Finite Group Databasebeta

← #198260

Subgroups — #198260

Order 512.

1115 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3104C211yesmaximal, characteristic
2C2×Q8×M16C211yesmaximal, characteristic
3#21337C211yesmaximal, characteristic
4#17708C211yesmaximal, characteristic
5C2×#1487C211yesmaximal, characteristic
6#9250C214yesmaximal, complemented
7#9248C214yesmaximal, complemented
8#8001C211yesmaximal, characteristic
9#7932C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12#3104—21no—
13#3104—21no—
14#3104—22no—
15#3104—22no—
16#3104—21no—
17#3104—21no—
18C2×#484C412yes—
19C2×#477C2212yes—
20C2×#462C2212yes—
21C2×#451C412yes—
22Q8×M16—22no—
23Q8×M16—22no—
24Q8×M16—22no—
25Q8×M16C2214yes—
26C2×#446—22no—
27C2×#446—21no—
28C2×C8×Q8—21no—
29C2×C4×Q16C414yes—
30C2×#431C2211yescharacteristic
31C2×#431C2211yescharacteristic
32C2×#423—21no—
33C2×#423C2211yescharacteristic
34C2×#423C2211yescharacteristic
35C2×C4×M16—21no—
36C2×C4×M16C2211yescharacteristic
37#1998—22no—
38#1997C2212yes—
39#1997C2212yes—
40#1742—21no—
41#1742—21no—
42#1550—22no—
43#1549C2212yes—
44#1548C2212yes—
45#1542—22no—
46#1487—21no—
47#1487—21no—
48#1487C2214yes—
49C2×Q8⋊C8—22no—
50#1292C2214yes—
51#1283C2214yes—
52C23×Q8C2×C411yescharacteristic
53C22×Q16C2×C414yes—
54C22×M16—21no—
55C22×M16—21no—
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58#533—21no—
59#533—21no—
60#533—21no—
61#533—21no—
62#533—21no—
63#533—21no—
64#533—21no—
65#533—21no—
66C2×C4⋊Q8C2311yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8C2×C412yes—
71C2×C4×Q8—21no—
72C2×C4×Q8—21no—
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×Q8—21no—
77C2×C4×Q8—21no—
78C2×C4×Q8C2×C412yes—
79C2×C4×Q8C2×C412yes—
80C2×C4×Q8D411yescharacteristic
81C2×C4×Q8D411yescharacteristic
82C2×C4×Q8D412yes—
83C2×C4×Q8D412yes—
84C2×C4×Q8D411yescharacteristic
85C2×C4×Q8D411yescharacteristic
86C2×C4×Q8C2312yes—
87C2×#117—21no—
88C22×C4⋊C4—21no—
89C22×C4⋊C4C2×C411yescharacteristic
90C22×C42—21no—
91C22×C42C2311yescharacteristic
92#484—22no—
93#484—22no—
94#484—24no—
95#477—22no—
96#477—22no—
97#477—22no—
98#477—22no—
99#462—22no—
100#462—22no—
101#462—22no—
102#462—22no—
103#451—22no—
104#451—24no—
105#451—22no—
106#446—28no—
107#446—24no—
108#446—24no—
109#446—28no—
110C8×Q8—24no—
111C8×Q8—24no—
112C4×Q16—24no—
113C4×Q16—24no—
114C4×Q16—24no—
115C4×Q16—24no—
116#431—21no—
117#431—21no—
118#431—21no—
119#431—21no—
120#431—21no—
121#431—21no—
122#431—21no—
123#431C2312yes—
124C2×#107C2×C414yes—
125#423—24no—
126#423—24no—
127#423—21no—
128#423—21no—
129#423—21no—
130#423—21no—
131#423—21no—
132#423D412yes—
133#423C2312yes—
134#423C2312yes—
135C2×C4⋊C8—22no—
136C2×C4⋊C8—22no—
137C2×C4⋊C8—21no—
138C2×C4⋊C8—21no—
139C2×C4⋊C8—21no—
140C2×C4⋊C8—21no—
141C2×C4⋊C8D412yes—
142C2×C4⋊C8C2312yes—
143C2×#103C2×C414yes—
144C2×#103C2×C414yes—
145C2×C22⋊C8—21no—
146C2×C22⋊C8C2×C412yes—
147C4×M16—24no—
148C4×M16—24no—
149C4×M16—21no—
150C4×M16—21no—
151C4×M16D412yes—
152C4×M16C2312yes—
153C2×#97—22no—
154C2×#97—21no—
155C2×C4×C8—22no—
156C2×C4×C8—21no—
157C2×C4×C8—21no—
158C2×C4×C8C2×C412yes—
159#343—22no—
160#343—22no—
161#343—22no—
162#343—22no—
163#333—24no—
164#332—24no—
165Q8⋊C8—24no—
166Q8⋊C8—24no—
167C22×Q8—21no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8—21no—
171C22×Q8—44no—
172C22×Q8C22⋊C411yescharacteristic
173C22×Q8C22⋊C412yes—
174C22×Q8C22⋊C411yescharacteristic
175C22×Q8C22×C412yes—
176C22×Q8C22×C412yes—
177C23×C4—21no—
178C23×C4—21no—
179C23×C4C22×C411yescharacteristic
180C23×C4C2×D411yescharacteristic
181C23×C4#4111yescharacteristic
182C2×Q16—24no—
183C2×Q16—24no—
184C2×Q16—48no—
185C2×M16—21no—
186C2×M16—22no—
187C2×M16—22no—
188C2×M16—21no—
189C2×M16—22no—
190C2×M16—22no—
191C2×M16—22no—
192C2×M16—22no—
193C2×M16—42no—
194C2×M16—42no—
195C2×M16—41no—
196C2×M16—42no—
197C2×M16—42no—
198C2×M16Q1614yescomplemented
199C2×M16C2×D412yes—
200C2×M16C2×D412yes—
201C22×C8—21no—
202C22×C8—22no—
203C22×C8—21no—
204C22×C8—41no—
205C22×C8—41no—
206C22×C8C22×C414yes—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8—21no—
210C4⋊Q8—21no—
211C4⋊Q8—21no—
212C4⋊Q8—21no—
213C4⋊Q8—21no—
214C4⋊Q8—21no—
215#125—41no—
216#125—41no—
217#125—41no—
218#125—41no—
219C22⋊Q8—22no—
220C22⋊Q8—22no—
221C22⋊Q8—22no—
222C22⋊Q8—22no—
223C22⋊Q8—41no—
224C22⋊Q8—41no—
225C22⋊Q8—41no—
226C22⋊Q8—41no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—21no—
230C4×Q8—22no—
231C4×Q8—21no—
232C4×Q8—21no—
233C4×Q8—22no—
234C4×Q8—21no—
235C4×Q8—21no—
236C4×Q8—21no—
237C4×Q8—22no—
238C4×Q8—22no—
239C4×Q8—21no—
240C4×Q8—21no—
241C4×Q8—22no—
242C4×Q8—22no—
243C4×Q8—22no—
244C4×Q8—22no—
245C4×Q8—22no—
246C4×Q8—22no—
247C4×Q8—41no—
248C4×Q8—41no—
249C4×Q8—41no—
250C4×Q8—41no—
251C4×Q8C22⋊C412yes—
252C4×Q8C22⋊C412yes—
253C4×Q8C22⋊C412yes—
254C4×Q8C22⋊C412yes—
255C4×Q8C2×D412yes—
256C4×Q8C2×D412yes—
257C4×Q8C2×D412yes—
258C4×Q8C2×D412yes—
259#117—41no—
260#117—41no—
261#117—41no—
262#117—41no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—21no—
265C2×C4⋊C4—21no—
266C2×C4⋊C4—21no—
267C2×C4⋊C4—21no—
268C2×C4⋊C4—21no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4—21no—
277C2×C4⋊C4—21no—
278C2×C4⋊C4—21no—
279C2×C4⋊C4C22⋊C411yescharacteristic
280C2×C4⋊C4C22⋊C411yescharacteristic
281C2×C4⋊C4C22⋊C412yes—
282C2×C4⋊C4C22×C412yes—
283C2×C4⋊C4C22×C412yes—
284C2×C22⋊C4—22no—
285C2×C42—22no—
286C2×C42—21no—
287C2×C42—21no—
288C2×C42—22no—
289C2×C42—21no—
290C2×C42—21no—
291C2×C42—22no—
292C2×C42—22no—
293C2×C42—22no—
294C2×C42C22×C411yescharacteristic
295C2×C42C2×D411yescharacteristic
296C2×C42C2×D411yescharacteristic
297C2×C42C2×D411yescharacteristic
298C2×C42C2×D411yescharacteristic
299C2×C42C2×D411yescharacteristic
300C2×C42C2×D411yescharacteristic
301C2×C42C2×D411yescharacteristic
302C2×C42#4111yescharacteristic
303C2×C42#4111yescharacteristic
304C2×C42#4111yescharacteristic
305C2×C42C2411yescharacteristic
306#107—24no—
307#107—24no—
308C4⋊C8—28no—
309C4⋊C8—28no—
310C4⋊C8—22no—
311C4⋊C8—22no—
312C4⋊C8—22no—
313C4⋊C8—22no—
314C4⋊C8—22no—
315C4⋊C8—22no—
316C4⋊C8—22no—
317C4⋊C8—22no—
318C4⋊C8—41no—
319C4⋊C8—41no—
320C4⋊C8—41no—
321C4⋊C8C2×D414yes—
322#103—24no—
323#103—24no—
324#103—24no—
325#103—24no—
326C22⋊C8—22no—
327C22⋊C8—22no—
328C22⋊C8—22no—
329C22⋊C8—41no—
330#97—28no—
331#97—22no—
332#97—22no—
333C4×C8—28no—
334C4×C8—22no—
335C4×C8—22no—
336C4×C8—22no—
337C4×C8—22no—
338C4×C8—22no—
339C4×C8—41no—
340C24C4×D411yescharacteristic
341C2×Q8—24no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—22no—
345C2×Q8—21no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—22no—
349C2×Q8—22no—
350C2×Q8—42no—
351C2×Q8—42no—
352C2×Q8—42no—
353C2×Q8—44no—
354C2×Q8—44no—
355C2×Q8—44no—
356C2×Q8—44no—
357C2×Q8—41no—
358C2×Q8—41no—
359C2×Q8—82no—
360C2×Q8C2×C22⋊C412yes—
361C2×Q8C2×C22⋊C412yes—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—42no—
373C22×C4—44no—
374C22×C4—81no—
375C22×C4—82no—
376C22×C4C2×C22⋊C411yescharacteristic
377C22×C4C2×C22⋊C411yescharacteristic
378C22×C4C4×D411yescharacteristic
379C22×C4C4×D411yescharacteristic
380C22×C4C4×D411yescharacteristic
381C22×C4C4×D411yescharacteristic
382C22×C4C4×D411yescharacteristic
383C22×C4#12011yescharacteristic
384C22×C4#12011yescharacteristic
385C22×C4#12111yescharacteristic
386C22×C4#12111yescharacteristic
387C22×C4#12111yescharacteristic
388C22×C4#12111yescharacteristic
389C22×C4#12311yescharacteristic
390C22×C4#12311yescharacteristic
391C22×C4C2×Q1612yes—
392C22×C4#13512yes—
393C22×C4#13712yes—
394C22×C4#13712yes—
395C22×C4C23×C411yescharacteristic
396C22×C4C22×D411yescharacteristic
397C22×C4C2×#4111yescharacteristic
398Q16—48no—
399Q16—48no—
400M16—22no—
401M16—24no—
402M16—22no—
403M16—22no—
404M16—42no—
405M16—42no—
406M16—42no—
407M16—44no—
408M16—42no—
409M16—42no—
410M16—82no—
411M16C2×Q1614yes—
412C2×C8—22no—
413C2×C8—24no—
414C2×C8—24no—
415C2×C8—22no—
416C2×C8—24no—
417C2×C8—42no—
418C2×C8—44no—
419C2×C8—44no—
420C2×C8—42no—
421C2×C8—42no—
422C2×C8—82no—
423C2×C8—82no—
424C4⋊C4—22no—
425C4⋊C4—22no—
426C4⋊C4—22no—
427C4⋊C4—22no—
428C4⋊C4—22no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—21no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—42no—
445C4⋊C4—42no—
446C4⋊C4—42no—
447C4⋊C4—42no—
448C4⋊C4C2×C22⋊C412yes—
449C4⋊C4C2×C22⋊C412yes—
450C22⋊C4—42no—
451C22⋊C4—42no—
452C42—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—42no—
461C42—42no—
462C42C2×C22⋊C411yescharacteristic
463C42C2×C22⋊C411yescharacteristic
464C42C4×D411yescharacteristic
465C42C4×D411yescharacteristic
466C42#12011yescharacteristic
467C42#12011yescharacteristic
468C42#12111yescharacteristic
469C42#12111yescharacteristic
470C42#12111yescharacteristic
471C42#12111yescharacteristic
472C42#12311yescharacteristic
473C42#12311yescharacteristic
474C42C22×D411yescharacteristic
475C42C22×D411yescharacteristic
476C42C22×D411yescharacteristic
477C42C2×#4111yescharacteristic
478C23—41no—
479C23—41no—
480C23#38611yescharacteristic
481C23#38611yescharacteristic
482C23C4×Q1612yes—
483C23#44112yes—
484C23C2×C4×D411yescharacteristic
485Q8—24no—
486Q8—22no—
487Q8—22no—
488Q8—22no—
489Q8—44no—
490Q8—44no—
491Q8—42no—
492Q8—44no—
493Q8—82no—
494Q8—82no—
495Q8—82no—
496Q8#41314yes—
497C2×C4—24no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—44no—
515C2×C4—44no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—82no—
525C2×C4#38611yescharacteristic
526C2×C4#38611yescharacteristic
527C2×C4#38611yescharacteristic
528C2×C4#38611yescharacteristic
529C2×C4#38611yescharacteristic
530C2×C4#38611yescharacteristic
531C2×C4C4×Q1612yes—
532C2×C4#44112yes—
533C2×C4#45112yes—
534C2×C4#45112yes—
535C2×C4#45212yes—
536C2×C4#45212yes—
537C2×C4#46212yes—
538C2×C4#46212yes—
539C2×C4#46412yes—
540C2×C4#46412yes—
541C2×C4C22×C22⋊C411yescharacteristic
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C2×#12011yescharacteristic
544C2×C4C2×#12111yescharacteristic
545C2×C4C2×#12111yescharacteristic
546C2×C4C2×#12311yescharacteristic
547C2×C4C22×Q1611yescharacteristic
548C2×C4C2×#13511yescharacteristic
549C2×C4C2×#13711yescharacteristic
550C2×C4C2×#13711yescharacteristic
551C8—24no—
552C8—44no—
553C8—42no—
554C8—44no—
555C8—82no—
556C8—82no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—42no—
564C22—42no—
565C22#187812yes—
566C22#187812yes—
567C22C2×#38611yescharacteristic
568C22C2×C4×Q1611yescharacteristic
569C22C2×#44111yescharacteristic
570C4—21no—
571C4—21no—
572C4—21no—
573C4—21no—
574C4—24no—
575C4—21no—
576C4—21no—
577C4—21no—
578C4—42no—
579C4—42no—
580C4—42no—
581C4—82no—
582C4—82no—
583C4#187812yes—
584C4#187812yes—
585C4C2×#41312yes—
586C4C2×#45111yescharacteristic
587C4C2×#45211yescharacteristic
588C4C2×#46211yescharacteristic
589C4C2×#46411yescharacteristic
590C2—22no—
591C2—22no—
592C2—21no—
593C2—21no—
594C2C2×#187811yescharacteristic
595C2#2098911yescharacteristic
596C2#2099511yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC22C2×#386
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9250C2semidirect
7. #9248C2semidirect
198. C2×M16Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#31042
C2×Q8×M162
#213372
#177082
C2×#14872
#92502
#92482
#80012
#79322