Finite Group Databasebeta

← #227560

Subgroups — #227560

Order 512.

707 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23067C211yesmaximal, characteristic
2#22971C211yesmaximal, characteristic, complemented
3#22544C211yesmaximal, characteristic
4#22490C211yesmaximal, characteristic
5C2×#1638C211yesmaximal, characteristic
6#9833C211yesmaximal, characteristic, complemented
7#9574C211yesmaximal, characteristic, complemented
8#9570C211yesmaximal, characteristic, complemented
9#9224C211yesmaximal, characteristic, complemented
10#9215C211yesmaximal, characteristic
11#9060C211yesmaximal, characteristic, complemented
12#8885C211yesmaximal, characteristic
13#8884C211yesmaximal, characteristic, complemented
14#8209C211yesmaximal, characteristic, complemented
15#8202C211yesmaximal, characteristic
16C2×#556C2211yescharacteristic
17C2×#487C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#476C2211yescharacteristic
20C2×#446C2211yescharacteristic
21C2×C8×D4C2211yescharacteristic
22#2369C2211yescharacteristic
23#2354C2211yescharacteristic
24Q8×C4⋊C4C2211yescharacteristic
25#1990C2211yescharacteristic
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1988C2211yescharacteristic
29#1983—21no—
30#1981C2211yescharacteristic
31#1959—21no—
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1947C2211yescharacteristic
35#1943—21no—
36#1936—21no—
37#1925—21no—
38#1919—21no—
39#1918—21no—
40#1892—21no—
41#1873—21no—
42#1865—21no—
43#1845C2211yescharacteristic
44#1825—21no—
45#1803—21no—
46#1803C2211yescharacteristic
47#1802C2211yescharacteristic
48#1776—21no—
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51#1763—21no—
52#1763—21no—
53C4×#103C2211yescharacteristic
54C4×#102C2211yescharacteristic
55#1735—21no—
56#1638C418yescomplemented
57C2×#333C2211yescharacteristic
58C2×#332C2211yescharacteristic
59C2×Q8⋊C8C2211yescharacteristic
60C2×D4⋊C8C2211yescharacteristic
61C2×#322C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C23×C8—21no—
69#556—22no—
70#556—22no—
71#556—41no—
72#556C2×C414yes—
73C2×C4⋊Q8C2311yescharacteristic
74C2×C4⋊Q8C2311yescharacteristic
75C2×#126—41no—
76C2×#124—21no—
77C2×C22⋊Q8—21no—
78C2×C4×Q8—21no—
79C2×C4×Q8—21no—
80C2×C4×Q8C2311yescharacteristic
81C2×C4×D4C2311yescharacteristic
82C2×#117—21no—
83C2×#117—21no—
84#487—41no—
85#487C2×C414yes—
86#477—22no—
87#477C2×C414yes—
88#476—22no—
89#476C2×C414yes—
90#446—22no—
91#446—41no—
92C8×D4—22no—
93C8×D4—41no—
94C2×#107—21no—
95C2×#107—21no—
96C2×#107—41no—
97C2×#106—21no—
98C2×#106—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#103—21no—
105C2×#103—21no—
106C2×#103—21no—
107C2×#103—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—41no—
111C2×#103D411yescharacteristic
112C2×#103D411yescharacteristic
113C2×#102—21no—
114C2×#102—21no—
115C2×#102D411yescharacteristic
116C2×#102D411yescharacteristic
117C2×C22⋊C8—21no—
118C2×#97—21no—
119C2×#97—21no—
120C2×#97—21no—
121C2×C4×C8C2311yescharacteristic
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—41no—
125#390—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—41no—
130#388—41no—
131#384—21no—
132#384C2311yescharacteristic
133#384C2311yescharacteristic
134#380—21no—
135C4×C4⋊C4—21no—
136C4×C4⋊C4—21no—
137C4×C4⋊C4—21no—
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C4⋊C4C2311yescharacteristic
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340D411yescharacteristic
154#340D411yescharacteristic
155#340D411yescharacteristic
156#340D411yescharacteristic
157#336—21no—
158#336—21no—
159#333—41no—
160#332C2×C414yes—
161Q8⋊C8C2×C414yes—
162D4⋊C8C2×C414yes—
163#322—41no—
164C22×Q8—21no—
165C22×Q8—21no—
166C22×Q8C2×D411yescharacteristic
167C22×D4C2×D411yescharacteristic
168C23×C4—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—42no—
175C22×C8—42no—
176C22×C8—81no—
177C22×C8C2×D411yescharacteristic
178C22×C8C2×D411yescharacteristic
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8C22×C412yes—
188C4⋊Q8C22×C412yes—
189#126—44no—
190#126—44no—
191#124—24no—
192#124—42no—
193C22⋊Q8—24no—
194C22⋊Q8—42no—
195C4×Q8—21no—
196C4×Q8—21no—
197C4×Q8—24no—
198C4×Q8—42no—
199C4×Q8—42no—
200C4×Q8—81no—
201C4×Q8C22×C412yes—
202C4×Q8#4112yes—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4C22×C412yes—
206C4×D4#4112yes—
207#117—24no—
208#117—42no—
209#117—42no—
210#117—81no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4—41no—
224C2×C4⋊C4—41no—
225C2×C4⋊C4—41no—
226C2×C4⋊C4C2×D411yescharacteristic
227C2×C4⋊C4C2×D411yescharacteristic
228C2×C4⋊C4C2×D411yescharacteristic
229C2×C4⋊C4C2×D411yescharacteristic
230C2×C4⋊C4C2×D411yescharacteristic
231C2×C4⋊C4C2×D411yescharacteristic
232C2×C22⋊C4—21no—
233C2×C22⋊C4—41no—
234C2×C22⋊C4—41no—
235C2×C22⋊C4—41no—
236C2×C22⋊C4—41no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—41no—
241C2×C42—41no—
242C2×C42—41no—
243C2×C42—41no—
244C2×C42C2411yescharacteristic
245#107—24no—
246#107—42no—
247#107—82no—
248#106—24no—
249#106—81no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—42no—
257C4⋊C8C22×C412yes—
258C4⋊C8C22×C412yes—
259#103—24no—
260#103—22no—
261#103—44no—
262#103—41no—
263#103—42no—
264#103—42no—
265#103—81no—
266#103—81no—
267#103—81no—
268#102—24no—
269#102—41no—
270#102—41no—
271#102—81no—
272C22⋊C8—42no—
273#97—42no—
274#97—42no—
275#97—81no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—21no—
279C4×C8—21no—
280C4×C8—21no—
281C4×C8C22×C412yes—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288C24—21no—
289C2×Q8—21no—
290C2×Q8—22no—
291C2×Q8—22no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×Q8—21no—
295C2×Q8—21no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8—41no—
302C2×Q8—81no—
303C2×Q8—81no—
304C2×Q8C4×D412yes—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4C4×D412yes—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D411yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8—41no—
353C2×C8—41no—
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—84no—
358C2×C8#13712yes—
359C4⋊C4—22no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—21no—
374C4⋊C4—21no—
375C4⋊C4—44no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—81no—
389C4⋊C4—81no—
390C4⋊C4—82no—
391C4⋊C4—161no—
392C4⋊C4—161no—
393C4⋊C4C4×D412yes—
394C4⋊C4C4×D412yes—
395C22⋊C4—22no—
396C22⋊C4—22no—
397C22⋊C4—44no—
398C22⋊C4—44no—
399C22⋊C4—82no—
400C22⋊C4—161no—
401C42—22no—
402C42—22no—
403C42—44no—
404C42—41no—
405C42—41no—
406C42—41no—
407C42—41no—
408C42—82no—
409C42—82no—
410C42—161no—
411C42C23×C411yescharacteristic
412C42C2×#4111yescharacteristic
413C42C2×#4111yescharacteristic
414C42H(5,𝔽2)11yescharacteristic
415C23—22no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—41no—
422C23D4211yescharacteristic
423Q8—22no—
424Q8—22no—
425Q8—41no—
426Q8—41no—
427Q8—42no—
428Q8—42no—
429Q8—82no—
430Q8—81no—
431Q8—81no—
432Q8—161no—
433D4—22no—
434D4—22no—
435D4—41no—
436D4—41no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—41no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4C2×C4×D411yescharacteristic
478C2×C4#51811yescharacteristic
479C2×C4#54611yescharacteristic
480C2×C4#54611yescharacteristic
481C2×C4#55111yescharacteristic
482C2×C4C2×#13711yescharacteristic
483C2×C4#57511yescharacteristic
484C2×C4#57511yescharacteristic
485C2×C4#57511yescharacteristic
486C2×C4#57611yescharacteristic
487C2×C4#57711yescharacteristic
488C2×C4#57811yescharacteristic
489C2×C4#57811yescharacteristic
490C8—22no—
491C8—42no—
492C8—42no—
493C8—42no—
494C8—41no—
495C8—41no—
496C8—41no—
497C8—81no—
498C8—81no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22—81no—
503C22—81no—
504C22—81no—
505C22—81no—
506C22#235811yescharacteristic
507C22#326911yescharacteristic
508C22#327111yescharacteristic
509C22#327311yescharacteristic
510C22#327611yescharacteristic
511C22#328811yescharacteristic
512C22#328911yescharacteristic
513C4—21no—
514C4—21no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—81no—
519C4—81no—
520C4—82no—
521C4—82no—
522C4—81no—
523C4—81no—
524C4—161no—
525C4—161no—
526C4#294311yescharacteristic
527C4C4×#13711yescharacteristic
528C4#294711yescharacteristic
529C4#295111yescharacteristic
530C4#331311yescharacteristic
531C4#331411yescharacteristic
532C4#331511yescharacteristic
533C4#331711yescharacteristic
534C2—82no—
535C2#2022311yescharacteristic
536C2#2022911yescharacteristic
537C2#2027411yescharacteristic
538C2#2027711yescharacteristic
539C2#2286111yescharacteristic
540C2#2286211yescharacteristic
541C2#2289311yescharacteristic
subgroupG/H
center, socleC23D42
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ 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. #22971C2semidirect
6. #9833C2semidirect
7. #9574C2semidirect
8. #9570C2semidirect
9. #9224C2semidirect
11. #9060C2semidirect
13. #8884C2semidirect
14. #8209C2semidirect
56. #1638C4semidirect

Maximal subgroups

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

subgroupindex
#230672
#229712
#225442
#224902
C2×#16382
#98332
#95742
#95702
#92242
#92152
#90602
#88852
#88842
#82092
#82022