Finite Group Databasebeta

← #285849

Subgroups — #285849

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#27972C211yesmaximal, characteristic
2#27680C211yesmaximal, characteristic
3#24845C211yesmaximal, characteristic
4#23050C211yesmaximal, characteristic
5#10656C211yesmaximal, characteristic, complemented
6#10624C211yesmaximal, characteristic, complemented
7#10600C211yesmaximal, characteristic, complemented
8#10487C211yesmaximal, characteristic, complemented
9#9616C211yesmaximal, characteristic, complemented
10#9562C211yesmaximal, characteristic, complemented
11#9220C211yesmaximal, characteristic, complemented
12#9132C211yesmaximal, characteristic
13#8832C211yesmaximal, characteristic
14#8649C211yesmaximal, characteristic, complemented
15#8043C211yesmaximal, characteristic
16C2×#475C2211yescharacteristic
17C2×C8×Q8C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C2×#423C2211yescharacteristic
20#2857C2211yescharacteristic
21C4×C22⋊Q8C2211yescharacteristic
22C2×#384C2211yescharacteristic
23#2099—21no—
24#2096—21no—
25#2088—21no—
26#2072—21no—
27#2072—21no—
28#2068—21no—
29#1988C2211yescharacteristic
30#1984C2211yescharacteristic
31#1956C2211yescharacteristic
32#1956C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1939C2211yescharacteristic
37#1923—21no—
38#1923—21no—
39#1916—21no—
40#1894—21no—
41#1872—21no—
42#1872—21no—
43#1872—21no—
44#1853—21no—
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1809—21no—
48#1803C2211yescharacteristic
49#1803C2211yescharacteristic
50#1803C2211yescharacteristic
51#1797C2211yescharacteristic
52C8×C4⋊C4—21no—
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8D411yescharacteristic
71C2×C4×Q8C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#475—22no—
77#475—41no—
78C8×Q8—22no—
79C8×Q8—41no—
80#431—22no—
81#431—41no—
82C2×#107—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86#423—22no—
87#423—41no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#103—21no—
103C2×#103—21no—
104C2×#103—21no—
105C2×C22⋊C8—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8D411yescharacteristic
108C2×C22⋊C8D411yescharacteristic
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114#391—21no—
115#391D411yescharacteristic
116#389—21no—
117#389—21no—
118#389D411yescharacteristic
119#387—21no—
120#385—21no—
121#384—21no—
122#384—21no—
123#384—21no—
124#384C2311yescharacteristic
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127#380—21no—
128#380—21no—
129#380D411yescharacteristic
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#335—41no—
154#332—22no—
155#332—41no—
156C22×Q8—21no—
157C22×Q8C2×D411yescharacteristic
158C23×C4—41no—
159C23×C4C2×D411yescharacteristic
160C23×C4C2×D411yescharacteristic
161C23×C4#4111yescharacteristic
162C2×M16—42no—
163C2×M16—42no—
164C2×M16—81no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—21no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C22⋊Q8—22no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C22⋊Q8Q1614yes—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×Q8—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—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—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4—81no—
206C2×C4⋊C4—81no—
207C2×C4⋊C4—81no—
208C2×C4⋊C4—81no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4C2×D411yescharacteristic
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C22⋊C4—21no—
214C2×C22⋊C4—21no—
215C2×C22⋊C4—41no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42C2411yescharacteristic
226#107—81no—
227#106—42no—
228#106—81no—
229#106—81no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—42no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8#4112yes—
244C4⋊C8#4112yes—
245#103—42no—
246#103—81no—
247#103—81no—
248#103—81no—
249#103—81no—
250#103—81no—
251#103—81no—
252#103—81no—
253C22⋊C8—41no—
254C22⋊C8—41no—
255C22⋊C8—42no—
256C22⋊C8—42no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—42no—
262C4×C8—42no—
263C4×C8—42no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271C24#12111yescharacteristic
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8C2×Q1612yes—
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—22no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4#12111yescharacteristic
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4C2×Q1612yes—
313C22×C4#13512yes—
314C22×C4C22×D411yescharacteristic
315C22×C4C22×D411yescharacteristic
316C22×C4C2×#4111yescharacteristic
317M16—84no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—21no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
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—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C4⋊C4—21no—
349C4⋊C4—24no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—41no—
355C4⋊C4—41no—
356C4⋊C4—42no—
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—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—81no—
374C4⋊C4—81no—
375C4⋊C4—81no—
376C4⋊C4—82no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C4⋊C4—82no—
380C4⋊C4C2×Q1612yes—
381C22⋊C4—24no—
382C22⋊C4—81no—
383C22⋊C4—82no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C42H(5,𝔽2)11yescharacteristic
397C422−1+411yescharacteristic
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23#46712yes—
405C23C2×#12111yescharacteristic
406Q8—22no—
407Q8—22no—
408Q8—41no—
409Q8—41no—
410Q8—81no—
411Q8—81no—
412Q8—81no—
413Q8—81no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
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—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—164no—
458C2×C4#46712yes—
459C2×C4#47112yes—
460C2×C4#52911yescharacteristic
461C2×C4#53911yescharacteristic
462C2×C4#54611yescharacteristic
463C2×C4#54611yescharacteristic
464C2×C4#54711yescharacteristic
465C2×C4#54711yescharacteristic
466C2×C4C22×Q1611yescharacteristic
467C2×C4C2×#13511yescharacteristic
468C2×C4#57511yescharacteristic
469C2×C4#57511yescharacteristic
470C2×C4#57611yescharacteristic
471C2×C4#57711yescharacteristic
472C2×C4#57711yescharacteristic
473C2×C4#57711yescharacteristic
474C8—41no—
475C8—41no—
476C8—42no—
477C8—41no—
478C8—42no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C22—22no—
485C22—22no—
486C22—22no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22#242511yescharacteristic
492C22#304011yescharacteristic
493C22#304311yescharacteristic
494C22C2×#46711yescharacteristic
495C22#306211yescharacteristic
496C22C2×#47111yescharacteristic
497C22#306911yescharacteristic
498C4—22no—
499C4—21no—
500C4—21no—
501C4—22no—
502C4—42no—
503C4—41no—
504C4—42no—
505C4—81no—
506C4—81no—
507C4—161no—
508C4—161no—
509C4—161no—
510C4—161no—
511C4#329711yescharacteristic
512C4#329911yescharacteristic
513C4#330711yescharacteristic
514C4#332611yescharacteristic
515C4#333711yescharacteristic
516C4#333811yescharacteristic
517C4#335011yescharacteristic
518C4#335111yescharacteristic
519C2—42no—
520C2#1927711yescharacteristic
521C2#1938311yescharacteristic
522C2#1939911yescharacteristic
523C2#1956411yescharacteristic
524C2#2523111yescharacteristic
525C2#2523411yescharacteristic
526C2#2526811yescharacteristic
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. #10656C2semidirect
6. #10624C2semidirect
7. #10600C2semidirect
8. #10487C2semidirect
9. #9616C2semidirect
10. #9562C2semidirect
11. #9220C2semidirect
14. #8649C2semidirect

Maximal subgroups

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

subgroupindex
#279722
#276802
#248452
#230502
#106562
#106242
#106002
#104872
#96162
#95622
#92202
#91322
#88322
#86492
#80432