Finite Group Databasebeta

← #225584

Subgroups — #225584

Order 512.

709 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22813C211yesmaximal, characteristic
2#22807C211yesmaximal, characteristic
3#22708C211yesmaximal, characteristic, complemented
4#22658C211yesmaximal, characteristic, complemented
5C2×#1580C211yesmaximal, characteristic
6#9621C211yesmaximal, characteristic, complemented
7#9612C211yesmaximal, characteristic, complemented
8#8893C211yesmaximal, characteristic, complemented
9#8893C211yesmaximal, characteristic, complemented
10#8885C211yesmaximal, characteristic, complemented
11#8885C211yesmaximal, characteristic, complemented
12#8820C211yesmaximal, characteristic, complemented
13#8648C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic, complemented
15#8291C211yesmaximal, characteristic, complemented
16C2×C4×D8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19C2×#427—21no—
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22#2354C2211yescharacteristic
23#2354C2211yescharacteristic
24#1972—21no—
25#1970—21no—
26#1948C2211yescharacteristic
27#1948C2211yescharacteristic
28#1948C2211yescharacteristic
29#1948C2211yescharacteristic
30#1926—21no—
31#1926—21no—
32#1925—21no—
33#1925—21no—
34#1854—21no—
35#1854—21no—
36#1854—21no—
37#1854—21no—
38#1845C2211yescharacteristic
39#1845C2211yescharacteristic
40#1845C2211yescharacteristic
41#1844C2211yescharacteristic
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1808—21no—
45#1807—21no—
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1776—21no—
51C4×#107—21no—
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1735—21no—
56#1580C418yescomplemented
57C2×#335C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1279C2211yescharacteristic
67#1279C2211yescharacteristic
68C22×D8D411yescharacteristic
69C23×C8—21no—
70C23×C8—21no—
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75C2×#117—21no—
76C2×#117—21no—
77C4×D8C2×C414yes—
78C4×D8Q814yes—
79C8×D4—22no—
80C8×D4—22no—
81C8×D4C2×C414yes—
82C8×D4C2×C414yes—
83#427—44no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#107—21no—
87C2×#107—21no—
88C2×#107—41no—
89C2×#107D411yescharacteristic
90C2×#107D411yescharacteristic
91C2×#107D411yescharacteristic
92C2×#106—21no—
93C2×#106—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×#106—21no—
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102D411yescharacteristic
107C2×#102D411yescharacteristic
108C2×C22⋊C8—21no—
109C2×C22⋊C8—21no—
110C2×#97—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390—21no—
116#390—21no—
117#390—21no—
118#390—21no—
119#388—41no—
120#388—41no—
121#388—41no—
122#388—41no—
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#380—21no—
128#380—21no—
129#380—21no—
130#380—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4—21no—
134C4×C4⋊C4—21no—
135C4×C4⋊C4C2311yescharacteristic
136C4×C4⋊C4C2311yescharacteristic
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—41no—
148#340D411yescharacteristic
149#340D411yescharacteristic
150#336—21no—
151#336—21no—
152#335—22no—
153#334C2×C414yes—
154D4⋊C8C2×C414yes—
155D4⋊C8C2×C414yes—
156#322C2×C414yes—
157C22×D4C2×D411yescharacteristic
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C2×D8—22no—
162C2×D8—22no—
163C2×D8C4⋊C414yes—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—24no—
168C22×C8—24no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—42no—
172C22×C8—42no—
173C22×C8—81no—
174C22×C8—81no—
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4C22×C412yes—
184C4×D4C22×C412yes—
185C4×D4C2×Q812yes—
186C4×D4C2×Q812yes—
187#117—42no—
188#117—42no—
189#117—42no—
190#117—42no—
191#117—81no—
192#117—81no—
193#117—81no—
194#117—81no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C4⋊C4C2×D411yescharacteristic
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42—41no—
229C2×C42C2411yescharacteristic
230#107—22no—
231#107—41no—
232#107—42no—
233#107—42no—
234#107—81no—
235#107—81no—
236#107—82no—
237#107C4⋊C414yes—
238#106—42no—
239#106—42no—
240#106—42no—
241#106—81no—
242#106—81no—
243#106—81no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8C22×C412yes—
251C4⋊C8C22×C412yes—
252#102—81no—
253#102—81no—
254#102—81no—
255#102—81no—
256#102C4⋊C414yes—
257#102C4⋊C414yes—
258C22⋊C8—24no—
259C22⋊C8—24no—
260#97—42no—
261C4×C8—21no—
262C4×C8—42no—
263C4×C8C22×C412yes—
264C4×C8C22×C412yes—
265C4×C8C22×C412yes—
266C4×C8C2×Q812yes—
267C4×C8#4112yes—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—21no—
275C24—21no—
276C2×D4—21no—
277C2×D4—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×D4C2×C4⋊C412yes—
287C2×D4C2×C4⋊C412yes—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
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×C4C22×D411yescharacteristic
311C22×C4C22×D411yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313D8—28no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—48no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—84no—
337C2×C8C2×C4⋊C412yes—
338C2×C8C2×C4⋊C412yes—
339C2×C8C4×D412yes—
340C2×C8C4×D412yes—
341C2×C8#13512yes—
342C2×C8#13512yes—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4C2×C4⋊C412yes—
373C4⋊C4C2×C4⋊C412yes—
374C4⋊C4AGL(1,ℤ/8)12yes—
375C22⋊C4—22no—
376C22⋊C4—22no—
377C22⋊C4—22no—
378C22⋊C4—22no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C22⋊C4—161no—
384C22⋊C4—161no—
385C22⋊C4—161no—
386C22⋊C4—161no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—41no—
391C42—82no—
392C42—82no—
393C42—82no—
394C42—82no—
395C42C23×C411yescharacteristic
396C42C22×Q811yescharacteristic
397C42C2×#4111yescharacteristic
398C422−1+411yescharacteristic
399C23—22no—
400C23—22no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23—41no—
405C23—41no—
406C23—41no—
407C23—41no—
408C23—41no—
409C23—41no—
410C23—41no—
411C23—41no—
412C23—41no—
413C23D4211yescharacteristic
414D4—22no—
415D4—22no—
416D4—22no—
417D4—22no—
418D4—41no—
419D4—41no—
420D4—41no—
421D4—41no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4C22×C4⋊C411yescharacteristic
463C2×C4C2×C4×D411yescharacteristic
464C2×C4#52011yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4D4×Q811yescharacteristic
467C2×C4#55411yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×#13511yescharacteristic
470C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
471C2×C4#57511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57611yescharacteristic
474C2×C4#57711yescharacteristic
475C2×C4#57811yescharacteristic
476C8—22no—
477C8—22no—
478C8—22no—
479C8—42no—
480C8—42no—
481C8—42no—
482C8—81no—
483C8—81no—
484C8#42714yescomplemented
485C22—42no—
486C22—42no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—81no—
492C22—81no—
493C22—81no—
494C22—81no—
495C22—81no—
496C22—81no—
497C22—81no—
498C22—81no—
499C22D4×C4⋊C411yescharacteristic
500C22#326411yescharacteristic
501C22#326711yescharacteristic
502C22#328311yescharacteristic
503C22#328711yescharacteristic
504C22#328811yescharacteristic
505C22#328911yescharacteristic
506C4—21no—
507C4—21no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4C2×#42711yescharacteristic
519C4#291311yescharacteristic
520C4C4×AGL(1,ℤ/8)11yescharacteristic
521C4#294611yescharacteristic
522C4#334111yescharacteristic
523C4#334211yescharacteristic
524C4#338411yescharacteristic
525C4#338911yescharacteristic
526C2—82no—
527C2—82no—
528C2#1985911yescharacteristic
529C2#1986211yescharacteristic
530C2#1986711yescharacteristic
531C2#1987011yescharacteristic
532C2#2265811yescharacteristic
533C2#2267611yescharacteristic
534C2#2269311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D8D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #22708C2semidirect
4. #22658C2semidirect
6. #9621C2semidirect
7. #9612C2semidirect
8. #8893C2semidirect
9. #8893C2semidirect
10. #8885C2semidirect
11. #8885C2semidirect
12. #8820C2semidirect
13. #8648C2semidirect
14. #8644C2semidirect
15. #8291C2semidirect
56. #1580C4semidirect
484. C8#427semidirect

Maximal subgroups

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

subgroupindex
#228132
#228072
#227082
#226582
C2×#15802
#96212
#96122
#88932
#88852
#88202
#86482
#86442
#82912