Finite Group Databasebeta

← #212121

Subgroups — #212121

Order 512.

605 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22104C211yesmaximal, characteristic
2C4×#485C211yesmaximal, characteristic
3#20547C211yesmaximal, characteristic
4#9548C211yesmaximal, characteristic, complemented
5#9548C211yesmaximal, characteristic, complemented
6#9539C211yesmaximal, characteristic, complemented
7#9537C211yesmaximal, characteristic, complemented
8#8937C211yesmaximal, characteristic
9#8829C211yesmaximal, characteristic
10#8708C211yesmaximal, characteristic
11#8286C211yesmaximal, characteristic, complemented
12#8282C211yesmaximal, characteristic, complemented
13#8209C211yesmaximal, characteristic, complemented
14#8207C211yesmaximal, characteristic, complemented
15#8148C211yesmaximal, characteristic
16C2×#485C411yescharacteristic
17C2×#446C2211yescharacteristic
18C2×#431C2211yescharacteristic
19C22×C4⋊C8C2211yescharacteristic
20C4×C22⋊Q8C2211yescharacteristic
21C4×#117C2211yescharacteristic
22#1984C2211yescharacteristic
23#1940—21no—
24#1939C2211yescharacteristic
25#1928C411yescharacteristic
26#1927C411yescharacteristic
27#1924—21no—
28#1924—21no—
29#1919C411yescharacteristic
30#1918C411yescharacteristic
31#1917—21no—
32#1873C411yescharacteristic
33#1854C411yescharacteristic
34#1825C411yescharacteristic
35#1797C2211yescharacteristic
36C4×#107C2211yescharacteristic
37C4×#106C2211yescharacteristic
38#1770C2211yescharacteristic
39#1770C2211yescharacteristic
40#1770C2211yescharacteristic
41#1770C2211yescharacteristic
42#1770C2211yescharacteristic
43#1767—21no—
44#1767—21no—
45C4×#103C2211yescharacteristic
46C4×#103C2211yescharacteristic
47#1752—21no—
48#1752—21no—
49C4×C22⋊C8C2211yescharacteristic
50C4×C22⋊C8C2211yescharacteristic
51C2×#335C2211yescharacteristic
52C2×#335C2211yescharacteristic
53C2×Q8⋊C8C2211yescharacteristic
54C2×Q8⋊C8C2211yescharacteristic
55C2×#322C2211yescharacteristic
56C2×C82C2211yescharacteristic
57#1279C2211yescharacteristic
58#1279C2211yescharacteristic
59#1279C2211yescharacteristic
60#1279C2211yescharacteristic
61#1279C2211yescharacteristic
62#1279C2211yescharacteristic
63#1277C2211yescharacteristic
64#1277C2211yescharacteristic
65#1273C2211yescharacteristic
66#1273C2211yescharacteristic
67C23×C8—21no—
68C2×C22⋊Q8C2×C411yescharacteristic
69C2×C4×Q8C2311yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C411yescharacteristic
72C22×C42C2311yescharacteristic
73#485—24no—
74#446—22no—
75#446—22no—
76#431—22no—
77#431—22no—
78C2×#107C2×C411yescharacteristic
79C2×#107C2×C411yescharacteristic
80C2×#106C2×C411yescharacteristic
81C2×#106C2×C411yescharacteristic
82C2×C4⋊C8—21no—
83C2×C4⋊C8—22no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8D411yescharacteristic
88C2×C4⋊C8D411yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×#103C2×C411yescharacteristic
94C2×#103C2×C411yescharacteristic
95C2×#103C2×C411yescharacteristic
96C2×#103C2×C411yescharacteristic
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8C2×C411yescharacteristic
100C2×C22⋊C8C2×C411yescharacteristic
101C2×C22⋊C8D411yescharacteristic
102C2×C22⋊C8D411yescharacteristic
103C2×#97—21no—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113C2×C4×C8C2311yescharacteristic
114C2×C4×C8C2311yescharacteristic
115#391C2×C411yescharacteristic
116#389C2×C411yescharacteristic
117#387—21no—
118#385—21no—
119#380C2×C411yescharacteristic
120C4×C4⋊C4—21no—
121C4×C4⋊C4C2×C411yescharacteristic
122C4×C4⋊C4C2×C411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
125C4×C4⋊C4C2311yescharacteristic
126C4×C22⋊C4—21no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129#376—21no—
130C43—21no—
131#340C2×C411yescharacteristic
132#340C2×C411yescharacteristic
133#340C2×C411yescharacteristic
134#340C2×C411yescharacteristic
135#340C2×C411yescharacteristic
136#340C2×C411yescharacteristic
137#340C2×C411yescharacteristic
138#340C2×C411yescharacteristic
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336C2×C411yescharacteristic
150#336C2×C411yescharacteristic
151#335—22no—
152#335—22no—
153Q8⋊C8—22no—
154Q8⋊C8—22no—
155#322—22no—
156C82—22no—
157C22×Q8C22×C411yescharacteristic
158C23×C4C22×C411yescharacteristic
159C23×C4C2×D411yescharacteristic
160C23×C4#4111yescharacteristic
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—24no—
165C22×C8—21no—
166C22×C8—21no—
167C22×C8—21no—
168C22×C8—81no—
169C22×C8—81no—
170C22×C8C22×C411yescharacteristic
171C22×C8C22×C411yescharacteristic
172C22×C8C22×C411yescharacteristic
173C22×C8C22×C411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C22×C8C2×D411yescharacteristic
176C22×C8C2×D411yescharacteristic
177C22×C8C2×D411yescharacteristic
178C22⋊Q8—22no—
179C22⋊Q8—22no—
180C4×Q8—21no—
181C4×Q8—21no—
182C4×Q8—21no—
183C4×Q8—21no—
184#117—22no—
185#117—22no—
186#117—41no—
187#117—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4C22×C411yescharacteristic
192C2×C4⋊C4C22×C411yescharacteristic
193C2×C4⋊C4C22×C411yescharacteristic
194C2×C4⋊C4C22×C411yescharacteristic
195C2×C4⋊C4C22×C411yescharacteristic
196C2×C4⋊C4C22×C411yescharacteristic
197C2×C4⋊C4C22×C411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C22⋊C4—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—21no—
208C2×C42—21no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42C22×C411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42C2411yescharacteristic
218#107—22no—
219#107—41no—
220#106—22no—
221#106—41no—
222C4⋊C8—24no—
223C4⋊C8—22no—
224C4⋊C8—22no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—42no—
232C4⋊C8—81no—
233C4⋊C8#4112yes—
234C4⋊C8#4112yes—
235#103—22no—
236#103—22no—
237#103—41no—
238#103—41no—
239C22⋊C8—22no—
240C22⋊C8—22no—
241C22⋊C8—22no—
242C22⋊C8—42no—
243C22⋊C8—81no—
244C22⋊C8M1614yes—
245#97—42no—
246#97—81no—
247#97—81no—
248#97—81no—
249#97—81no—
250#97—81no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8#4112yes—
262C4×C8#4112yes—
263#95—21no—
264#95—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—21no—
271#95—21no—
272C24C4×D411yescharacteristic
273C2×Q8—21no—
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
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—41no—
301C22×C4C4×D411yescharacteristic
302C22×C4C4×D411yescharacteristic
303C22×C4C4×D411yescharacteristic
304C22×C4C2×M1612yes—
305C22×C4#13112yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
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—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—84no—
348C2×C8#12312yes—
349C2×C8#12312yes—
350C2×C8C2×M1612yes—
351C2×C8C2×M1612yes—
352C4⋊C4—21no—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—21no—
360C4⋊C4—21no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—81no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—42no—
373C22⋊C4—81no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—21no—
379C42—21no—
380C42—42no—
381C42—41no—
382C42—41no—
383C42—41no—
384C42—41no—
385C42—41no—
386C42—41no—
387C42—41no—
388C42—41no—
389C42—42no—
390C42—81no—
391C42—81no—
392C42C2×#4111yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23#43612yes—
403C23C2×C4×D411yescharacteristic
404Q8—41no—
405Q8—41no—
406Q8—41no—
407Q8—41no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—41no—
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—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—81no—
451C2×C4—81no—
452C2×C4#43612yes—
453C2×C4C4×#4111yescharacteristic
454C2×C4C4×#4111yescharacteristic
455C2×C4C2×#12311yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#52911yescharacteristic
459C2×C4C22×M1611yescharacteristic
460C2×C4C2×#13111yescharacteristic
461C2×C4#56811yescharacteristic
462C2×C4#56811yescharacteristic
463C2×C4#57611yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C2×C4#57811yescharacteristic
467C8—22no—
468C8—22no—
469C8—22no—
470C8—22no—
471C8—42no—
472C8—41no—
473C8—41no—
474C8—42no—
475C8—42no—
476C8—41no—
477C8—41no—
478C8—81no—
479C8—81no—
480C8—81no—
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22C4×#12311yescharacteristic
489C22C2×#43611yescharacteristic
490C22#293011yescharacteristic
491C22#294711yescharacteristic
492C22#295111yescharacteristic
493C22#295611yescharacteristic
494C22#295711yescharacteristic
495C4—22no—
496C4—21no—
497C4—21no—
498C4—41no—
499C4—41no—
500C4—81no—
501C4—81no—
502C4—81no—
503C4—81no—
504C4—81no—
505C4—81no—
506C4—81no—
507C4—81no—
508C4#296811yescharacteristic
509C4#296811yescharacteristic
510C4#297011yescharacteristic
511C4#297011yescharacteristic
512C4#309711yescharacteristic
513C4#309811yescharacteristic
514C4#311811yescharacteristic
515C4#312711yescharacteristic
516C2—42no—
517C2#1789411yescharacteristic
518C2#1789811yescharacteristic
519C2#1794911yescharacteristic
520C2#1795411yescharacteristic
521C2#2156411yescharacteristic
522C2#2158911yescharacteristic
523C2#2159411yescharacteristic
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
socleC23C2×C4×D4
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×#431 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #9548C2semidirect
5. #9548C2semidirect
6. #9539C2semidirect
7. #9537C2semidirect
11. #8286C2semidirect
12. #8282C2semidirect
13. #8209C2semidirect
14. #8207C2semidirect

Maximal subgroups

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

subgroupindex
#221042
C4×#4852
#205472
#95482
#95392
#95372
#89372
#88292
#87082
#82862
#82822
#82092
#82072
#81482