Finite Group Databasebeta

← #185563

Subgroups — #185563

Order 512.

2234 conjugacy classes of subgroups.

The subgroup lattice has 567 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#557C211yesmaximal, characteristic
2#20727C211yesmaximal, characteristic
3#17667C212yesmaximal
4#17449C212yesmaximal
5#17365C211yesmaximal, characteristic
6#8374C218yesmaximal, complemented
7C2×#557—24no—
8C2×#557—24no—
9C2×#557—24no—
10C2×#557—24no—
11C2×#557—22no—
12C2×#557—22no—
13C2×#557C414yes—
14C2×#557C2214yes—
15C22×C4⋊Q8—21no—
16C22×C4⋊Q8C411yescharacteristic
17C22×#125—21no—
18C22×#125—21no—
19C22×#125C411yescharacteristic
20C22×#125C2211yescharacteristic
21C22×C4×Q8—21no—
22C22×C4×Q8C411yescharacteristic
23C22×C4×Q8C411yescharacteristic
24C22×C4×Q8C2211yescharacteristic
25C22×C4×Q8C2211yescharacteristic
26#2920C2212yes—
27#2920C2212yes—
28#1821—21no—
29#1801—22no—
30#1784—21no—
31C4×#101C2214yes—
32#1534—24no—
33#1528—24no—
34#1528C2218yes—
35#1491C2218yes—
36#1487—24no—
37#1479—24no—
38#1467C2214yes—
39#1463—21no—
40#1463—21no—
41#1461—22no—
42C23×Q8—21no—
43C23×Q8C2×C411yescharacteristic
44C23×Q8D411yescharacteristic
45#557—216no—
46#557—216no—
47#557—216no—
48#557—216no—
49#557—216no—
50#557—216no—
51#557—216no—
52#557—216no—
53C2×C4⋊Q8—28no—
54C2×C4⋊Q8—28no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—24no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8C2×C414yes—
63C2×#125—28no—
64C2×#125—28no—
65C2×#125—28no—
66C2×#125—28no—
67C2×#125—24no—
68C2×#125—24no—
69C2×#125—22no—
70C2×#125—22no—
71C2×#125—22no—
72C2×#125—22no—
73C2×#125—24no—
74C2×#125—24no—
75C2×#125—22no—
76C2×#125—22no—
77C2×#125—22no—
78C2×#125—21no—
79C2×#125—21no—
80C2×#125C2×C414yes—
81C2×#125C2×C412yes—
82C2×#125D414yes—
83C2×#125D414yes—
84C2×#125C2312yes—
85C2×C4×Q8—28no—
86C2×C4×Q8—28no—
87C2×C4×Q8—22no—
88C2×C4×Q8—22no—
89C2×C4×Q8—22no—
90C2×C4×Q8—22no—
91C2×C4×Q8—24no—
92C2×C4×Q8—22no—
93C2×C4×Q8—22no—
94C2×C4×Q8—22no—
95C2×C4×Q8—22no—
96C2×C4×Q8—24no—
97C2×C4×Q8—22no—
98C2×C4×Q8—22no—
99C2×C4×Q8—22no—
100C2×C4×Q8—22no—
101C2×C4×Q8—22no—
102C2×C4×Q8—22no—
103C2×C4×Q8—21no—
104C2×C4×Q8—22no—
105C2×C4×Q8—22no—
106C2×C4×Q8—21no—
107C2×C4×Q8C2×C414yes—
108C2×C4×Q8C2×C414yes—
109C2×C4×Q8C2×C412yes—
110C2×C4×Q8C2×C412yes—
111C2×C4×Q8D414yes—
112C2×C4×Q8D414yes—
113C2×C4×Q8D412yes—
114C2×C4×Q8D412yes—
115C2×C4×Q8C2312yes—
116C2×C4×Q8C2312yes—
117C22×C4⋊C4—21no—
118C22×C4⋊C4—21no—
119C22×C4⋊C4—21no—
120C22×C4⋊C4—21no—
121C22×C4⋊C4—21no—
122C22×C4⋊C4—21no—
123C22×C4⋊C4—21no—
124C22×C4⋊C4—21no—
125C22×C4⋊C4C2×C411yescharacteristic
126C22×C4⋊C4C2×C411yescharacteristic
127C22×C4⋊C4C2×C411yescharacteristic
128C22×C4⋊C4D411yescharacteristic
129C22×C4⋊C4D411yescharacteristic
130C22×C4⋊C4D411yescharacteristic
131C22×C42—21no—
132C22×C42—21no—
133C22×C42—21no—
134C22×C42C2×C411yescharacteristic
135C22×C42C2×C411yescharacteristic
136C22×C42C2311yescharacteristic
137#432—22no—
138#432—22no—
139#432—22no—
140#432C2314yes—
141#431—22no—
142#431—22no—
143#431—22no—
144#431—22no—
145#423—22no—
146#423—22no—
147C2×#101—24no—
148#407—22no—
149#407—22no—
150C4×M16—22no—
151C4×M16C2314yes—
152#343—24no—
153#330—216no—
154Q8⋊C8—216no—
155#324—24no—
156#324—24no—
157C22×Q8—28no—
158C22×Q8—24no—
159C22×Q8—24no—
160C22×Q8—21no—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×Q8—24no—
164C22×Q8—22no—
165C22×Q8—22no—
166C22×Q8—44no—
167C22×Q8—48no—
168C22×Q8C22⋊C412yes—
169C22×Q8C22⋊C412yes—
170C22×Q8C22⋊C412yes—
171C22×Q8C22×C412yes—
172C22×Q8C2×D412yes—
173C23×C4—21no—
174C23×C4—21no—
175C23×C4—21no—
176C23×C4—21no—
177C23×C4C22⋊C411yescharacteristic
178C23×C4C22⋊C411yescharacteristic
179C23×C4C22⋊C411yescharacteristic
180C23×C4C22⋊C411yescharacteristic
181C23×C4C22×C411yescharacteristic
182C23×C4C2×D411yescharacteristic
183C23×C4C2×D411yescharacteristic
184C2×M16—24no—
185C2×M16—42no—
186C4⋊Q8—232no—
187C4⋊Q8—232no—
188C4⋊Q8—28no—
189C4⋊Q8—28no—
190C4⋊Q8—28no—
191C4⋊Q8—28no—
192#125—232no—
193#125—232no—
194#125—232no—
195#125—232no—
196#125—28no—
197#125—28no—
198#125—28no—
199#125—28no—
200#125—24no—
201#125—24no—
202#125—24no—
203#125—24no—
204#125—24no—
205#125—24no—
206#125C22⋊C418yes—
207#125C2×D418yes—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—232no—
211C4×Q8—232no—
212C4×Q8—28no—
213C4×Q8—28no—
214C4×Q8—28no—
215C4×Q8—28no—
216C4×Q8—28no—
217C4×Q8—28no—
218C4×Q8—24no—
219C4×Q8—24no—
220C4×Q8—24no—
221C4×Q8—24no—
222C4×Q8—24no—
223C4×Q8—24no—
224C4×Q8—24no—
225C4×Q8—24no—
226C4×Q8—24no—
227C4×Q8—24no—
228C4×Q8—24no—
229C4×Q8—24no—
230C4×Q8—24no—
231C4×Q8—24no—
232C4×Q8C22⋊C418yes—
233C4×Q8C2×D418yes—
234C2×C4⋊C4—24no—
235C2×C4⋊C4—24no—
236C2×C4⋊C4—24no—
237C2×C4⋊C4—24no—
238C2×C4⋊C4—24no—
239C2×C4⋊C4—24no—
240C2×C4⋊C4—24no—
241C2×C4⋊C4—24no—
242C2×C4⋊C4—24no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—24no—
245C2×C4⋊C4—22no—
246C2×C4⋊C4—24no—
247C2×C4⋊C4—24no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—24no—
250C2×C4⋊C4—24no—
251C2×C4⋊C4—22no—
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⋊C4—21no—
261C2×C4⋊C4—24no—
262C2×C4⋊C4—24no—
263C2×C4⋊C4—24no—
264C2×C4⋊C4—24no—
265C2×C4⋊C4—24no—
266C2×C4⋊C4—24no—
267C2×C4⋊C4—22no—
268C2×C4⋊C4—22no—
269C2×C4⋊C4—22no—
270C2×C4⋊C4—22no—
271C2×C4⋊C4—22no—
272C2×C4⋊C4—22no—
273C2×C4⋊C4—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C22⋊C412yes—
277C2×C4⋊C4C22⋊C412yes—
278C2×C4⋊C4C22⋊C412yes—
279C2×C4⋊C4C22⋊C412yes—
280C2×C4⋊C4C22⋊C412yes—
281C2×C4⋊C4C22⋊C412yes—
282C2×C4⋊C4C22⋊C412yes—
283C2×C4⋊C4C22⋊C412yes—
284C2×C4⋊C4C22⋊C412yes—
285C2×C4⋊C4C22×C412yes—
286C2×C4⋊C4C22×C412yes—
287C2×C4⋊C4C22×C412yes—
288C2×C4⋊C4C2×D412yes—
289C2×C4⋊C4C2×D412yes—
290C2×C4⋊C4C2×D412yes—
291C2×C42—24no—
292C2×C42—24no—
293C2×C42—24no—
294C2×C42—24no—
295C2×C42—21no—
296C2×C42—21no—
297C2×C42—21no—
298C2×C42—21no—
299C2×C42—21no—
300C2×C42—21no—
301C2×C42—21no—
302C2×C42—21no—
303C2×C42—24no—
304C2×C42—22no—
305C2×C42—24no—
306C2×C42—24no—
307C2×C42—24no—
308C2×C42—22no—
309C2×C42C22⋊C412yes—
310C2×C42C22⋊C412yes—
311C2×C42C22×C412yes—
312C2×C42C22×C412yes—
313C2×C42C22×C411yescharacteristic
314C2×C42C2×D412yes—
315C2×C42C2×D412yes—
316C2×C42C2×D412yes—
317C2×C42C2×D412yes—
318C2×C42C2×D411yescharacteristic
319C2×C42C2×D411yescharacteristic
320C2×C42C2411yescharacteristic
321C4⋊C8—24no—
322C4⋊C8—42no—
323C4⋊C8—42no—
324C4⋊C8—42no—
325#101—48no—
326C22⋊C8—24no—
327C22⋊C8—42no—
328C22⋊C8—42no—
329C22⋊C8—42no—
330#97—24no—
331#97—24no—
332C4×C8—24no—
333C4×C8—42no—
334C24C2×C22⋊C411yescharacteristic
335C2×Q8—28no—
336C2×Q8—28no—
337C2×Q8—28no—
338C2×Q8—28no—
339C2×Q8—28no—
340C2×Q8—216no—
341C2×Q8—22no—
342C2×Q8—24no—
343C2×Q8—22no—
344C2×Q8—22no—
345C2×Q8—24no—
346C2×Q8—24no—
347C2×Q8—24no—
348C2×Q8—24no—
349C2×Q8—42no—
350C2×Q8—42no—
351C2×Q8—48no—
352C2×Q8—48no—
353C2×Q8—416no—
354C2×Q8—416no—
355C2×Q8—48no—
356C2×Q8—48no—
357C2×Q8#10118yescomplemented
358C2×Q8C2×C22⋊C414yes—
359C22×C4—28no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—24no—
366C22×C4—24no—
367C22×C4—22no—
368C22×C4—22no—
369C22×C4—22no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—42no—
383C22×C4—42no—
384C22×C4—42no—
385C22×C4—44no—
386C22×C4—44no—
387C22×C4—44no—
388C22×C4—42no—
389C22×C4—42no—
390C22×C4C2×C22⋊C412yes—
391C22×C4C2×C22⋊C411yescharacteristic
392C22×C4C2×C22⋊C411yescharacteristic
393C22×C4C2×C22⋊C411yescharacteristic
394C22×C4C2×C22⋊C411yescharacteristic
395C22×C4C2×C22⋊C411yescharacteristic
396C22×C4C2×C22⋊C412yes—
397C22×C4C2×C22⋊C412yes—
398C22×C4C2×C22⋊C412yes—
399C22×C4C2×C22⋊C411yescharacteristic
400C22×C4C2×C22⋊C411yescharacteristic
401C22×C4#12012yes—
402C22×C4#12012yes—
403C22×C4#12012yes—
404C22×C4#12012yes—
405C22×C4C23×C411yescharacteristic
406C22×C4C22×D411yescharacteristic
407C22×C4C22×D411yescharacteristic
408M16—48no—
409M16—84no—
410C2×C8—44no—
411C2×C8—44no—
412C4⋊C4—216no—
413C4⋊C4—216no—
414C4⋊C4—216no—
415C4⋊C4—216no—
416C4⋊C4—216no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—28no—
421C4⋊C4—28no—
422C4⋊C4—28no—
423C4⋊C4—24no—
424C4⋊C4—24no—
425C4⋊C4—24no—
426C4⋊C4—28no—
427C4⋊C4—28no—
428C4⋊C4—28no—
429C4⋊C4—24no—
430C4⋊C4—24no—
431C4⋊C4—24no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4C2×C22⋊C414yes—
439C4⋊C4C2×C22⋊C414yes—
440C4⋊C4C2×C22⋊C414yes—
441C42—216no—
442C42—216no—
443C42—22no—
444C42—22no—
445C42—22no—
446C42—22no—
447C42—28no—
448C42—28no—
449C42—22no—
450C42—22no—
451C42—22no—
452C42—22no—
453C42C2×C22⋊C412yes—
454C42C2×C22⋊C412yes—
455C42#12012yes—
456C42#12012yes—
457C42#12012yes—
458C42#12012yes—
459C42C22×D412yes—
460C42C22×D412yes—
461C23—22no—
462C23—22no—
463C23—22no—
464C23#37911yescharacteristic
465C23#37911yescharacteristic
466C23C22×C22⋊C411yescharacteristic
467Q8—28no—
468Q8—28no—
469Q8—24no—
470Q8—28no—
471Q8—48no—
472Q8—48no—
473Q8—432no—
474Q8C2×#10118yes—
475C2×C4—28no—
476C2×C4—28no—
477C2×C4—24no—
478C2×C4—24no—
479C2×C4—24no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—22no—
485C2×C4—28no—
486C2×C4—22no—
487C2×C4—22no—
488C2×C4—22no—
489C2×C4—22no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—24no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—41no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—44no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—44no—
511C2×C4—44no—
512C2×C4—44no—
513C2×C4—44no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—44no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4#37912yes—
522C2×C4#37911yescharacteristic
523C2×C4#37912yes—
524C2×C4#37911yescharacteristic
525C2×C4C2×#10114yes—
526C2×C4#42114yes—
527C2×C4C22×C22⋊C411yescharacteristic
528C2×C4C22×C22⋊C411yescharacteristic
529C2×C4C2×#12012yes—
530C2×C4C2×#12012yes—
531C8—84no—
532C8—84no—
533C22—28no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—21no—
539C22—21no—
540C22#178412yes—
541C22#179612yes—
542C22C2×#37911yescharacteristic
543C22C2×#42112yes—
544C4—28no—
545C4—21no—
546C4—21no—
547C4—22no—
548C4—24no—
549C4—22no—
550C4—22no—
551C4—22no—
552C4—42no—
553C4—44no—
554C4—44no—
555C4—42no—
556C4—42no—
557C4—44no—
558C4#178412yes—
559C4C22×#10112yes—
560C2—22no—
561C2—22no—
562C2—22no—
563C2C2×#178411yescharacteristic
564C2#2060511yescharacteristic
565C2C2×#179611yescharacteristic
subgroupG/H
center, socleC22C2×#379
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8374C2semidirect
357. C2×Q8#101semidirect

Maximal subgroups

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

subgroupindex
C22×#5572
#207272
#176672
#174492
#173652
#83742