Finite Group Databasebeta

← #226692

Subgroups — #226692

Order 512.

629 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22836C211yesmaximal, characteristic, complemented
2#22700C211yesmaximal, characteristic, complemented
3#21113C211yesmaximal, characteristic
4#9558C211yesmaximal, characteristic, complemented
5#9554C211yesmaximal, characteristic, complemented
6#9540C211yesmaximal, characteristic, complemented
7#9536C211yesmaximal, characteristic, complemented
8#8951C211yesmaximal, characteristic
9#8951C211yesmaximal, characteristic
10#8788C211yesmaximal, characteristic, complemented
11#8569C211yesmaximal, characteristic, complemented
12#8565C211yesmaximal, characteristic, complemented
13#8264C211yesmaximal, characteristic, complemented
14#8210C211yesmaximal, characteristic, complemented
15#8206C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×#436C2211yescharacteristic
18C2×#436C2211yescharacteristic
19#2293C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#1954C2211yescharacteristic
22#1954C2211yescharacteristic
23#1939C2211yescharacteristic
24#1939C2211yescharacteristic
25#1925C411yescharacteristic
26#1925C411yescharacteristic
27#1924—21no—
28#1923—21no—
29#1918C411yescharacteristic
30#1918C411yescharacteristic
31#1879C411yescharacteristic
32#1879C411yescharacteristic
33#1862C411yescharacteristic
34#1862C411yescharacteristic
35#1851—21no—
36#1851—21no—
37#1831C2211yescharacteristic
38#1831C2211yescharacteristic
39#1818—21no—
40#1797—21no—
41C4×#107C2211yescharacteristic
42#1770C2211yescharacteristic
43C8×C4⋊C4C2211yescharacteristic
44C4×C4⋊C8C2211yescharacteristic
45C4×C4⋊C8C2211yescharacteristic
46C4×#102C2211yescharacteristic
47C4×#102C2211yescharacteristic
48#1750—21no—
49#1744C2211yescharacteristic
50#1728—21no—
51C2×D4⋊C8C2211yescharacteristic
52C2×D4⋊C8C2211yescharacteristic
53C2×D4⋊C8C2211yescharacteristic
54C2×D4⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1277C2211yescharacteristic
62#1277C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×D8C2×C411yescharacteristic
68C22×M16—21no—
69C22×M16—21no—
70C2×C4×D4C2×C411yescharacteristic
71C2×C4×D4C2×C411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75C22×C42—21no—
76C4×D8—22no—
77C4×D8—22no—
78#436—22no—
79#436—22no—
80#436—41no—
81#436—41no—
82C2×#107C2×C411yescharacteristic
83C2×#107C2×C411yescharacteristic
84C2×#107C2×C411yescharacteristic
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×#102C2×C411yescharacteristic
95C2×#102C2×C411yescharacteristic
96C2×#102C2×C411yescharacteristic
97C2×#102C2×C411yescharacteristic
98C2×#102C2×C411yescharacteristic
99C2×#102C2×C411yescharacteristic
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#390C2×C411yescharacteristic
115#390C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#389C2×C411yescharacteristic
118#388—21no—
119#388—21no—
120C4×C4⋊C4C2×C411yescharacteristic
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C43C2311yescharacteristic
129#340C2×C411yescharacteristic
130#340C2×C411yescharacteristic
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336C2×C411yescharacteristic
145#336C2×C411yescharacteristic
146#336C2×C411yescharacteristic
147#336C2×C411yescharacteristic
148#336D411yescharacteristic
149#336D411yescharacteristic
150D4⋊C8—22no—
151D4⋊C8—22no—
152D4⋊C8—41no—
153D4⋊C8—41no—
154#322—41no—
155C82—41no—
156C22×D4C22×C411yescharacteristic
157C22×D4C22×C411yescharacteristic
158C23×C4—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C2×D8—22no—
163C2×D8—41no—
164C2×D8—41no—
165C2×M16—42no—
166C2×M16—42no—
167C2×M16—42no—
168C2×M16—42no—
169C2×M16—81no—
170C2×M16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8C22×C411yescharacteristic
178C22×C8C22×C411yescharacteristic
179C22×C8C22×C411yescharacteristic
180C22×C8C22×C411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C4×D4—22no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—41no—
193C4×D4—41no—
194C4×D4—41no—
195C4×D4#4112yes—
196C4×D4#4112yes—
197#117—81no—
198#117—81no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4C22×C411yescharacteristic
202C2×C4⋊C4C22×C411yescharacteristic
203C2×C4⋊C4C22×C411yescharacteristic
204C2×C4⋊C4C22×C411yescharacteristic
205C2×C4⋊C4C22×C411yescharacteristic
206C2×C4⋊C4C22×C411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C42—21no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—81no—
220C2×C42—81no—
221C2×C42C22×C411yescharacteristic
222C2×C42C22×C411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42C2411yescharacteristic
228#107—22no—
229#107—22no—
230#107—41no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—42no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244#102—22no—
245#102—22no—
246#102—22no—
247#102—41no—
248#102—41no—
249#102—41no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254C22⋊C8—81no—
255C22⋊C8—81no—
256#97—81no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—81no—
270#95—21no—
271#95—21no—
272#95—21no—
273#95—21no—
274#95—21no—
275#95—21no—
276C24—21no—
277C24—21no—
278C2×D4—21no—
279C2×D4—21no—
280C2×D4—21no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4#11712yes—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4C4×D411yescharacteristic
315C22×C4C4×D411yescharacteristic
316C22×C4C4×D411yescharacteristic
317C22×C4C4×D411yescharacteristic
318C22×C4C23×C411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C2×#4111yescharacteristic
321D8—44no—
322M16—84no—
323M16—84no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—22no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
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#13512yes—
365C2×C8#13512yes—
366C4⋊C4—21no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—81no—
377C4⋊C4—81no—
378C4⋊C4#11712yes—
379C4⋊C4#13112yes—
380C22⋊C4—22no—
381C22⋊C4—22no—
382C22⋊C4—42no—
383C22⋊C4—42no—
384C22⋊C4—42no—
385C22⋊C4—41no—
386C22⋊C4—41no—
387C22⋊C4—81no—
388C22⋊C4—81no—
389C22⋊C4—81no—
390C42—21no—
391C42—21no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C42—41no—
404C42—81no—
405C42—81no—
406C42C2×#4111yescharacteristic
407C42C2×#4111yescharacteristic
408C42H(5,𝔽2)11yescharacteristic
409C422−1+411yescharacteristic
410C23—22no—
411C23—41no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23C2×C4×D411yescharacteristic
425D4—22no—
426D4—22no—
427D4—41no—
428D4—41no—
429D4—41no—
430D4—41no—
431D4—41no—
432D4—41no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
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—84no—
470C2×C4—84no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4C2×#11711yescharacteristic
476C2×C4#52011yescharacteristic
477C2×C4#54611yescharacteristic
478C2×C4#54811yescharacteristic
479C2×C4#55211yescharacteristic
480C2×C4#55311yescharacteristic
481C2×C4C2×#13111yescharacteristic
482C2×C4#56811yescharacteristic
483C2×C4#56811yescharacteristic
484C2×C4#56811yescharacteristic
485C2×C4C2×#13511yescharacteristic
486C2×C4C2×#13511yescharacteristic
487C2×C4#57611yescharacteristic
488C2×C4#57611yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—41no—
493C8—41no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—41no—
498C8—81no—
499C8—81no—
500C8—81no—
501C8—81no—
502C8—81no—
503C22—42no—
504C22—42no—
505C22—42no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—81no—
512C22—81no—
513C22—81no—
514C22—81no—
515C22—81no—
516C22—81no—
517C22#235211yescharacteristic
518C22#293011yescharacteristic
519C22#293311yescharacteristic
520C22C4×#13511yescharacteristic
521C22#294811yescharacteristic
522C22#295511yescharacteristic
523C22#295611yescharacteristic
524C4—21no—
525C4—21no—
526C4—41no—
527C4—41no—
528C4—41no—
529C4—41no—
530C4—42no—
531C4—42no—
532C4—81no—
533C4—81no—
534C4—81no—
535C4—81no—
536C4#292111yescharacteristic
537C4#292511yescharacteristic
538C4#299311yescharacteristic
539C4#299511yescharacteristic
540C4#329711yescharacteristic
541C4#329711yescharacteristic
542C4#336111yescharacteristic
543C4#336111yescharacteristic
544C2—82no—
545C2—161no—
546C2#1793211yescharacteristic
547C2#1795011yescharacteristic
548C2#1799911yescharacteristic
549C2#1802611yescharacteristic
550C2#2275511yescharacteristic
551C2#2277211yescharacteristic
552C2#2278411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC22×D8C2×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
1. #22836C2semidirect
2. #22700C2semidirect
4. #9558C2semidirect
5. #9554C2semidirect
6. #9540C2semidirect
7. #9536C2semidirect
10. #8788C2semidirect
11. #8569C2semidirect
12. #8565C2semidirect
13. #8264C2semidirect
14. #8210C2semidirect
15. #8206C2semidirect

Maximal subgroups

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

subgroupindex
#228362
#227002
#211132
#95582
#95542
#95402
#95362
#89512
#87882
#85692
#85652
#82642
#82102
#82062