Finite Group Databasebeta

← #279306

Subgroups — #279306

Order 512.

627 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33072C211yesmaximal, characteristic
2#29191C211yesmaximal, characteristic, complemented
3#22696C211yesmaximal, characteristic
4C4×#474C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10653C211yesmaximal, characteristic, complemented
7#10596C211yesmaximal, characteristic, complemented
8#10592C211yesmaximal, characteristic, complemented
9#9901C211yesmaximal, characteristic, complemented
10#9603C211yesmaximal, characteristic, complemented
11#9219C211yesmaximal, characteristic
12#9219C211yesmaximal, characteristic
13#8648C211yesmaximal, characteristic
14#8642C211yesmaximal, characteristic
15#8593C211yesmaximal, characteristic, complemented
16C2×#474C2211yescharacteristic
17C2×C8×D4C2211yescharacteristic
18C2×C8×D4C2211yescharacteristic
19#2859C2211yescharacteristic
20C4×C4⋊Q8C2211yescharacteristic
21C42×D4C2211yescharacteristic
22#2066—21no—
23#2066—21no—
24#1988C2211yescharacteristic
25#1956C2211yescharacteristic
26#1956C2211yescharacteristic
27#1948C2211yescharacteristic
28#1947C2211yescharacteristic
29#1947C2211yescharacteristic
30#1939C2211yescharacteristic
31#1939C2211yescharacteristic
32#1921—21no—
33#1919—21no—
34#1918—21no—
35#1865—21no—
36#1865—21no—
37#1864—21no—
38#1862—21no—
39#1862—21no—
40#1850—21no—
41#1850—21no—
42#1844C2211yescharacteristic
43#1834C2211yescharacteristic
44#1831C2211yescharacteristic
45#1831C2211yescharacteristic
46#1802C2211yescharacteristic
47#1802C2211yescharacteristic
48#1802C2211yescharacteristic
49#1797—21no—
50C4×#107—21no—
51C4×C4⋊C8C2211yescharacteristic
52C4×C4⋊C8C2211yescharacteristic
53C4×#102—21no—
54C2×#336—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#332C2211yescharacteristic
58C2×#332C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C23×C8—21no—
69C2×C4⋊Q8C2311yescharacteristic
70C2×#125—21no—
71C2×C4×Q8—21no—
72C2×C4×D4D411yescharacteristic
73C2×C4×D4D411yescharacteristic
74C2×C4×D4C2311yescharacteristic
75C22×C42—21no—
76#474—22no—
77#474Q814yes—
78C8×D4—22no—
79C8×D4—22no—
80C8×D4—41no—
81C8×D4—41no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107—21no—
85C2×#106—21no—
86C2×#106—21no—
87C2×#106—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8D411yescharacteristic
91C2×C4⋊C8D411yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×C4⋊C8C2311yescharacteristic
96C2×C4⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×#103—21no—
99C2×#103—21no—
100C2×#103—21no—
101C2×#103—21no—
102C2×#102—21no—
103C2×#102—21no—
104C2×#102—21no—
105C2×#102—21no—
106C2×C22⋊C8—21no—
107C2×C22⋊C8—21no—
108C2×C4×C8—21no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#391—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#389—21no—
117#389—21no—
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#384C2311yescharacteristic
121#382—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C22⋊C4—21no—
128C4×C22⋊C4—21no—
129C4×C22⋊C4—21no—
130C43C2311yescharacteristic
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#340—21no—
141#336—21no—
142#336—21no—
143#336—21no—
144#336—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336D411yescharacteristic
149#336D411yescharacteristic
150#335—41no—
151#334—41no—
152#332—41no—
153#332—41no—
154C22×Q8—21no—
155C22×D4C2×D411yescharacteristic
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8—21no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—42no—
166C22×C8—42no—
167C22×C8—42no—
168C22×C8—42no—
169C22×C8—81no—
170C22×C8—81no—
171C22×C8C2×D411yescharacteristic
172C22×C8C2×D411yescharacteristic
173C22×C8C2×D411yescharacteristic
174C22×C8C2×D411yescharacteristic
175C4⋊Q8—21no—
176C4⋊Q8—21no—
177C4⋊Q8—21no—
178C4⋊Q8C2×Q812yes—
179#125—82no—
180C4×Q8—81no—
181C4×Q8—81no—
182C4×D4—22no—
183C4×D4—22no—
184C4×D4—22no—
185C4×D4—21no—
186C4×D4—21no—
187C4×D4—41no—
188C4×D4C2×Q812yes—
189C4×D4#4112yes—
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—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C42—21no—
211C2×C42—21no—
212C2×C42—21no—
213C2×C42—41no—
214C2×C42—81no—
215C2×C42—81no—
216C2×C42C2×D411yescharacteristic
217C2×C42C2×D411yescharacteristic
218C2×C42C2×D411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42#4111yescharacteristic
221C2×C42#4111yescharacteristic
222C2×C42C2411yescharacteristic
223#107—24no—
224#107—42no—
225#107—42no—
226#106—42no—
227#106—81no—
228#106—81no—
229C4⋊C8—22no—
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—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—41no—
242C4⋊C8—81no—
243C4⋊C8—81no—
244C4⋊C8C2×Q812yes—
245#103—42no—
246#103—42no—
247#103—81no—
248#103—81no—
249#102—24no—
250#102—42no—
251#102—42no—
252#102—42no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—42no—
260#95—21no—
261#95—21no—
262#95—21no—
263#95—41no—
264#95—41no—
265#95—41no—
266#95—41no—
267C24—21no—
268C2×Q8—22no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4C22⋊Q812yes—
278C2×D4#13512yes—
279C2×D4#13712yes—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—21no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—81no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4#12111yescharacteristic
300C22×C4#12111yescharacteristic
301C22×C4#12111yescharacteristic
302C22×C4#12111yescharacteristic
303C22×C4C22×D411yescharacteristic
304C22×C4C22×D411yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C2×C8—21no—
307C2×C8—21no—
308C2×C8—22no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—21no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8—21no—
317C2×C8—21no—
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×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—84no—
336C2×C8—84no—
337C2×C8#13512yes—
338C4⋊C4—21no—
339C4⋊C4—21no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—21no—
343C4⋊C4—21no—
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—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—81no—
359C4⋊C4—81no—
360C4⋊C4—81no—
361C4⋊C4—82no—
362C4⋊C4—82no—
363C4⋊C4—82no—
364C4⋊C4C22⋊Q812yes—
365C4⋊C4#13512yes—
366C22⋊C4—24no—
367C22⋊C4—22no—
368C22⋊C4—22no—
369C22⋊C4—42no—
370C42—21no—
371C42—21no—
372C42—21no—
373C42—21no—
374C42—21no—
375C42—21no—
376C42—21no—
377C42—21no—
378C42—22no—
379C42—41no—
380C42—41no—
381C42—41no—
382C42—81no—
383C42—81no—
384C42—82no—
385C42C22×Q811yescharacteristic
386C42C2×#4111yescharacteristic
387C42H(5,𝔽2)11yescharacteristic
388C422−1+411yescharacteristic
389C23—22no—
390C23—22no—
391C23—22no—
392C23—41no—
393C23—41no—
394C23—41no—
395C23—41no—
396C23C2×#12111yescharacteristic
397Q8—42no—
398Q8—42no—
399Q8—81no—
400Q8—81no—
401D4—22no—
402D4—22no—
403D4—22no—
404D4#47914yes—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—21no—
413C2×C4—21no—
414C2×C4—21no—
415C2×C4—22no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—21no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—41no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—84no—
433C2×C4—84no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—81no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4—81no—
445C2×C4#47112yes—
446C2×C4C2×C22⋊Q811yescharacteristic
447C2×C4#53711yescharacteristic
448C2×C4#54611yescharacteristic
449C2×C4D4×Q811yescharacteristic
450C2×C4#55211yescharacteristic
451C2×C4#55411yescharacteristic
452C2×C4C2×#13511yescharacteristic
453C2×C4C2×#13511yescharacteristic
454C2×C4C2×#13511yescharacteristic
455C2×C4C2×#13711yescharacteristic
456C2×C4#57511yescharacteristic
457C2×C4#57511yescharacteristic
458C2×C4#57611yescharacteristic
459C2×C4#57611yescharacteristic
460C8—22no—
461C8—42no—
462C8—41no—
463C8—41no—
464C8—41no—
465C8—81no—
466C8—81no—
467C8—81no—
468C8—81no—
469C8—81no—
470C22—24no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22#241711yescharacteristic
478C22#303511yescharacteristic
479C22#304111yescharacteristic
480C22#305011yescharacteristic
481C22#305611yescharacteristic
482C22C2×#47111yescharacteristic
483C22#306911yescharacteristic
484C4—22no—
485C4—21no—
486C4—21no—
487C4—41no—
488C4—41no—
489C4—42no—
490C4—41no—
491C4—42no—
492C4—81no—
493C4—82no—
494C4—161no—
495C4—161no—
496C4—161no—
497C4C2×#47911yescharacteristic
498C4#307611yescharacteristic
499C4#324811yescharacteristic
500C4#325011yescharacteristic
501C4#329711yescharacteristic
502C4#332211yescharacteristic
503C4#338011yescharacteristic
504C4#338911yescharacteristic
505C2—44no—
506C2#1912711yescharacteristic
507C2#1918111yescharacteristic
508C2#1938811yescharacteristic
509C2#1945711yescharacteristic
510C2#2482011yescharacteristic
511C2#2484011yescharacteristic
512C2#2485811yescharacteristic
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. #29191C2semidirect
5. #10655C2semidirect
6. #10653C2semidirect
7. #10596C2semidirect
8. #10592C2semidirect
9. #9901C2semidirect
10. #9603C2semidirect
15. #8593C2semidirect

Maximal subgroups

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

subgroupindex
#330722
#291912
#226962
C4×#4742
#106552
#106532
#105962
#105922
#99012
#96032
#92192
#86482
#86422
#85932