Finite Group Databasebeta

← #325390

Subgroups — #325390

Order 512.

566 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#30304C211yesmaximal, characteristic, complemented
2#29148C211yesmaximal, characteristic
3#29140C211yesmaximal, characteristic
4#22208C211yesmaximal, characteristic
5#10624C211yesmaximal, characteristic, complemented
6#10622C211yesmaximal, characteristic
7#10618C211yesmaximal, characteristic, complemented
8#10616C211yesmaximal, characteristic
9#9902C211yesmaximal, characteristic, complemented
10#9896C211yesmaximal, characteristic, complemented
11#8649C211yesmaximal, characteristic, complemented
12#8648C211yesmaximal, characteristic
13#8645C211yesmaximal, characteristic, complemented
14#8644C211yesmaximal, characteristic
15#8278C211yesmaximal, characteristic, complemented
16C2×C8×Q8C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×#124C2211yescharacteristic
21#2083—21no—
22#2083—21no—
23#2057—21no—
24#2057—21no—
25#2026—21no—
26#2025—21no—
27#2024—21no—
28#2023—21no—
29#1984C2211yescharacteristic
30#1959—21no—
31#1958—21no—
32#1954C2211yescharacteristic
33#1923—21no—
34#1923—21no—
35#1850—21no—
36#1850—21no—
37#1845C2211yescharacteristic
38#1845C2211yescharacteristic
39#1842C2211yescharacteristic
40#1836—21no—
41#1835—21no—
42#1834C2211yescharacteristic
43#1834C2211yescharacteristic
44#1803C2211yescharacteristic
45#1803C2211yescharacteristic
46#1803C2211yescharacteristic
47#1803C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57#1296C2211yescharacteristic
58#1296C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×#125—21no—
72C2×#124D411yescharacteristic
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C8×Q8—22no—
76C8×Q8—22no—
77C8×D4—22no—
78C8×D4—22no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8—21no—
85C2×C4⋊C8—21no—
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×C4⋊C8C2311yescharacteristic
91C2×C4⋊C8C2311yescharacteristic
92C2×#103—21no—
93C2×#103—21no—
94C2×#103—21no—
95C2×#103—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×#97—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391D411yescharacteristic
109#390D411yescharacteristic
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#388—21no—
115#388—21no—
116#384—21no—
117#384—21no—
118#384—21no—
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#384C2311yescharacteristic
122#384C2311yescharacteristic
123#382—21no—
124#382D411yescharacteristic
125C4×C22⋊C4—21no—
126C4×C22⋊C4—21no—
127C43C2311yescharacteristic
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#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—22no—
149#335—22no—
150C22×Q8—41no—
151C22×Q8C2×D411yescharacteristic
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—42no—
161C22×C8—81no—
162C22×C8—81no—
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C4⋊Q8—81no—
168C4⋊Q8—81no—
169C4⋊Q8—81no—
170C4⋊Q8—81no—
171C4⋊Q8—82no—
172#125—81no—
173#125—81no—
174#124—22no—
175#124Q1614yes—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
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—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C2×D411yescharacteristic
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42#4111yescharacteristic
215C2×C42#4111yescharacteristic
216C2×C42C2411yescharacteristic
217#107—81no—
218#107—81no—
219#107—81no—
220#107—81no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8—81no—
236#103—81no—
237#103—81no—
238#103—81no—
239#103—81no—
240#102—81no—
241#102—81no—
242#102—81no—
243#102—81no—
244C22⋊C8—42no—
245C22⋊C8—81no—
246C22⋊C8—81no—
247#97—42no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8—42no—
252C4×C8#4112yes—
253C4×C8#4112yes—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—21no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—21no—
262C2×Q8—21no—
263C2×Q8—21no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—82no—
267C2×Q8—82no—
268C2×Q8—82no—
269C2×Q8C2×Q1612yes—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4C2×Q1612yes—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4C22×D411yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8—41no—
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—84no—
325C2×C8#12412yes—
326C2×C8#12412yes—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
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—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—81no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C4⋊C4—82no—
357C4⋊C4—82no—
358C22⋊C4—24no—
359C22⋊C4—24no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—81no—
363C22⋊C4—81no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—41no—
371C42—41no—
372C42—41no—
373C42—41no—
374C42C2×Q1612yes—
375C42#13512yes—
376C42C2×#4111yescharacteristic
377C42C2×#4111yescharacteristic
378C422−1+411yescharacteristic
379C422−1+411yescharacteristic
380C23—22no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23C2×#12111yescharacteristic
388Q8—22no—
389Q8—22no—
390Q8—41no—
391Q8—41no—
392Q8—164no—
393D4—22no—
394D4—22no—
395D4—41no—
396D4—41no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
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#46712yes—
435C2×C4#46712yes—
436C2×C4C2×#12411yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#54711yescharacteristic
439C2×C4#54711yescharacteristic
440C2×C4#55611yescharacteristic
441C2×C4#55611yescharacteristic
442C2×C4C22×Q1611yescharacteristic
443C2×C4C2×#13511yescharacteristic
444C2×C4#57611yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57611yescharacteristic
447C2×C4#57711yescharacteristic
448C2×C4#57711yescharacteristic
449C2×C4#57811yescharacteristic
450C8—22no—
451C8—22no—
452C8—41no—
453C8—41no—
454C8—81no—
455C8—81no—
456C8—81no—
457C8—81no—
458C8—81no—
459C8—81no—
460C22—42no—
461C22—42no—
462C22—42no—
463C22—81no—
464C22—81no—
465C22—81no—
466C22—81no—
467C22#254711yescharacteristic
468C22#304511yescharacteristic
469C22#304611yescharacteristic
470C22C2×#46711yescharacteristic
471C22#305811yescharacteristic
472C22#306811yescharacteristic
473C22#306911yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—22no—
478C4—42no—
479C4—41no—
480C4—41no—
481C4—81no—
482C4—81no—
483C4—161no—
484C4—161no—
485C4—161no—
486C4—161no—
487C4#313611yescharacteristic
488C4#313711yescharacteristic
489C4#324311yescharacteristic
490C4#324411yescharacteristic
491C4#333811yescharacteristic
492C4#333811yescharacteristic
493C4#334111yescharacteristic
494C4#334111yescharacteristic
495C2—82no—
496C2#1941111yescharacteristic
497C2#1941711yescharacteristic
498C2#1953811yescharacteristic
499C2#1954611yescharacteristic
500C2#2870511yescharacteristic
501C2#2873011yescharacteristic
502C2#2873511yescharacteristic
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. #30304C2semidirect
5. #10624C2semidirect
7. #10618C2semidirect
9. #9902C2semidirect
10. #9896C2semidirect
11. #8649C2semidirect
13. #8645C2semidirect
15. #8278C2semidirect

Maximal subgroups

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

subgroupindex
#303042
#291482
#291402
#222082
#106242
#106222
#106182
#106162
#99022
#98962
#86492
#86482
#86452
#86442
#82782