Finite Group Databasebeta

← #276408

Subgroups — #276408

Order 512.

600 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#33068C211yesmaximal, characteristic
2#29207C211yesmaximal, characteristic, complemented
3#22795C211yesmaximal, characteristic
4C4×#476C211yesmaximal, characteristic
5#10619C211yesmaximal, characteristic
6#10615C211yesmaximal, characteristic
7#10481C211yesmaximal, characteristic
8#10480C211yesmaximal, characteristic
9#9900C211yesmaximal, characteristic, complemented
10#9615C211yesmaximal, characteristic, complemented
11#9612C211yesmaximal, characteristic, complemented
12#9603C211yesmaximal, characteristic, complemented
13#9567C211yesmaximal, characteristic, complemented
14#9567C211yesmaximal, characteristic, complemented
15#8590C211yesmaximal, characteristic, complemented
16C2×#476C2211yescharacteristic
17C2×#431—21no—
18#2859C2211yescharacteristic
19C4×C4⋊Q8C2211yescharacteristic
20C42×D4C2211yescharacteristic
21#2059—21no—
22#2059—21no—
23#1990C2211yescharacteristic
24#1990C2211yescharacteristic
25#1987C2211yescharacteristic
26#1954C2211yescharacteristic
27#1954C2211yescharacteristic
28#1949—21no—
29#1949—21no—
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1923—21no—
34#1923—21no—
35#1920—21no—
36#1920—21no—
37#1920—21no—
38#1864—21no—
39#1864—21no—
40#1864—21no—
41#1845C2211yescharacteristic
42#1844C2211yescharacteristic
43#1842C2211yescharacteristic
44#1831C2211yescharacteristic
45#1802C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1797—21no—
49C4×#106—21no—
50C8×C4⋊C4C2211yescharacteristic
51C8×C4⋊C4C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×#102—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—
76#431—44no—
77C2×#107—21no—
78C2×#107—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×#102—21no—
96C2×#102—21no—
97C2×#102—21no—
98C2×#102—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C4×C8—21no—
104C2×C4×C8—21no—
105C2×C4×C8—21no—
106C2×C4×C8C2311yescharacteristic
107C2×C4×C8C2311yescharacteristic
108#391—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#384C2311yescharacteristic
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#382—21no—
119C4×C4⋊C4—21no—
120C4×C4⋊C4—21no—
121C4×C4⋊C4D411yescharacteristic
122C4×C4⋊C4D411yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C22⋊C4—21no—
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#336—21no—
145#336D411yescharacteristic
146#336D411yescharacteristic
147#335—22no—
148#334—22no—
149#333—22no—
150#333—22no—
151C22×Q8—21no—
152C22×D4C2×D411yescharacteristic
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—21no—
169C4⋊Q8#4112yes—
170#125—81no—
171#125—81no—
172C4×Q8—82no—
173C4×D4—22no—
174C4×D4—22no—
175C4×D4—22no—
176C4×D4—21no—
177C4×D4—21no—
178C4×D4—21no—
179C4×D4—21no—
180C4×D4—41no—
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—21no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4C2×D411yescharacteristic
196C2×C4⋊C4C2×D411yescharacteristic
197C2×C4⋊C4C2×D411yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—21no—
204C2×C42—42no—
205C2×C42—42no—
206C2×C42—41no—
207C2×C42C2×D411yescharacteristic
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D411yescharacteristic
211C2×C42#4111yescharacteristic
212C2×C42#4111yescharacteristic
213C2×C42C2411yescharacteristic
214#107—81no—
215#107—81no—
216#106—24no—
217#106—42no—
218#106—42no—
219#106—42no—
220#106—81no—
221#106—81no—
222C4⋊C8—22no—
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—21no—
234C4⋊C8—42no—
235C4⋊C8—81no—
236C4⋊C8SD1614yes—
237C4⋊C8#4112yes—
238#102—24no—
239#102—42no—
240#102—42no—
241#102—42no—
242C22⋊C8—42no—
243C22⋊C8—42no—
244C22⋊C8—81no—
245C22⋊C8—81no—
246C4×C8—21no—
247C4×C8—21no—
248C4×C8—21no—
249C4×C8—21no—
250C4×C8—42no—
251C4×C8—42no—
252C4×C8—42no—
253#95—21no—
254#95—21no—
255#95—21no—
256#95—41no—
257#95—41no—
258#95—41no—
259#95—41no—
260C24—21no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×Q8—81no—
264C2×Q8—81no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—21no—
268C2×D4—21no—
269C2×D4—21no—
270C2×D4#13512yes—
271C22×C4—24no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—81no—
290C22×C4#12111yescharacteristic
291C22×C4#12111yescharacteristic
292C22×C4#12111yescharacteristic
293C22×C4#12111yescharacteristic
294C22×C4C22×D411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C2×C8—21no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—22no—
302C2×C8—21no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—21no—
306C2×C8—21no—
307C2×C8—21no—
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—41no—
325C2×C8C2×SD1612yes—
326C2×C8C2×SD1612yes—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4—22no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—22no—
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—81no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—82no—
351C4⋊C4—82no—
352C4⋊C4—82no—
353C4⋊C4#13512yes—
354C4⋊C4#13712yes—
355C22⋊C4—24no—
356C22⋊C4—42no—
357C22⋊C4—41no—
358C22⋊C4—41no—
359C42—21no—
360C42—21no—
361C42—21no—
362C42—21no—
363C42—21no—
364C42—21no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42—42no—
370C42—42no—
371C42—82no—
372C42C2×SD1612yes—
373C42#13712yes—
374C42C2×#4111yescharacteristic
375C42C2×#4111yescharacteristic
376C422−1+411yescharacteristic
377C422−1+411yescharacteristic
378C23—22no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23C2×#12111yescharacteristic
386Q8—82no—
387Q8—82no—
388D4—22no—
389D4—22no—
390D4—41no—
391D4—41no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—22no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—48no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—84no—
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#46112yes—
432C2×C4#46112yes—
433C2×C4#52511yescharacteristic
434C2×C4#53611yescharacteristic
435C2×C4#54711yescharacteristic
436C2×C4#54811yescharacteristic
437C2×C4#55211yescharacteristic
438C2×C4#55611yescharacteristic
439C2×C4C22×SD1611yescharacteristic
440C2×C4C2×#13511yescharacteristic
441C2×C4C2×#13511yescharacteristic
442C2×C4C2×#13711yescharacteristic
443C2×C4C2×#13711yescharacteristic
444C2×C4#57511yescharacteristic
445C2×C4#57611yescharacteristic
446C2×C4#57711yescharacteristic
447C8—22no—
448C8—22no—
449C8—41no—
450C8—41no—
451C8—42no—
452C8—81no—
453C8—81no—
454C8—81no—
455C8—81no—
456C8—81no—
457C22—42no—
458C22—42no—
459C22—42no—
460C22—81no—
461C22—81no—
462C22—81no—
463C22—81no—
464C22#241511yescharacteristic
465C22C2×#46111yescharacteristic
466C22#304511yescharacteristic
467C22#305011yescharacteristic
468C22#305611yescharacteristic
469C22#306711yescharacteristic
470C22#306811yescharacteristic
471C4—21no—
472C4—21no—
473C4—22no—
474C4—22no—
475C4—41no—
476C4—42no—
477C4—41no—
478C4—42no—
479C4—81no—
480C4—82no—
481C4—161no—
482C4—161no—
483C4—161no—
484C4#310211yescharacteristic
485C4#310811yescharacteristic
486C4#323111yescharacteristic
487C4#323211yescharacteristic
488C4#333511yescharacteristic
489C4#335711yescharacteristic
490C4#336211yescharacteristic
491C4#336911yescharacteristic
492C2—82no—
493C2#1910511yescharacteristic
494C2#1916511yescharacteristic
495C2#1941211yescharacteristic
496C2#1949311yescharacteristic
497C2#2468011yescharacteristic
498C2#2470511yescharacteristic
499C2#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. #29207C2semidirect
9. #9900C2semidirect
10. #9615C2semidirect
11. #9612C2semidirect
12. #9603C2semidirect
13. #9567C2semidirect
14. #9567C2semidirect
15. #8590C2semidirect

Maximal subgroups

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

subgroupindex
#330682
#292072
#227952
C4×#4762
#106192
#106152
#104812
#104802
#99002
#96152
#96122
#96032
#95672
#85902