Finite Group Databasebeta

← #351522

Subgroups — #351522

Order 512.

607 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33149C211yesmaximal, characteristic
2#33108C211yesmaximal, characteristic
3#29877C211yesmaximal, characteristic
4#20535C211yesmaximal, characteristic
5#9741C211yesmaximal, characteristic, complemented
6#9729C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9616C211yesmaximal, characteristic, complemented
9#9575C211yesmaximal, characteristic, complemented
10#9574C211yesmaximal, characteristic, complemented
11#9567C211yesmaximal, characteristic, complemented
12#9566C211yesmaximal, characteristic, complemented
13#9138C211yesmaximal, characteristic
14#9136C211yesmaximal, characteristic
15#8600C211yesmaximal, characteristic
16C2×#498C2211yescharacteristic
17C2×#483—21no—
18C2×#432C2211yescharacteristic
19C2×#431C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#2088—21no—
24#2088—21no—
25#2077—21no—
26#2077—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1987C2211yescharacteristic
30#1987C2211yescharacteristic
31#1981C2211yescharacteristic
32#1981C2211yescharacteristic
33#1948C2211yescharacteristic
34#1948C2211yescharacteristic
35#1947C2211yescharacteristic
36#1947C2211yescharacteristic
37#1927—21no—
38#1927—21no—
39#1921—21no—
40#1921—21no—
41#1872—21no—
42#1852—21no—
43#1834C2211yescharacteristic
44#1824—21no—
45#1797C2211yescharacteristic
46#1797C2211yescharacteristic
47#1771—21no—
48#1771—21no—
49#1770C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53#1749—21no—
54C4×C22⋊C8—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×C4⋊Q8C2311yescharacteristic
69C2×C22⋊Q8—21no—
70C2×C22⋊Q8—21no—
71C22×C4⋊C4D411yescharacteristic
72C22×C4⋊C4D411yescharacteristic
73C22×C42—21no—
74C22×C42C2311yescharacteristic
75#498—22no—
76#498—22no—
77#483—44no—
78#432—22no—
79#432—22no—
80#431—22no—
81#431—22no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8D411yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#103—21no—
97C2×#103—21no—
98C2×#103—21no—
99C2×#103—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8D411yescharacteristic
104C2×C22⋊C8D411yescharacteristic
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112#391—21no—
113#391—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#380—21no—
121#380—21no—
122#380—21no—
123#380—21no—
124C4×C4⋊C4—21no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4D411yescharacteristic
127C4×C4⋊C4D411yescharacteristic
128C4×C4⋊C4C2311yescharacteristic
129C4×C4⋊C4C2311yescharacteristic
130C2×#95—21no—
131#340—21no—
132#340—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#336—21no—
151#335—22no—
152#334—22no—
153#333—22no—
154#333—22no—
155#332—22no—
156#332—22no—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
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—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C22⋊Q8—42no—
183C22⋊Q8—42no—
184C22⋊Q8—82no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
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—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
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—21no—
216C2×C42—21no—
217C2×C42—81no—
218C2×C42—81no—
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—42no—
224#107—42no—
225#106—42no—
226#106—42no—
227#106—42no—
228#106—42no—
229C4⋊C8—22no—
230C4⋊C8—22no—
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#4112yes—
242C4⋊C8#4112yes—
243#103—42no—
244#103—42no—
245#103—42no—
246#103—42no—
247C22⋊C8—24no—
248C22⋊C8—22no—
249C22⋊C8—22no—
250C22⋊C8—42no—
251C22⋊C8—42no—
252#97—24no—
253#97—81no—
254#97—81no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—81no—
258C4×C8—81no—
259C4×C8#4112yes—
260C4×C8#4112yes—
261#95—21no—
262#95—21no—
263#95—21no—
264#95—21no—
265#95—41no—
266#95—41no—
267#95—41no—
268#95—41no—
269C24#12111yescharacteristic
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—81no—
279C2×Q8—81no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—22no—
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—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4#12111yescharacteristic
311C22×C4#12111yescharacteristic
312C22×C4#12111yescharacteristic
313C22×C4C22×D411yescharacteristic
314C22×C4C22×D411yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C2×C8—21no—
317C2×C8—21no—
318C2×C8—21no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
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—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8#12312yes—
345C2×C8#12312yes—
346C4⋊C4—21no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—42no—
353C4⋊C4—42no—
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—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—81no—
371C4⋊C4—81no—
372C4⋊C4—82no—
373C4⋊C4—82no—
374C4⋊C4AGL(1,ℤ/8)12yes—
375C4⋊C4#13712yes—
376C22⋊C4—82no—
377C22⋊C4—82no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—81no—
389C42—81no—
390C42C2×#4111yescharacteristic
391C42C2×#4111yescharacteristic
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23#47112yes—
401C23C2×#12111yescharacteristic
402Q8—81no—
403Q8—81no—
404Q8—81no—
405Q8—81no—
406Q8—161no—
407Q8—161no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—42no—
422C2×C4—42no—
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—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—84no—
442C2×C4—84no—
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—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4C2×#12311yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4#53211yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
462C2×C4C2×#13711yescharacteristic
463C2×C4#57511yescharacteristic
464C2×C4#57511yescharacteristic
465C2×C4#57711yescharacteristic
466C2×C4#57711yescharacteristic
467C2×C4#57811yescharacteristic
468C2×C4#57811yescharacteristic
469C8—22no—
470C8—22no—
471C8—41no—
472C8—41no—
473C8—42no—
474C8—42no—
475C8—81no—
476C8—81no—
477C8—81no—
478C8—81no—
479C22—22no—
480C22—22no—
481C22—22no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22#285411yescharacteristic
487C22#304211yescharacteristic
488C22#304411yescharacteristic
489C22#305511yescharacteristic
490C22#305611yescharacteristic
491C22C2×#47111yescharacteristic
492C22#306711yescharacteristic
493C4—21no—
494C4—21no—
495C4—42no—
496C4—42no—
497C4—41no—
498C4—41no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4—161no—
504C4—161no—
505C4—161no—
506C4#309711yescharacteristic
507C4#309711yescharacteristic
508C4#312011yescharacteristic
509C4#312011yescharacteristic
510C4#317111yescharacteristic
511C4#317211yescharacteristic
512C4#318211yescharacteristic
513C4#318311yescharacteristic
514C2—42no—
515C2#1946211yescharacteristic
516C2#1946611yescharacteristic
517C2#1947511yescharacteristic
518C2#1947911yescharacteristic
519C2#3311911yescharacteristic
520C2#3312211yescharacteristic
521C2#3314511yescharacteristic
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. #9741C2semidirect
6. #9729C2semidirect
7. #9622C2semidirect
8. #9616C2semidirect
9. #9575C2semidirect
10. #9574C2semidirect
11. #9567C2semidirect
12. #9566C2semidirect

Maximal subgroups

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

subgroupindex
#331492
#331082
#298772
#205352
#97412
#97292
#96222
#96162
#95752
#95742
#95672
#95662
#91382
#91362
#86002