Finite Group Databasebeta

← #276526

Subgroups — #276526

Order 512.

565 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33068C211yesmaximal, characteristic
2#29220C211yesmaximal, characteristic, complemented
3#22835C211yesmaximal, characteristic
4#21614C211yesmaximal, characteristic
5#10625C211yesmaximal, characteristic
6#10621C211yesmaximal, characteristic
7#10481C211yesmaximal, characteristic
8#10480C211yesmaximal, characteristic
9#9907C211yesmaximal, characteristic, complemented
10#9616C211yesmaximal, characteristic, complemented
11#9613C211yesmaximal, characteristic, complemented
12#9606C211yesmaximal, characteristic, complemented
13#9567C211yesmaximal, characteristic, complemented
14#9567C211yesmaximal, characteristic, complemented
15#8594C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17#2859C2211yescharacteristic
18C4×C4⋊Q8C2211yescharacteristic
19C42×D4C2211yescharacteristic
20#2101—21no—
21#2101—21no—
22#1990C2211yescharacteristic
23#1990C2211yescharacteristic
24#1989—21no—
25#1989—21no—
26#1987C2211yescharacteristic
27#1954C2211yescharacteristic
28#1954C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1923—21no—
33#1923—21no—
34#1920—21no—
35#1920—21no—
36#1919—21no—
37#1864—21no—
38#1864—21no—
39#1862—21no—
40#1834C2211yescharacteristic
41#1834C2211yescharacteristic
42#1831C2211yescharacteristic
43#1831C2211yescharacteristic
44#1817—21no—
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1776—21no—
49C8×C4⋊C4C2211yescharacteristic
50C8×C4⋊C4C2211yescharacteristic
51#1767C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53#1762—21no—
54C4×C22⋊C8—21no—
55C2×#335C2211yescharacteristic
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×D4D411yescharacteristic
71C2×C4×D4D411yescharacteristic
72C2×C4×D4C2311yescharacteristic
73C22×C42—21no—
74#476—22no—
75#476—22no—
76C2×#107—21no—
77C2×#106—21no—
78C2×#106—21no—
79C2×#106—21no—
80C2×C4⋊C8D411yescharacteristic
81C2×C4⋊C8D411yescharacteristic
82C2×C4⋊C8C2311yescharacteristic
83C2×C4⋊C8C2311yescharacteristic
84C2×C4⋊C8C2311yescharacteristic
85C2×C4⋊C8C2311yescharacteristic
86C2×C4⋊C8C2311yescharacteristic
87C2×C4⋊C8C2311yescharacteristic
88C2×#102—21no—
89C2×#102—21no—
90C2×#102—21no—
91C2×#102—21no—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—21no—
95C2×C22⋊C8—21no—
96C2×#97—21no—
97C2×C4×C8—21no—
98C2×C4×C8—21no—
99C2×C4×C8C2311yescharacteristic
100C2×C4×C8C2311yescharacteristic
101#391—21no—
102#389—21no—
103#389—21no—
104#389—21no—
105#389—21no—
106#389—21no—
107#389—21no—
108#384C2311yescharacteristic
109#384C2311yescharacteristic
110#384C2311yescharacteristic
111#382—21no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4—21no—
114C4×C4⋊C4D411yescharacteristic
115C4×C4⋊C4D411yescharacteristic
116C4×C4⋊C4C2311yescharacteristic
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120C43C2311yescharacteristic
121#340—21no—
122#340—21no—
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#336—21no—
138#336—21no—
139#336—21no—
140#336—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336D411yescharacteristic
145#336D411yescharacteristic
146#335—22no—
147#334—22no—
148#333—22no—
149#333—22no—
150C22×Q8—21no—
151C22×D4C2×D411yescharacteristic
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8C2×D411yescharacteristic
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8#4112yes—
169#125—42no—
170#125—42no—
171C4×Q8—82no—
172C4×D4—22no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—21no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—41no—
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—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—41no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C2×D411yescharacteristic
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C42—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—41no—
204C2×C42—81no—
205C2×C42—81no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42#4111yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42C2411yescharacteristic
213#107—42no—
214#106—24no—
215#106—42no—
216#106—42no—
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—21no—
227C4⋊C8—21no—
228C4⋊C8—41no—
229C4⋊C8—41no—
230C4⋊C8#4112yes—
231#102—24no—
232#102—42no—
233#102—42no—
234#102—42no—
235C22⋊C8—42no—
236C22⋊C8—42no—
237C22⋊C8—81no—
238C22⋊C8—81no—
239#97—42no—
240C4×C8—21no—
241C4×C8—21no—
242C4×C8—21no—
243C4×C8—21no—
244C4×C8—81no—
245C4×C8—81no—
246#95—21no—
247#95—21no—
248#95—21no—
249#95—41no—
250#95—41no—
251#95—41no—
252#95—41no—
253C24—21no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—81no—
257C2×Q8—81no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4—21no—
261C2×D4—21no—
262C2×D4—21no—
263C2×D4#13512yes—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—24no—
267C22×C4—21no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—81no—
281C22×C4—81no—
282C22×C4—81no—
283C22×C4#12111yescharacteristic
284C22×C4#12111yescharacteristic
285C22×C4#12111yescharacteristic
286C22×C4#12111yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C2×#4111yescharacteristic
290C2×C8—22no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
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—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—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—22no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—22no—
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—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—81no—
341C4⋊C4—81no—
342C4⋊C4—81no—
343C4⋊C4—82no—
344C4⋊C4—82no—
345C4⋊C4—82no—
346C4⋊C4#13512yes—
347C4⋊C4#13712yes—
348C22⋊C4—24no—
349C22⋊C4—42no—
350C22⋊C4—41no—
351C22⋊C4—41no—
352C42—21no—
353C42—21no—
354C42—21no—
355C42—21no—
356C42—21no—
357C42—21no—
358C42—21no—
359C42—21no—
360C42—41no—
361C42—41no—
362C42—41no—
363C42—41no—
364C42—81no—
365C42—81no—
366C42—82no—
367C42C2×#4111yescharacteristic
368C42C2×#4111yescharacteristic
369C422−1+411yescharacteristic
370C422−1+411yescharacteristic
371C23—22no—
372C23—41no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—41no—
378C23C2×#12111yescharacteristic
379Q8—82no—
380Q8—82no—
381D4—22no—
382D4—22no—
383D4—41no—
384D4—41no—
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—22no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—84no—
410C2×C4—84no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
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#52511yescharacteristic
427C2×C4#53611yescharacteristic
428C2×C4#54711yescharacteristic
429C2×C4#54811yescharacteristic
430C2×C4#55211yescharacteristic
431C2×C4#55611yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×#13511yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57511yescharacteristic
437C2×C4#57511yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C8—41no—
441C8—41no—
442C8—41no—
443C8—41no—
444C8—42no—
445C8—81no—
446C8—81no—
447C8—81no—
448C8—81no—
449C8—81no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—81no—
454C22—81no—
455C22—81no—
456C22—81no—
457C22#241511yescharacteristic
458C22#304011yescharacteristic
459C22#304211yescharacteristic
460C22#305011yescharacteristic
461C22#305611yescharacteristic
462C22#306711yescharacteristic
463C22#306811yescharacteristic
464C4—21no—
465C4—21no—
466C4—41no—
467C4—42no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—42no—
472C4—81no—
473C4—82no—
474C4—161no—
475C4—161no—
476C4—161no—
477C4#310211yescharacteristic
478C4#310811yescharacteristic
479C4#323111yescharacteristic
480C4#323211yescharacteristic
481C4#335011yescharacteristic
482C4#335011yescharacteristic
483C4#337111yescharacteristic
484C4#337111yescharacteristic
485C2—82no—
486C2#1927811yescharacteristic
487C2#1932611yescharacteristic
488C2#1939411yescharacteristic
489C2#1946611yescharacteristic
490C2#2469711yescharacteristic
491C2#2470511yescharacteristic
492C2#2472411yescharacteristic
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. #29220C2semidirect
9. #9907C2semidirect
10. #9616C2semidirect
11. #9613C2semidirect
12. #9606C2semidirect
13. #9567C2semidirect
14. #9567C2semidirect
15. #8594C2semidirect

Maximal subgroups

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

subgroupindex
#330682
#292202
#228352
#216142
#106252
#106212
#104812
#104802
#99072
#96162
#96132
#96062
#95672
#85942