Finite Group Databasebeta

← #108089

Subgroups — #108089

Order 512.

918 conjugacy classes of subgroups.

The subgroup lattice has 572 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34213C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3C2×#1493C211yesmaximal, characteristic
4#10847C211yesmaximal, characteristic
5#10834C211yesmaximal, characteristic
6#8045C211yesmaximal, characteristic
7#7897C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#423C2211yescharacteristic
10C2×#423C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4×C8C2211yescharacteristic
13#2370—22no—
14#2369—21no—
15#2365—21no—
16#2355—21no—
17#2355—22no—
18Q8×C4⋊C4—21no—
19#2329C414yes—
20#2317—21no—
21C4×C22⋊Q8—22no—
22#2285C2211yescharacteristic
23#2274—21no—
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×#343—21no—
26C2×#343—21no—
27C2×#336—22no—
28#1550—22no—
29#1550C414yescomplemented
30C2×#333—21no—
31C2×#332—21no—
32#1493—22no—
33#1493C414yescomplemented
34C2×Q8⋊C8—22no—
35#1296—21no—
36#1296—21no—
37#1279—22no—
38C22×M16—21no—
39C22×M16—21no—
40C23×C8—21no—
41C23×C8—21no—
42#533—21no—
43#533—21no—
44#533—21no—
45#533—21no—
46#533C2×C412yes—
47#533C2×C412yes—
48#533D412yes—
49#533Q812yes—
50C2×C4⋊Q8D411yescharacteristic
51C2×C4⋊Q8D411yescharacteristic
52C2×#125—21no—
53C2×C22⋊Q8—21no—
54C2×C22⋊Q8C2×C412yes—
55C2×C4×Q8—22no—
56C2×C4×Q8—21no—
57C2×C4×Q8D412yes—
58C2×#117—21no—
59C2×#117—21no—
60C22×C4⋊C4D411yescharacteristic
61C22×C4⋊C4D411yescharacteristic
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—21no—
69#423—21no—
70#423C2×C412yes—
71#423C2×C412yes—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77C2×C4⋊C8—21no—
78C2×C4⋊C8—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8C2×C412yes—
82C2×C4×C8—21no—
83C2×C4×C8—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87C2×C4×C8C2×C412yes—
88#391—22no—
89#391—24no—
90#389—22no—
91#389—22no—
92#389—22no—
93#389—24no—
94#387—24no—
95#387—24no—
96#387—24no—
97#385—24no—
98#384—22no—
99#384D411yescharacteristic
100#384D411yescharacteristic
101#382—22no—
102#382—21no—
103#380—21no—
104#380C2×C412yes—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—24no—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C4×C4⋊C4—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4—21no—
116C4×C4⋊C4D412yes—
117C4×C22⋊C4—24no—
118C2×#95C2×C412yes—
119#343—42no—
120#343—42no—
121#336—44no—
122#333—24no—
123#332—24no—
124Q8⋊C8—28no—
125C22×Q8—21no—
126C22×Q8C22⋊C412yes—
127C23×C4C22⋊C412yes—
128C23×C4C22⋊C412yes—
129C23×C4C22×C411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—22no—
133C2×M16—22no—
134C2×M16—41no—
135C2×M16—41no—
136C2×M16—41no—
137C2×M16—41no—
138C2×M16—41no—
139C2×M16—41no—
140C2×M16—41no—
141C2×M16—41no—
142C2×M16—41no—
143C2×M16—41no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—41no—
149C22×C8—41no—
150C22×C8—41no—
151C22×C8—41no—
152C22×C8—42no—
153C22×C8—42no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—41no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8C22⋊C412yes—
163C4⋊Q8C22⋊C412yes—
164C4⋊Q8C4⋊C412yes—
165C4⋊Q8C4⋊C412yes—
166#125—22no—
167#125—22no—
168#125—22no—
169#125—22no—
170C22⋊Q8—22no—
171C22⋊Q8—22no—
172C22⋊Q8—22no—
173C22⋊Q8—22no—
174C22⋊Q8—22no—
175C22⋊Q8—22no—
176C22⋊Q8C4214yes—
177C22⋊Q8C4⋊C414yes—
178C4×Q8—22no—
179C4×Q8—22no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—82no—
185C4×Q8—82no—
186C4×Q8C22⋊C414yes—
187C4×Q8C4⋊C414yes—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4C22⋊C412yes—
218C2×C4⋊C4C22⋊C412yes—
219C2×C4⋊C4C22⋊C412yes—
220C2×C4⋊C4SD1612yes—
221C2×C4⋊C4Q1612yes—
222C2×C22⋊C4—22no—
223C2×C22⋊C4—22no—
224C2×C42—22no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—24no—
229C2×C42—22no—
230C2×C42—42no—
231C2×C42—42no—
232C2×C42C22×C411yescharacteristic
233C2×C42C22×C411yescharacteristic
234C2×C42C2×D411yescharacteristic
235C2×C42C2×D411yescharacteristic
236C2×C42C2×D411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42C2×Q811yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42#4111yescharacteristic
242C2×C42#4111yescharacteristic
243C2×C42#4111yescharacteristic
244C4⋊C8—22no—
245C4⋊C8—22no—
246C4⋊C8—22no—
247C4⋊C8—22no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—41no—
251C4⋊C8—41no—
252C4⋊C8—41no—
253C4⋊C8—41no—
254C4⋊C8—41no—
255C4⋊C8—41no—
256C4×C8—22no—
257C4×C8—22no—
258C4×C8—41no—
259C4×C8—41no—
260#95—24no—
261#95—24no—
262#95—24no—
263#95—24no—
264#95—22no—
265#95—22no—
266#95—22no—
267#95—22no—
268C24C2×C22⋊C411yescharacteristic
269C2×Q8—22no—
270C2×Q8—22no—
271C2×Q8—22no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8#9514yes—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4#10314yes—
299C22×C4C2×C4211yescharacteristic
300C22×C4C2×C22⋊C411yescharacteristic
301C22×C4C2×C22⋊C411yescharacteristic
302C22×C4C2×C4⋊C411yescharacteristic
303C22×C4#11711yescharacteristic
304C22×C4#11711yescharacteristic
305C22×C4C4×D411yescharacteristic
306C22×C4C4×D411yescharacteristic
307C22×C4C4×D411yescharacteristic
308C22×C4C4×D411yescharacteristic
309C22×C4#12011yescharacteristic
310C22×C4#12011yescharacteristic
311C22×C4#12011yescharacteristic
312C22×C4#12011yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4C22⋊Q811yescharacteristic
316C22×C4C22⋊Q811yescharacteristic
317C22×C4C2×SD1611yescharacteristic
318C22×C4C2×Q1611yescharacteristic
319C22×C4#13511yescharacteristic
320C22×C4#13511yescharacteristic
321C22×C4AGL(1,ℤ/8)11yescharacteristic
322C22×C4#13711yescharacteristic
323C22×C4#13711yescharacteristic
324C22×C4#13711yescharacteristic
325M16—42no—
326M16—42no—
327M16—42no—
328M16—42no—
329M16—81no—
330M16—81no—
331M16—81no—
332M16—81no—
333C2×C8—24no—
334C2×C8—42no—
335C2×C8—42no—
336C2×C8—42no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—44no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—82no—
345C2×C8—81no—
346C2×C8—81no—
347C2×C8—81no—
348C2×C8—81no—
349C2×C8—81no—
350C2×C8—81no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—24no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—84no—
381C4⋊C4—84no—
382C4⋊C4#9514yes—
383C22⋊C4—24no—
384C22⋊C4—24no—
385C22⋊C4—42no—
386C22⋊C4—42no—
387C42—24no—
388C42—24no—
389C42—42no—
390C42—42no—
391C42—42no—
392C42—42no—
393C42—82no—
394C42—84no—
395C42—84no—
396C42C2×C22⋊C411yescharacteristic
397C42C2×C22⋊C411yescharacteristic
398C42C2×C4⋊C411yescharacteristic
399C42C2×C4⋊C411yescharacteristic
400C42C4×D411yescharacteristic
401C42C4×D411yescharacteristic
402C42C4×D411yescharacteristic
403C42C4×D411yescharacteristic
404C42#12111yescharacteristic
405C42#12111yescharacteristic
406C42C22⋊Q811yescharacteristic
407C42C22⋊Q811yescharacteristic
408C42#12311yescharacteristic
409C42#12311yescharacteristic
410C42#12311yescharacteristic
411C42#12311yescharacteristic
412C23—21no—
413C23—21no—
414C23—21no—
415C23—21no—
416C23C4×C22⋊C411yescharacteristic
417C23#37911yescharacteristic
418C23#38011yescharacteristic
419C23C2×#10311yescharacteristic
420C23#41611yescharacteristic
421C23#41711yescharacteristic
422C23#41911yescharacteristic
423Q8—44no—
424Q8—42no—
425Q8—42no—
426Q8—42no—
427Q8—42no—
428Q8—82no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4C2×#9511yescharacteristic
470C2×C4C4×C22⋊C411yescharacteristic
471C2×C4#38011yescharacteristic
472C2×C4#38111yescharacteristic
473C2×C4#38511yescharacteristic
474C2×C4#38511yescharacteristic
475C2×C4#38511yescharacteristic
476C2×C4#38511yescharacteristic
477C2×C4#38611yescharacteristic
478C2×C4#38611yescharacteristic
479C2×C4#38611yescharacteristic
480C2×C4#38611yescharacteristic
481C2×C4C4×SD1611yescharacteristic
482C2×C4C4×SD1611yescharacteristic
483C2×C4C4×Q1611yescharacteristic
484C2×C4C4×Q1611yescharacteristic
485C2×C4#44011yescharacteristic
486C2×C4#44011yescharacteristic
487C2×C4#44111yescharacteristic
488C2×C4#44111yescharacteristic
489C2×C4#44811yescharacteristic
490C2×C4#44811yescharacteristic
491C2×C4#44911yescharacteristic
492C2×C4#44911yescharacteristic
493C2×C4#45111yescharacteristic
494C2×C4#45111yescharacteristic
495C2×C4#45211yescharacteristic
496C2×C4#45211yescharacteristic
497C2×C4#45311yescharacteristic
498C2×C4#45411yescharacteristic
499C2×C4#45511yescharacteristic
500C2×C4#45611yescharacteristic
501C2×C4#46011yescharacteristic
502C2×C4#46211yescharacteristic
503C2×C4#46411yescharacteristic
504C2×C4#46411yescharacteristic
505C2×C4#47511yescharacteristic
506C2×C4#47711yescharacteristic
507C2×C4#47911yescharacteristic
508C2×C4#47911yescharacteristic
509C8—44no—
510C8—82no—
511C8—82no—
512C8—82no—
513C22—21no—
514C22—21no—
515C22—21no—
516C22—21no—
517C22—21no—
518C22—21no—
519C22—21no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22C22⋊#9511yescharacteristic
526C22C4×#10311yescharacteristic
527C22#176311yescharacteristic
528C22#178711yescharacteristic
529C22#178811yescharacteristic
530C22#179111yescharacteristic
531C22#179111yescharacteristic
532C22#179211yescharacteristic
533C22#179311yescharacteristic
534C22#180311yescharacteristic
535C22#180511yescharacteristic
536C4—21no—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—44no—
543C4—44no—
544C4—44no—
545C4—82no—
546C4—82no—
547C4—82no—
548C4#185911yescharacteristic
549C4#186011yescharacteristic
550C4#186311yescharacteristic
551C4#186311yescharacteristic
552C4#186511yescharacteristic
553C4#186511yescharacteristic
554C4#187811yescharacteristic
555C4#187811yescharacteristic
556C4#188011yescharacteristic
557C4#188011yescharacteristic
558C4#188211yescharacteristic
559C4#188311yescharacteristic
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2#835511yescharacteristic
565C2#835611yescharacteristic
566C2#835711yescharacteristic
567C2#835811yescharacteristic
568C2#1074711yescharacteristic
569C2#1078111yescharacteristic
570C2#1078911yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
29. #1550C4semidirect
33. #1493C4semidirect

Maximal subgroups

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

subgroupindex
#342132
C2×#15502
C2×#14932
#108472
#108342
#80452
#78972