Finite Group Databasebeta

← #308431

Subgroups — #308431

Order 512.

584 conjugacy classes of subgroups.

The subgroup lattice has 499 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26506C211yesmaximal, characteristic, complemented
2#26021C211yesmaximal, characteristic
3#23100C211yesmaximal, characteristic
4#22708C211yesmaximal, characteristic
5#10681C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10596C211yesmaximal, characteristic, complemented
8#10580C211yesmaximal, characteristic, complemented
9#9622C211yesmaximal, characteristic, complemented
10#9537C211yesmaximal, characteristic, complemented
11#9217C211yesmaximal, characteristic
12#8988C211yesmaximal, characteristic
13#8925C211yesmaximal, characteristic, complemented
14#8889C211yesmaximal, characteristic
15#8845C211yesmaximal, characteristic
16C2×#461C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2552C2211yescharacteristic
19#2365C2211yescharacteristic
20#2354C2211yescharacteristic
21#2354C2211yescharacteristic
22#2099—21no—
23#2086—21no—
24#2065—21no—
25#2025—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1989—21no—
29#1988C2211yescharacteristic
30#1973—21no—
31#1956C2211yescharacteristic
32#1948C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947—21no—
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1928—21no—
41#1919—21no—
42#1918—21no—
43#1880—21no—
44#1854—21no—
45#1854—21no—
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1807—21no—
50#1802C2211yescharacteristic
51C4×#107C2211yescharacteristic
52#1770C2211yescharacteristic
53#1763—21no—
54C4×#102C2211yescharacteristic
55#1735—21no—
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C22×SD16—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74C2×#117—21no—
75#461—22no—
76#461Q814yes—
77C8×D4—22no—
78C8×D4—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107D411yescharacteristic
82C2×#107D411yescharacteristic
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—41no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—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#398—41no—
105#391—21no—
106#390—21no—
107#390—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#388—41no—
112#388—41no—
113#387—41no—
114#384—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#380—21no—
119#380—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
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—21no—
140#340—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#334—22no—
149#332—22no—
150D4⋊C8—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C2×SD16—24no—
155C2×SD16—81no—
156C2×SD16—81no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—24no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—81no—
164C22×C8—81no—
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8C2×Q812yes—
173#125—42no—
174#125—81no—
175C4×Q8—42no—
176C4×Q8—81no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4C2×Q812yes—
180C4×D4#4112yes—
181#117—42no—
182#117—42no—
183#117—81no—
184#117—81no—
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—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
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×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42C2411yescharacteristic
219#107—41no—
220#107—41no—
221#107—42no—
222#107—81no—
223#106—42no—
224#106—81no—
225#106—81no—
226#106—82no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8C2×Q812yes—
236C4⋊C8#4112yes—
237#103—24no—
238#103—42no—
239#103—81no—
240#102—22no—
241#102—41no—
242#102—81no—
243C22⋊C8—24no—
244#97—81no—
245C4×C8—21no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—21no—
251#95—21no—
252#95—21no—
253#95—21no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—21no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4C22⋊Q812yes—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4C22×D411yescharacteristic
290C22×C4C22×D411yescharacteristic
291C22×C4H(5,𝔽2)11yescharacteristic
292SD16—84no—
293C2×C8—22no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
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—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—84no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—22no—
324C4⋊C4—22no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
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—41no—
349C4⋊C4—82no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—161no—
356C4⋊C4C22⋊Q812yes—
357C4⋊C4#13512yes—
358C22⋊C4—22no—
359C22⋊C4—22no—
360C22⋊C4—82no—
361C22⋊C4—82no—
362C22⋊C4—161no—
363C22⋊C4—161no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—82no—
371C42—82no—
372C42—82no—
373C42C22×Q811yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C422−1+411yescharacteristic
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23D4211yescharacteristic
385Q8—42no—
386Q8—42no—
387Q8—81no—
388Q8—81no—
389D4—22no—
390D4—22no—
391D4—41no—
392D4—41no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4#52511yescharacteristic
434C2×C4#53911yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4D4×Q811yescharacteristic
437C2×C4#55411yescharacteristic
438C2×C4C2×#13511yescharacteristic
439C2×C4#57511yescharacteristic
440C2×C4#57511yescharacteristic
441C2×C4#57511yescharacteristic
442C2×C4#57611yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57711yescharacteristic
445C2×C4#57711yescharacteristic
446C8—42no—
447C8—42no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—41no—
452C8—42no—
453C8—81no—
454C8—81no—
455C22—42no—
456C22—42no—
457C22—42no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22#244911yescharacteristic
463C22#326711yescharacteristic
464C22#327111yescharacteristic
465C22#327511yescharacteristic
466C22#328011yescharacteristic
467C22#328911yescharacteristic
468C22#329111yescharacteristic
469C4—21no—
470C4—21no—
471C4—42no—
472C4—42no—
473C4—81no—
474C4—81no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—82no—
479C4—161no—
480C4—161no—
481C4—161no—
482C4#307611yescharacteristic
483C4#308111yescharacteristic
484C4#311111yescharacteristic
485C4#312311yescharacteristic
486C4#334011yescharacteristic
487C4#334711yescharacteristic
488C4#338811yescharacteristic
489C4#338811yescharacteristic
490C2—82no—
491C2#1977611yescharacteristic
492C2#1990411yescharacteristic
493C2#2014711yescharacteristic
494C2#2027611yescharacteristic
495C2#2634111yescharacteristic
496C2#2638611yescharacteristic
497C2#2640111yescharacteristic
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. #26506C2semidirect
5. #10681C2semidirect
6. #10656C2semidirect
7. #10596C2semidirect
8. #10580C2semidirect
9. #9622C2semidirect
10. #9537C2semidirect
13. #8925C2semidirect

Maximal subgroups

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

subgroupindex
#265062
#260212
#231002
#227082
#106812
#106562
#105962
#105802
#96222
#95372
#92172
#89882
#89252
#88892
#88452