Finite Group Databasebeta

← #276739

Subgroups — #276739

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 509 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29981C211yesmaximal, characteristic
2#29206C211yesmaximal, characteristic, complemented
3#22796C211yesmaximal, characteristic
4#22146C211yesmaximal, characteristic
5#10619C211yesmaximal, characteristic
6#10619C211yesmaximal, characteristic
7#10477C211yesmaximal, characteristic
8#10476C211yesmaximal, characteristic
9#9898C211yesmaximal, characteristic, complemented
10#9618C211yesmaximal, characteristic, complemented
11#9612C211yesmaximal, characteristic, complemented
12#9603C211yesmaximal, characteristic, complemented
13#9573C211yesmaximal, characteristic, complemented
14#9565C211yesmaximal, characteristic, complemented
15#8591C211yesmaximal, characteristic, complemented
16C2×#461C2211yescharacteristic
17C2×#433—21no—
18#2868C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20#2293C2211yescharacteristic
21#2100—21no—
22#2094—21no—
23#2058—21no—
24#2058—21no—
25#2058—21no—
26#2023—21no—
27#1990C2211yescharacteristic
28#1990C2211yescharacteristic
29#1981C2211yescharacteristic
30#1968—21no—
31#1954C2211yescharacteristic
32#1954C2211yescharacteristic
33#1948—21no—
34#1948—21no—
35#1948C2211yescharacteristic
36#1948C2211yescharacteristic
37#1948C2211yescharacteristic
38#1933—21no—
39#1924—21no—
40#1923—21no—
41#1880—21no—
42#1865—21no—
43#1844C2211yescharacteristic
44#1844C2211yescharacteristic
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1798—21no—
51#1770C2211yescharacteristic
52C8×C4⋊C4C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×SD16—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8—21no—
70C2×C4⋊Q8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4×Q8—21no—
73C2×C4×D4C2311yescharacteristic
74C2×#117—21no—
75#461—22no—
76#461—22no—
77#433—44no—
78C2×#106—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
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×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#103—21no—
96C2×#103—21no—
97C2×#103—21no—
98C2×#102—21no—
99C2×#102—21no—
100C2×#102—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C2×C22⋊C8—21no—
105C2×#97—21no—
106C2×C4×C8—21no—
107C2×C4×C8—21no—
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#390D411yescharacteristic
112#390D411yescharacteristic
113#389—21no—
114#389—21no—
115#389D411yescharacteristic
116#389D411yescharacteristic
117#388—21no—
118#388—21no—
119#384—21no—
120#384—21no—
121#384—21no—
122#384C2311yescharacteristic
123#384C2311yescharacteristic
124#384C2311yescharacteristic
125C4×C4⋊C4—21no—
126C4×C4⋊C4—21no—
127C4×C4⋊C4C2311yescharacteristic
128C4×C22⋊C4—21no—
129C43C2311yescharacteristic
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#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#334—22no—
149#332—22no—
150#332—22no—
151C22×Q8—21no—
152C22×Q8—41no—
153C22×D4C2×D411yescharacteristic
154C23×C4—21no—
155C2×SD16—24no—
156C2×SD16—81no—
157C2×SD16—81no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C4⋊Q8—21no—
169C4⋊Q8—21no—
170C4⋊Q8—21no—
171C4⋊Q8—81no—
172C4⋊Q8—81no—
173C4⋊Q8—81no—
174C4⋊Q8—81no—
175C4⋊Q8—82no—
176C4⋊Q8#4112yes—
177C4×Q8—81no—
178C4×Q8—81no—
179C4×D4—21no—
180C4×D4—21no—
181C4×D4—21no—
182C4×D4—21no—
183#117—42no—
184#117—42no—
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—21no—
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×C4⋊C4C2×D411yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—41no—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D411yescharacteristic
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42#4111yescharacteristic
217C2×C42#4111yescharacteristic
218C2×C42C2411yescharacteristic
219#106—81no—
220#106—81no—
221#106—81no—
222#106—81no—
223#106—81no—
224#106—81no—
225#106—81no—
226C4⋊C8—22no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—42no—
239C4⋊C8—81no—
240C4⋊C8SD1614yescomplemented
241C4⋊C8#4112yes—
242#103—24no—
243#103—81no—
244#103—81no—
245#102—81no—
246#102—81no—
247#102—81no—
248C22⋊C8—42no—
249C22⋊C8—42no—
250C22⋊C8—81no—
251C22⋊C8—81no—
252#97—42no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—42no—
258C4×C8—81no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—41no—
263#95—41no—
264C24—21no—
265C2×Q8—22no—
266C2×Q8—41no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—82no—
272C2×Q8—82no—
273C2×Q8—82no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—21no—
280C22×C4—21no—
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—41no—
292C22×C4—41no—
293C22×C4#12111yescharacteristic
294C22×C4#12111yescharacteristic
295C22×C4#12111yescharacteristic
296C22×C4#12111yescharacteristic
297C22×C4C22×D411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C2×#4111yescharacteristic
300SD16—84no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—22no—
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
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—41no—
325C2×C8—41no—
326C2×C8—41no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8C2×SD1612yes—
330C2×C8C2×SD1612yes—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
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—41no—
352C4⋊C4—41no—
353C4⋊C4—41no—
354C4⋊C4—41no—
355C4⋊C4—81no—
356C4⋊C4—81no—
357C4⋊C4—81no—
358C4⋊C4—82no—
359C4⋊C4—82no—
360C4⋊C4—82no—
361C4⋊C4—82no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—42no—
365C22⋊C4—42no—
366C22⋊C4—42no—
367C22⋊C4—81no—
368C42—21no—
369C42—21no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—22no—
375C42—41no—
376C42—41no—
377C42—41no—
378C42—82no—
379C42C2×SD1612yes—
380C42#13712yes—
381C42C2×#4111yescharacteristic
382C42C2×#4111yescharacteristic
383C422−1+411yescharacteristic
384C422−1+411yescharacteristic
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23C2×#12111yescharacteristic
393Q8—42no—
394Q8—42no—
395Q8—81no—
396Q8—81no—
397Q8—164no—
398D4—41no—
399D4—41no—
400D4—41no—
401D4—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—41no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
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—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4#46112yes—
439C2×C4#46112yes—
440C2×C4#47112yes—
441C2×C4#52511yescharacteristic
442C2×C4#53611yescharacteristic
443C2×C4#54711yescharacteristic
444C2×C4#54811yescharacteristic
445C2×C4#55211yescharacteristic
446C2×C4#55611yescharacteristic
447C2×C4C22×SD1611yescharacteristic
448C2×C4C2×#13711yescharacteristic
449C2×C4#57611yescharacteristic
450C2×C4#57611yescharacteristic
451C2×C4#57711yescharacteristic
452C2×C4#57711yescharacteristic
453C2×C4#57711yescharacteristic
454C2×C4#57711yescharacteristic
455C8—22no—
456C8—22no—
457C8—41no—
458C8—41no—
459C8—42no—
460C8—81no—
461C8—81no—
462C8—81no—
463C8—81no—
464C8—81no—
465C22—81no—
466C22—81no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#241511yescharacteristic
473C22C2×#46111yescharacteristic
474C22#304511yescharacteristic
475C22#305711yescharacteristic
476C22#306211yescharacteristic
477C22C2×#47111yescharacteristic
478C22#306911yescharacteristic
479C4—22no—
480C4—22no—
481C4—21no—
482C4—21no—
483C4—22no—
484C4—41no—
485C4—81no—
486C4—81no—
487C4—82no—
488C4—81no—
489C4—161no—
490C4—161no—
491C4—161no—
492C4#311811yescharacteristic
493C4#312511yescharacteristic
494C4#324111yescharacteristic
495C4#324411yescharacteristic
496C4#333511yescharacteristic
497C4#335711yescharacteristic
498C4#336211yescharacteristic
499C4#336911yescharacteristic
500C2—161no—
501C2#1919111yescharacteristic
502C2#1923211yescharacteristic
503C2#1953611yescharacteristic
504C2#1958111yescharacteristic
505C2#2468011yescharacteristic
506C2#2471611yescharacteristic
507C2#2472311yescharacteristic
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
2. #29206C2semidirect
9. #9898C2semidirect
10. #9618C2semidirect
11. #9612C2semidirect
12. #9603C2semidirect
13. #9573C2semidirect
14. #9565C2semidirect
15. #8591C2semidirect
240. C4⋊C8SD16semidirect

Maximal subgroups

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

subgroupindex
#299812
#292062
#227962
#221462
#106192
#104772
#104762
#98982
#96182
#96122
#96032
#95732
#95652
#85912