Finite Group Databasebeta

← #339001

Subgroups — #339001

Order 512.

545 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28809C211yesmaximal, characteristic, complemented
2#27821C211yesmaximal, characteristic
3#26299C211yesmaximal, characteristic
4#26024C211yesmaximal, characteristic
5#10681C211yesmaximal, characteristic, complemented
6#10619C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10575C211yesmaximal, characteristic, complemented
9#10412C211yesmaximal, characteristic
10#10349C211yesmaximal, characteristic, complemented
11#9619C211yesmaximal, characteristic, complemented
12#9567C211yesmaximal, characteristic, complemented
13#9541C211yesmaximal, characteristic
14#9539C211yesmaximal, characteristic, complemented
15#8893C211yesmaximal, characteristic
16C2×#498—21no—
17#2563C2211yescharacteristic
18#2551C2211yescharacteristic
19#2551C2211yescharacteristic
20#2354C2211yescharacteristic
21#2101—21no—
22#2077—21no—
23#2075—21no—
24#2065—21no—
25#2058—21no—
26#2058—21no—
27#2023—21no—
28#2023—21no—
29#1993—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1987—21no—
33#1987C2211yescharacteristic
34#1987C2211yescharacteristic
35#1956C2211yescharacteristic
36#1954C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1941—21no—
42#1939C2211yescharacteristic
43#1936—21no—
44#1920—21no—
45#1854—21no—
46#1844C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1825—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×D4⋊C8C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#498—44no—
75C2×#107—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106D411yescharacteristic
82C2×#106D411yescharacteristic
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—41no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102D411yescharacteristic
92C2×#102D411yescharacteristic
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2311yescharacteristic
98C2×C4×C8C2311yescharacteristic
99C2×C4×C8C2311yescharacteristic
100#395—41no—
101#395—41no—
102#393—41no—
103#390—21no—
104#390—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#388—41no—
110#384C2311yescharacteristic
111#384C2311yescharacteristic
112#384C2311yescharacteristic
113#382—21no—
114#380—21no—
115#380—21no—
116#380—21no—
117C4×C4⋊C4—21no—
118C4×C4⋊C4C2311yescharacteristic
119C4×C4⋊C4C2311yescharacteristic
120C4×C4⋊C4C2311yescharacteristic
121#340—21no—
122#340—21no—
123#340—21no—
124#340—21no—
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—41no—
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#340D411yescharacteristic
142#336—21no—
143#336—21no—
144#336—21no—
145#335—22no—
146#333—22no—
147D4⋊C8—22no—
148C22×Q8—21no—
149C22×Q8—41no—
150C22×Q8—41no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—81no—
171C4⋊Q8—81no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C4×D4—21no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178#117—42no—
179#117—81no—
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—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
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×C4⋊C4C2×D411yescharacteristic
202C2×C22⋊C4—21no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—41no—
211C2×C42C2411yescharacteristic
212#107—81no—
213#106—22no—
214#106—22no—
215#106—42no—
216#106—42no—
217#106—42no—
218#106—42no—
219#106—81no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—81no—
228C4⋊C8#4112yes—
229#103—82no—
230#102—22no—
231#102—22no—
232#102—81no—
233#102—81no—
234C22⋊C8—42no—
235C22⋊C8—81no—
236#97—81no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—42no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—41no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—21no—
253C2×Q8—41no—
254C2×Q8—41no—
255C2×Q8—82no—
256C2×Q8—82no—
257C2×Q8—81no—
258C2×Q8—81no—
259C2×Q8—82no—
260C2×Q8—82no—
261C2×Q8—161no—
262C2×Q8—161no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—22no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
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×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8AGL(1,ℤ/8)12yes—
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—82no—
344C4⋊C4—82no—
345C4⋊C4—81no—
346C4⋊C4—81no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—82no—
350C4⋊C4#13512yes—
351C4⋊C4#13712yes—
352C4⋊C4#13712yes—
353C22⋊C4—41no—
354C22⋊C4—41no—
355C22⋊C4—82no—
356C22⋊C4—161no—
357C22⋊C4—161no—
358C22⋊C4—161no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—41no—
365C42—82no—
366C42C2×#4111yescharacteristic
367C422−1+411yescharacteristic
368C422−1+411yescharacteristic
369C422−1+411yescharacteristic
370C23—41no—
371C23—41no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23D4211yescharacteristic
378Q8—82no—
379Q8—161no—
380Q8—164no—
381Q8—164no—
382D4—41no—
383D4—41no—
384D4—41no—
385D4—41no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
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—41no—
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—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4#53611yescharacteristic
426C2×C4#53611yescharacteristic
427C2×C4#54211yescharacteristic
428C2×C4#54711yescharacteristic
429C2×C4#54811yescharacteristic
430C2×C4#56311yescharacteristic
431C2×C4C2×#13511yescharacteristic
432C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
433C2×C4C2×#13711yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57711yescharacteristic
436C2×C4#57711yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57711yescharacteristic
439C8—22no—
440C8—41no—
441C8—41no—
442C8—42no—
443C8—41no—
444C8—81no—
445C8—81no—
446C8—81no—
447C8—81no—
448C22—81no—
449C22—81no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22—81no—
455C22#267411yescharacteristic
456C22#326611yescharacteristic
457C22#326611yescharacteristic
458C22#327211yescharacteristic
459C22#328211yescharacteristic
460C22#328811yescharacteristic
461C22#329011yescharacteristic
462C4—21no—
463C4—21no—
464C4—42no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—161no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4#323111yescharacteristic
476C4#323811yescharacteristic
477C4#324411yescharacteristic
478C4#324411yescharacteristic
479C4#334011yescharacteristic
480C4#335611yescharacteristic
481C4#336911yescharacteristic
482C4#337311yescharacteristic
483C2—161no—
484C2#1982811yescharacteristic
485C2#1991011yescharacteristic
486C2#2004611yescharacteristic
487C2#2017411yescharacteristic
488C2#3077411yescharacteristic
489C2#3082911yescharacteristic
490C2#3083911yescharacteristic
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. #28809C2semidirect
5. #10681C2semidirect
7. #10601C2semidirect
8. #10575C2semidirect
10. #10349C2semidirect
11. #9619C2semidirect
12. #9567C2semidirect
14. #9539C2semidirect

Maximal subgroups

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

subgroupindex
#288092
#278212
#262992
#260242
#106812
#106192
#106012
#105752
#104122
#103492
#96192
#95672
#95412
#95392
#88932