Finite Group Databasebeta

← #338697

Subgroups — #338697

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28791C211yesmaximal, characteristic, complemented
2#27814C211yesmaximal, characteristic
3#26299C211yesmaximal, characteristic
4#26020C211yesmaximal, characteristic
5#10682C211yesmaximal, characteristic, complemented
6#10625C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10581C211yesmaximal, characteristic, complemented
9#10412C211yesmaximal, characteristic
10#10349C211yesmaximal, characteristic, complemented
11#9616C211yesmaximal, characteristic, complemented
12#9568C211yesmaximal, characteristic, complemented
13#9541C211yesmaximal, characteristic
14#9535C211yesmaximal, characteristic, complemented
15#8893C211yesmaximal, characteristic
16#2563C2211yescharacteristic
17#2552C2211yescharacteristic
18#2551C2211yescharacteristic
19#2354C2211yescharacteristic
20#2101—21no—
21#2084—21no—
22#2077—21no—
23#2075—21no—
24#2067—21no—
25#2058—21no—
26#2025—21no—
27#2023—21no—
28#1991—21no—
29#1990—21no—
30#1990C2211yescharacteristic
31#1990C2211yescharacteristic
32#1988—21no—
33#1987C2211yescharacteristic
34#1987C2211yescharacteristic
35#1956C2211yescharacteristic
36#1954C2211yescharacteristic
37#1948C2211yescharacteristic
38#1948C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1941—21no—
42#1939C2211yescharacteristic
43#1934—21no—
44#1918—21no—
45#1854—21no—
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1834C2211yescharacteristic
49#1825—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
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⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×D4C2311yescharacteristic
72C2×#117—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106D411yescharacteristic
78C2×#106D411yescharacteristic
79C2×C4⋊C8—21no—
80C2×C4⋊C8C2311yescharacteristic
81C2×C4⋊C8C2311yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×#103—41no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102D411yescharacteristic
88C2×#102D411yescharacteristic
89C2×C22⋊C8—21no—
90C2×C22⋊C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8C2311yescharacteristic
94C2×C4×C8C2311yescharacteristic
95C2×C4×C8C2311yescharacteristic
96#398—41no—
97#395—41no—
98#393—41no—
99#390—21no—
100#390—21no—
101#389—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#388—41no—
106#384—21no—
107#384C2311yescharacteristic
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#382—21no—
111#380—21no—
112#380—21no—
113#380—21no—
114C4×C4⋊C4—21no—
115C4×C4⋊C4C2311yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C4⋊C4C2311yescharacteristic
118#340—21no—
119#340—21no—
120#340—21no—
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—41no—
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#340D411yescharacteristic
141#336—21no—
142#336—21no—
143#336—21no—
144#335—22no—
145#333—22no—
146D4⋊C8—22no—
147C22×Q8—21no—
148C22×Q8—41no—
149C22×D4C2×D411yescharacteristic
150C23×C4—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158C22×C8C2×D411yescharacteristic
159C22×C8C2×D411yescharacteristic
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—42no—
165C4⋊Q8—42no—
166C4⋊Q8—81no—
167C4⋊Q8—81no—
168#125—42no—
169#125—81no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173C4×D4—21no—
174#117—42no—
175#117—81no—
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—41no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C22⋊C4—41no—
202C2×C22⋊C4—41no—
203C2×C22⋊C4—41no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—41no—
208C2×C42C2411yescharacteristic
209#107—42no—
210#107—81no—
211#106—22no—
212#106—41no—
213#106—42no—
214#106—81no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—81no—
223C4⋊C8#4112yes—
224#103—82no—
225#102—22no—
226#102—22no—
227#102—81no—
228#102—81no—
229C22⋊C8—42no—
230C22⋊C8—81no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—21no—
236C4×C8—21no—
237C4×C8—81no—
238C4×C8—81no—
239#95—21no—
240#95—21no—
241#95—21no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247C24—21no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—82no—
251C2×Q8—82no—
252C2×Q8—81no—
253C2×Q8—81no—
254C2×Q8—161no—
255C2×D4—21no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4C22×D411yescharacteristic
279C22×C4C22×D411yescharacteristic
280C22×C4H(5,𝔽2)11yescharacteristic
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
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—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
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—82no—
336C4⋊C4—82no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—81no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—161no—
343C4⋊C4#13512yes—
344C4⋊C4#13512yes—
345C4⋊C4#13712yes—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C22⋊C4—82no—
349C22⋊C4—161no—
350C22⋊C4—161no—
351C22⋊C4—161no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—82no—
359C42C2×#4111yescharacteristic
360C422−1+411yescharacteristic
361C422−1+411yescharacteristic
362C422−1+411yescharacteristic
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23D4211yescharacteristic
371Q8—82no—
372Q8—161no—
373Q8—164no—
374D4—41no—
375D4—41no—
376D4—41no—
377D4—41no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—41no—
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—81no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
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#53611yescharacteristic
418C2×C4#53611yescharacteristic
419C2×C4#54211yescharacteristic
420C2×C4#54711yescharacteristic
421C2×C4#54811yescharacteristic
422C2×C4#56311yescharacteristic
423C2×C4C2×#13511yescharacteristic
424C2×C4C2×#13511yescharacteristic
425C2×C4C2×#13711yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57611yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57811yescharacteristic
431C8—41no—
432C8—41no—
433C8—42no—
434C8—41no—
435C8—41no—
436C8—81no—
437C8—81no—
438C8—81no—
439C8—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22#267411yescharacteristic
448C22#326611yescharacteristic
449C22#326911yescharacteristic
450C22#328011yescharacteristic
451C22#328211yescharacteristic
452C22#328811yescharacteristic
453C22#329011yescharacteristic
454C4—21no—
455C4—21no—
456C4—42no—
457C4—42no—
458C4—42no—
459C4—81no—
460C4—81no—
461C4—81no—
462C4—81no—
463C4—161no—
464C4—161no—
465C4—161no—
466C4—161no—
467C4#323111yescharacteristic
468C4#323311yescharacteristic
469C4#324311yescharacteristic
470C4#324411yescharacteristic
471C4#334011yescharacteristic
472C4#335611yescharacteristic
473C4#336411yescharacteristic
474C4#337011yescharacteristic
475C2—161no—
476C2#1977411yescharacteristic
477C2#1980011yescharacteristic
478C2#1999111yescharacteristic
479C2#2011011yescharacteristic
480C2#3077411yescharacteristic
481C2#3080911yescharacteristic
482C2#3084211yescharacteristic
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. #28791C2semidirect
5. #10682C2semidirect
7. #10601C2semidirect
8. #10581C2semidirect
10. #10349C2semidirect
11. #9616C2semidirect
12. #9568C2semidirect
14. #9535C2semidirect

Maximal subgroups

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

subgroupindex
#287912
#278142
#262992
#260202
#106822
#106252
#106012
#105812
#104122
#103492
#96162
#95682
#95412
#95352
#88932