Finite Group Databasebeta

← #351240

Subgroups — #351240

Order 512.

585 conjugacy classes of subgroups.

The subgroup lattice has 507 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30228C211yesmaximal, characteristic, complemented
2#29995C211yesmaximal, characteristic
3#29868C211yesmaximal, characteristic
4#20562C211yesmaximal, characteristic
5#9901C211yesmaximal, characteristic, complemented
6#9691C211yesmaximal, characteristic, complemented
7#9576C211yesmaximal, characteristic, complemented
8#9575C211yesmaximal, characteristic, complemented
9#9568C211yesmaximal, characteristic, complemented
10#9567C211yesmaximal, characteristic, complemented
11#9223C211yesmaximal, characteristic
12#9223C211yesmaximal, characteristic
13#9215C211yesmaximal, characteristic
14#9215C211yesmaximal, characteristic
15#8789C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#436C2211yescharacteristic
19C2×C8×D4C2211yescharacteristic
20#2868C2211yescharacteristic
21#2859C2211yescharacteristic
22#2293C2211yescharacteristic
23#2099—21no—
24#2099—21no—
25#2099—21no—
26#2099—21no—
27#2088—21no—
28#2087—21no—
29#2084—21no—
30#2084—21no—
31#2071—21no—
32#2071—21no—
33#2067—21no—
34#2065—21no—
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1990C2211yescharacteristic
38#1990C2211yescharacteristic
39#1987C2211yescharacteristic
40#1987C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1842C2211yescharacteristic
44#1831C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1770C2211yescharacteristic
48C8×C4⋊C4C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50#1753—21no—
51C8×C22⋊C4—21no—
52#1750—21no—
53#1744C2211yescharacteristic
54#1729—21no—
55C2×#333C2211yescharacteristic
56C2×#333C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C23×C8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×#125—21no—
74C2×C4×D4C2311yescharacteristic
75C2×#117—21no—
76#476—22no—
77#476—22no—
78#476—22no—
79#476—22no—
80#436—22no—
81#436—22no—
82C8×D4—22no—
83C8×D4—22no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×#97—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8C2311yescharacteristic
106C2×C4×C8C2311yescharacteristic
107#390D411yescharacteristic
108#390D411yescharacteristic
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389D411yescharacteristic
114#389D411yescharacteristic
115#388—21no—
116#388—21no—
117#384—21no—
118#384—21no—
119#384—21no—
120#384—21no—
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124C4×C4⋊C4C2311yescharacteristic
125C4×C22⋊C4—21no—
126C43C2311yescharacteristic
127#340—21no—
128#340—21no—
129#340—21no—
130#340—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#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#336D411yescharacteristic
151#336D411yescharacteristic
152#333—22no—
153#333—22no—
154#333—22no—
155#333—22no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C2×M16—24no—
161C2×M16—81no—
162C2×M16—81no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—24no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—81no—
171C22×C8—81no—
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—21no—
182C4⋊Q8—21no—
183C4⋊Q8—21no—
184C4⋊Q8—82no—
185C4⋊Q8—82no—
186#125—82no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4#4112yes—
190C4×D4#4112yes—
191#117—81no—
192#117—81no—
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—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4C2×D411yescharacteristic
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C22⋊C4—21no—
215C2×C22⋊C4—21no—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—21no—
219C2×C42—21no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42#4111yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42C2411yescharacteristic
227#107—81no—
228#106—42no—
229#106—42no—
230#106—81no—
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—81no—
242C4⋊C8—81no—
243C4⋊C8#4112yes—
244C4⋊C8#4112yes—
245#102—42no—
246#102—42no—
247#102—81no—
248#102—81no—
249C22⋊C8—24no—
250C22⋊C8—24no—
251#97—81no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—81no—
257C4×C8—81no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—41no—
262#95—41no—
263C24—21no—
264C2×Q8—41no—
265C2×Q8—41no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—81no—
269C2×Q8—81no—
270C2×Q8—81no—
271C2×Q8—81no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4#12412yes—
278C22×C4—21no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C2×#4111yescharacteristic
298M16—84no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—84no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—41no—
349C4⋊C4—41no—
350C4⋊C4—41no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—81no—
354C4⋊C4—81no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4#12412yes—
362C22⋊C4—22no—
363C22⋊C4—22no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—81no—
367C22⋊C4—81no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—22no—
377C42—22no—
378C42—41no—
379C42—41no—
380C42C2×#4111yescharacteristic
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C42C2×#4111yescharacteristic
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23C2×#12111yescharacteristic
392Q8—161no—
393Q8—161no—
394Q8—161no—
395Q8—161no—
396D4—22no—
397D4—22no—
398D4—41no—
399D4—41no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4#47112yes—
438C2×C4#47212yes—
439C2×C4#52511yescharacteristic
440C2×C4#52511yescharacteristic
441C2×C4C2×#12411yescharacteristic
442C2×C4#52911yescharacteristic
443C2×C4#52911yescharacteristic
444C2×C4#53211yescharacteristic
445C2×C4#57711yescharacteristic
446C2×C4#57711yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57711yescharacteristic
450C2×C4#57811yescharacteristic
451C2×C4#57811yescharacteristic
452C2×C4#57811yescharacteristic
453C8—41no—
454C8—41no—
455C8—41no—
456C8—41no—
457C8—42no—
458C8—42no—
459C8—81no—
460C8—81no—
461C8—81no—
462C8—81no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22#285011yescharacteristic
471C22#304211yescharacteristic
472C22#304611yescharacteristic
473C22#306311yescharacteristic
474C22#306311yescharacteristic
475C22C2×#47111yescharacteristic
476C22C2×#47211yescharacteristic
477C4—21no—
478C4—21no—
479C4—22no—
480C4—22no—
481C4—42no—
482C4—41no—
483C4—41no—
484C4—81no—
485C4—81no—
486C4—161no—
487C4—161no—
488C4—161no—
489C4—161no—
490C4#312011yescharacteristic
491C4#312011yescharacteristic
492C4#312311yescharacteristic
493C4#312311yescharacteristic
494C4#313711yescharacteristic
495C4#313711yescharacteristic
496C4#317411yescharacteristic
497C4#317711yescharacteristic
498C2—82no—
499C2#1955911yescharacteristic
500C2#1956111yescharacteristic
501C2#1958811yescharacteristic
502C2#1959111yescharacteristic
503C2#3304011yescharacteristic
504C2#3305611yescharacteristic
505C2#3306011yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #30228C2semidirect
5. #9901C2semidirect
6. #9691C2semidirect
7. #9576C2semidirect
8. #9575C2semidirect
9. #9568C2semidirect
10. #9567C2semidirect
15. #8789C2semidirect

Maximal subgroups

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

subgroupindex
#302282
#299952
#298682
#205622
#99012
#96912
#95762
#95752
#95682
#95672
#92232
#92152
#87892