Finite Group Databasebeta

← #306312

Subgroups — #306312

Order 512.

575 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28836C211yesmaximal, characteristic, complemented
2#26087C211yesmaximal, characteristic
3#26045C211yesmaximal, characteristic
4#22713C211yesmaximal, characteristic
5#10653C211yesmaximal, characteristic, complemented
6#10651C211yesmaximal, characteristic, complemented
7#10582C211yesmaximal, characteristic, complemented
8#10577C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9598C211yesmaximal, characteristic, complemented
11#8897C211yesmaximal, characteristic
12#8886C211yesmaximal, characteristic
13#8819C211yesmaximal, characteristic, complemented
14#8650C211yesmaximal, characteristic
15#8208C211yesmaximal, characteristic
16C2×#478—21no—
17C2×#436C2211yescharacteristic
18#2571C2211yescharacteristic
19#2552C2211yescharacteristic
20#2354C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2092—21no—
23#2088—21no—
24#2085—21no—
25#2084—21no—
26#2067—21no—
27#2059—21no—
28#1949—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1921—21no—
39#1864—21no—
40#1854—21no—
41#1852—21no—
42#1847—21no—
43#1845C2211yescharacteristic
44#1844C2211yescharacteristic
45#1844C2211yescharacteristic
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1809—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#107C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1736—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×#125—21no—
69C2×C4×D4—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72C22×C4⋊C4—21no—
73#478—44no—
74#436—22no—
75#436Q814yes—
76C2×#107—21no—
77C2×#107—21no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107D411yescharacteristic
82C2×#107D411yescharacteristic
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—41no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#102—21no—
98C2×#102—21no—
99C2×#102D411yescharacteristic
100C2×#102D411yescharacteristic
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8C2311yescharacteristic
104C2×C4×C8C2311yescharacteristic
105C2×C4×C8C2311yescharacteristic
106#400—41no—
107#398—41no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#388—41no—
115#385—41no—
116#384—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122#382—21no—
123#380—21no—
124#380—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
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⋊C8Q814yes—
149#322—22no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—41no—
153C2×M16—24no—
154C2×M16—81no—
155C2×M16—81no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165#125—42no—
166#125—81no—
167C4×D4—21no—
168C4×D4—21no—
169C4×D4—42no—
170C4×D4—81no—
171C4×D4C2×Q812yes—
172C4×D4C2×Q812yes—
173#117—42no—
174#117—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
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—42no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C22⋊C4—21no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—41no—
209C2×C42C2411yescharacteristic
210#107—41no—
211#107—41no—
212#107—42no—
213#107—42no—
214#107—81no—
215#107—81no—
216#107—81no—
217#106—42no—
218#106—42no—
219#106—81no—
220#106—81no—
221#106—81no—
222#106—81no—
223#106—81no—
224#106—82no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—42no—
232C4⋊C8—42no—
233C4⋊C8C2×Q812yes—
234C4⋊C8C2×Q812yes—
235#102—22no—
236#102—22no—
237#102—42no—
238#102—81no—
239C22⋊C8—24no—
240#97—42no—
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×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4C4⋊Q812yes—
263C22×C4—21no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—82no—
281C22×C4—82no—
282C22×C4—161no—
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4H(5,𝔽2)11yescharacteristic
286M16—84no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8C22⋊Q812yes—
311C2×C8C22⋊Q812yes—
312C2×C8AGL(1,ℤ/8)12yes—
313C2×C8#13712yes—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
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—81no—
344C4⋊C4—81no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—82no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4—161no—
351C4⋊C4—161no—
352C4⋊C4C4⋊Q812yes—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—82no—
356C22⋊C4—82no—
357C22⋊C4—161no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—82no—
365C42C22×Q811yescharacteristic
366C42C22×Q811yescharacteristic
367C42C2×#4111yescharacteristic
368C422−1+411yescharacteristic
369C23—22no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23D4211yescharacteristic
377D4—22no—
378D4—22no—
379D4—41no—
380D4—41no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
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—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—164no—
421C2×C4C2×C22⋊Q811yescharacteristic
422C2×C4C2×C4⋊Q811yescharacteristic
423C2×C4#54411yescharacteristic
424C2×C4#54711yescharacteristic
425C2×C4D4×Q811yescharacteristic
426C2×C4#55411yescharacteristic
427C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
428C2×C4C2×#13711yescharacteristic
429C2×C4#57511yescharacteristic
430C2×C4#57511yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57711yescharacteristic
434C2×C4#57711yescharacteristic
435C8—22no—
436C8—22no—
437C8—22no—
438C8—22no—
439C8—42no—
440C8—42no—
441C8—42no—
442C8—81no—
443C8—81no—
444C22—42no—
445C22—42no—
446C22—42no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22#244011yescharacteristic
452C22#326711yescharacteristic
453C22#326811yescharacteristic
454C22#327211yescharacteristic
455C22#327811yescharacteristic
456C22#328811yescharacteristic
457C22#328911yescharacteristic
458C4—21no—
459C4—21no—
460C4—42no—
461C4—81no—
462C4—81no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—81no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#308011yescharacteristic
472C4#308111yescharacteristic
473C4#316311yescharacteristic
474C4#316511yescharacteristic
475C4#334411yescharacteristic
476C4#334511yescharacteristic
477C4#338811yescharacteristic
478C4#338911yescharacteristic
479C2—82no—
480C2#2024311yescharacteristic
481C2#2025211yescharacteristic
482C2#2025411yescharacteristic
483C2#2026511yescharacteristic
484C2#2605911yescharacteristic
485C2#2607811yescharacteristic
486C2#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. #28836C2semidirect
5. #10653C2semidirect
6. #10651C2semidirect
7. #10582C2semidirect
8. #10577C2semidirect
9. #9619C2semidirect
10. #9598C2semidirect
13. #8819C2semidirect

Maximal subgroups

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

subgroupindex
#288362
#260872
#260452
#227132
#106532
#106512
#105822
#105772
#96192
#95982
#88972
#88862
#88192
#86502
#82082