Finite Group Databasebeta

← #229656

Subgroups — #229656

Order 512.

702 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23205C211yesmaximal, characteristic
2Q8×C4⋊C8C211yesmaximal, characteristic, complemented
3C4×#464C211yesmaximal, characteristic
4C2×#1588C211yesmaximal, characteristic
5C2×C8×Q16C211yesmaximal, characteristic, complemented
6#9646C211yesmaximal, characteristic
7#9604C211yesmaximal, characteristic, complemented
8#9559C211yesmaximal, characteristic, complemented
9#9554C211yesmaximal, characteristic
10#8948C211yesmaximal, characteristic
11#8946C211yesmaximal, characteristic, complemented
12#8847C211yesmaximal, characteristic
13#8844C211yesmaximal, characteristic, complemented
14#8262C211yesmaximal, characteristic, complemented
15#8173C211yesmaximal, characteristic, complemented
16C2×#464C411yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×C8×Q8C2211yescharacteristic
19C2×C8×Q8C2211yescharacteristic
20C2×C4×Q16C2211yescharacteristic
21C2×C4×SD16C2211yescharacteristic
22C2×#436C2211yescharacteristic
23C4×#124C2211yescharacteristic
24C42×Q8C2211yescharacteristic
25#1984C2211yescharacteristic
26#1983C411yescharacteristic
27#1954C2211yescharacteristic
28#1949C411yescharacteristic
29#1939—21no—
30#1939—21no—
31#1939C2211yescharacteristic
32#1939C2211yescharacteristic
33#1927C411yescharacteristic
34#1925C411yescharacteristic
35#1924—21no—
36#1923—21no—
37#1880C411yescharacteristic
38#1878C411yescharacteristic
39#1851—21no—
40#1842—21no—
41#1842C2211yescharacteristic
42#1805C411yescharacteristic
43#1767C2211yescharacteristic
44C4×C4⋊C8—21no—
45C4×C4⋊C8C2211yescharacteristic
46C4×C4⋊C8C2211yescharacteristic
47C4×#103C2211yescharacteristic
48C4×#102C2211yescharacteristic
49#1753—21no—
50C4×#97C2211yescharacteristic
51#1588—24no—
52C8×Q16—24no—
53C2×#335C2211yescharacteristic
54C2×#335C2211yescharacteristic
55C2×Q8⋊C8C2211yescharacteristic
56C2×Q8⋊C8C2211yescharacteristic
57C2×Q8⋊C8C2211yescharacteristic
58C2×Q8⋊C8C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62C2×C82C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67#1273C2211yescharacteristic
68#1273C2211yescharacteristic
69C22×Q16C2×C411yescharacteristic
70C22×SD16C2×C411yescharacteristic
71C22×M16—21no—
72C2×#124C2×C411yescharacteristic
73C2×C4×Q8C2×C411yescharacteristic
74C2×C4×Q8C2×C411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C2×C4×Q8C2311yescharacteristic
77C2×C4×D4C2311yescharacteristic
78#464—24no—
79C8×Q8—22no—
80C8×Q8—22no—
81C8×Q8—22no—
82C8×Q8—41no—
83C8×Q8—41no—
84C8×Q8—41no—
85C4×Q16—22no—
86C4×Q16—22no—
87C4×SD16—22no—
88C4×SD16—22no—
89#436—22no—
90#436—41no—
91C2×#107C2×C411yescharacteristic
92C2×#106C2×C411yescharacteristic
93C2×C4⋊C8—21no—
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⋊C8C2×C411yescharacteristic
101C2×C4⋊C8C2×C411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8D411yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103C2×C411yescharacteristic
109C2×#103C2×C411yescharacteristic
110C2×#103C2×C411yescharacteristic
111C2×#103C2×C411yescharacteristic
112C2×#103C2×C411yescharacteristic
113C2×#102C2×C411yescharacteristic
114C2×#102C2×C411yescharacteristic
115C2×#102C2×C411yescharacteristic
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×#97—21no—
122C2×C4×C8—21no—
123C2×C4×C8—21no—
124C2×C4×C8—21no—
125C2×C4×C8C2311yescharacteristic
126C2×C4×C8C2311yescharacteristic
127C2×C4×C8C2311yescharacteristic
128C2×C4×C8C2311yescharacteristic
129C2×C4×C8C2311yescharacteristic
130C2×C4×C8C2311yescharacteristic
131#391C2×C411yescharacteristic
132#390C2×C411yescharacteristic
133#388—21no—
134#388—21no—
135#382C2×C411yescharacteristic
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2×C411yescharacteristic
140C4×C4⋊C4C2×C411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143C4×C22⋊C4—21no—
144C43—21no—
145C43C2311yescharacteristic
146#340C2×C411yescharacteristic
147#340C2×C411yescharacteristic
148#340C2×C411yescharacteristic
149#340C2×C411yescharacteristic
150#336—21no—
151#336—21no—
152#336—21no—
153#336—21no—
154#336—21no—
155#336C2×C411yescharacteristic
156#336C2×C411yescharacteristic
157#336D411yescharacteristic
158#336D411yescharacteristic
159#335—22no—
160#335—22no—
161Q8⋊C8—22no—
162Q8⋊C8—22no—
163Q8⋊C8—41no—
164Q8⋊C8D414yes—
165D4⋊C8—22no—
166D4⋊C8—41no—
167#322—22no—
168C82—22no—
169C22×Q8C22×C411yescharacteristic
170C22×Q8C22×C411yescharacteristic
171C22×D4C22×C411yescharacteristic
172C23×C4—21no—
173C2×Q16—22no—
174C2×Q16—22no—
175C2×Q16M1614yescomplemented
176C2×SD16—22no—
177C2×SD16—22no—
178C2×SD16—22no—
179C2×M16—42no—
180C2×M16—42no—
181C2×M16—81no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—21no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8C22×C411yescharacteristic
189C22×C8C22×C411yescharacteristic
190C22×C8C22×C411yescharacteristic
191C22×C8C22×C411yescharacteristic
192C22×C8C2×D411yescharacteristic
193C22×C8C2×D411yescharacteristic
194C22×C8C2×D411yescharacteristic
195C22×C8C2×D411yescharacteristic
196#124—22no—
197#124—41no—
198C4×Q8—22no—
199C4×Q8—22no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—22no—
203C4×Q8—21no—
204C4×Q8—21no—
205C4×Q8—21no—
206C4×Q8—21no—
207C4×Q8—41no—
208C4×Q8C2×D412yes—
209C4×Q8#4112yes—
210C4×D4—21no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4C22×C411yescharacteristic
219C2×C4⋊C4C22×C411yescharacteristic
220C2×C4⋊C4C22×C411yescharacteristic
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—21no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42C22×C411yescharacteristic
236C2×C42C22×C411yescharacteristic
237C2×C42C2×D411yescharacteristic
238C2×C42C2×D411yescharacteristic
239C2×C42#4111yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42C2411yescharacteristic
242#107—22no—
243#106—22no—
244C4⋊C8—22no—
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—41no—
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⋊C8C2×D412yes—
263#103—22no—
264#103—22no—
265#103—22no—
266#103—22no—
267#103—22no—
268#102—22no—
269#102—22no—
270#102—22no—
271C22⋊C8—42no—
272C22⋊C8—42no—
273C22⋊C8—81no—
274#97—24no—
275#97—42no—
276#97—81no—
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—21no—
287C4×C8—21no—
288C4×C8—42no—
289C4×C8—42no—
290C4×C8—42no—
291C4×C8C2×D412yes—
292#95—21no—
293#95—21no—
294#95—21no—
295#95—21no—
296#95—21no—
297#95—21no—
298#95—21no—
299C24—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—21no—
308C2×Q8C4×D412yes—
309C2×Q8C2×M1612yes—
310C2×Q8C2×M1612yes—
311C2×Q8#13112yes—
312C2×D4—21no—
313C2×D4—21no—
314C2×D4—21no—
315C2×D4—21no—
316C2×D4—21no—
317C2×D4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4C4×D411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4C4×D411yescharacteristic
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338Q16—28no—
339SD16—44no—
340M16—84no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—22no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—22no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—22no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
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—41no—
381C2×C8C2×M1612yes—
382C2×C8#13712yes—
383C4⋊C4—24no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—21no—
390C4⋊C4—21no—
391C4⋊C4—21no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—42no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4C4×D412yes—
398C4⋊C4#13112yes—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C22⋊C4—42no—
404C22⋊C4—81no—
405C42—24no—
406C42—22no—
407C42—22no—
408C42—21no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—21no—
416C42—22no—
417C42—41no—
418C42—42no—
419C42—41no—
420C42—41no—
421C42—41no—
422C42—41no—
423C42—42no—
424C42—81no—
425C42C22×D411yescharacteristic
426C42C2×#4111yescharacteristic
427C42H(5,𝔽2)11yescharacteristic
428C422−1+411yescharacteristic
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—41no—
435C23—41no—
436C23C2×C4×D411yescharacteristic
437Q8—22no—
438Q8—22no—
439Q8—22no—
440Q8—22no—
441Q8—22no—
442Q8—41no—
443Q8—41no—
444Q8#43514yes—
445D4—41no—
446D4—41no—
447D4—41no—
448D4—41no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—22no—
452C2×C4—22no—
453C2×C4—21no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—22no—
457C2×C4—21no—
458C2×C4—22no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—22no—
462C2×C4—21no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—21no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4#44312yes—
488C2×C4C2×C4×D411yescharacteristic
489C2×C4#52011yescharacteristic
490C2×C4D4211yescharacteristic
491C2×C4#54711yescharacteristic
492C2×C4#54811yescharacteristic
493C2×C4#55311yescharacteristic
494C2×C4C22×M1611yescharacteristic
495C2×C4C2×#13111yescharacteristic
496C2×C4C2×#13111yescharacteristic
497C2×C4#56811yescharacteristic
498C2×C4C2×#13711yescharacteristic
499C2×C4#57511yescharacteristic
500C2×C4#57611yescharacteristic
501C2×C4#57811yescharacteristic
502C8—24no—
503C8—22no—
504C8—22no—
505C8—42no—
506C8—41no—
507C8—41no—
508C8—41no—
509C8—41no—
510C8—41no—
511C8—42no—
512C8—41no—
513C8—81no—
514C8—81no—
515C8—81no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—81no—
522C22—81no—
523C22#236111yescharacteristic
524C22#292911yescharacteristic
525C22D4×M1611yescharacteristic
526C22C4×#13711yescharacteristic
527C22#295111yescharacteristic
528C22C2×#44311yescharacteristic
529C22#295611yescharacteristic
530C4—24no—
531C4—21no—
532C4—21no—
533C4—22no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—41no—
538C4—42no—
539C4—42no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4C2×#43511yescharacteristic
544C4#293511yescharacteristic
545C4#299011yescharacteristic
546C4#299211yescharacteristic
547C4#326811yescharacteristic
548C4#327011yescharacteristic
549C4#337511yescharacteristic
550C4#337711yescharacteristic
551C2—161no—
552C2#1791111yescharacteristic
553C2#1792611yescharacteristic
554C2#1804911yescharacteristic
555C2#1805911yescharacteristic
556C2#2298711yescharacteristic
557C2#2300911yescharacteristic
558C2#2301511yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D4C22×C4
LCSG ▷ C22×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
2. Q8×C4⋊C8C2semidirect
5. C2×C8×Q16C2semidirect
7. #9604C2semidirect
8. #9559C2semidirect
11. #8946C2semidirect
13. #8844C2semidirect
14. #8262C2semidirect
15. #8173C2semidirect
175. C2×Q16M16semidirect

Maximal subgroups

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

subgroupindex
#232052
Q8×C4⋊C82
C4×#4642
C2×#15882
C2×C8×Q162
#96462
#96042
#95592
#95542
#89482
#89462
#88472
#88442
#82622
#81732