Finite Group Databasebeta

← #305921

Subgroups — #305921

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28799C211yesmaximal, characteristic, complemented
2#26111C211yesmaximal, characteristic
3#26044C211yesmaximal, characteristic
4#22703C211yesmaximal, characteristic
5#10657C211yesmaximal, characteristic, complemented
6#10654C211yesmaximal, characteristic, complemented
7#10576C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#8894C211yesmaximal, characteristic
12#8882C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15#8202C211yesmaximal, characteristic
16C2×#476—21no—
17C2×C8×D4C2211yescharacteristic
18#2571C2211yescharacteristic
19#2551C2211yescharacteristic
20D4×C4⋊C4C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2101—21no—
23#2088—21no—
24#2087—21no—
25#2085—21no—
26#2058—21no—
27#2058—21no—
28#1993—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1943—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1864—21no—
39#1862—21no—
40#1853—21no—
41#1853—21no—
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1807—21no—
49#1804—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75#476—44no—
76C8×D4—22no—
77C8×D4Q814yescomplemented
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—41no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106D411yescharacteristic
88C2×#106D411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#102—21no—
96C2×#102—21no—
97C2×#102D411yescharacteristic
98C2×#102D411yescharacteristic
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#395—41no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#385—41no—
113#385—41no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119#382—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125#340—21no—
126#340—21no—
127#340—21no—
128#340—21no—
129#340—21no—
130#340—21no—
131#340—21no—
132#340—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—41no—
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#335—22no—
148D4⋊C8Q814yescomplemented
149#322—22no—
150C22×Q8—41no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—41no—
154C23×C4—41no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—24no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—81no—
162C22×C8—81no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—81no—
170C4⋊Q8—81no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—42no—
174C4×D4—42no—
175C4×D4—81no—
176C4×D4—81no—
177C4×D4C2×Q812yes—
178C4×D4C2×Q812yes—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42C2411yescharacteristic
214#107—42no—
215#107—81no—
216#107—82no—
217#106—22no—
218#106—41no—
219#106—42no—
220#106—42no—
221#106—81no—
222#106—81no—
223#106—81no—
224#106—81no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8C2×Q812yes—
234C4⋊C8C2×Q812yes—
235#102—22no—
236#102—22no—
237#102—42no—
238#102—42no—
239C22⋊C8—24no—
240#97—81no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8C2×Q812yes—
245C4×C8C2×Q812yes—
246C4×C8#4112yes—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—21no—
257C2×Q8—82no—
258C2×Q8—82no—
259C2×Q8—161no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4C4⋊Q812yes—
264C2×D4AGL(1,ℤ/8)12yes—
265C2×D4#13712yes—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—82no—
284C22×C4—82no—
285C22×C4—82no—
286C22×C4—82no—
287C22×C4—161no—
288C22×C4—161no—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—84no—
318C2×C8C22⋊Q812yes—
319C2×C8C22⋊Q812yes—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—82no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—161no—
357C4⋊C4C4⋊Q812yes—
358C4⋊C4#13512yes—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42C22×Q811yescharacteristic
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C23—22no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23D4211yescharacteristic
381Q8—164no—
382D4—22no—
383D4—22no—
384D4—22no—
385D4#50114yes—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—164no—
426C2×C4—164no—
427C2×C4C2×C22⋊Q811yescharacteristic
428C2×C4C2×C4⋊Q811yescharacteristic
429C2×C4#54411yescharacteristic
430C2×C4#54711yescharacteristic
431C2×C4D4×Q811yescharacteristic
432C2×C4#55411yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
435C2×C4C2×#13711yescharacteristic
436C2×C4#57511yescharacteristic
437C2×C4#57611yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C8—22no—
442C8—22no—
443C8—42no—
444C8—42no—
445C8—41no—
446C8—41no—
447C8—42no—
448C8—81no—
449C8—81no—
450C22—24no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22—42no—
457C22#244011yescharacteristic
458C22#326511yescharacteristic
459C22#326611yescharacteristic
460C22#327311yescharacteristic
461C22#328211yescharacteristic
462C22#328811yescharacteristic
463C22#328911yescharacteristic
464C4—21no—
465C4—21no—
466C4—42no—
467C4—42no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—81no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#307611yescharacteristic
478C4#308111yescharacteristic
479C4C2×#50111yescharacteristic
480C4#316311yescharacteristic
481C4#334411yescharacteristic
482C4#334511yescharacteristic
483C4#338011yescharacteristic
484C4#338811yescharacteristic
485C2—44no—
486C2#1978911yescharacteristic
487C2#1979611yescharacteristic
488C2#2017511yescharacteristic
489C2#2019411yescharacteristic
490C2#2605111yescharacteristic
491C2#2606811yescharacteristic
492C2#2608511yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×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
1. #28799C2semidirect
5. #10657C2semidirect
6. #10654C2semidirect
7. #10576C2semidirect
8. #10575C2semidirect
9. #9619C2semidirect
10. #9596C2semidirect
13. #8821C2semidirect
77. C8×D4Q8semidirect
148. D4⋊C8Q8semidirect

Maximal subgroups

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

subgroupindex
#287992
#261112
#260442
#227032
#106572
#106542
#105762
#105752
#96192
#95962
#88942
#88822
#88212
#86482
#82022