Finite Group Databasebeta

← #225438

Subgroups — #225438

Order 512.

781 conjugacy classes of subgroups.

The subgroup lattice has 555 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C4⋊C8C211yesmaximal, characteristic, complemented
2#22696C211yesmaximal, characteristic
3C4×#440C211yesmaximal, characteristic
4C2×SD16⋊C8C211yesmaximal, characteristic
5C2×C8×SD16C211yesmaximal, characteristic
6#9595C211yesmaximal, characteristic, complemented
7#9594C211yesmaximal, characteristic, complemented
8#8846C211yesmaximal, characteristic, complemented
9#8845C211yesmaximal, characteristic
10#8844C211yesmaximal, characteristic, complemented
11#8843C211yesmaximal, characteristic
12#8808C211yesmaximal, characteristic, complemented
13#8566C211yesmaximal, characteristic, complemented
14#8563C211yesmaximal, characteristic
15#8262C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#440C411yescharacteristic
19C2×#440C411yescharacteristic
20C2×C4×SD16C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22C2×C8×D4C2211yescharacteristic
23C42×Q8C2211yescharacteristic
24C42×D4C2211yescharacteristic
25#1939—21no—
26#1939—21no—
27#1939C2211yescharacteristic
28#1939C2211yescharacteristic
29#1939C2211yescharacteristic
30#1939C2211yescharacteristic
31#1850—21no—
32#1850—21no—
33#1842—21no—
34#1842C2211yescharacteristic
35#1831C2211yescharacteristic
36#1797—21no—
37#1776C411yescharacteristic
38#1776C411yescharacteristic
39C4×#107C2211yescharacteristic
40C4×C4⋊C8—21no—
41C4×C4⋊C8C2211yescharacteristic
42C4×C4⋊C8C2211yescharacteristic
43#1763C411yescharacteristic
44#1763C411yescharacteristic
45#1762C411yescharacteristic
46#1762C411yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49C2×#336—21no—
50C4×#97C2211yescharacteristic
51SD16⋊C8—24no—
52C8×SD16—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1279C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×SD16C2×C411yescharacteristic
70C23×C8—21no—
71C23×C8—21no—
72C2×C4×Q8C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2311yescharacteristic
75C2×C4×D4C2×C411yescharacteristic
76C2×C4×D4C2×C411yescharacteristic
77C2×C4×D4C2311yescharacteristic
78C22×C42—21no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8—41no—
82C8×Q8—41no—
83#440—24no—
84#440C818yescomplemented
85C4×SD16D414yes—
86C4×SD16Q814yes—
87C8×D4—22no—
88C8×D4—22no—
89C8×D4—41no—
90C8×D4—41no—
91C2×#107C2×C411yescharacteristic
92C2×#107C2×C411yescharacteristic
93C2×#106C2×C411yescharacteristic
94C2×C4⋊C8—21no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8D411yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#102C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×#97C2×C411yescharacteristic
118C2×#97C2×C411yescharacteristic
119C2×#97C2×C411yescharacteristic
120C2×#97C2×C411yescharacteristic
121C2×C4×C8—21no—
122C2×C4×C8—21no—
123C2×C4×C8C2311yescharacteristic
124C2×C4×C8C2311yescharacteristic
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4C2×C411yescharacteristic
133C4×C4⋊C4C2×C411yescharacteristic
134C4×C4⋊C4C2×C411yescharacteristic
135C4×C4⋊C4C2×C411yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2311yescharacteristic
138C4×C22⋊C4—21no—
139C4×C22⋊C4—21no—
140C4×C22⋊C4—21no—
141C43—21no—
142C43C2311yescharacteristic
143#340C2×C411yescharacteristic
144#340C2×C411yescharacteristic
145#340C2×C411yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#336—21no—
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336—21no—
156#336D411yescharacteristic
157#336D411yescharacteristic
158#335—22no—
159#335—41no—
160#334—22no—
161#334—41no—
162Q8⋊C8—22no—
163Q8⋊C8—22no—
164D4⋊C8—22no—
165D4⋊C8—22no—
166#322—22no—
167C82—22no—
168C22×Q8C22×C411yescharacteristic
169C22×D4C22×C411yescharacteristic
170C23×C4—21no—
171C23×C4—21no—
172C23×C4—21no—
173C2×SD16C4⋊C414yes—
174C2×SD16C2×C814yes—
175C2×SD16M1614yes—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—42no—
183C22×C8—42no—
184C22×C8—42no—
185C22×C8—42no—
186C22×C8—81no—
187C22×C8—81no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196C4×Q8—22no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—22no—
200C4×Q8—41no—
201C4×Q8C2×C814yescomplemented
202C4×Q8C2×D412yes—
203C4×Q8C2×Q812yes—
204C4×D4—22no—
205C4×D4—22no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—41no—
209C4×D4C2×C814yes—
210C4×D4C2×D412yes—
211C4×D4C2×Q812yes—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4C22×C411yescharacteristic
216C2×C4⋊C4C22×C411yescharacteristic
217C2×C4⋊C4C22×C411yescharacteristic
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—81no—
234C2×C42—81no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42C2411yescharacteristic
242#107—22no—
243#107C2×C814yescomplemented
244#106C4⋊C414yes—
245C4⋊C8—22no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—41no—
255C4⋊C8—42no—
256C4⋊C8—42no—
257C4⋊C8—42no—
258C4⋊C8—42no—
259C4⋊C8—81no—
260C4⋊C8—81no—
261C4⋊C8—81no—
262C4⋊C8—81no—
263C4⋊C8—81no—
264#103—22no—
265#103C4⋊C414yes—
266#103C2×C814yescomplemented
267#102—22no—
268#102C4⋊C414yes—
269#102C2×C814yes—
270C22⋊C8—42no—
271C22⋊C8—42no—
272#97—22no—
273#97—22no—
274#97—41no—
275#97C2×C814yescomplemented
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8—21no—
282C4×C8—21no—
283C4×C8—21no—
284C4×C8—21no—
285C4×C8—21no—
286C4×C8—42no—
287C4×C8—42no—
288C4×C8C2×D412yes—
289C4×C8C2×Q812yes—
290#95—21no—
291#95—21no—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296C24—21no—
297C2×Q8—21no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8C2×C4⋊C412yes—
301C2×Q8C22×C812yes—
302C2×Q8C2×M1612yes—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C2×C4⋊C412yes—
307C2×D4C22×C812yes—
308C2×D4C2×M1612yes—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—24no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—81no—
326C22×C4—81no—
327C22×C4—81no—
328C22×C4C4×D411yescharacteristic
329C22×C4C4×D411yescharacteristic
330C22×C4C4×D411yescharacteristic
331C22×C4C4×D411yescharacteristic
332C22×C4C23×C411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C2×#4111yescharacteristic
335SD16C4⋊C8116yescomplemented
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
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—84no—
373C2×C8—84no—
374C2×C8C2×C4⋊C412yes—
375C2×C8C2×C4⋊C412yes—
376C2×C8C22×C812yes—
377C2×C8C22×C812yes—
378C2×C8C2×M1612yes—
379C2×C8#13712yes—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—21no—
385C4⋊C4—21no—
386C4⋊C4—24no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—42no—
392C4⋊C4C2×C4⋊C412yes—
393C4⋊C4C2×C4⋊C412yes—
394C4⋊C4C22×C812yes—
395C4⋊C4C22×C812yes—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—24no—
399C22⋊C4—42no—
400C42—22no—
401C42—22no—
402C42—21no—
403C42—21no—
404C42—21no—
405C42—21no—
406C42—21no—
407C42—21no—
408C42—24no—
409C42—41no—
410C42—41no—
411C42—42no—
412C42—42no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—81no—
418C42—81no—
419C42—81no—
420C42C22×C812yes—
421C42#13112yes—
422C42C22×D411yescharacteristic
423C42C22×Q811yescharacteristic
424C42H(5,𝔽2)11yescharacteristic
425C422−1+411yescharacteristic
426C23—22no—
427C23—22no—
428C23—22no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23C2×C4×D411yescharacteristic
434Q8—22no—
435Q8—22no—
436Q8—22no—
437Q8C2×C4⋊C814yes—
438D4—22no—
439D4—22no—
440D4—22no—
441D4C2×C4⋊C814yes—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—21no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4C8×D412yes—
482C2×C4C8×D412yes—
483C2×C4C22×C4⋊C411yescharacteristic
484C2×C4#52011yescharacteristic
485C2×C4D4211yescharacteristic
486C2×C4#54711yescharacteristic
487C2×C4D4×Q811yescharacteristic
488C2×C4#55411yescharacteristic
489C2×C4C23×C811yescharacteristic
490C2×C4C22×M1611yescharacteristic
491C2×C4C2×#13111yescharacteristic
492C2×C4#56811yescharacteristic
493C2×C4C2×#13711yescharacteristic
494C2×C4#57511yescharacteristic
495C2×C4#57611yescharacteristic
496C2×C4#57711yescharacteristic
497C8—22no—
498C8—22no—
499C8—22no—
500C8—42no—
501C8—41no—
502C8—41no—
503C8—41no—
504C8—41no—
505C8—81no—
506C8—81no—
507C8—81no—
508C8—81no—
509C8—81no—
510C8C2×C4⋊C814yes—
511C22—24no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22—42no—
518C22D4×C4⋊C411yescharacteristic
519C22C2×C8×D411yescharacteristic
520C22D4×M1611yescharacteristic
521C22C4×#13711yescharacteristic
522C22#294611yescharacteristic
523C22#295511yescharacteristic
524C22#295611yescharacteristic
525C4—24no—
526C4—21no—
527C4—21no—
528C4—22no—
529C4—22no—
530C4—41no—
531C4—41no—
532C4—42no—
533C4—42no—
534C4—42no—
535C4—42no—
536C4—81no—
537C4—81no—
538C4C22×C4⋊C811yescharacteristic
539C4#290611yescharacteristic
540C4#298811yescharacteristic
541C4#299011yescharacteristic
542C4#326611yescharacteristic
543C4#326811yescharacteristic
544C4#338511yescharacteristic
545C4#338811yescharacteristic
546C2—44no—
547C2C8×#13711yescharacteristic
548C2#1785211yescharacteristic
549C2#1804711yescharacteristic
550C2#1805211yescharacteristic
551C2D4×C4⋊C811yescharacteristic
552C2#2267811yescharacteristic
553C2#2268711yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC2×C4C23×C8
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. Q8×C4⋊C8C2semidirect
6. #9595C2semidirect
7. #9594C2semidirect
8. #8846C2semidirect
10. #8844C2semidirect
12. #8808C2semidirect
13. #8566C2semidirect
15. #8262C2semidirect
84. #440C8semidirect
201. C4×Q8C2×C8semidirect
243. #107C2×C8semidirect
266. #103C2×C8semidirect
275. #97C2×C8semidirect
335. SD16C4⋊C8semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊C82
#226962
C4×#4402
C2×SD16⋊C82
C2×C8×SD162
#95952
#95942
#88462
#88452
#88442
#88432
#88082
#85662
#85632
#82622