Finite Group Databasebeta

← #275552

Subgroups — #275552

Order 512.

628 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29920C211yesmaximal, characteristic
2#27907C211yesmaximal, characteristic
3#22600C211yesmaximal, characteristic, complemented
4#21961C211yesmaximal, characteristic, complemented
5#10653C211yesmaximal, characteristic, complemented
6#10650C211yesmaximal, characteristic, complemented
7#10591C211yesmaximal, characteristic, complemented
8#10591C211yesmaximal, characteristic, complemented
9#9617C211yesmaximal, characteristic, complemented
10#9614C211yesmaximal, characteristic, complemented
11#9571C211yesmaximal, characteristic, complemented
12#9563C211yesmaximal, characteristic, complemented
13#9133C211yesmaximal, characteristic
14#8829C211yesmaximal, characteristic, complemented
15#8726C211yesmaximal, characteristic, complemented
16C2×#482—21no—
17C2×#482—21no—
18C2×#433—21no—
19C22×C4⋊C8C2211yescharacteristic
20#2856C2211yescharacteristic
21#2285C2211yescharacteristic
22#2285C2211yescharacteristic
23#2093—21no—
24#2092—21no—
25#2092—21no—
26#2071—21no—
27#2033—21no—
28#2033—21no—
29#1982C2211yescharacteristic
30#1982C2211yescharacteristic
31#1957C2211yescharacteristic
32#1957C2211yescharacteristic
33#1947—21no—
34#1947—21no—
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1947C2211yescharacteristic
38#1947C2211yescharacteristic
39#1947C2211yescharacteristic
40#1947C2211yescharacteristic
41#1939C2211yescharacteristic
42#1939C2211yescharacteristic
43#1917—21no—
44#1916—21no—
45#1845C2211yescharacteristic
46#1844C2211yescharacteristic
47#1818—21no—
48#1808—21no—
49#1808—21no—
50#1797C2211yescharacteristic
51#1797C2211yescharacteristic
52#1770C2211yescharacteristic
53C8×C4⋊C4C2211yescharacteristic
54C4×C22⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×#127C2311yescharacteristic
69C2×#121—21no—
70C2×#121—21no—
71C2×#121—21no—
72C2×#117—21no—
73C2×#117—21no—
74C22×C42C2311yescharacteristic
75#482—44no—
76#482—44no—
77#433—44no—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#107—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×C4⋊C8—21no—
92C2×C4⋊C8—21no—
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×#102—21no—
100C2×#102—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×#102—21no—
107C2×C22⋊C8—21no—
108C2×C22⋊C8—21no—
109C2×C22⋊C8D411yescharacteristic
110C2×C22⋊C8D411yescharacteristic
111C2×#97—21no—
112C2×C4×C8—21no—
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#390—21no—
116#390—21no—
117#390—21no—
118#384D411yescharacteristic
119#384D411yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#382—21no—
125#380—21no—
126#380—21no—
127#380—21no—
128#380D411yescharacteristic
129#380D411yescharacteristic
130C4×C4⋊C4—21no—
131C4×C4⋊C4—21no—
132C4×C4⋊C4—21no—
133C4×C4⋊C4C2311yescharacteristic
134C4×C4⋊C4C2311yescharacteristic
135#340—21no—
136#340—21no—
137#340—21no—
138#340—21no—
139#340—21no—
140#340—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336D411yescharacteristic
148#336D411yescharacteristic
149#335—22no—
150#335—22no—
151#331—22no—
152#331—22no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4—41no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D411yescharacteristic
159C23×C4#4111yescharacteristic
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—24no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—81no—
168C22×C8—81no—
169C22×C8C2×D411yescharacteristic
170C22×C8C2×D411yescharacteristic
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173#127—21no—
174#127—21no—
175#127—21no—
176#127—21no—
177#121—42no—
178#121—42no—
179#121—81no—
180#121—81no—
181#121—82no—
182#117—42no—
183#117—42no—
184#117—42no—
185#117—42no—
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—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4C2×D411yescharacteristic
200C2×C4⋊C4C2×D411yescharacteristic
201C2×C4⋊C4C2×D411yescharacteristic
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—41no—
206C2×C22⋊C4—41no—
207C2×C22⋊C4—41no—
208C2×C22⋊C4—41no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42—41no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42C2411yescharacteristic
220#107—42no—
221#107—42no—
222#107—81no—
223#107—81no—
224#107—81no—
225#107—81no—
226#107—81no—
227#107—81no—
228#106—42no—
229#106—42no—
230C4⋊C8—24no—
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—21no—
241C4⋊C8—42no—
242C4⋊C8—42no—
243C4⋊C8#4112yes—
244C4⋊C8#4112yes—
245#102—42no—
246#102—42no—
247#102—42no—
248#102—42no—
249#102—81no—
250#102—81no—
251#102—81no—
252#102—81no—
253C22⋊C8—22no—
254C22⋊C8—42no—
255C22⋊C8—81no—
256C22⋊C8D814yes—
257#97—42no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—42no—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—41no—
268#95—41no—
269C24—41no—
270C24#12111yescharacteristic
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—41no—
284C2×D4—41no—
285C2×D4—81no—
286C2×D4—81no—
287C2×D4—82no—
288C2×D4—82no—
289C2×D4—82no—
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—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4C2×D812yes—
315C22×C4AGL(1,ℤ/8)12yes—
316C22×C4C22×D411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
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—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—84no—
348C2×C8C2×D812yes—
349C2×C8C2×D812yes—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
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—81no—
377C22⋊C4—42no—
378C22⋊C4—42no—
379C22⋊C4—82no—
380C22⋊C4—82no—
381C22⋊C4—82no—
382C22⋊C4—82no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—41no—
390C42—41no—
391C42—41no—
392C42—41no—
393C42—82no—
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C422−1+411yescharacteristic
397C422−1+411yescharacteristic
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—21no—
404C23—81no—
405C23—81no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23—81no—
411C23#45912yes—
412C23C2×#12111yescharacteristic
413D4—81no—
414D4—81no—
415D4—81no—
416D4—81no—
417D4—81no—
418D4—81no—
419D4—81no—
420D4—81no—
421D4—161no—
422D4—161no—
423D4—164no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
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#45912yes—
466C2×C4#52511yescharacteristic
467C2×C4#53611yescharacteristic
468C2×C4#54711yescharacteristic
469C2×C4#54711yescharacteristic
470C2×C4#55211yescharacteristic
471C2×C4#55211yescharacteristic
472C2×C4C22×D811yescharacteristic
473C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
474C2×C4#57611yescharacteristic
475C2×C4#57611yescharacteristic
476C2×C4#57711yescharacteristic
477C2×C4#57811yescharacteristic
478C2×C4#57811yescharacteristic
479C2×C4#57811yescharacteristic
480C8—22no—
481C8—22no—
482C8—41no—
483C8—41no—
484C8—42no—
485C8—42no—
486C8—81no—
487C8—81no—
488C8—81no—
489C8—81no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—41no—
494C22—41no—
495C22—41no—
496C22—41no—
497C22—161no—
498C22—161no—
499C22—161no—
500C22—161no—
501C22—161no—
502C22—161no—
503C22—161no—
504C22#241311yescharacteristic
505C22C2×#45911yescharacteristic
506C22#304411yescharacteristic
507C22#306111yescharacteristic
508C22#306211yescharacteristic
509C22#306711yescharacteristic
510C22#306811yescharacteristic
511C4—22no—
512C4—21no—
513C4—21no—
514C4—41no—
515C4—41no—
516C4—81no—
517C4—81no—
518C4—81no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4#312111yescharacteristic
524C4#312211yescharacteristic
525C4#324111yescharacteristic
526C4#324311yescharacteristic
527C4#333411yescharacteristic
528C4#333411yescharacteristic
529C4#334311yescharacteristic
530C4#334311yescharacteristic
531C2—42no—
532C2—321no—
533C2#1919011yescharacteristic
534C2#1923011yescharacteristic
535C2#1953511yescharacteristic
536C2#1957511yescharacteristic
537C2#2457611yescharacteristic
538C2#2461511yescharacteristic
539C2#2462511yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radical#2856C22
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
3. #22600C2semidirect
4. #21961C2semidirect
5. #10653C2semidirect
6. #10650C2semidirect
7. #10591C2semidirect
8. #10591C2semidirect
9. #9617C2semidirect
10. #9614C2semidirect
11. #9571C2semidirect
12. #9563C2semidirect
14. #8829C2semidirect
15. #8726C2semidirect

Maximal subgroups

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

subgroupindex
#299202
#279072
#226002
#219612
#106532
#106502
#105912
#96172
#96142
#95712
#95632
#91332
#88292
#87262