Finite Group Databasebeta

← #279127

Subgroups — #279127

Order 512.

612 conjugacy classes of subgroups.

The subgroup lattice has 535 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29876C211yesmaximal, characteristic
2#24846C211yesmaximal, characteristic
3#24702C211yesmaximal, characteristic
4#22102C211yesmaximal, characteristic
5C2×#1696C211yesmaximal, characteristic, complemented
6C2×#1574C211yesmaximal, characteristic
7#10601C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#10483C211yesmaximal, characteristic, complemented
10#10482C211yesmaximal, characteristic, complemented
11#9730C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9135C211yesmaximal, characteristic
14#9128C211yesmaximal, characteristic
15#8655C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#477C2211yescharacteristic
18C2×#477C2211yescharacteristic
19C2×#446C2211yescharacteristic
20C2×#431C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C2×#423C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×C22⋊Q8C2211yescharacteristic
26#2086—21no—
27#2086—21no—
28#2086—21no—
29#2076—21no—
30#2075—21no—
31#2075—21no—
32#2066—21no—
33#2036—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1984C2211yescharacteristic
39#1948C2211yescharacteristic
40#1947C2211yescharacteristic
41#1942—21no—
42#1939—21no—
43#1939C2211yescharacteristic
44#1933—21no—
45#1924—21no—
46#1923—21no—
47#1917—21no—
48#1894—21no—
49#1834C2211yescharacteristic
50#1809—21no—
51#1803C2211yescharacteristic
52#1803C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1696—24no—
56#1574—24no—
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#333C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×M16—21no—
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8D411yescharacteristic
75C2×C4×Q8C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#500—22no—
78#500—22no—
79#477—22no—
80#477—22no—
81#477—41no—
82#477—41no—
83#446—22no—
84#446—22no—
85#431—22no—
86#431—22no—
87#431—22no—
88#431—41no—
89C2×#107—21no—
90C2×#107—21no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95#423—22no—
96#423—41no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8D411yescharacteristic
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×C22⋊C8—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8D411yescharacteristic
115C2×C22⋊C8D411yescharacteristic
116C2×#97—21no—
117C2×C4×C8—21no—
118C2×C4×C8—21no—
119C2×C4×C8C2311yescharacteristic
120C2×C4×C8C2311yescharacteristic
121#391—21no—
122#391—21no—
123#391D411yescharacteristic
124#389—21no—
125#389—21no—
126#389—21no—
127#389—21no—
128#389D411yescharacteristic
129#387—21no—
130#385—21no—
131#384C2311yescharacteristic
132#384C2311yescharacteristic
133#380—21no—
134#380—21no—
135#380—21no—
136#380—21no—
137#380D411yescharacteristic
138C4×C4⋊C4—21no—
139C4×C4⋊C4C2311yescharacteristic
140C4×C22⋊C4—21no—
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#340—21no—
154#340—21no—
155#336—21no—
156#336—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334D414yes—
164#333—22no—
165#333—22no—
166#333—41no—
167#333—41no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×Q8C2×D411yescharacteristic
171C23×C4C2×D411yescharacteristic
172C23×C4C2×D411yescharacteristic
173C23×C4#4111yescharacteristic
174C2×M16—42no—
175C2×M16—42no—
176C2×M16—81no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8C2×D411yescharacteristic
184C22×C8C2×D411yescharacteristic
185C22×C8C2×D411yescharacteristic
186C22×C8C2×D411yescharacteristic
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C4⋊Q8—21no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C22⋊Q8—22no—
196C22⋊Q8—41no—
197C22⋊Q8—82no—
198C22⋊Q8—82no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—21no—
202C4×Q8—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C4⋊C4C2×D411yescharacteristic
219C2×C4⋊C4C2×D411yescharacteristic
220C2×C4⋊C4C2×D411yescharacteristic
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—41no—
224C2×C22⋊C4—41no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—21no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42C2×D411yescharacteristic
232C2×C42C2×D411yescharacteristic
233C2×C42#4111yescharacteristic
234C2×C42C2411yescharacteristic
235#107—42no—
236#107—42no—
237#107—42no—
238#107—42no—
239#107—42no—
240#106—81no—
241C4⋊C8—22no—
242C4⋊C8—22no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—42no—
254C4⋊C8—42no—
255C4⋊C8—81no—
256C4⋊C8C2×D412yes—
257C4⋊C8C2×D412yes—
258#103—42no—
259#103—42no—
260#103—42no—
261#103—81no—
262C22⋊C8—22no—
263C22⋊C8—22no—
264C22⋊C8—42no—
265C22⋊C8—81no—
266#97—42no—
267C4×C8—21no—
268C4×C8—21no—
269C4×C8—42no—
270C4×C8—81no—
271C4×C8C2×D412yes—
272C4×C8#4112yes—
273#95—21no—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—41no—
279#95—41no—
280#95—41no—
281#95—41no—
282C24#12111yescharacteristic
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—21no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—81no—
294C2×Q8—81no—
295C2×Q8—81no—
296C2×Q8—81no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—22no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4#12111yescharacteristic
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4C22×D411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C2×#4111yescharacteristic
327M16—84no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
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#12112yes—
357C2×C8#12112yes—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—42no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—41no—
378C4⋊C4—41no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—81no—
384C4⋊C4—81no—
385C4⋊C4—82no—
386C4⋊C4—82no—
387C4⋊C4#13712yes—
388C22⋊C4—42no—
389C22⋊C4—42no—
390C22⋊C4—82no—
391C22⋊C4—82no—
392C42—21no—
393C42—21no—
394C42—21no—
395C42—21no—
396C42—21no—
397C42—21no—
398C42—41no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42C22×D411yescharacteristic
403C42C2×#4111yescharacteristic
404C42H(5,𝔽2)11yescharacteristic
405C422−1+411yescharacteristic
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23C2×#12111yescharacteristic
414Q8—41no—
415Q8—41no—
416Q8—41no—
417Q8—41no—
418Q8—161no—
419Q8—161no—
420Q8—161no—
421Q8—161no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4#47212yes—
467C2×C4C2×#12111yescharacteristic
468C2×C4#53711yescharacteristic
469C2×C4D4211yescharacteristic
470C2×C4#54711yescharacteristic
471C2×C4#54811yescharacteristic
472C2×C4#55311yescharacteristic
473C2×C4C2×#13711yescharacteristic
474C2×C4#57511yescharacteristic
475C2×C4#57711yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C2×C4#57811yescharacteristic
479C2×C4#57811yescharacteristic
480C2×C4#57811yescharacteristic
481C8—22no—
482C8—22no—
483C8—42no—
484C8—42no—
485C8—41no—
486C8—41no—
487C8—81no—
488C8—81no—
489C8—81no—
490C8—81no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—41no—
498C22#241611yescharacteristic
499C22#304411yescharacteristic
500C22#304611yescharacteristic
501C22#305611yescharacteristic
502C22#306311yescharacteristic
503C22C2×#47211yescharacteristic
504C22#306711yescharacteristic
505C4—21no—
506C4—21no—
507C4—22no—
508C4—42no—
509C4—41no—
510C4—41no—
511C4—81no—
512C4—81no—
513C4—81no—
514C4—161no—
515C4—161no—
516C4—161no—
517C4—161no—
518C4#305911yescharacteristic
519C4#306311yescharacteristic
520C4#325811yescharacteristic
521C4#326111yescharacteristic
522C4#327511yescharacteristic
523C4#327611yescharacteristic
524C4#335211yescharacteristic
525C4#335611yescharacteristic
526C2—81no—
527C2#1947811yescharacteristic
528C2#1949111yescharacteristic
529C2#1957311yescharacteristic
530C2#1958711yescharacteristic
531C2#2476411yescharacteristic
532C2#2479311yescharacteristic
533C2#2480311yescharacteristic
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. C2×#1696C2semidirect
7. #10601C2semidirect
8. #10589C2semidirect
9. #10483C2semidirect
10. #10482C2semidirect
11. #9730C2semidirect
12. #9613C2semidirect
15. #8655C2semidirect

Maximal subgroups

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

subgroupindex
#298762
#248462
#247022
#221022
C2×#16962
C2×#15742
#106012
#105892
#104832
#104822
#97302
#96132
#91352
#91282
#86552