Finite Group Databasebeta

← #220141

Subgroups — #220141

Order 512.

1107 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×C4×#137C211yesmaximal, characteristic
2#22532C211yesmaximal, characteristic
3#22510C211yesmaximal, characteristic, complemented
4C2×#1593C211yesmaximal, characteristic, complemented
5C2×#1591C211yesmaximal, characteristic, complemented
6#9660C211yesmaximal, characteristic, complemented
7#9655C211yesmaximal, characteristic, complemented
8#8952C211yesmaximal, characteristic, complemented
9#8952C211yesmaximal, characteristic, complemented
10#8950C211yesmaximal, characteristic, complemented
11#8950C211yesmaximal, characteristic, complemented
12#8597C211yesmaximal, characteristic, complemented
13#8312C211yesmaximal, characteristic
14#8311C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic, complemented
16C22×#137C411yescharacteristic
17C2×C4×#41C2211yescharacteristic
18C22×C4×Q8C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×#446C2211yescharacteristic
22C2×#446C2211yescharacteristic
23C4×#137—24no—
24C4×#137—24no—
25C2×#441C411yescharacteristic
26C2×#441C411yescharacteristic
27C2×#440C411yescharacteristic
28C2×#440C411yescharacteristic
29C2×C4×Q16C2211yescharacteristic
30C2×C4×Q16C2211yescharacteristic
31C2×C4×SD16C2211yescharacteristic
32C2×C4×SD16C2211yescharacteristic
33C2×#428C411yescharacteristic
34C2×#417C411yescharacteristic
35C2×#419C411yescharacteristic
36C2×C4×M16C2211yescharacteristic
37#1985—21no—
38#1985—21no—
39#1984C2211yescharacteristic
40#1984C2211yescharacteristic
41#1955—21no—
42#1955—21no—
43#1954C2211yescharacteristic
44#1954C2211yescharacteristic
45#1797—21no—
46#1797—21no—
47#1797C2211yescharacteristic
48#1797C2211yescharacteristic
49#1749—21no—
50#1749—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1593—24no—
54#1591—24no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×D4⋊C8C2211yescharacteristic
64C2×D4⋊C8C2211yescharacteristic
65C2×#322C2211yescharacteristic
66C2×C82C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×#41C2×C411yescharacteristic
72C23×Q8C2×C411yescharacteristic
73C2×#137—24no—
74C2×#137—24no—
75C2×#137—24no—
76C22×Q16C2×C411yescharacteristic
77C22×Q16C2×C411yescharacteristic
78C22×SD16C2×C411yescharacteristic
79C22×SD16C2×C411yescharacteristic
80C22×M16C2×C411yescharacteristic
81C22×M16C2×C411yescharacteristic
82C4×#41—22no—
83C4×#41—22no—
84C4×#41—41no—
85C4×#41—41no—
86C2×C4×Q8—21no—
87C2×C4×Q8—21no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—41no—
91C2×C4×Q8—41no—
92C2×C4×Q8C2×C411yescharacteristic
93C2×C4×Q8C2×C411yescharacteristic
94C2×C4×Q8C2×C411yescharacteristic
95C2×C4×Q8C2311yescharacteristic
96C2×C4×Q8C2311yescharacteristic
97C2×C4×Q8C2311yescharacteristic
98C2×C4×D4—21no—
99C2×C4×D4—21no—
100C2×C4×D4C2×C411yescharacteristic
101C2×C4×D4C2311yescharacteristic
102C2×#117—21no—
103C2×#117C2×C411yescharacteristic
104C22×C4⋊C4—21no—
105C22×C4⋊C4C2×C411yescharacteristic
106C22×C42—21no—
107C22×C42—21no—
108C22×C42C2311yescharacteristic
109#446—22no—
110#446—22no—
111#446—22no—
112#446—22no—
113#446—41no—
114#446—41no—
115#446—41no—
116#446—41no—
117#441—24no—
118#441—24no—
119#440—24no—
120#440—24no—
121C4×Q16—22no—
122C4×Q16—22no—
123C4×Q16D414yes—
124C4×Q16D414yes—
125C4×SD16—22no—
126C4×SD16—22no—
127C4×SD16—22no—
128C4×SD16—22no—
129#428—24no—
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⋊C8C2311yescharacteristic
139C2×C4⋊C8C2311yescharacteristic
140C2×C4⋊C8C2311yescharacteristic
141C2×C4⋊C8C2311yescharacteristic
142#419—24no—
143#417—24no—
144C2×#103C2×C411yescharacteristic
145C2×#103C2×C411yescharacteristic
146C2×#103C2×C411yescharacteristic
147C2×#103C2×C411yescharacteristic
148C2×#103C2×C411yescharacteristic
149C2×#103C2×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⋊C8D411yescharacteristic
160C2×C22⋊C8D411yescharacteristic
161C4×M16—22no—
162C4×M16—22no—
163C2×#97—21no—
164C2×#97—21no—
165C2×#97—21no—
166C2×#97—21no—
167C2×#97C2×C411yescharacteristic
168C2×#97C2×C411yescharacteristic
169C2×C4×C8C2311yescharacteristic
170C2×C4×C8C2311yescharacteristic
171C2×C4×C8C2311yescharacteristic
172C2×C4×C8C2311yescharacteristic
173C2×C4×C8C2311yescharacteristic
174C2×C4×C8C2311yescharacteristic
175#336—21no—
176#336—21no—
177#336—21no—
178#336—21no—
179#336—21no—
180#336—21no—
181#336—21no—
182#336D411yescharacteristic
183#336D411yescharacteristic
184#335—22no—
185#334—22no—
186Q8⋊C8—22no—
187Q8⋊C8—22no—
188Q8⋊C8—22no—
189Q8⋊C8—22no—
190Q8⋊C8—41no—
191Q8⋊C8—41no—
192D4⋊C8—41no—
193D4⋊C8—41no—
194#322—22no—
195C82—22no—
196C2×#41—22no—
197C2×#41—22no—
198C2×#41—22no—
199C2×#41—22no—
200C2×#41—41no—
201C2×#41—41no—
202C22×Q8—21no—
203C22×Q8—22no—
204C22×Q8—22no—
205C22×Q8—22no—
206C22×Q8—22no—
207C22×Q8—41no—
208C22×Q8—41no—
209C22×Q8C22×C411yescharacteristic
210C22×Q8C22×C411yescharacteristic
211C22×Q8C22×C411yescharacteristic
212C22×D4—21no—
213C22×D4C22×C411yescharacteristic
214C23×C4—21no—
215C23×C4—21no—
216C23×C4—21no—
217C23×C4—21no—
218C23×C4—21no—
219C23×C4C22×C411yescharacteristic
220C23×C4C2×D411yescharacteristic
221C23×C4#4111yescharacteristic
222#137—232no—
223C2×Q16—22no—
224C2×Q16—22no—
225C2×Q16—22no—
226C2×Q16—22no—
227C2×Q16—22no—
228C2×Q16C22⋊C414yes—
229C2×SD16—22no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—22no—
233C2×SD16—22no—
234C2×SD16M1614yes—
235C2×M16—22no—
236C2×M16—22no—
237C2×M16—22no—
238C2×M16—22no—
239C2×M16—41no—
240C2×M16M1614yes—
241C22×C8—21no—
242C22×C8—21no—
243C22×C8—21no—
244C22×C8—21no—
245C22×C8—21no—
246C22×C8—21no—
247C22×C8C22×C411yescharacteristic
248C22×C8C22×C411yescharacteristic
249C22×C8C22×C411yescharacteristic
250C22×C8C22×C411yescharacteristic
251C22×C8C2×D411yescharacteristic
252C22×C8C2×D411yescharacteristic
253C22×C8C2×D411yescharacteristic
254C22×C8C2×D411yescharacteristic
255C4×Q8—21no—
256C4×Q8—21no—
257C4×Q8—21no—
258C4×Q8—21no—
259C4×Q8—22no—
260C4×Q8—22no—
261C4×Q8—22no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8—42no—
267C4×Q8—42no—
268C4×Q8—41no—
269C4×Q8—41no—
270C4×Q8—41no—
271C4×Q8—42no—
272C4×Q8—42no—
273C4×Q8C2×D412yes—
274C4×Q8C2×D412yes—
275C4×Q8C2×D412yes—
276C4×Q8C2×D412yes—
277C4×D4—22no—
278C4×D4—21no—
279C4×D4—21no—
280C4×D4—21no—
281C4×D4—21no—
282C4×D4—41no—
283C4×D4—42no—
284C4×D4—42no—
285C4×D4—42no—
286C4×D4—42no—
287#117—22no—
288#117—42no—
289#117—42no—
290#117—41no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—22no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—21no—
298C2×C4⋊C4—21no—
299C2×C4⋊C4—41no—
300C2×C4⋊C4—42no—
301C2×C4⋊C4—42no—
302C2×C4⋊C4C22×C411yescharacteristic
303C2×C4⋊C4C22×C411yescharacteristic
304C2×C4⋊C4C22×C411yescharacteristic
305C2×C4⋊C4C22×C411yescharacteristic
306C2×C22⋊C4—21no—
307C2×C22⋊C4—21no—
308C2×C22⋊C4—21no—
309C2×C42—21no—
310C2×C42—21no—
311C2×C42—21no—
312C2×C42—21no—
313C2×C42—21no—
314C2×C42—21no—
315C2×C42—21no—
316C2×C42—21no—
317C2×C42—21no—
318C2×C42—21no—
319C2×C42—21no—
320C2×C42—42no—
321C2×C42—42no—
322C2×C42—42no—
323C2×C42—42no—
324C2×C42C22×C411yescharacteristic
325C2×C42C2×D411yescharacteristic
326C2×C42#4111yescharacteristic
327C2×C42C2411yescharacteristic
328#107—22no—
329#107C22⋊C414yes—
330#106—22no—
331#106—22no—
332C4⋊C8—21no—
333C4⋊C8—21no—
334C4⋊C8—21no—
335C4⋊C8—21no—
336C4⋊C8—21no—
337C4⋊C8—21no—
338C4⋊C8—21no—
339C4⋊C8—21no—
340C4⋊C8—42no—
341C4⋊C8—42no—
342C4⋊C8—42no—
343C4⋊C8—42no—
344#103—22no—
345#103—22no—
346#103—22no—
347#103—22no—
348#103C22⋊C414yes—
349#103C22⋊C414yes—
350#102—22no—
351#102—22no—
352C22⋊C8—22no—
353C22⋊C8—41no—
354C22⋊C8—42no—
355C22⋊C8—42no—
356C22⋊C8—42no—
357C22⋊C8—42no—
358C22⋊C8—81no—
359C22⋊C8—81no—
360C22⋊C8—81no—
361#97—22no—
362#97—22no—
363#97—42no—
364#97—42no—
365#97—42no—
366#97—42no—
367C4×C8—21no—
368C4×C8—21no—
369C4×C8—21no—
370C4×C8—21no—
371C4×C8—21no—
372C4×C8—21no—
373C4×C8—21no—
374C4×C8—21no—
375C4×C8—21no—
376C4×C8—21no—
377C4×C8C2×D412yes—
378C4×C8C2×D412yes—
379C24—21no—
380C24C4×D411yescharacteristic
381#41—28no—
382#41—44no—
383#41—44no—
384#41—44no—
385C2×Q8—21no—
386C2×Q8—21no—
387C2×Q8—21no—
388C2×Q8—28no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—21no—
392C2×Q8—21no—
393C2×Q8—21no—
394C2×Q8—21no—
395C2×Q8—21no—
396C2×Q8—21no—
397C2×Q8—21no—
398C2×Q8—21no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×Q8—42no—
403C2×Q8—44no—
404C2×Q8—44no—
405C2×Q8—44no—
406C2×Q8—42no—
407C2×Q8C2×C22⋊C412yes—
408C2×Q8C2×C22⋊C412yes—
409C2×Q8C2×M1612yes—
410C2×D4—21no—
411C2×D4—21no—
412C2×D4—21no—
413C2×D4—21no—
414C2×D4—21no—
415C2×D4—42no—
416C2×D4—42no—
417C2×D4—42no—
418C2×D4C2×M1612yes—
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—21no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—21no—
434C22×C4—21no—
435C22×C4—21no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—42no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—42no—
442C22×C4—42no—
443C22×C4—42no—
444C22×C4—42no—
445C22×C4—42no—
446C22×C4—42no—
447C22×C4—42no—
448C22×C4—42no—
449C22×C4—42no—
450C22×C4—41no—
451C22×C4—41no—
452C22×C4—41no—
453C22×C4—41no—
454C22×C4—41no—
455C22×C4—41no—
456C22×C4—81no—
457C22×C4—81no—
458C22×C4C4×D411yescharacteristic
459C22×C4C4×D411yescharacteristic
460C22×C4C4×D411yescharacteristic
461C22×C4C2×M1612yes—
462C22×C4#13112yes—
463C22×C4C23×C411yescharacteristic
464C22×C4C22×D411yescharacteristic
465C22×C4C2×#4111yescharacteristic
466Q16—28no—
467Q16—28no—
468SD16—28no—
469SD16—28no—
470M16—28no—
471M16—44no—
472C2×C8—21no—
473C2×C8—21no—
474C2×C8—21no—
475C2×C8—21no—
476C2×C8—21no—
477C2×C8—21no—
478C2×C8—22no—
479C2×C8—21no—
480C2×C8—21no—
481C2×C8—21no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—41no—
497C2×C8—41no—
498C2×C8—41no—
499C2×C8—41no—
500C2×C8—41no—
501C2×C8—41no—
502C2×C8—41no—
503C2×C8—41no—
504C2×C8—41no—
505C2×C8—41no—
506C2×C8—41no—
507C2×C8—41no—
508C2×C8C2×C22⋊C412yes—
509C2×C8C2×C22⋊C412yes—
510C2×C8C2×M1612yes—
511C2×C8C2×M1612yes—
512C2×C8C2×M1612yes—
513C2×C8#13712yescomplemented
514C4⋊C4—21no—
515C4⋊C4—21no—
516C4⋊C4—22no—
517C4⋊C4—22no—
518C4⋊C4—22no—
519C4⋊C4—22no—
520C4⋊C4—21no—
521C4⋊C4—21no—
522C4⋊C4—21no—
523C4⋊C4—21no—
524C4⋊C4—42no—
525C4⋊C4—42no—
526C4⋊C4—42no—
527C4⋊C4—42no—
528C4⋊C4—41no—
529C4⋊C4—41no—
530C4⋊C4C2×C22⋊C412yes—
531C4⋊C4C2×C22⋊C412yes—
532C22⋊C4—42no—
533C22⋊C4—42no—
534C22⋊C4—41no—
535C22⋊C4—41no—
536C42—21no—
537C42—21no—
538C42—22no—
539C42—22no—
540C42—22no—
541C42—22no—
542C42—21no—
543C42—21no—
544C42—21no—
545C42—21no—
546C42—42no—
547C42—42no—
548C42—42no—
549C42—42no—
550C42—41no—
551C42—41no—
552C42C22×D411yescharacteristic
553C42C22×D411yescharacteristic
554C42H(5,𝔽2)11yescharacteristic
555C42H(5,𝔽2)11yescharacteristic
556C23—21no—
557C23—21no—
558C23—21no—
559C23—21no—
560C23—21no—
561C23—21no—
562C23—22no—
563C23—41no—
564C23—41no—
565C23—41no—
566C23—41no—
567C23—41no—
568C23—41no—
569C23#43612yes—
570C23C2×C4×D411yescharacteristic
571Q8—22no—
572Q8—22no—
573Q8—22no—
574Q8—22no—
575Q8—22no—
576Q8—22no—
577Q8—48no—
578Q8—41no—
579Q8—41no—
580Q8—41no—
581Q8—41no—
582Q8—41no—
583Q8—41no—
584D4—22no—
585D4—22no—
586D4—48no—
587D4—41no—
588D4—41no—
589C2×C4—21no—
590C2×C4—21no—
591C2×C4—21no—
592C2×C4—21no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—21no—
598C2×C4—21no—
599C2×C4—22no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—21no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—48no—
609C2×C4—48no—
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—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—41no—
627C2×C4—41no—
628C2×C4—41no—
629C2×C4—41no—
630C2×C4—81no—
631C2×C4—81no—
632C2×C4—84no—
633C2×C4—84no—
634C2×C4—81no—
635C2×C4—81no—
636C2×C4—81no—
637C2×C4—81no—
638C2×C4#43612yes—
639C2×C4C22×C22⋊C411yescharacteristic
640C2×C4#51811yescharacteristic
641C2×C4D4211yescharacteristic
642C2×C4D4211yescharacteristic
643C2×C4#54611yescharacteristic
644C2×C4#54611yescharacteristic
645C2×C4C22×M1611yescharacteristic
646C2×C4C22×M1611yescharacteristic
647C2×C4C2×#13111yescharacteristic
648C2×C4#56811yescharacteristic
649C2×C4C2×#13711yescharacteristic
650C2×C4#57511yescharacteristic
651C2×C4#57711yescharacteristic
652C2×C4#57811yescharacteristic
653C8—22no—
654C8—22no—
655C8—22no—
656C8—42no—
657C8—41no—
658C8—41no—
659C8—41no—
660C8—41no—
661C8—81no—
662C8—81no—
663C8—81no—
664C8—81no—
665C8—81no—
666C8#40714yes—
667C22—22no—
668C22—22no—
669C22—22no—
670C22—41no—
671C22—41no—
672C22—41no—
673C22—41no—
674C22—42no—
675C22—42no—
676C22—42no—
677C22—81no—
678C22—81no—
679C22—81no—
680C22—81no—
681C22D4×C22⋊C411yescharacteristic
682C22C2×#43611yescharacteristic
683C22D4×M1611yescharacteristic
684C22C4×#13711yescharacteristic
685C22#294711yescharacteristic
686C22#295511yescharacteristic
687C22#295611yescharacteristic
688C4—22no—
689C4—21no—
690C4—21no—
691C4—42no—
692C4—42no—
693C4—42no—
694C4—41no—
695C4—41no—
696C4—42no—
697C4—42no—
698C4—81no—
699C4—81no—
700C4C2×#40711yescharacteristic
701C4#288011yescharacteristic
702C4#297411yescharacteristic
703C4#297511yescharacteristic
704C4#327011yescharacteristic
705C4#327111yescharacteristic
706C4#327711yescharacteristic
707C4#327811yescharacteristic
708C2—42no—
709C2—82no—
710C2#1789211yescharacteristic
711C2#1789411yescharacteristic
712C2#1804711yescharacteristic
713C2#1805111yescharacteristic
714C2#2240111yescharacteristic
715C2#2244511yescharacteristic
716C2#2245311yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×#41C2×C4
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
3. #22510C2semidirect
4. C2×#1593C2semidirect
5. C2×#1591C2semidirect
6. #9660C2semidirect
7. #9655C2semidirect
8. #8952C2semidirect
9. #8952C2semidirect
10. #8950C2semidirect
11. #8950C2semidirect
12. #8597C2semidirect
14. #8311C2semidirect
15. #8146C2semidirect
513. C2×C8#137semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#1372
#225322
#225102
C2×#15932
C2×#15912
#96602
#96552
#89522
#89502
#85972
#83122
#83112
#81462