Finite Group Databasebeta

← #211398

Subgroups — #211398

Order 512.

662 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22246C211yesmaximal, characteristic, complemented
2#22146C211yesmaximal, characteristic
3#22117C211yesmaximal, characteristic
4#20562C211yesmaximal, characteristic
5C2×#1671C211yesmaximal, characteristic
6#9741C211yesmaximal, characteristic, complemented
7#9622C211yesmaximal, characteristic, complemented
8#9567C211yesmaximal, characteristic, complemented
9#9566C211yesmaximal, characteristic, complemented
10#9225C211yesmaximal, characteristic
11#9223C211yesmaximal, characteristic
12#8648C211yesmaximal, characteristic
13#8279C211yesmaximal, characteristic, complemented
14#8236C211yesmaximal, characteristic, complemented
15C4×#333C211yesmaximal, characteristic, complemented
16C2×#498C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#461C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×C8×D4C2211yescharacteristic
21C4×C4⋊Q8C2211yescharacteristic
22C4×C4⋊Q8C2211yescharacteristic
23#2293C2211yescharacteristic
24#1990C2211yescharacteristic
25#1990C2211yescharacteristic
26#1987C2211yescharacteristic
27#1981C2211yescharacteristic
28#1968—21no—
29#1948C2211yescharacteristic
30#1947C2211yescharacteristic
31#1944—21no—
32#1936—21no—
33#1933—21no—
34#1925—21no—
35#1920—21no—
36#1920—21no—
37#1920—21no—
38#1919—21no—
39#1919—21no—
40#1880—21no—
41#1865—21no—
42#1834C2211yescharacteristic
43#1802C2211yescharacteristic
44#1802C2211yescharacteristic
45#1770C2211yescharacteristic
46C8×C4⋊C4C2211yescharacteristic
47#1767C2211yescharacteristic
48C4×C4⋊C8C2211yescharacteristic
49C4×C4⋊C8C2211yescharacteristic
50#1753—21no—
51C8×C22⋊C4—21no—
52#1750—21no—
53#1744C2211yescharacteristic
54#1729—21no—
55#1671C418yescomplemented
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#332C2211yescharacteristic
61C2×#332C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1277C2211yescharacteristic
67#1277C2211yescharacteristic
68C22×SD16—21no—
69C22×M16—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C4×Q8—21no—
74C2×C4×Q8—21no—
75C2×C4×D4C2311yescharacteristic
76C2×#117—21no—
77#498—22no—
78#498C2×C414yes—
79#476—22no—
80#476C2×C414yes—
81#461—22no—
82#461C2×C414yes—
83#436—22no—
84#436—22no—
85C8×D4—22no—
86C8×D4C2×C414yes—
87C2×#107—21no—
88C2×#106—21no—
89C2×#106—21no—
90C2×#106—21no—
91C2×#106—21no—
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#102—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×C22⋊C8—21no—
105C2×C22⋊C8—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8C2311yescharacteristic
112C2×C4×C8C2311yescharacteristic
113#391—21no—
114#391—21no—
115#390D411yescharacteristic
116#390D411yescharacteristic
117#389—21no—
118#389—21no—
119#389—21no—
120#389—21no—
121#389D411yescharacteristic
122#389D411yescharacteristic
123#388—21no—
124#388—21no—
125#384C2311yescharacteristic
126#384C2311yescharacteristic
127C4×C4⋊C4—21no—
128C4×C4⋊C4—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—21no—
131C4×C4⋊C4C2311yescharacteristic
132C4×C22⋊C4—21no—
133C43C2311yescharacteristic
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#340—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#336D411yescharacteristic
152#336D411yescharacteristic
153#336D411yescharacteristic
154#336D411yescharacteristic
155#335—22no—
156#334C2×C414yes—
157#333—22no—
158#333C2×C414yes—
159#332—22no—
160#332C2×C414yes—
161C22×Q8—21no—
162C22×Q8—21no—
163C22×D4C2×D411yescharacteristic
164C23×C4—21no—
165C2×SD16—24no—
166C2×SD16—81no—
167C2×SD16—81no—
168C2×M16—24no—
169C2×M16—81no—
170C2×M16—81no—
171C22×C8—21no—
172C22×C8—21no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—24no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—81no—
179C22×C8—81no—
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C22×C8C2×D411yescharacteristic
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8C22×C412yes—
191C4⋊Q8C22×C412yes—
192C4×Q8—81no—
193C4×Q8—81no—
194C4×Q8—82no—
195C4×D4—21no—
196C4×D4—21no—
197C4×D4C22×C412yes—
198C4×D4#4112yes—
199#117—81no—
200#117—81no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—21no—
209C2×C4⋊C4—21no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—41no—
214C2×C4⋊C4—41no—
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C4⋊C4C2×D411yescharacteristic
217C2×C4⋊C4C2×D411yescharacteristic
218C2×C22⋊C4—21no—
219C2×C22⋊C4—21no—
220C2×C22⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C42—21no—
224C2×C42—41no—
225C2×C42—41no—
226C2×C42C2×D411yescharacteristic
227C2×C42C2×D411yescharacteristic
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42#4111yescharacteristic
232C2×C42C2411yescharacteristic
233#107—81no—
234#106—24no—
235#106—24no—
236#106—24no—
237#106—81no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8C22×C412yes—
247C4⋊C8C22×C412yes—
248C4⋊C8C22×C412yes—
249C4⋊C8#4112yes—
250#103—24no—
251#103—81no—
252#103—81no—
253#102—24no—
254#102—81no—
255#102—81no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258#97—81no—
259#97—81no—
260C4×C8—21no—
261C4×C8—21no—
262C4×C8—81no—
263C4×C8—81no—
264C4×C8—81no—
265C4×C8C22×C412yes—
266C4×C8#4112yes—
267#95—21no—
268#95—21no—
269#95—21no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274C24—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—81no—
284C2×Q8—81no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4C4×D412yes—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4#12111yescharacteristic
305C22×C4#12111yescharacteristic
306C22×C4#12111yescharacteristic
307C22×C4#12111yescharacteristic
308C22×C4C22×D411yescharacteristic
309C22×C4C22×D411yescharacteristic
310C22×C4C2×#4111yescharacteristic
311SD16—84no—
312M16—84no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—41no—
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—84no—
342C2×C8C4×D412yes—
343C2×C8C4×D412yes—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—22no—
347C4⋊C4—22no—
348C4⋊C4—22no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—41no—
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—81no—
366C4⋊C4—81no—
367C4⋊C4—81no—
368C4⋊C4—81no—
369C4⋊C4—82no—
370C4⋊C4—82no—
371C4⋊C4C4×D412yes—
372C22⋊C4—22no—
373C22⋊C4—22no—
374C22⋊C4—42no—
375C22⋊C4—42no—
376C22⋊C4—81no—
377C22⋊C4—81no—
378C42—21no—
379C42—21no—
380C42—21no—
381C42—21no—
382C42—21no—
383C42—21no—
384C42—21no—
385C42—21no—
386C42—22no—
387C42—41no—
388C42—41no—
389C42—41no—
390C42—82no—
391C42—82no—
392C42C23×C411yescharacteristic
393C42C2×#4111yescharacteristic
394C42C2×#4111yescharacteristic
395C42C2×#4111yescharacteristic
396C23—22no—
397C23—41no—
398C23—41no—
399C23—41no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23C2×#12111yescharacteristic
404Q8—42no—
405Q8—42no—
406Q8—82no—
407Q8—82no—
408Q8—81no—
409Q8—81no—
410D4—22no—
411D4—22no—
412D4—41no—
413D4—41no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—21no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—41no—
429C2×C4—41no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—81no—
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#47112yes—
453C2×C4C2×C4×D411yescharacteristic
454C2×C4C2×C4×D411yescharacteristic
455C2×C4C4×#4111yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52911yescharacteristic
458C2×C4#53211yescharacteristic
459C2×C4#57611yescharacteristic
460C2×C4#57711yescharacteristic
461C2×C4#57711yescharacteristic
462C2×C4#57711yescharacteristic
463C2×C4#57711yescharacteristic
464C2×C4#57711yescharacteristic
465C2×C4#57811yescharacteristic
466C2×C4#57811yescharacteristic
467C8—22no—
468C8—22no—
469C8—41no—
470C8—41no—
471C8—42no—
472C8—42no—
473C8—42no—
474C8—42no—
475C8—81no—
476C8—81no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—81no—
481C22—81no—
482C22—81no—
483C22—81no—
484C22C4×#12111yescharacteristic
485C22#304211yescharacteristic
486C22#304611yescharacteristic
487C22#306211yescharacteristic
488C22#306311yescharacteristic
489C22C2×#47111yescharacteristic
490C22#306711yescharacteristic
491C4—21no—
492C4—21no—
493C4—22no—
494C4—42no—
495C4—41no—
496C4—41no—
497C4—41no—
498C4—82no—
499C4—82no—
500C4—81no—
501C4—81no—
502C4—161no—
503C4—161no—
504C4#294711yescharacteristic
505C4#294911yescharacteristic
506C4#294911yescharacteristic
507C4#295111yescharacteristic
508C4#312011yescharacteristic
509C4#312311yescharacteristic
510C4#317811yescharacteristic
511C4#317911yescharacteristic
512C2—82no—
513C2#1955811yescharacteristic
514C2#1956211yescharacteristic
515C2#1958411yescharacteristic
516C2#1958811yescharacteristic
517C2#2151411yescharacteristic
518C2#2153111yescharacteristic
519C2C4×#47111yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ 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. #22246C2semidirect
6. #9741C2semidirect
7. #9622C2semidirect
8. #9567C2semidirect
9. #9566C2semidirect
13. #8279C2semidirect
14. #8236C2semidirect
15. C4×#333C2semidirect
55. #1671C4semidirect

Maximal subgroups

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

subgroupindex
#222462
#221462
#221172
#205622
C2×#16712
#97412
#96222
#95672
#95662
#92252
#92232
#86482
#82792
#82362
C4×#3332