Finite Group Databasebeta

← #290199

Subgroups — #290199

Order 512.

553 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#29321C211yesmaximal, characteristic, complemented
2#29256C211yesmaximal, characteristic
3#29143C211yesmaximal, characteristic
4#23253C211yesmaximal, characteristic, complemented
5#10658C211yesmaximal, characteristic, complemented
6#10656C211yesmaximal, characteristic, complemented
7#10595C211yesmaximal, characteristic
8#10591C211yesmaximal, characteristic
9#9899C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9613C211yesmaximal, characteristic, complemented
12#9606C211yesmaximal, characteristic, complemented
13#9564C211yesmaximal, characteristic
14#9563C211yesmaximal, characteristic
15#8234C211yesmaximal, characteristic
16#2859C2211yescharacteristic
17C4×#127C2211yescharacteristic
18C4×#125C2211yescharacteristic
19#2092—21no—
20#2085—21no—
21#2059—21no—
22#2025—21no—
23#2025—21no—
24#2025—21no—
25#1982C2211yescharacteristic
26#1982C2211yescharacteristic
27#1957C2211yescharacteristic
28#1957C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1947C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1943—21no—
37#1939—21no—
38#1939C2211yescharacteristic
39#1939C2211yescharacteristic
40#1916—21no—
41#1916—21no—
42#1864—21no—
43#1862—21no—
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1831C2211yescharacteristic
47#1804—21no—
48#1804—21no—
49C8×C4⋊C4—21no—
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×#127C2311yescharacteristic
68C2×#125—21no—
69C2×#125D411yescharacteristic
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4—21no—
73C2×#107—21no—
74C2×#107—21no—
75C2×#107—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×C4⋊C8—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×#102—21no—
97C2×#97—21no—
98C2×C4×C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#390—21no—
104#390—21no—
105#390—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389D411yescharacteristic
111#389D411yescharacteristic
112#387—21no—
113#387—21no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#382—21no—
118#382D411yescharacteristic
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2311yescharacteristic
122C4×C4⋊C4C2311yescharacteristic
123C43C2311yescharacteristic
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#336—21no—
135#336—21no—
136#336—21no—
137#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336D411yescharacteristic
143#336D411yescharacteristic
144#335—22no—
145#334—22no—
146#331—22no—
147#331Q814yes—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C23×C4—41no—
152C22×C8—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
162#127—21no—
163#127—21no—
164#127—21no—
165#127C2×Q812yes—
166#125—22no—
167#125—22no—
168#125—81no—
169#125—81no—
170C4×D4—42no—
171C4×D4—42no—
172C4×D4—81no—
173C4×D4—81no—
174C4×D4—82no—
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—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4C2×D411yescharacteristic
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—41no—
195C2×C22⋊C4—41no—
196C2×C22⋊C4—41no—
197C2×C42—21no—
198C2×C42—21no—
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42#4111yescharacteristic
204C2×C42#4111yescharacteristic
205C2×C42C2411yescharacteristic
206#107—42no—
207#107—81no—
208#107—81no—
209#106—42no—
210#106—81no—
211#106—81no—
212C4⋊C8—21no—
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—41no—
222C4⋊C8—41no—
223C4⋊C8—42no—
224C4⋊C8—81no—
225C4⋊C8C2×Q812yes—
226C4⋊C8C2×Q812yes—
227C4⋊C8#4112yes—
228#102—42no—
229#102—42no—
230#102—81no—
231#102—81no—
232#102—81no—
233#102—81no—
234#102—81no—
235#102—81no—
236#97—81no—
237C4×C8—21no—
238C4×C8—21no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—42no—
242C4×C8—81no—
243C4×C8—81no—
244#95—21no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—21no—
250#95—41no—
251#95—41no—
252C24—41no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—41no—
262C2×D4—41no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—81no—
268C2×D4—81no—
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—82no—
279C22×C4#12111yescharacteristic
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4C22×D411yescharacteristic
284C22×C4C22×D411yescharacteristic
285C22×C4C2×#4111yescharacteristic
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—21no—
294C2×C8—21no—
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×C8—41no—
316C4⋊C4—24no—
317C4⋊C4—24no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—42no—
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—81no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C4⋊C4#13512yes—
344C4⋊C4#13512yes—
345C22⋊C4—82no—
346C22⋊C4—82no—
347C22⋊C4—82no—
348C42—21no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—41no—
362C42C22×Q811yescharacteristic
363C42C2×#4111yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C23—42no—
367C23—81no—
368C23—81no—
369C23—81no—
370C23—81no—
371C23—81no—
372C23—81no—
373C23C2×#12111yescharacteristic
374D4—42no—
375D4—42no—
376D4—42no—
377D4—42no—
378D4—82no—
379D4—82no—
380D4—81no—
381D4—81no—
382D4—81no—
383D4—81no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
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—41no—
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—164no—
420C2×C4#53311yescharacteristic
421C2×C4#54111yescharacteristic
422C2×C4#54711yescharacteristic
423C2×C4#54811yescharacteristic
424C2×C4D4×Q811yescharacteristic
425C2×C4D4×Q811yescharacteristic
426C2×C4C2×#13511yescharacteristic
427C2×C4C2×#13511yescharacteristic
428C2×C4#57511yescharacteristic
429C2×C4#57511yescharacteristic
430C2×C4#57611yescharacteristic
431C2×C4#57611yescharacteristic
432C2×C4#57711yescharacteristic
433C2×C4#57711yescharacteristic
434C8—41no—
435C8—41no—
436C8—41no—
437C8—41no—
438C8—42no—
439C8—42no—
440C8—42no—
441C8—81no—
442C8—81no—
443C8—81no—
444C22—82no—
445C22—82no—
446C22—82no—
447C22—161no—
448C22—161no—
449C22—161no—
450C22—161no—
451C22#242911yescharacteristic
452C22#304011yescharacteristic
453C22#304511yescharacteristic
454C22#305011yescharacteristic
455C22#306211yescharacteristic
456C22#306711yescharacteristic
457C22#306811yescharacteristic
458C4—21no—
459C4—21no—
460C4—41no—
461C4—42no—
462C4—41no—
463C4—41no—
464C4—41no—
465C4—42no—
466C4—81no—
467C4—81no—
468C4—161no—
469C4—161no—
470C4—161no—
471C4#334711yescharacteristic
472C4#334811yescharacteristic
473C4#336111yescharacteristic
474C4#336511yescharacteristic
475C4#338011yescharacteristic
476C4#338111yescharacteristic
477C4#338811yescharacteristic
478C4#338911yescharacteristic
479C2—162no—
480C2#1927811yescharacteristic
481C2#1938011yescharacteristic
482C2#1941211yescharacteristic
483C2#1957811yescharacteristic
484C2#2546611yescharacteristic
485C2#2547311yescharacteristic
486C2#2549311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #29321C2semidirect
4. #23253C2semidirect
5. #10658C2semidirect
6. #10656C2semidirect
9. #9899C2semidirect
10. #9616C2semidirect
11. #9613C2semidirect
12. #9606C2semidirect

Maximal subgroups

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

subgroupindex
#293212
#292562
#291432
#232532
#106582
#106562
#105952
#105912
#98992
#96162
#96132
#96062
#95642
#95632
#82342