Finite Group Databasebeta

← #209309

Subgroups — #209309

Order 512.

910 conjugacy classes of subgroups.

The subgroup lattice has 617 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2962C211yesmaximal, characteristic
2#23571C211yesmaximal, characteristic
3#21596C211yesmaximal, characteristic
4C2×#1584C211yesmaximal, characteristic, complemented
5C2×C8×Q16C211yesmaximal, characteristic, complemented
6#8950C211yesmaximal, characteristic, complemented
7#8946C211yesmaximal, characteristic, complemented
8#8937C211yesmaximal, characteristic
9#8848C211yesmaximal, characteristic, complemented
10#8844C211yesmaximal, characteristic, complemented
11#8829C211yesmaximal, characteristic
12#8314C211yesmaximal, characteristic
13#8207C211yesmaximal, characteristic, complemented
14#8203C211yesmaximal, characteristic, complemented
15#8146C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17#2962—24no—
18#2962—24no—
19C2×#446—21no—
20C2×#446—21no—
21C2×#446C2211yescharacteristic
22C2×#446C2211yescharacteristic
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×C8×Q8C2211yescharacteristic
26C2×C4×Q16C2211yescharacteristic
27C2×C4×Q16C2211yescharacteristic
28C2×#424—21no—
29C22×C4⋊C8C2211yescharacteristic
30C2×#405—21no—
31C2×C4×M16C2211yescharacteristic
32C4×C22⋊Q8C2211yescharacteristic
33#2057C411yescharacteristic
34#2043C411yescharacteristic
35#2036C411yescharacteristic
36#2036C411yescharacteristic
37#2024C411yescharacteristic
38#2016C411yescharacteristic
39#2001C411yescharacteristic
40#2001C411yescharacteristic
41#1984C2211yescharacteristic
42#1939C2211yescharacteristic
43#1917—21no—
44#1851—21no—
45#1797C2211yescharacteristic
46#1771—21no—
47#1770C2211yescharacteristic
48C4×#103C2211yescharacteristic
49C4×#103C2211yescharacteristic
50#1753—21no—
51C4×C22⋊C8C2211yescharacteristic
52C4×C22⋊C8C2211yescharacteristic
53#1584—24no—
54C8×Q16—24no—
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×Q8⋊C8C2211yescharacteristic
61C2×Q8⋊C8C2211yescharacteristic
62C2×Q8⋊C8C2211yescharacteristic
63C2×#322C2211yescharacteristic
64C2×C82C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1277C2211yescharacteristic
68#1277C2211yescharacteristic
69#1273C2211yescharacteristic
70#1273C2211yescharacteristic
71C22×Q16C2×C411yescharacteristic
72C22×Q16C2×C411yescharacteristic
73C22×M16D411yescharacteristic
74C22×M16D411yescharacteristic
75C23×C8—21no—
76#533—22no—
77#533—22no—
78#533—22no—
79#533—22no—
80C2×C4⋊Q8C2×C411yescharacteristic
81C2×C4⋊Q8C2×C411yescharacteristic
82C2×#125—21no—
83C2×C22⋊Q8—21no—
84C2×C22⋊Q8C2×C411yescharacteristic
85C2×C22⋊Q8C2×C411yescharacteristic
86C2×C22⋊Q8C2×C411yescharacteristic
87C2×C4×Q8—21no—
88C2×C4×Q8C2311yescharacteristic
89C2×C4×Q8C2311yescharacteristic
90C2×C4×Q8C2311yescharacteristic
91C2×#117—21no—
92C22×C42C2311yescharacteristic
93#446—28no—
94#446—28no—
95#446—22no—
96#446—22no—
97#446—22no—
98#446D414yes—
99C8×Q8—22no—
100C8×Q8—22no—
101C8×Q8—22no—
102C8×Q8—22no—
103C8×Q8—22no—
104C8×Q8D414yes—
105C4×Q16—22no—
106C4×Q16—22no—
107C4×Q16—22no—
108C4×Q16—22no—
109C2×#107C2×C411yescharacteristic
110C2×#107C2×C411yescharacteristic
111#424—28no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—21no—
115C2×C4⋊C8—21no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8—22no—
118C2×C4⋊C8—22no—
119C2×C4⋊C8—21no—
120C2×C4⋊C8—21no—
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103C2×C411yescharacteristic
126C2×#103C2×C411yescharacteristic
127C2×#103C2×C411yescharacteristic
128C2×#103C2×C411yescharacteristic
129C2×#103C2×C411yescharacteristic
130C2×#103C2×C411yescharacteristic
131C2×#103C2×C411yescharacteristic
132C2×#103C2×C411yescharacteristic
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8C2×C411yescharacteristic
136C2×C22⋊C8C2×C411yescharacteristic
137#405—28no—
138C4×M16—22no—
139C4×M16—22no—
140C2×#97—21no—
141C2×#97—21no—
142C2×#97—21no—
143C2×#97D411yescharacteristic
144C2×#97D411yescharacteristic
145C2×C4×C8—21no—
146C2×C4×C8—21no—
147C2×C4×C8—21no—
148C2×C4×C8C2311yescharacteristic
149C2×C4×C8C2311yescharacteristic
150C2×C4×C8C2311yescharacteristic
151C2×C4×C8C2311yescharacteristic
152C2×C4×C8C2311yescharacteristic
153C2×C4×C8C2311yescharacteristic
154#391C2×C411yescharacteristic
155#389C2×C411yescharacteristic
156#387—21no—
157#385—21no—
158#380C2×C411yescharacteristic
159C4×C4⋊C4—21no—
160C4×C4⋊C4C2311yescharacteristic
161C4×C22⋊C4—21no—
162#340C2×C411yescharacteristic
163#340C2×C411yescharacteristic
164#340C2×C411yescharacteristic
165#340C2×C411yescharacteristic
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336C2×C411yescharacteristic
173#336C2×C411yescharacteristic
174#335—22no—
175#335—22no—
176Q8⋊C8—22no—
177Q8⋊C8—22no—
178Q8⋊C8—22no—
179Q8⋊C8—22no—
180Q8⋊C8—22no—
181Q8⋊C8—22no—
182#322—22no—
183C82—22no—
184C22×Q8—21no—
185C22×Q8C22×C411yescharacteristic
186C22×Q8C22×C411yescharacteristic
187C22×Q8C22×C411yescharacteristic
188C23×C4C22×C411yescharacteristic
189C23×C4C2×D411yescharacteristic
190C23×C4#4111yescharacteristic
191C2×Q16—22no—
192C2×Q16—22no—
193C2×Q16—22no—
194C2×Q16—22no—
195C2×Q16—22no—
196C2×Q16—41no—
197C2×M16—22no—
198C2×M16—22no—
199C2×M16—22no—
200C2×M16—22no—
201C2×M16—41no—
202C2×M16Q1614yescomplemented
203C22×C8—24no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—42no—
211C22×C8—42no—
212C22×C8C22×C411yescharacteristic
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C22×C411yescharacteristic
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8C2×D411yescharacteristic
219C22×C8C2×D411yescharacteristic
220C4⋊Q8—22no—
221C4⋊Q8—22no—
222C4⋊Q8—22no—
223C4⋊Q8—22no—
224#125—24no—
225#125—24no—
226C22⋊Q8—24no—
227C22⋊Q8—24no—
228C22⋊Q8—22no—
229C22⋊Q8—22no—
230C22⋊Q8—22no—
231C22⋊Q8—22no—
232C22⋊Q8—22no—
233C22⋊Q8—41no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—21no—
237C4×Q8—21no—
238C4×Q8—21no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—21no—
242C4×Q8—21no—
243C4×Q8—21no—
244C4×Q8C2×D412yes—
245C4×Q8C2×D412yes—
246C4×Q8#4112yes—
247C4×Q8#4112yes—
248#117—24no—
249#117—24no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22×C411yescharacteristic
261C2×C4⋊C4C22×C411yescharacteristic
262C2×C4⋊C4C22×C411yescharacteristic
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C42—21no—
270C2×C42—21no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—21no—
277C2×C42C22×C411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42#4111yescharacteristic
280C2×C42C2411yescharacteristic
281#107—22no—
282#107—22no—
283C4⋊C8—24no—
284C4⋊C8—24no—
285C4⋊C8—24no—
286C4⋊C8—24no—
287C4⋊C8—24no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—42no—
295C4⋊C8—42no—
296C4⋊C8C2×D412yes—
297C4⋊C8C2×D412yes—
298#103—22no—
299#103—22no—
300#103—22no—
301#103—22no—
302#103—22no—
303#103—41no—
304#103—41no—
305#103—41no—
306C22⋊C8—22no—
307C22⋊C8—41no—
308C22⋊C8—42no—
309C22⋊C8—42no—
310#97—24no—
311#97—24no—
312#97—22no—
313#97—22no—
314#97—42no—
315C4×C8—24no—
316C4×C8—24no—
317C4×C8—21no—
318C4×C8—21no—
319C4×C8—21no—
320C4×C8—21no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—21no—
326C4×C8—21no—
327C4×C8—42no—
328C4×C8C2×D412yes—
329C4×C8C2×D412yes—
330#95—21no—
331#95—21no—
332#95—21no—
333#95—21no—
334#95—21no—
335C24C4×D411yescharacteristic
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—21no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—81no—
353C2×Q8—81no—
354C2×Q8C4×D412yes—
355C2×Q8C4×D412yes—
356C2×Q8#13112yes—
357C22×C4—21no—
358C22×C4—21no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4C2×Q1612yes—
383C22×C4#13512yes—
384C22×C4C23×C411yescharacteristic
385C22×C4C22×D411yescharacteristic
386C22×C4C2×#4111yescharacteristic
387Q16—28no—
388Q16—44no—
389M16—28no—
390M16—44no—
391C2×C8—21no—
392C2×C8—21no—
393C2×C8—21no—
394C2×C8—21no—
395C2×C8—21no—
396C2×C8—21no—
397C2×C8—22no—
398C2×C8—21no—
399C2×C8—21no—
400C2×C8—21no—
401C2×C8—22no—
402C2×C8—21no—
403C2×C8—22no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—41no—
413C2×C8—48no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8#12012yes—
429C2×C8#12012yes—
430C2×C8#13112yes—
431C2×C8C2×Q1612yes—
432C2×C8C2×Q1612yes—
433C2×C8#13712yes—
434C4⋊C4—24no—
435C4⋊C4—24no—
436C4⋊C4—24no—
437C4⋊C4—24no—
438C4⋊C4—24no—
439C4⋊C4—21no—
440C4⋊C4—21no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—22no—
444C4⋊C4—22no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—42no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—81no—
455C4⋊C4C4×D412yes—
456C4⋊C4C4×D412yes—
457C22⋊C4—24no—
458C22⋊C4—24no—
459C22⋊C4—42no—
460C22⋊C4—81no—
461C42—21no—
462C42—21no—
463C42—22no—
464C42—22no—
465C42—21no—
466C42—21no—
467C42—21no—
468C42—21no—
469C42—22no—
470C42—22no—
471C42—41no—
472C42—41no—
473C42—41no—
474C42—41no—
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C42C2×#4111yescharacteristic
479C23—21no—
480C23—21no—
481C23—21no—
482C23—21no—
483C23—21no—
484C23—21no—
485C23C4×Q1612yes—
486C23C2×C4×D411yescharacteristic
487Q8—22no—
488Q8—22no—
489Q8—22no—
490Q8—22no—
491Q8—22no—
492Q8—22no—
493Q8—41no—
494Q8—41no—
495Q8—41no—
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—84no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—22no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—41no—
523C2×C4—41no—
524C2×C4—41no—
525C2×C4—41no—
526C2×C4—41no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—41no—
541C2×C4—41no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4C4×Q1612yes—
545C2×C4C2×C4×D411yescharacteristic
546C2×C4C2×C4×D411yescharacteristic
547C2×C4C2×#12011yescharacteristic
548C2×C4#52511yescharacteristic
549C2×C4#52511yescharacteristic
550C2×C4#52511yescharacteristic
551C2×C4C2×#13111yescharacteristic
552C2×C4C2×#13111yescharacteristic
553C2×C4#56811yescharacteristic
554C2×C4#56811yescharacteristic
555C2×C4C22×Q1611yescharacteristic
556C2×C4C2×#13511yescharacteristic
557C2×C4C2×#13711yescharacteristic
558C2×C4#57511yescharacteristic
559C8—22no—
560C8—22no—
561C8—22no—
562C8—42no—
563C8—41no—
564C8—41no—
565C8—41no—
566C8—41no—
567C8—42no—
568C8—42no—
569C8—81no—
570C8—81no—
571C8—81no—
572C8#45114yes—
573C22—22no—
574C22—22no—
575C22—22no—
576C22—41no—
577C22—41no—
578C22—41no—
579C22—41no—
580C22C4×#12011yescharacteristic
581C22#293311yescharacteristic
582C22#293511yescharacteristic
583C22C2×C4×Q1611yescharacteristic
584C22C4×#13711yescharacteristic
585C22#295611yescharacteristic
586C22#295711yescharacteristic
587C4—22no—
588C4—21no—
589C4—21no—
590C4—41no—
591C4—42no—
592C4—42no—
593C4—42no—
594C4—41no—
595C4—42no—
596C4—42no—
597C4—81no—
598C4—81no—
599C4—81no—
600C4#292911yescharacteristic
601C4#293211yescharacteristic
602C4#293311yescharacteristic
603C4#293511yescharacteristic
604C4C2×#45111yescharacteristic
605C4#300211yescharacteristic
606C4#310411yescharacteristic
607C4#310811yescharacteristic
608C2—42no—
609C2#1799811yescharacteristic
610C2#1801311yescharacteristic
611C2#1806811yescharacteristic
612C2#1807911yescharacteristic
613C2#2144311yescharacteristic
614C2C4×#45111yescharacteristic
615C2#2146811yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1584C2semidirect
5. C2×C8×Q16C2semidirect
6. #8950C2semidirect
7. #8946C2semidirect
9. #8848C2semidirect
10. #8844C2semidirect
13. #8207C2semidirect
14. #8203C2semidirect
202. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#29622
#235712
#215962
C2×#15842
C2×C8×Q162
#89502
#89462
#89372
#88482
#88442
#88292
#83142
#82072
#82032
#81462