Finite Group Databasebeta

← #393127

Subgroups — #393127

Order 512.

1058 conjugacy classes of subgroups.

The subgroup lattice has 551 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic
3C2×#1726C211yesmaximal, characteristic
4#14266C212yesmaximal
5#10881C212yesmaximal
6#10881C212yesmaximal
7#10862C211yesmaximal, characteristic
8#10862C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10862C211yesmaximal, characteristic
11#10821C212yesmaximal
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2211yescharacteristic
16C22×#97C2211yescharacteristic
17C22×#97C2211yescharacteristic
18C22×#97C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C22×C4×C8C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22C2×#389C2212yes—
23C2×#389C2212yes—
24C2×#376C2211yescharacteristic
25#1942—24no—
26#1942C418yescomplemented
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33#1726—24no—
34#1445—22no—
35#1443—22no—
36#1442—22no—
37#1441—22no—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—22no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107C2×C414yes—
66C2×#107C2×C414yes—
67C2×#97—22no—
68C2×#97—22no—
69C2×#97—22no—
70C2×#97—22no—
71C2×#97—22no—
72C2×#97—22no—
73C2×#97—22no—
74C2×#97—22no—
75C2×#97Q814yes—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8C2×C414yes—
83C2×C4×C8D414yes—
84C2×C4×C8Q814yes—
85#389—24no—
86#389—24no—
87#389—24no—
88#389C2×C418yes—
89#376—22no—
90#376—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95D411yescharacteristic
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105#340—24no—
106#340—24no—
107#340—24no—
108#340—24no—
109#340—24no—
110#340C2×C418yes—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2411yescharacteristic
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8C22×C412yes—
135C22×C8C22×C412yes—
136C22×C8C2×D412yes—
137C22×C8C2×D412yes—
138C22×C8C2×Q812yes—
139C22×C8C2×Q812yes—
140C22×C8C2×Q812yes—
141C22×C8C2×Q812yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144C22×C8#4112yes—
145C22×C8#4112yes—
146C22×C8#4112yes—
147C22×C8#4112yes—
148C22×C8#4112yes—
149C22×C8#4112yes—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—28no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4—82no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C22×C414yes—
179C2×C4⋊C4C22×C414yes—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42C22×C412yes—
193C2×C42C2×D412yes—
194C2×C42C2×Q812yes—
195C2×C42C2×Q812yes—
196C2×C42#4112yes—
197C2×C42#4112yes—
198#107—28no—
199#107—28no—
200#107—44no—
201#107—44no—
202#97—28no—
203#97—28no—
204#97—28no—
205C4×C8—28no—
206C4×C8—28no—
207C4×C8—28no—
208#95—28no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95—22no—
213#95—44no—
214#95—82no—
215#95—82no—
216#95—82no—
217#95—82no—
218#95—82no—
219#95—82no—
220#95—82no—
221#95—82no—
222C24C22×D411yescharacteristic
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4C4×D412yes—
288C22×C4C4×D412yes—
289C22×C4#12112yes—
290C22×C4#12112yes—
291C22×C4C4⋊Q812yes—
292C22×C4C4⋊Q812yes—
293C22×C4C23×C411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×Q811yescharacteristic
296C22×C4C22×Q811yescharacteristic
297C22×C4C2×#4111yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4111yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4111yescharacteristic
303C22×C4C2×#4111yescharacteristic
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—22no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8#11714yes—
326C2×C8C4×D414yes—
327C2×C8C4×Q814yes—
328C2×C8C4×Q814yes—
329C2×C8C22⋊Q814yes—
330C2×C8C22⋊Q814yes—
331C2×C8#12314yes—
332C2×C8#12514yes—
333C2×C8#12514yes—
334C2×C8#12614yes—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—88no—
352C4⋊C4—88no—
353C42—22no—
354C42—22no—
355C42—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42—22no—
360C42—22no—
361C42—22no—
362C42—22no—
363C42—22no—
364C42—22no—
365C23C2×C4×D411yescharacteristic
366C23C2×#12111yescharacteristic
367C23C2×C4⋊Q811yescharacteristic
368C23#53211yescharacteristic
369C23#53211yescharacteristic
370C23#53211yescharacteristic
371C23#53211yescharacteristic
372C23#57611yescharacteristic
373C23#57611yescharacteristic
374C23#57611yescharacteristic
375C23#57611yescharacteristic
376C23#57811yescharacteristic
377C23#57811yescharacteristic
378C23#57811yescharacteristic
379C23#57811yescharacteristic
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4C2×#11711yescharacteristic
451C2×C4C2×C4×D411yescharacteristic
452C2×C4C2×C4×Q811yescharacteristic
453C2×C4C2×C4×Q811yescharacteristic
454C2×C4C4×#4111yescharacteristic
455C2×C4C4×#4111yescharacteristic
456C2×C4C2×C22⋊Q811yescharacteristic
457C2×C4C2×C22⋊Q811yescharacteristic
458C2×C4C2×#12311yescharacteristic
459C2×C4#52511yescharacteristic
460C2×C4#52511yescharacteristic
461C2×C4C2×#12511yescharacteristic
462C2×C4C2×#12511yescharacteristic
463C2×C4C2×#12611yescharacteristic
464C2×C4#52911yescharacteristic
465C2×C4#52911yescharacteristic
466C2×C4#52911yescharacteristic
467C2×C4#52911yescharacteristic
468C2×C4#52911yescharacteristic
469C2×C4#52911yescharacteristic
470C2×C4#52911yescharacteristic
471C2×C4#52911yescharacteristic
472C2×C4#53311yescharacteristic
473C2×C4#53311yescharacteristic
474C2×C4#53311yescharacteristic
475C2×C4#53311yescharacteristic
476C2×C4#53311yescharacteristic
477C2×C4#53311yescharacteristic
478C8—24no—
479C8—24no—
480C8—24no—
481C8#38718yes—
482C22#229311yescharacteristic
483C22C4×#12111yescharacteristic
484C22C4×C4⋊Q811yescharacteristic
485C22#285411yescharacteristic
486C22#285711yescharacteristic
487C22#285911yescharacteristic
488C22#286011yescharacteristic
489C22#294811yescharacteristic
490C22#295011yescharacteristic
491C22#295111yescharacteristic
492C22#295111yescharacteristic
493C22#305811yescharacteristic
494C22#305911yescharacteristic
495C22#306111yescharacteristic
496C22#306111yescharacteristic
497C22#316411yescharacteristic
498C22#316411yescharacteristic
499C22#316411yescharacteristic
500C22#316411yescharacteristic
501C22#317511yescharacteristic
502C22#317511yescharacteristic
503C22#317511yescharacteristic
504C22#317511yescharacteristic
505C22#317511yescharacteristic
506C22#317511yescharacteristic
507C22#317511yescharacteristic
508C22#317511yescharacteristic
509C22#317611yescharacteristic
510C22#317611yescharacteristic
511C22#317611yescharacteristic
512C22#317611yescharacteristic
513C22#317711yescharacteristic
514C22#317711yescharacteristic
515C22#317711yescharacteristic
516C22#317711yescharacteristic
517C4—42no—
518C4—42no—
519C4—42no—
520C4—42no—
521C4—42no—
522C4—42no—
523C4—84no—
524C4—84no—
525C4—162no—
526C4—162no—
527C4C2×#38711yescharacteristic
528C4#229511yescharacteristic
529C4C4×C22⋊Q811yescharacteristic
530C4C4×#12511yescharacteristic
531C4#285111yescharacteristic
532C4#285511yescharacteristic
533C4#286311yescharacteristic
534C4#286411yescharacteristic
535C2#1469911yescharacteristic
536C2#2128111yescharacteristic
537C2#2128211yescharacteristic
538C2#2152411yescharacteristic
539C2#2152511yescharacteristic
540C2#2172311yescharacteristic
541C2#2172511yescharacteristic
542C2#3313811yescharacteristic
543C2#3313911yescharacteristic
544C2#3321511yescharacteristic
545C2#3321611yescharacteristic
546C2#3326111yescharacteristic
547C2#3326111yescharacteristic
548C2#3327211yescharacteristic
549C2#3327311yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
26. #1942C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#17262
#142662
#108812
#108622
#108212