Finite Group Databasebeta

← #341538

Subgroups — #341538

Order 512.

524 conjugacy classes of subgroups.

The subgroup lattice has 479 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28830C211yesmaximal, characteristic, complemented
2#28828C211yesmaximal, characteristic, complemented
3#26533C211yesmaximal, characteristic
4#26529C211yesmaximal, characteristic
5#10672C211yesmaximal, characteristic
6#10654C211yesmaximal, characteristic, complemented
7#10649C211yesmaximal, characteristic, complemented
8#10578C211yesmaximal, characteristic, complemented
9#10578C211yesmaximal, characteristic, complemented
10#10351C211yesmaximal, characteristic
11#9564C211yesmaximal, characteristic
12#9563C211yesmaximal, characteristic
13#9536C211yesmaximal, characteristic, complemented
14#9535C211yesmaximal, characteristic, complemented
15#8921C211yesmaximal, characteristic
16#2571C2211yescharacteristic
17#2571C2211yescharacteristic
18#2571C2211yescharacteristic
19#2366C2211yescharacteristic
20#2092—21no—
21#2092—21no—
22#2085—21no—
23#2085—21no—
24#2085—21no—
25#2085—21no—
26#2067—21no—
27#2065—21no—
28#1982C2211yescharacteristic
29#1982C2211yescharacteristic
30#1957C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1948C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1942—21no—
40#1941—21no—
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1864—21no—
44#1862—21no—
45#1846—21no—
46#1846—21no—
47#1845C2211yescharacteristic
48#1844C2211yescharacteristic
49#1834C2211yescharacteristic
50#1804—21no—
51#1804—21no—
52C4×#107C2211yescharacteristic
53C4×#106C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55C8×C4⋊C4C2211yescharacteristic
56C2×#331C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1279C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1273C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×C4×D4—21no—
69C2×C4×D4—21no—
70C2×#107—21no—
71C2×#107—21no—
72C2×#107—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107D411yescharacteristic
76C2×#107D411yescharacteristic
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106D411yescharacteristic
83C2×#106D411yescharacteristic
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—41no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#400—41no—
106#400—41no—
107#390—21no—
108#390—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#386—41no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120#382—21no—
121#382—21no—
122#382—21no—
123#382—21no—
124#382—21no—
125C4×C4⋊C4C2311yescharacteristic
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—41no—
136#340D411yescharacteristic
137#340D411yescharacteristic
138#340D411yescharacteristic
139#340D411yescharacteristic
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#331Q814yes—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C23×C4—41no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D411yescharacteristic
155C22×C8C2×D411yescharacteristic
156C22×C8C2×D411yescharacteristic
157C22×C8C2×D411yescharacteristic
158#127—21no—
159#127—21no—
160#127—21no—
161#127C2×Q812yes—
162C4×D4—42no—
163C4×D4—42no—
164C4×D4—81no—
165C4×D4—81no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—41no—
176C2×C4⋊C4—41no—
177C2×C4⋊C4—41no—
178C2×C4⋊C4C2×D411yescharacteristic
179C2×C4⋊C4C2×D411yescharacteristic
180C2×C4⋊C4C2×D411yescharacteristic
181C2×C4⋊C4C2×D411yescharacteristic
182C2×C4⋊C4C2×D411yescharacteristic
183C2×C4⋊C4C2×D411yescharacteristic
184C2×C4⋊C4C2×D411yescharacteristic
185C2×C4⋊C4C2×D411yescharacteristic
186C2×C22⋊C4—41no—
187C2×C22⋊C4—41no—
188C2×C22⋊C4—41no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42C2411yescharacteristic
194#107—22no—
195#107—41no—
196#107—42no—
197#107—81no—
198#107—81no—
199#107—81no—
200#107—81no—
201#106—22no—
202#106—41no—
203#106—42no—
204#106—81no—
205#106—81no—
206#106—81no—
207#106—81no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—21no—
211C4⋊C8—21no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—81no—
215C4⋊C8—81no—
216C4⋊C8C2×Q812yes—
217C4⋊C8C2×Q812yes—
218#102—42no—
219#102—42no—
220#102—81no—
221#102—81no—
222#102—81no—
223#102—81no—
224#102—81no—
225#102—81no—
226#102—82no—
227C4×C8—21no—
228C4×C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C4×C8—21no—
233C4×C8—42no—
234C4×C8—42no—
235#95—21no—
236#95—21no—
237#95—21no—
238#95—21no—
239#95—41no—
240#95—41no—
241#95—41no—
242#95—41no—
243#95—41no—
244#95—41no—
245#95—41no—
246#95—41no—
247#95—41no—
248C24—41no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—81no—
264C2×D4—81no—
265C22×C4—21no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—82no—
277C22×C4—82no—
278C22×C4—161no—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4H(5,𝔽2)11yescharacteristic
282C2×C8—21no—
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—21no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—22no—
289C2×C8—22no—
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#13512yes—
308C2×C8#13512yes—
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—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
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—41no—
336C4⋊C4—41no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—161no—
341C4⋊C4—161no—
342C4⋊C4—161no—
343C22⋊C4—82no—
344C22⋊C4—82no—
345C22⋊C4—161no—
346C42—41no—
347C42—41no—
348C42—41no—
349C42—41no—
350C42—41no—
351C42—41no—
352C42—41no—
353C42—41no—
354C42C22×Q811yescharacteristic
355C422−1+411yescharacteristic
356C422−1+411yescharacteristic
357C422−1+411yescharacteristic
358C23—42no—
359C23—81no—
360C23—81no—
361C23—81no—
362C23—81no—
363C23—81no—
364C23—81no—
365C23D4211yescharacteristic
366D4—42no—
367D4—42no—
368D4—42no—
369D4—42no—
370D4—82no—
371D4—81no—
372D4—81no—
373D4—81no—
374D4—81no—
375D4—161no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
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—41no—
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—164no—
412C2×C4#53611yescharacteristic
413C2×C4#53611yescharacteristic
414C2×C4#54411yescharacteristic
415C2×C4D4×Q811yescharacteristic
416C2×C4D4×Q811yescharacteristic
417C2×C4#56311yescharacteristic
418C2×C4C2×#13511yescharacteristic
419C2×C4C2×#13511yescharacteristic
420C2×C4#57511yescharacteristic
421C2×C4#57511yescharacteristic
422C2×C4#57611yescharacteristic
423C2×C4#57611yescharacteristic
424C2×C4#57711yescharacteristic
425C2×C4#57711yescharacteristic
426C8—22no—
427C8—22no—
428C8—42no—
429C8—42no—
430C8—41no—
431C8—41no—
432C8—81no—
433C8—81no—
434C8—81no—
435C22—82no—
436C22—82no—
437C22—82no—
438C22—161no—
439C22—161no—
440C22—161no—
441C22—161no—
442C22#267711yescharacteristic
443C22#326711yescharacteristic
444C22#326811yescharacteristic
445C22#328411yescharacteristic
446C22#328711yescharacteristic
447C22#328811yescharacteristic
448C22#329011yescharacteristic
449C4—21no—
450C4—21no—
451C4—81no—
452C4—81no—
453C4—81no—
454C4—81no—
455C4—81no—
456C4—81no—
457C4—81no—
458C4—81no—
459C4—161no—
460C4—161no—
461C4—161no—
462C4#323211yescharacteristic
463C4#323211yescharacteristic
464C4#324111yescharacteristic
465C4#324111yescharacteristic
466C4#338011yescharacteristic
467C4#338111yescharacteristic
468C4#338811yescharacteristic
469C4#338911yescharacteristic
470C2—162no—
471C2#1985411yescharacteristic
472C2#1987111yescharacteristic
473C2#2001811yescharacteristic
474C2#2013311yescharacteristic
475C2#3095111yescharacteristic
476C2#3095911yescharacteristic
477C2#3098111yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
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. #28830C2semidirect
2. #28828C2semidirect
6. #10654C2semidirect
7. #10649C2semidirect
8. #10578C2semidirect
9. #10578C2semidirect
13. #9536C2semidirect
14. #9535C2semidirect

Maximal subgroups

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

subgroupindex
#288302
#288282
#265332
#265292
#106722
#106542
#106492
#105782
#103512
#95642
#95632
#95362
#95352
#89212