Finite Group Databasebeta

← #189167

Subgroups — #189167

Order 512.

860 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2986C211yesmaximal, characteristic
2#21984C211yesmaximal, characteristic
3#21885C211yesmaximal, characteristic
4#21870C211yesmaximal, characteristic
5C2×#1573C211yesmaximal, characteristic
6#8657C211yesmaximal, characteristic, complemented
7#8655C211yesmaximal, characteristic, complemented
8#8653C211yesmaximal, characteristic, complemented
9#8651C211yesmaximal, characteristic, complemented
10#8649C211yesmaximal, characteristic, complemented
11#8647C211yesmaximal, characteristic, complemented
12#8645C211yesmaximal, characteristic, complemented
13#8643C211yesmaximal, characteristic, complemented
14#8046C211yesmaximal, characteristic
15#8039C211yesmaximal, characteristic
16C2×#533C2211yescharacteristic
17#2986—24no—
18#2986C418yescomplemented
19C2×#446—21no—
20C2×#446C2211yescharacteristic
21C2×#446C2211yescharacteristic
22C2×C8×Q8—21no—
23C2×C8×Q8C2211yescharacteristic
24C2×C8×Q8C2211yescharacteristic
25C2×#433—21no—
26C2×#433—21no—
27C2×#432C2211yescharacteristic
28C2×#431C2211yescharacteristic
29C2×#431C2211yescharacteristic
30C22×C4⋊C8C2211yescharacteristic
31#2285C2211yescharacteristic
32#2285C2211yescharacteristic
33#1993—21no—
34#1987—21no—
35#1981—21no—
36#1960—21no—
37#1949—21no—
38#1949—21no—
39#1892—21no—
40#1892—21no—
41#1873—21no—
42#1873—21no—
43#1854—21no—
44#1854—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1803C2211yescharacteristic
54#1803C2211yescharacteristic
55#1803C2211yescharacteristic
56#1803C2211yescharacteristic
57#1573—24no—
58C2×#335C2211yescharacteristic
59C2×#335C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#334C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70C23×C8—21no—
71#533—22no—
72#533—22no—
73#533C2×C414yes—
74#533Q814yes—
75C2×C4⋊Q8D411yescharacteristic
76C2×C4⋊Q8D411yescharacteristic
77C2×#125—21no—
78C2×C22⋊Q8—21no—
79C2×C22⋊Q8D411yescharacteristic
80C2×C22⋊Q8D411yescharacteristic
81C2×C4×Q8—21no—
82C2×C4×Q8C2311yescharacteristic
83C2×C4×Q8C2311yescharacteristic
84C2×#117—21no—
85C2×#117—21no—
86C2×#117—21no—
87C22×C42C2311yescharacteristic
88#446—28no—
89#446—22no—
90#446—22no—
91#446C2×C414yes—
92#446C2×C414yes—
93C8×Q8—28no—
94C8×Q8—22no—
95C8×Q8—22no—
96C8×Q8C2×C414yes—
97C8×Q8C2×C414yes—
98#433—28no—
99#433—28no—
100#432—22no—
101#432—22no—
102#431—22no—
103#431C2×C414yes—
104#431C2×C414yes—
105#431D414yes—
106C2×#107—21no—
107C2×#107—21no—
108C2×#106—21no—
109C2×#106—21no—
110C2×C4⋊C8—21no—
111C2×C4⋊C8—21no—
112C2×C4⋊C8—21no—
113C2×C4⋊C8—21no—
114C2×C4⋊C8—22no—
115C2×C4⋊C8—22no—
116C2×C4⋊C8—21no—
117C2×C4⋊C8D411yescharacteristic
118C2×C4⋊C8D411yescharacteristic
119C2×C4⋊C8C2311yescharacteristic
120C2×C4⋊C8C2311yescharacteristic
121C2×C4⋊C8C2311yescharacteristic
122C2×C4⋊C8C2311yescharacteristic
123C2×C4⋊C8C2311yescharacteristic
124C2×C4⋊C8C2311yescharacteristic
125C2×#103—21no—
126C2×#103—21no—
127C2×#103—21no—
128C2×#103—21no—
129C2×#103—21no—
130C2×#103—21no—
131C2×#103—21no—
132C2×#103—21no—
133C2×C22⋊C8—21no—
134C2×C22⋊C8—21no—
135C2×C22⋊C8D411yescharacteristic
136C2×C22⋊C8D411yescharacteristic
137C2×#97—21no—
138C2×#97—21no—
139C2×#97—21no—
140C2×C4×C8—21no—
141C2×C4×C8—21no—
142C2×C4×C8—21no—
143C2×C4×C8C2311yescharacteristic
144C2×C4×C8C2311yescharacteristic
145#384C2311yescharacteristic
146#384C2311yescharacteristic
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#382—21no—
150#382—21no—
151#380—21no—
152#380—21no—
153#380—21no—
154#380—21no—
155C4×C4⋊C4—21no—
156C4×C4⋊C4—21no—
157C4×C4⋊C4—21no—
158C4×C4⋊C4—21no—
159#340—21no—
160#340—21no—
161#340—21no—
162#340—21no—
163#340—21no—
164#340—21no—
165#340—21no—
166#340—21no—
167#340—21no—
168#340—21no—
169#340—21no—
170#340—21no—
171#335—22no—
172#335—22no—
173#334—22no—
174#334—22no—
175C22×Q8—21no—
176C22×Q8C2×D411yescharacteristic
177C22×Q8C2×D411yescharacteristic
178C23×C4C2×D411yescharacteristic
179C23×C4C2×D411yescharacteristic
180C23×C4#4111yescharacteristic
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—42no—
188C22×C8—81no—
189C22×C8—81no—
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C4⋊Q8—22no—
195C4⋊Q8—22no—
196C4⋊Q8C4⋊C414yes—
197C4⋊Q8C4⋊C414yes—
198#125—24no—
199#125—24no—
200C22⋊Q8—24no—
201C22⋊Q8—22no—
202C22⋊Q8—22no—
203C22⋊Q8—24no—
204C22⋊Q8C4⋊C414yes—
205C22⋊Q8C4⋊C414yes—
206C4×Q8—24no—
207C4×Q8—24no—
208C4×Q8—21no—
209C4×Q8—21no—
210C4×Q8—21no—
211C4×Q8—21no—
212C4×Q8C22×C412yes—
213C4×Q8C22×C412yes—
214C4×Q8C2×Q812yes—
215C4×Q8C2×Q812yes—
216#117—24no—
217#117—24no—
218#117—81no—
219#117—81no—
220#117—81no—
221#117—81no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×D411yescharacteristic
242C2×C4⋊C4C2×D411yescharacteristic
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—41no—
247C2×C42—21no—
248C2×C42—21no—
249C2×C42—41no—
250C2×C42—41no—
251C2×C42C22×C412yes—
252C2×C42C2×D411yescharacteristic
253C2×C42C2×D412yes—
254C2×C42C2×D411yescharacteristic
255C2×C42C2×Q812yes—
256C2×C42#4111yescharacteristic
257C2×C42#4112yes—
258C2×C42C2411yescharacteristic
259#107—81no—
260#107—81no—
261#106—81no—
262#106—81no—
263C4⋊C8—24no—
264C4⋊C8—24no—
265C4⋊C8—24no—
266C4⋊C8—24no—
267C4⋊C8—21no—
268C4⋊C8—21no—
269C4⋊C8—21no—
270C4⋊C8—21no—
271C4⋊C8—21no—
272C4⋊C8—21no—
273C4⋊C8—21no—
274C4⋊C8—21no—
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—42no—
278C4⋊C8C22⋊C414yes—
279C4⋊C8C22⋊C414yes—
280C4⋊C8C22×C412yes—
281C4⋊C8C22×C412yes—
282#103—81no—
283#103—81no—
284#103—81no—
285#103—81no—
286#103—81no—
287#103—81no—
288#103—81no—
289#103—81no—
290C22⋊C8—24no—
291C22⋊C8—42no—
292C22⋊C8C22⋊C414yes—
293C22⋊C8C22⋊C414yes—
294#97—24no—
295#97—24no—
296#97—42no—
297C4×C8—24no—
298C4×C8—24no—
299C4×C8—24no—
300C4×C8C22×C412yes—
301C4×C8C22×C412yes—
302C4×C8C2×D412yes—
303C4×C8C2×D412yes—
304#95—41no—
305#95—41no—
306#95—41no—
307#95—41no—
308C24#12111yescharacteristic
309C2×Q8—21no—
310C2×Q8—21no—
311C2×Q8—24no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—42no—
321C2×Q8—42no—
322C2×Q8C2×C4⋊C412yes—
323C2×Q8C2×C4⋊C412yes—
324C22×C4—21no—
325C22×C4—21no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4C2×C22⋊C412yes—
343C22×C4C2×C4⋊C412yes—
344C22×C4#11712yes—
345C22×C4#12112yes—
346C22×C4#12111yescharacteristic
347C22×C4#12111yescharacteristic
348C22×C4#12111yescharacteristic
349C22×C4C22⋊Q812yes—
350C22×C4C22⋊Q812yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C22×D411yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—84no—
381C2×C8C2×C22⋊C412yes—
382C2×C8C2×C22⋊C412yes—
383C2×C8C2×C22⋊C412yes—
384C2×C8C2×C22⋊C412yes—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—24no—
392C4⋊C4—24no—
393C4⋊C4—24no—
394C4⋊C4—24no—
395C4⋊C4—24no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—81no—
413C4⋊C4—81no—
414C4⋊C4—81no—
415C4⋊C4—81no—
416C4⋊C4C2×C4⋊C412yes—
417C4⋊C4C2×C4⋊C412yes—
418C22⋊C4—24no—
419C22⋊C4—24no—
420C22⋊C4—82no—
421C22⋊C4—82no—
422C42—22no—
423C42—22no—
424C42—22no—
425C42—22no—
426C42—82no—
427C42—82no—
428C42C2×C22⋊C412yes—
429C42C2×C4⋊C412yes—
430C42#11712yes—
431C42#12112yes—
432C42C22⋊Q812yes—
433C42C22⋊Q812yes—
434C42C23×C411yescharacteristic
435C42C22×D411yescharacteristic
436C42C22×Q811yescharacteristic
437C42C2×#4111yescharacteristic
438C23—21no—
439C23—21no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23#38012yes—
445C23C2×#12111yescharacteristic
446Q8—22no—
447Q8—22no—
448Q8—22no—
449Q8—22no—
450Q8—48no—
451Q8—41no—
452Q8—41no—
453Q8—41no—
454Q8—41no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—21no—
463C2×C4—22no—
464C2×C4—22no—
465C2×C4—22no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—42no—
476C2×C4—42no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—81no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4#38012yes—
498C2×C4#38012yes—
499C2×C4#38012yes—
500C2×C4C22×C22⋊C411yescharacteristic
501C2×C4C22×C4⋊C411yescharacteristic
502C2×C4C2×#11711yescharacteristic
503C2×C4C2×#12111yescharacteristic
504C2×C4C2×C22⋊Q811yescharacteristic
505C2×C4C2×C22⋊Q811yescharacteristic
506C2×C4#57611yescharacteristic
507C2×C4#57711yescharacteristic
508C2×C4#57711yescharacteristic
509C2×C4#57711yescharacteristic
510C2×C4#57711yescharacteristic
511C2×C4#57811yescharacteristic
512C2×C4#57811yescharacteristic
513C2×C4#57811yescharacteristic
514C8—22no—
515C8—22no—
516C8—22no—
517C8—22no—
518C8—42no—
519C8—42no—
520C8—81no—
521C8—81no—
522C8—81no—
523C8—81no—
524C22—22no—
525C22—22no—
526C22—22no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22C2×#38011yescharacteristic
532C22#304311yescharacteristic
533C22#304411yescharacteristic
534C22#306211yescharacteristic
535C22#306311yescharacteristic
536C22#306711yescharacteristic
537C22#306711yescharacteristic
538C4—22no—
539C4—22no—
540C4—21no—
541C4—21no—
542C4—22no—
543C4—42no—
544C4—42no—
545C4—42no—
546C4—42no—
547C4—161no—
548C4—161no—
549C4—161no—
550C4—161no—
551C4#289811yescharacteristic
552C4#289811yescharacteristic
553C4#291211yescharacteristic
554C4#291311yescharacteristic
555C4#306211yescharacteristic
556C4#306311yescharacteristic
557C4#308211yescharacteristic
558C4#308211yescharacteristic
559C2—42no—
560C2#1956811yescharacteristic
561C2#1956911yescharacteristic
562C2#1957411yescharacteristic
563C2#1957511yescharacteristic
564C2#2065111yescharacteristic
565C2#2066411yescharacteristic
566C2#2067811yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #8657C2semidirect
7. #8655C2semidirect
8. #8653C2semidirect
9. #8651C2semidirect
10. #8649C2semidirect
11. #8647C2semidirect
12. #8645C2semidirect
13. #8643C2semidirect
18. #2986C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29862
#219842
#218852
#218702
C2×#15732
#86572
#86552
#86532
#86512
#86492
#86472
#86452
#86432
#80462
#80392