Finite Group Databasebeta

← #330704

Subgroups — #330704

Order 512.

522 conjugacy classes of subgroups.

The subgroup lattice has 501 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#22009C211yesmaximal, characteristic
5#10601C211yesmaximal, characteristic, complemented
6#10597C211yesmaximal, characteristic, complemented
7#10593C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#10481C211yesmaximal, characteristic
10#10481C211yesmaximal, characteristic
11#10480C211yesmaximal, characteristic
12#10480C211yesmaximal, characteristic
13#9901C211yesmaximal, characteristic, complemented
14#9691C211yesmaximal, characteristic, complemented
15#8812C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#476C2211yescharacteristic
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20#2293C2211yescharacteristic
21#2099—21no—
22#2099—21no—
23#2099—21no—
24#2099—21no—
25#2088—21no—
26#2087—21no—
27#2084—21no—
28#2084—21no—
29#2071—21no—
30#2071—21no—
31#2067—21no—
32#2065—21no—
33#1990C2211yescharacteristic
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1990C2211yescharacteristic
37#1987C2211yescharacteristic
38#1987C2211yescharacteristic
39#1954C2211yescharacteristic
40#1954C2211yescharacteristic
41#1948C2211yescharacteristic
42#1947C2211yescharacteristic
43#1939C2211yescharacteristic
44#1939C2211yescharacteristic
45#1923—21no—
46#1923—21no—
47#1842C2211yescharacteristic
48#1831C2211yescharacteristic
49#1818—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52#1798—21no—
53C4×C4⋊C8C2211yescharacteristic
54#1744C2211yescharacteristic
55C2×#335C2211yescharacteristic
56C2×#335C2211yescharacteristic
57C2×#334C2211yescharacteristic
58C2×#334C2211yescharacteristic
59C2×#333C2211yescharacteristic
60C2×#333C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67C2×C4⋊Q8—21no—
68C2×C4⋊Q8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×#125—21no—
72C2×C4×D4C2311yescharacteristic
73C2×#117—21no—
74#476—41no—
75#476—41no—
76#476—41no—
77#476—41no—
78C2×#107—21no—
79C2×#106—21no—
80C2×#106—21no—
81C2×#106—21no—
82C2×C4⋊C8—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×#102—21no—
93C2×#102—21no—
94C2×#102—21no—
95C2×#102—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#390D411yescharacteristic
104#390D411yescharacteristic
105#389—21no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389D411yescharacteristic
110#389D411yescharacteristic
111#388—21no—
112#388—21no—
113#384—21no—
114#384—21no—
115#384—21no—
116#384—21no—
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120C4×C4⋊C4C2311yescharacteristic
121C4×C22⋊C4—21no—
122C43C2311yescharacteristic
123#340—21no—
124#340—21no—
125#340—21no—
126#340—21no—
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#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#336D411yescharacteristic
147#336D411yescharacteristic
148#335—41no—
149#335—41no—
150#334—41no—
151#334—41no—
152#333—41no—
153#333—41no—
154#333—41no—
155#333—41no—
156C22×Q8—21no—
157C22×Q8—21no—
158C22×D4C2×D411yescharacteristic
159C23×C4—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—21no—
166C22×C8C2×D411yescharacteristic
167C22×C8C2×D411yescharacteristic
168C22×C8C2×D411yescharacteristic
169C22×C8C2×D411yescharacteristic
170C4⋊Q8—21no—
171C4⋊Q8—21no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8—82no—
179C4⋊Q8—82no—
180#125—82no—
181C4×D4—21no—
182C4×D4—21no—
183C4×D4—21no—
184C4×D4—21no—
185#117—81no—
186#117—81no—
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—21no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—41no—
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4—21no—
212C2×C22⋊C4—21no—
213C2×C42—21no—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D411yescharacteristic
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42#4111yescharacteristic
219C2×C42#4111yescharacteristic
220C2×C42C2411yescharacteristic
221#107—81no—
222#106—42no—
223#106—42no—
224#106—81no—
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—21no—
234C4⋊C8—21no—
235C4⋊C8—21no—
236C4⋊C8—21no—
237C4⋊C8—81no—
238C4⋊C8—81no—
239C4⋊C8—81no—
240C4⋊C8—81no—
241#102—42no—
242#102—42no—
243#102—81no—
244#102—81no—
245C22⋊C8—81no—
246C22⋊C8—81no—
247C22⋊C8—81no—
248C22⋊C8—81no—
249C4×C8—21no—
250C4×C8—21no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—81no—
254#95—21no—
255#95—21no—
256#95—21no—
257#95—41no—
258#95—41no—
259C24—21no—
260C2×Q8—41no—
261C2×Q8—41no—
262C2×Q8—41no—
263C2×Q8—41no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×Q8—81no—
267C2×Q8—81no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—21no—
277C22×C4—41no—
278C22×C4—41no—
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#12111yescharacteristic
288C22×C4#12111yescharacteristic
289C22×C4#12111yescharacteristic
290C22×C4#12111yescharacteristic
291C22×C4C22×D411yescharacteristic
292C22×C4C22×D411yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C2×C8—21no—
295C2×C8—21no—
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—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—41no—
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—81no—
347C4⋊C4—81no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—82no—
356C22⋊C4—41no—
357C22⋊C4—41no—
358C22⋊C4—42no—
359C22⋊C4—42no—
360C22⋊C4—81no—
361C22⋊C4—81no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—21no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—21no—
370C42—22no—
371C42—22no—
372C42—41no—
373C42—41no—
374C42H(5,𝔽2)11yescharacteristic
375C42H(5,𝔽2)11yescharacteristic
376C422−1+411yescharacteristic
377C422−1+411yescharacteristic
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23C2×#12111yescharacteristic
386Q8—161no—
387Q8—161no—
388Q8—161no—
389Q8—161no—
390D4—41no—
391D4—41no—
392D4—41no—
393D4—41no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
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—21no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
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#47112yes—
432C2×C4#47212yes—
433C2×C4#53511yescharacteristic
434C2×C4#53611yescharacteristic
435C2×C4#53711yescharacteristic
436C2×C4#53711yescharacteristic
437C2×C4#54211yescharacteristic
438C2×C4#54211yescharacteristic
439C2×C4#57711yescharacteristic
440C2×C4#57711yescharacteristic
441C2×C4#57711yescharacteristic
442C2×C4#57711yescharacteristic
443C2×C4#57711yescharacteristic
444C2×C4#57811yescharacteristic
445C2×C4#57811yescharacteristic
446C2×C4#57811yescharacteristic
447C8—41no—
448C8—41no—
449C8—41no—
450C8—41no—
451C8—81no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8—81no—
457C22—81no—
458C22—81no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#262011yescharacteristic
465C22#304211yescharacteristic
466C22#304611yescharacteristic
467C22#306311yescharacteristic
468C22#306311yescharacteristic
469C22C2×#47111yescharacteristic
470C22C2×#47211yescharacteristic
471C4—22no—
472C4—22no—
473C4—21no—
474C4—21no—
475C4—41no—
476C4—41no—
477C4—81no—
478C4—81no—
479C4—81no—
480C4—161no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4#322111yescharacteristic
485C4#322311yescharacteristic
486C4#324211yescharacteristic
487C4#324211yescharacteristic
488C4#325811yescharacteristic
489C4#325811yescharacteristic
490C4#326011yescharacteristic
491C4#326011yescharacteristic
492C2—161no—
493C2#1955911yescharacteristic
494C2#1956111yescharacteristic
495C2#1958811yescharacteristic
496C2#1959111yescharacteristic
497C2#2983611yescharacteristic
498C2#2985611yescharacteristic
499C2#2985811yescharacteristic
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. #10601C2semidirect
6. #10597C2semidirect
7. #10593C2semidirect
8. #10589C2semidirect
13. #9901C2semidirect
14. #9691C2semidirect
15. #8812C2semidirect

Maximal subgroups

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

subgroupindex
#302282
#299952
#298682
#220092
#106012
#105972
#105932
#105892
#104812
#104802
#99012
#96912
#88122