21no—
312#117—21no—
313#117—21no—
314#117—21no—
315#117—21no—
316#117C2×C811yescharacteristic
317#117M1611yescharacteristic
318#117M1611yescharacteristic
319#117M1611yescharacteristic
320C2×C4⋊C4—21no—
321C2×C4⋊C4—21no—
322C2×C4⋊C4C2×C811yescharacteristic
323C2×C4⋊C4C2×C811yescharacteristic
324C2×C42—21no—
325C2×C42—21no—
326C2×C42—21no—
327C2×C42C22×C411yescharacteristic
328C2×C42C22×C411yescharacteristic
329C2×C42C22×C411yescharacteristic
330M32—42no—
331M32—42no—
332M32—81no—
333C2×C16—21no—
334C2×C16—21no—
335C2×C16—21no—
336C2×C16—21no—
337C2×C16—21no—
338C2×C16—21no—
339C2×C16—21no—
340C2×C16—21no—
341C2×C16—42no—
342C2×C16—81no—
343C2×C16—81no—
344C4⋊C8—21no—
345C4⋊C8—21no—
346C4⋊C8—21no—
347C4⋊C8—21no—
348C4⋊C8—21no—
349C4⋊C8—21no—
350C4⋊C8—21no—
351C4⋊C8—21no—
352C4⋊C8—21no—
353C4⋊C8—21no—
354C4⋊C8—21no—
355C4⋊C8—21no—
356C4⋊C8—21no—
357C4⋊C8—21no—
358C4⋊C8—21no—
359C4⋊C8C22⋊C411yescharacteristic
360C4⋊C8C22⋊C411yescharacteristic
361C4⋊C8C22⋊C411yescharacteristic
362C4⋊C8C22⋊C411yescharacteristic
363C4⋊C8C22⋊C411yescharacteristic
364C4⋊C8C22⋊C411yescharacteristic
365C4⋊C8C22×C411yescharacteristic
366C4⋊C8C22×C411yescharacteristic
367C4⋊C8C22×C411yescharacteristic
368C4⋊C8C22×C411yescharacteristic
369C22⋊C8—21no—
370C22⋊C8—21no—
371C22⋊C8—21no—
372C22⋊C8—21no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8C22⋊C411yescharacteristic
377C22⋊C8C22⋊C411yescharacteristic
378C22⋊C8C22⋊C411yescharacteristic
379C22⋊C8C22⋊C411yescharacteristic
380C22⋊C8C22⋊C411yescharacteristic
381C22⋊C8C22⋊C411yescharacteristic
382#97—21no—
383#97—21no—
384#97—21no—
385#97—21no—
386#97—21no—
387#97—21no—
388C4×C8—21no—
389C4×C8—21no—
390C4×C8—21no—
391C4×C8—21no—
392C4×C8—21no—
393C4×C8—21no—
394C4×C8—21no—
395C4×C8—21no—
396C4×C8—21no—
397C4×C8C22×C411yescharacteristic
398C4×C8C22×C411yescharacteristic
399C4×C8C22×C411yescharacteristic
400C4×C8C2×D411yescharacteristic
401C4×C8C2×D411yescharacteristic
402C4×C8C2×D411yescharacteristic
403C4×C8C2×D411yescharacteristic
404C4×C8C2×D411yescharacteristic
405C4×C8C2×D411yescharacteristic
406C4×C8C2411yescharacteristic
407#41—22no—
408#41—42no—
409#41—41no—
410#41—44no—
411C2×Q8—21no—
412C2×Q8—21no—
413C2×Q8—21no—
414C2×Q8—22no—
415C2×Q8—22no—
416C2×Q8—41no—
417C2×Q8—41no—
418C2×Q8C22⋊C811yescharacteristic
419C2×Q8C22×C811yescharacteristic
420C2×Q8C22×C811yescharacteristic
421C2×Q8C22×C811yescharacteristic
422C2×D4—21no—
423C2×D4—21no—
424C2×D4—21no—
425C2×D4C22⋊C811yescharacteristic
426C2×D4C22⋊C811yescharacteristic
427C2×D4C22⋊C811yescharacteristic
428C2×D4C22×C811yescharacteristic
429C22×C4—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4C2×C22⋊C411yescharacteristic
437C22×C4C2×C22⋊C411yescharacteristic
438C22×C4C2×C22⋊C411yescharacteristic
439C22×C4C22×C811yescharacteristic
440C22×C4C22×C811yescharacteristic
441C22×C4C22×C811yescharacteristic
442C22×C4C2×M1611yescharacteristic
443C22×C4C2×M1611yescharacteristic
444C22×C4C2×M1611yescharacteristic
445M16—81no—
446M16—81no—
447M16—81no—
448C2×C8—21no—
449C2×C8—21no—
450C2×C8—21no—
451C2×C8—21no—
452C2×C8—21no—
453C2×C8—21no—
454C2×C8—21no—
455C2×C8—21no—
456C2×C8—21no—
457C2×C8—21no—
458C2×C8—41no—
459C2×C8—41no—
460C2×C8—41no—
461C2×C8—41no—
462C2×C8—41no—
463C2×C8—81no—
464C2×C8—81no—
465C2×C8—81no—
466C2×C8C2×C22⋊C411yescharacteristic
467C2×C8C2×C22⋊C411yescharacteristic
468C2×C8C2×C22⋊C411yescharacteristic
469C2×C8C2×C22⋊C411yescharacteristic
470C2×C8C2×C22⋊C411yescharacteristic
471C2×C8C2×C22⋊C411yescharacteristic
472C2×C8C22×D411yescharacteristic
473C2×C8C22×D411yescharacteristic
474C4⋊C4—21no—
475C4⋊C4—21no—
476C4⋊C4—21no—
477C4⋊C4—21no—
478C4⋊C4—21no—
479C4⋊C4—21no—
480C4⋊C4—21no—
481C4⋊C4—21no—
482C4⋊C4—21no—
483C4⋊C4—21no—
484C4⋊C4—21no—
485C4⋊C4—21no—
486C4⋊C4—21no—
487C4⋊C4—21no—
488C4⋊C4—21no—
489C4⋊C4—21no—
490C4⋊C4—21no—
491C4⋊C4—21no—
492C4⋊C4—21no—
493C4⋊C4—21no—
494C4⋊C4—21no—
495C4⋊C4C22⋊C811yescharacteristic
496C4⋊C4C22⋊C811yescharacteristic
497C4⋊C4C22⋊C811yescharacteristic
498C4⋊C4C22⋊C811yescharacteristic
499C4⋊C4C22⋊C811yescharacteristic
500C4⋊C4C22⋊C811yescharacteristic
501C4⋊C4C22×C811yescharacteristic
502C4⋊C4C22×C811yescharacteristic
503C4⋊C4C22×C811yescharacteristic
504C4⋊C4C22×C811yescharacteristic
505C22⋊C4—21no—
506C22⋊C4—21no—
507C22⋊C4—21no—
508C22⋊C4—21no—
509C22⋊C4—21no—
510C22⋊C4—21no—
511C22⋊C4—21no—
512C22⋊C4—21no—
513C22⋊C4—21no—
514C22⋊C4—21no—
515C22⋊C4—21no—
516C22⋊C4—21no—
517C22⋊C4—21no—
518C22⋊C4C22⋊C811yescharacteristic
519C22⋊C4C22⋊C811yescharacteristic
520C22⋊C4C22⋊C811yescharacteristic
521C22⋊C4C22⋊C811yescharacteristic
522C22⋊C4C22⋊C811yescharacteristic
523C22⋊C4C22⋊C811yescharacteristic
524C42—21no—
525C42—21no—
526C42—21no—
527C42—21no—
528C42—21no—
529C42—21no—
530C42—21no—
531C42—21no—
532C42—21no—
533C42C2×C22⋊C411yescharacteristic
534C42C2×C22⋊C411yescharacteristic
535C42C2×C22⋊C411yescharacteristic
536C42C22×C811yescharacteristic
537C42C22×C811yescharacteristic
538C42C22×C811yescharacteristic
539C42C2×M1611yescharacteristic
540C42C2×M1611yescharacteristic
541C42C2×M1611yescharacteristic
542C42C23×C411yescharacteristic
543C16—22no—
544C16—41no—
545C16—41no—
546C16—41no—
547C16—41no—
548C16—41no—
549C16—41no—
550C16—41no—
551C23—21no—
552C23C2×C22⋊C811yescharacteristic
553C23C2×C22⋊C811yescharacteristic
554C23C2×C22⋊C811yescharacteristic
555Q8—21no—
556Q8—21no—
557Q8—21no—
558Q8—21no—
559Q8—21no—
560Q8—41no—
561Q8—42no—
562Q8—42no—
563Q8—42no—
564Q8#50412yes—
565D4—22no—
566D4—22no—
567D4—21no—
568D4—21no—
569D4—41no—
570D4—42no—
571D4—42no—
572D4—42no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—21no—
576C2×C4—21no—
577C2×C4—21no—
578C2×C4—21no—
579C2×C4—21no—
580C2×C4—21no—
581C2×C4—21no—
582C2×C4—21no—
583C2×C4—21no—
584C2×C4—21no—
585C2×C4—21no—
586C2×C4—21no—
587C2×C4—21no—
588C2×C4—21no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—41no—
594C2×C4—41no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—42no—
599C2×C4—42no—
600C2×C4—42no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4C2×C22⋊C811yescharacteristic
604C2×C4C2×C22⋊C811yescharacteristic
605C2×C4C2×C22⋊C811yescharacteristic
606C2×C4C2×C22⋊C811yescharacteristic
607C2×C4C2×C22⋊C811yescharacteristic
608C2×C4C2×C22⋊C811yescharacteristic
609C2×C4C2×C22⋊C811yescharacteristic
610C2×C4C2×C22⋊C811yescharacteristic
611C2×C4C2×C22⋊C811yescharacteristic
612C2×C4C22×C22⋊C411yescharacteristic
613C2×C4C23×C811yescharacteristic
614C2×C4C22×M1611yescharacteristic
615C8—21no—
616C8—21no—
617C8—21no—
618C8—21no—
619C8—21no—
620C8—21no—
621C8—21no—
622C8—41no—
623C8—41no—
624C8—41no—
625C8—41no—
626C8#57711yescharacteristic
627C8#57811yescharacteristic
628C22—21no—
629C22—21no—
630C22—21no—
631C22—21no—
632C22—21no—
633C22—21no—
634C22—21no—
635C22—21no—
636C22—41no—
637C22—41no—
638C22—41no—
639C22#211712yes—
640C22C22×C22⋊C811yescharacteristic
641C4—21no—
642C4—21no—
643C4—21no—
644C4—21no—
645C4—21no—
646C4—21no—
647C4—22no—
648C4—21no—
649C4—21no—
650C4—21no—
651C4—21no—
652C4—41no—
653C4—41no—
654C4—41no—
655C4—41no—
656C4—41no—
657C4—41no—
658C4#289811yescharacteristic
659C4#290111yescharacteristic
660C4C2×#50411yescharacteristic
661C4#340711yescharacteristic
662C2—22no—
663C2—41no—
664C2—41no—
665C2—81no—
666C2#1743911yescharacteristic
667C2#3333411yescharacteristic
668C2C2×#211711yescharacteristic
subgroupG/H
centerC2×C8C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC4×C8C24
socleC22C22×C22⋊C8
prime radical#3482C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C8 ◁ C8×#41 ◁ G
LECSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #33504C2semidirect
3. #33479C2semidirect
4. #11640C2semidirect
5. #11639C2semidirect
6. #11637C2semidirect
7. #11624C2semidirect
8. #11622C2semidirect
9. #11621C2semidirect
10. #11616C2semidirect
11. #11615C2semidirect
12. #11614C2semidirect
13. #11607C2semidirect
14. #11606C2semidirect
15. C2×Q8⋊C16C2semidirect

Maximal subgroups

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

subgroupindex
#603312
#335042
#334792
#116402
#116392
#116372
#116242
#116222
#116212
#116162
#116152
#116142
#116072
#116062
C2×Q8⋊C162