Finite Group Databasebeta

← #276746

Subgroups — #276746

Order 512.

527 conjugacy classes of subgroups.

The subgroup lattice has 487 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29995C211yesmaximal, characteristic
2#29176C211yesmaximal, characteristic, complemented
3#22836C211yesmaximal, characteristic
4#22111C211yesmaximal, characteristic
5#10622C211yesmaximal, characteristic
6#10619C211yesmaximal, characteristic
7#10480C211yesmaximal, characteristic
8#10480C211yesmaximal, characteristic
9#9901C211yesmaximal, characteristic, complemented
10#9622C211yesmaximal, characteristic, complemented
11#9612C211yesmaximal, characteristic, complemented
12#9602C211yesmaximal, characteristic, complemented
13#9575C211yesmaximal, characteristic, complemented
14#9568C211yesmaximal, characteristic, complemented
15#8590C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17#2859C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19#2293C2211yescharacteristic
20#2088—21no—
21#2088—21no—
22#2088—21no—
23#2087—21no—
24#2067—21no—
25#2065—21no—
26#1990C2211yescharacteristic
27#1990C2211yescharacteristic
28#1987C2211yescharacteristic
29#1954C2211yescharacteristic
30#1954C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1941—21no—
35#1934—21no—
36#1926—21no—
37#1924—21no—
38#1923—21no—
39#1920—21no—
40#1920—21no—
41#1919—21no—
42#1844C2211yescharacteristic
43#1842C2211yescharacteristic
44#1834C2211yescharacteristic
45#1831C2211yescharacteristic
46#1818—21no—
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1802C2211yescharacteristic
50#1770C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54#1750—21no—
55C2×#334C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C2×C4⋊Q8C2311yescharacteristic
68C2×#125—21no—
69C2×C4×Q8—21no—
70C2×C4×D4C2311yescharacteristic
71C2×#117—21no—
72#476—22no—
73#476—22no—
74C2×#107—21no—
75C2×#106—21no—
76C2×#106—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×C4⋊C8—21no—
81C2×C4⋊C8—21no—
82C2×C4⋊C8D411yescharacteristic
83C2×C4⋊C8D411yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×C4⋊C8C2311yescharacteristic
89C2×C4⋊C8C2311yescharacteristic
90C2×#102—21no—
91C2×#102—21no—
92C2×#102—21no—
93C2×#102—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×#97—21no—
99C2×C4×C8—21no—
100C2×C4×C8—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103#391—21no—
104#390D411yescharacteristic
105#390D411yescharacteristic
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389D411yescharacteristic
113#389D411yescharacteristic
114#388—21no—
115#388—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#382—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C22⋊C4—21no—
124C43C2311yescharacteristic
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#334—22no—
147#334—22no—
148#333—22no—
149#333—22no—
150C22×Q8—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C22×C8—21no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8C2×D411yescharacteristic
160C22×C8C2×D411yescharacteristic
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8#4112yes—
167#125—82no—
168C4×Q8—82no—
169C4×D4—21no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—21no—
173#117—81no—
174#117—81no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—41no—
189C2×C4⋊C4—41no—
190C2×C4⋊C4—41no—
191C2×C4⋊C4C2×D411yescharacteristic
192C2×C4⋊C4C2×D411yescharacteristic
193C2×C4⋊C4C2×D411yescharacteristic
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C42—21no—
200C2×C42—41no—
201C2×C42C2×D411yescharacteristic
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D411yescharacteristic
205C2×C42#4111yescharacteristic
206C2×C42#4111yescharacteristic
207C2×C42C2411yescharacteristic
208#107—81no—
209#106—24no—
210#106—42no—
211#106—42no—
212#106—81no—
213#106—81no—
214C4⋊C8—22no—
215C4⋊C8—21no—
216C4⋊C8—21no—
217C4⋊C8—21no—
218C4⋊C8—21no—
219C4⋊C8—21no—
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—41no—
227C4⋊C8—81no—
228C4⋊C8—81no—
229C4⋊C8#4112yes—
230#102—24no—
231#102—81no—
232#102—81no—
233#102—81no—
234C22⋊C8—42no—
235C22⋊C8—42no—
236C22⋊C8—81no—
237C22⋊C8—81no—
238#97—81no—
239C4×C8—21no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—81no—
244C4×C8—81no—
245#95—21no—
246#95—21no—
247#95—21no—
248#95—41no—
249#95—41no—
250#95—41no—
251#95—41no—
252C24—21no—
253C2×Q8—22no—
254C2×Q8—41no—
255C2×Q8—81no—
256C2×Q8—81no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4#12111yescharacteristic
277C22×C4#12111yescharacteristic
278C22×C4#12111yescharacteristic
279C22×C4#12111yescharacteristic
280C22×C4C22×D411yescharacteristic
281C22×C4C22×D411yescharacteristic
282C22×C4C2×#4111yescharacteristic
283C2×C8—21no—
284C2×C8—21no—
285C2×C8—22no—
286C2×C8—21no—
287C2×C8—21no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C2×C8—41no—
304C2×C8—41no—
305C2×C8—41no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—41no—
309C2×C8—41no—
310C2×C8—41no—
311C2×C8—41no—
312C2×C8#13512yes—
313C4⋊C4—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—22no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—41no—
322C4⋊C4—41no—
323C4⋊C4—41no—
324C4⋊C4—41no—
325C4⋊C4—41no—
326C4⋊C4—41no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—81no—
336C4⋊C4—81no—
337C4⋊C4—81no—
338C4⋊C4—81no—
339C4⋊C4—81no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C4⋊C4—82no—
343C22⋊C4—41no—
344C22⋊C4—41no—
345C22⋊C4—42no—
346C22⋊C4—81no—
347C22⋊C4—81no—
348C22⋊C4—81no—
349C42—21no—
350C42—21no—
351C42—21no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—22no—
358C42—41no—
359C42—41no—
360C42—41no—
361C42—82no—
362C42C2×#4111yescharacteristic
363C42C2×#4111yescharacteristic
364C422−1+411yescharacteristic
365C422−1+411yescharacteristic
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23—41no—
371C23—41no—
372C23—41no—
373C23C2×#12111yescharacteristic
374Q8—82no—
375Q8—82no—
376D4—41no—
377D4—41no—
378D4—41no—
379D4—41no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—81no—
406C2×C4—81no—
407C2×C4—81no—
408C2×C4—81no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4#47112yes—
419C2×C4#52511yescharacteristic
420C2×C4#53611yescharacteristic
421C2×C4#54711yescharacteristic
422C2×C4#54811yescharacteristic
423C2×C4#55211yescharacteristic
424C2×C4#55611yescharacteristic
425C2×C4C2×#13511yescharacteristic
426C2×C4#57511yescharacteristic
427C2×C4#57711yescharacteristic
428C2×C4#57711yescharacteristic
429C2×C4#57711yescharacteristic
430C2×C4#57711yescharacteristic
431C2×C4#57711yescharacteristic
432C2×C4#57811yescharacteristic
433C8—22no—
434C8—41no—
435C8—41no—
436C8—41no—
437C8—42no—
438C8—81no—
439C8—81no—
440C8—81no—
441C8—81no—
442C8—81no—
443C22—81no—
444C22—81no—
445C22—81no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22#241511yescharacteristic
451C22#303511yescharacteristic
452C22#304211yescharacteristic
453C22#306011yescharacteristic
454C22#306311yescharacteristic
455C22C2×#47111yescharacteristic
456C22#306811yescharacteristic
457C4—22no—
458C4—21no—
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—81no—
464C4—81no—
465C4—81no—
466C4—82no—
467C4—161no—
468C4—161no—
469C4—161no—
470C4#312011yescharacteristic
471C4#312611yescharacteristic
472C4#324211yescharacteristic
473C4#324411yescharacteristic
474C4#333711yescharacteristic
475C4#335111yescharacteristic
476C4#336511yescharacteristic
477C4#337111yescharacteristic
478C2—161no—
479C2#1922111yescharacteristic
480C2#1925311yescharacteristic
481C2#1951611yescharacteristic
482C2#1955911yescharacteristic
483C2#2468611yescharacteristic
484C2#2471911yescharacteristic
485C2#2472111yescharacteristic
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. #29176C2semidirect
9. #9901C2semidirect
10. #9622C2semidirect
11. #9612C2semidirect
12. #9602C2semidirect
13. #9575C2semidirect
14. #9568C2semidirect
15. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#299952
#291762
#228362
#221112
#106222
#106192
#104802
#99012
#96222
#96122
#96022
#95752
#95682
#85902