Finite Group Databasebeta

← #286401

Subgroups — #286401

Order 512.

564 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27877C211yesmaximal, characteristic
2#27682C211yesmaximal, characteristic
3#24652C211yesmaximal, characteristic
4#23055C211yesmaximal, characteristic
5#10650C211yesmaximal, characteristic, complemented
6#10623C211yesmaximal, characteristic, complemented
7#10598C211yesmaximal, characteristic, complemented
8#10478C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9566C211yesmaximal, characteristic, complemented
11#9226C211yesmaximal, characteristic, complemented
12#9138C211yesmaximal, characteristic
13#8831C211yesmaximal, characteristic
14#8653C211yesmaximal, characteristic, complemented
15#8046C211yesmaximal, characteristic
16C2×#462C2211yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22#2285C2211yescharacteristic
23#2101—21no—
24#2097—21no—
25#2075—21no—
26#2075—21no—
27#2036—21no—
28#2001—21no—
29#1990C2211yescharacteristic
30#1990C2211yescharacteristic
31#1984C2211yescharacteristic
32#1981C2211yescharacteristic
33#1969—21no—
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1939C2211yescharacteristic
38#1933—21no—
39#1924—21no—
40#1923—21no—
41#1918—21no—
42#1918—21no—
43#1917—21no—
44#1873—21no—
45#1854—21no—
46#1845C2211yescharacteristic
47#1834C2211yescharacteristic
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×Q16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C2×#117—21no—
73C22×C42C2311yescharacteristic
74#462—22no—
75#462—41no—
76#446—22no—
77#446—41no—
78#432—22no—
79#432—41no—
80#431—22no—
81#431—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8D411yescharacteristic
105C2×C22⋊C8D411yescharacteristic
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391D411yescharacteristic
114#389—21no—
115#389—21no—
116#389D411yescharacteristic
117#387—21no—
118#385—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#382—21no—
123#380—21no—
124#380—21no—
125#380—21no—
126#380—21no—
127#380D411yescharacteristic
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133#340—21no—
134#340—21no—
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336D411yescharacteristic
151#336D411yescharacteristic
152#335—22no—
153#334—41no—
154#332—22no—
155#332—41no—
156C22×Q8—21no—
157C22×Q8C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C2×Q16—42no—
162C2×Q16—42no—
163C2×Q16—81no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C22⋊Q8—22no—
179C22⋊Q8—41no—
180C22⋊Q8—81no—
181C22⋊Q8—81no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186#117—81no—
187#117—81no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—21no—
216C2×C42—41no—
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—42no—
222#107—42no—
223#107—81no—
224#107—81no—
225#106—81no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—42no—
237C4⋊C8—42no—
238C4⋊C8#4112yes—
239C4⋊C8#4112yes—
240#103—42no—
241#103—42no—
242#103—81no—
243#103—81no—
244#103—81no—
245#103—81no—
246C22⋊C8—22no—
247C22⋊C8—41no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—42no—
251#97—42no—
252#97—42no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C4×C8—42no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—41no—
265#95—41no—
266#95—41no—
267#95—41no—
268C24#12111yescharacteristic
269C2×Q8—21no—
270C2×Q8—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311Q16—84no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8#13512yes—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—42no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—81no—
370C4⋊C4—82no—
371C4⋊C4#13712yes—
372C22⋊C4—42no—
373C22⋊C4—42no—
374C22⋊C4—82no—
375C22⋊C4—82no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—82no—
387C42C2×#4111yescharacteristic
388C42C2×#4111yescharacteristic
389C42H(5,𝔽2)11yescharacteristic
390C422−1+411yescharacteristic
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23C2×#12111yescharacteristic
399Q8—41no—
400Q8—41no—
401Q8—41no—
402Q8—41no—
403Q8—81no—
404Q8—81no—
405Q8—81no—
406Q8—81no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—22no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4#52911yescharacteristic
453C2×C4#53911yescharacteristic
454C2×C4#54611yescharacteristic
455C2×C4#54611yescharacteristic
456C2×C4#54711yescharacteristic
457C2×C4#54711yescharacteristic
458C2×C4C2×#13511yescharacteristic
459C2×C4C2×#13711yescharacteristic
460C2×C4#57511yescharacteristic
461C2×C4#57511yescharacteristic
462C2×C4#57611yescharacteristic
463C2×C4#57611yescharacteristic
464C2×C4#57811yescharacteristic
465C2×C4#57811yescharacteristic
466C8—22no—
467C8—41no—
468C8—41no—
469C8—41no—
470C8—42no—
471C8—42no—
472C8—81no—
473C8—81no—
474C8—81no—
475C8—81no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22#242511yescharacteristic
484C22#303511yescharacteristic
485C22#304411yescharacteristic
486C22#305611yescharacteristic
487C22#305811yescharacteristic
488C22#306711yescharacteristic
489C22#306911yescharacteristic
490C4—21no—
491C4—21no—
492C4—41no—
493C4—41no—
494C4—41no—
495C4—42no—
496C4—81no—
497C4—81no—
498C4—81no—
499C4—161no—
500C4—161no—
501C4—161no—
502C4—161no—
503C4#330011yescharacteristic
504C4#331511yescharacteristic
505C4#332211yescharacteristic
506C4#333311yescharacteristic
507C4#334211yescharacteristic
508C4#334911yescharacteristic
509C4#335511yescharacteristic
510C4#335711yescharacteristic
511C2—81no—
512C2#1918111yescharacteristic
513C2#1922511yescharacteristic
514C2#1947811yescharacteristic
515C2#1953011yescharacteristic
516C2#2520111yescharacteristic
517C2#2525811yescharacteristic
518C2#2526511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10650C2semidirect
6. #10623C2semidirect
7. #10598C2semidirect
8. #10478C2semidirect
9. #9616C2semidirect
10. #9566C2semidirect
11. #9226C2semidirect
14. #8653C2semidirect

Maximal subgroups

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

subgroupindex
#278772
#276822
#246522
#230552
#106502
#106232
#105982
#104782
#96162
#95662
#92262
#91382
#88312
#86532
#80462