Finite Group Databasebeta

← #360741

Subgroups — #360741

Order 512.

1592 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q8×C4⋊C4C211yesmaximal, characteristic
2C2×#1865C212yesmaximal
3C2×#1863C212yesmaximal
4C2×#1847C211yesmaximal, characteristic
5C2×#1805C211yesmaximal, characteristic
6C2×#1763C211yesmaximal, characteristic
7#13954C211yesmaximal, characteristic
8#11328C212yesmaximal
9#11327C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10835C211yesmaximal, characteristic
12C22×C4×Q8C2212yes—
13C22×C4×Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×C4⋊C8C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C22×#103C2211yescharacteristic
19C22×#103C2212yes—
20C22×#97C2211yescharacteristic
21Q8×C4⋊C4—24no—
22Q8×C4⋊C4—24no—
23C2×#389—22no—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#389C2212yes—
27C2×#384—21no—
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×C4×C4⋊C4—21no—
31C2×C4×C4⋊C4C2211yescharacteristic
32C2×C4×C4⋊C4C2211yescharacteristic
33#1865—28no—
34#1863—28no—
35#1847—24no—
36#1805—24no—
37#1763C418yescomplemented
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2211yescharacteristic
41C2×#340C2212yes—
42C2×#340C2212yes—
43C2×#336C2212yes—
44C2×#336C2212yes—
45#1444—21no—
46#1444—21no—
47C4⋊#95—22no—
48C23×Q8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4×Q8—24no—
53C2×C4×Q8—24no—
54C2×C4×Q8—22no—
55C2×C4×Q8—22no—
56C2×C4×Q8—22no—
57C2×C4×Q8—24no—
58C2×C4×Q8—24no—
59C2×C4×Q8—42no—
60C2×C4×Q8—42no—
61C2×C4×Q8C2×C414yes—
62C2×C4×Q8D414yes—
63C2×C4×Q8Q814yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—21no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—22no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2312yes—
77C22×C4⋊C4C2311yescharacteristic
78C22×C42—21no—
79C22×C42—22no—
80C22×C42—21no—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—42no—
86C2×#106—24no—
87C2×#106—24no—
88C2×#106—42no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—22no—
92C2×#103—22no—
93C2×#103—22no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103C2×C418yes—
97C2×#103D418yes—
98C2×#103Q818yes—
99C2×#97—22no—
100C2×#97—22no—
101C2×#97C2×C414yes—
102#389—216no—
103#389—24no—
104#389—24no—
105#389—24no—
106#389—24no—
107#389—42no—
108#389—42no—
109#384—28no—
110#384—22no—
111#384D414yes—
112#382—22no—
113#382—22no—
114C4×C4⋊C4—28no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C4×C4⋊C4—22no—
118C4×C4⋊C4C2×C414yes—
119C2×#95—21no—
120C2×#95—22no—
121C2×#95—21no—
122C2×#95—22no—
123C2×#95—22no—
124C2×#95—21no—
125C2×#95D412yes—
126#340—22no—
127#340—22no—
128#340—24no—
129#340—24no—
130#340C2×C418yes—
131#336—24no—
132#336—24no—
133C22×Q8—21no—
134C22×Q8—21no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×D414yes—
141C22×Q8C2×Q812yes—
142C22×Q8C2×Q814yes—
143C23×C4—21no—
144C23×C4—21no—
145C23×C4—22no—
146C23×C4—22no—
147C23×C4—21no—
148C23×C4—21no—
149C23×C4C2×D412yes—
150C23×C4C2×D412yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8C22×C414yes—
169C22×C8C2×D414yes—
170C22×C8C2×Q814yes—
171C4×Q8—28no—
172C4×Q8—28no—
173C4×Q8—28no—
174C4×Q8—48no—
175C4×Q8—48no—
176C4×Q8—48no—
177C4×Q8—48no—
178C4×Q8C4⋊C4116yescomplemented
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—28no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—28no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—44no—
212C2×C4⋊C4—44no—
213C2×C4⋊C4—44no—
214C2×C4⋊C4—44no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—81no—
217C2×C4⋊C4—81no—
218C2×C4⋊C4—82no—
219C2×C4⋊C4—82no—
220C2×C4⋊C4—82no—
221C2×C4⋊C4—82no—
222C2×C4⋊C4C22×C412yes—
223C2×C4⋊C4C22×C414yes—
224C2×C4⋊C4C2×D414yes—
225C2×C4⋊C4C2×D412yes—
226C2×C4⋊C4C2×D414yes—
227C2×C4⋊C4C2×Q812yes—
228C2×C4⋊C4C2×Q814yes—
229C2×C42—24no—
230C2×C42—24no—
231C2×C42—24no—
232C2×C42—28no—
233C2×C42—22no—
234C2×C42—22no—
235C2×C42—22no—
236C2×C42—22no—
237C2×C42—22no—
238C2×C42—22no—
239C2×C42—24no—
240C2×C42—44no—
241C2×C42—44no—
242C2×C42—81no—
243C2×C42—81no—
244C2×C42C22×C412yes—
245C2×C42C2×D412yes—
246C2×C42C2×D412yes—
247C2×C42C2×Q812yes—
248C2×C42#4112yes—
249C2×C42#4112yes—
250#107—48no—
251#106—48no—
252C4⋊C8—44no—
253#103—44no—
254#103—44no—
255#103C4⋊C4132yescomplemented
256#97—28no—
257#95—28no—
258#95—24no—
259#95—24no—
260#95—44no—
261#95—44no—
262#95—42no—
263#95—42no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—24no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—24no—
275C2×Q8—24no—
276C2×Q8—24no—
277C2×Q8—24no—
278C2×Q8C2×C4⋊C418yes—
279C2×Q8C2×C4⋊C414yes—
280C2×Q8#12714yes—
281C2×Q8C4⋊Q814yes—
282C2×Q8C4⋊Q818yes—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—22no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—24no—
301C22×C4—22no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—24no—
308C22×C4—22no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4—42no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4C4×D412yes—
339C22×C4C4×D412yes—
340C22×C4#12012yes—
341C22×C4#12012yes—
342C22×C4C22⋊Q812yes—
343C22×C4C22⋊Q812yes—
344C22×C4#13714yes—
345C22×C4C23×C411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×Q811yescharacteristic
350C22×C4C22×Q812yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4C2×#4111yescharacteristic
353C22×C4H(5,𝔽2)12yes—
354C22×C42−1+412yes—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8C2×C4⋊C418yes—
372C4⋊C4—28no—
373C4⋊C4—22no—
374C4⋊C4—24no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—24no—
378C4⋊C4—24no—
379C4⋊C4—216no—
380C4⋊C4—24no—
381C4⋊C4—24no—
382C4⋊C4—44no—
383C4⋊C4—48no—
384C4⋊C4—416no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C4⋊C4—84no—
394C4⋊C4—88no—
395C4⋊C4—88no—
396C4⋊C4—88no—
397C4⋊C4C2×C4⋊C418yes—
398C4⋊C4C2×C4⋊C414yes—
399C42—216no—
400C42—416no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—42no—
405C42—84no—
406C42C2×C4⋊C414yes—
407C42C4×D414yes—
408C42C22⋊Q814yes—
409C42#12314yes—
410C23C2×C4×D411yescharacteristic
411C23C2×#12011yescharacteristic
412C23C2×C22⋊Q811yescharacteristic
413C23D4212yes—
414C23D4×Q812yes—
415C23C2×#13712yes—
416C23#57512yes—
417C23#57611yescharacteristic
418C23#57711yescharacteristic
419C23#57711yescharacteristic
420C23#57811yescharacteristic
421Q8—24no—
422Q8—28no—
423Q8—28no—
424Q8—24no—
425Q8—24no—
426Q8#38418yes—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—28no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—28no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—24no—
442C2×C4—28no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—44no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—81no—
483C2×C4—82no—
484C2×C4—81no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4#38514yes—
488C2×C4#38514yes—
489C2×C4C22×C4⋊C411yescharacteristic
490C2×C4C22×C4⋊C412yes—
491C2×C4C2×C4×D411yescharacteristic
492C2×C4#52012yes—
493C2×C4C2×C22⋊Q811yescharacteristic
494C2×C4C2×#12311yescharacteristic
495C2×C4C2×#12711yescharacteristic
496C2×C4C2×C4⋊Q812yes—
497C2×C4C2×C4⋊Q811yescharacteristic
498C2×C4#53511yescharacteristic
499C2×C4#53611yescharacteristic
500C2×C4#54412yes—
501C2×C4D4212yes—
502C2×C4#54712yes—
503C2×C4#54712yes—
504C2×C4D4×Q812yes—
505C2×C4#55412yes—
506C2×C4#55412yes—
507C8—28no—
508C8—81no—
509C8—81no—
510C22C2×#38511yescharacteristic
511C22D4×C4⋊C412yes—
512C22#243311yescharacteristic
513C22#243411yescharacteristic
514C22#244012yes—
515C22C4×#13712yes—
516C22#294611yescharacteristic
517C22#294711yescharacteristic
518C22#300411yescharacteristic
519C22#300511yescharacteristic
520C22#301211yescharacteristic
521C22#301311yescharacteristic
522C22#307812yes—
523C22#308311yescharacteristic
524C22#308411yescharacteristic
525C22#326612yes—
526C22#326812yes—
527C22#327012yes—
528C22#327112yes—
529C22#338512yes—
530C22#338712yes—
531C22#338812yes—
532C22#339012yes—
533C4—28no—
534C4—24no—
535C4—24no—
536C4—44no—
537C4—48no—
538C4—44no—
539C4—82no—
540C4—84no—
541C4—161no—
542C4—161no—
543C4C2×#38411yescharacteristic
544C4#231711yescharacteristic
545C4D4×C4⋊C412yes—
546C4#244012yes—
547C4#250111yescharacteristic
548C4#250211yescharacteristic
549C2#1388011yescharacteristic
550C2#2087411yescharacteristic
551C2#2087811yescharacteristic
552C2#2267812yes—
553C2#2267912yes—
554C2#2576511yescharacteristic
555C2#2576611yescharacteristic
556C2#2584611yescharacteristic
557C2#2584711yescharacteristic
558C2#2605412yes—
559C2#2605512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
37. #1763C4semidirect
178. C4×Q8C4⋊C4semidirect
255. #103C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×Q8×C4⋊C42
C2×#18652
C2×#18632
C2×#18472
C2×#18052
C2×#17632
#139542
#113282
#113272
#108492
#108352