Finite Group Databasebeta

← #311024

Subgroups — #311024

Order 512.

568 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27748C211yesmaximal, characteristic
2#26507C211yesmaximal, characteristic, complemented
3#26296C211yesmaximal, characteristic
4#22963C211yesmaximal, characteristic
5#10602C211yesmaximal, characteristic, complemented
6#10578C211yesmaximal, characteristic, complemented
7#10481C211yesmaximal, characteristic
8#10058C211yesmaximal, characteristic
9#9821C211yesmaximal, characteristic, complemented
10#9568C211yesmaximal, characteristic, complemented
11#9562C211yesmaximal, characteristic, complemented
12#9554C211yesmaximal, characteristic
13#9060C211yesmaximal, characteristic, complemented
14#8894C211yesmaximal, characteristic
15C8×#107C211yesmaximal, characteristic, complemented
16C2×#487C2211yescharacteristic
17C2×#476C2211yescharacteristic
18#2571C2211yescharacteristic
19#2547C2211yescharacteristic
20#2369C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2101—21no—
23#2086—21no—
24#2075—21no—
25#2075—21no—
26#2074—21no—
27#2042—21no—
28#1990C2211yescharacteristic
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1988C2211yescharacteristic
32#1987C2211yescharacteristic
33#1981C2211yescharacteristic
34#1958—21no—
35#1956C2211yescharacteristic
36#1954C2211yescharacteristic
37#1949—21no—
38#1948C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1941—21no—
42#1939C2211yescharacteristic
43#1933—21no—
44#1920—21no—
45#1919—21no—
46#1864—21no—
47#1852—21no—
48#1834C2211yescharacteristic
49#1823—21no—
50#1802C2211yescharacteristic
51C4×#107C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×#102—21no—
55C4×#102C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×D4⋊C8C2211yescharacteristic
60C2×C82C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×#124—21no—
70C2×C4×D4—21no—
71C2×C4×D4C2311yescharacteristic
72C22×C4⋊C4—21no—
73#487—22no—
74#487—22no—
75#476—22no—
76#476—22no—
77C2×#107—21no—
78C2×#107D411yescharacteristic
79C2×#107D411yescharacteristic
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#103—21no—
89C2×#103—21no—
90C2×#103—21no—
91C2×#103—21no—
92C2×#103—41no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102D411yescharacteristic
96C2×#102D411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#400—41no—
105#397—41no—
106#391—21no—
107#391—41no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#385—41no—
114#384C2311yescharacteristic
115#384C2311yescharacteristic
116#382—21no—
117#382—21no—
118#380—21no—
119#380—21no—
120#380—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
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—21no—
138#340—21no—
139#340—21no—
140#340—41no—
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#340D411yescharacteristic
145#336—21no—
146#336—21no—
147#336—21no—
148#335—22no—
149#333—22no—
150#332—22no—
151D4⋊C8—22no—
152C82—22no—
153C22×Q8—21no—
154C22×Q8—21no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C23×C4—41no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8#4112yes—
175#124—42no—
176#124—81no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—42no—
182C4×D4—81no—
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—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C2×D411yescharacteristic
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—41no—
213C2×C42C2411yescharacteristic
214#107—22no—
215#107—22no—
216#107—81no—
217#106—24no—
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—42no—
228C4⋊C8#4112yes—
229#103—44no—
230#103—42no—
231#103—81no—
232#103—81no—
233#103—81no—
234#102—24no—
235#102—22no—
236#102—22no—
237#102—81no—
238C22⋊C8—42no—
239C22⋊C8—81no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—81no—
246C4×C8—81no—
247C4×C8#4112yes—
248#95—21no—
249#95—21no—
250#95—21no—
251#95—41no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256#95—41no—
257#95—41no—
258#95—41no—
259C24—21no—
260C2×Q8—22no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4#13512yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—82no—
294C22×C4—82no—
295C22×C4—161no—
296C22×C4C22×D411yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—22no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8#13512yes—
325C2×C8#13712yes—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—22no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—21no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—82no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—161no—
359C4⋊C4—161no—
360C4⋊C4#13512yes—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—82no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—41no—
370C42—41no—
371C42—41no—
372C42—161no—
373C42C2×#4111yescharacteristic
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C422−1+411yescharacteristic
377C23—22no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23D4211yescharacteristic
385Q8—82no—
386Q8—81no—
387Q8—81no—
388Q8—81no—
389Q8—81no—
390Q8—161no—
391D4—22no—
392D4—22no—
393D4—41no—
394D4—41no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—22no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—42no—
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—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—164no—
435C2×C4#52511yescharacteristic
436C2×C4#53211yescharacteristic
437C2×C4#54211yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#54811yescharacteristic
440C2×C4#55111yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×#13511yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4C2×#13711yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57811yescharacteristic
449C8—22no—
450C8—22no—
451C8—42no—
452C8—41no—
453C8—42no—
454C8—41no—
455C8—81no—
456C8—81no—
457C8—81no—
458C22—42no—
459C22—42no—
460C22—42no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22#246111yescharacteristic
466C22#327811yescharacteristic
467C22#328011yescharacteristic
468C22#328211yescharacteristic
469C22#328311yescharacteristic
470C22#329011yescharacteristic
471C22#329011yescharacteristic
472C4—21no—
473C4—21no—
474C4—42no—
475C4—81no—
476C4—81no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—82no—
481C4—81no—
482C4—161no—
483C4—161no—
484C4—161no—
485C4#310811yescharacteristic
486C4#311911yescharacteristic
487C4#316811yescharacteristic
488C4#317811yescharacteristic
489C4#333711yescharacteristic
490C4#334611yescharacteristic
491C4#336411yescharacteristic
492C4#336511yescharacteristic
493C2—82no—
494C2#1992111yescharacteristic
495C2#1995911yescharacteristic
496C2#1998811yescharacteristic
497C2#1998911yescharacteristic
498C2#2679111yescharacteristic
499C2#2681711yescharacteristic
500C2#2685111yescharacteristic
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
2. #26507C2semidirect
5. #10602C2semidirect
6. #10578C2semidirect
9. #9821C2semidirect
10. #9568C2semidirect
11. #9562C2semidirect
13. #9060C2semidirect
15. C8×#107C2semidirect

Maximal subgroups

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

subgroupindex
#277482
#265072
#262962
#229632
#106022
#105782
#104812
#100582
#98212
#95682
#95622
#95542
#90602
#88942
C8×#1072