Finite Group Databasebeta

← #305907

Subgroups — #305907

Order 512.

603 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#28800C211yesmaximal, characteristic, complemented
2#26106C211yesmaximal, characteristic
3#26041C211yesmaximal, characteristic
4#22703C211yesmaximal, characteristic
5#10655C211yesmaximal, characteristic, complemented
6#10652C211yesmaximal, characteristic, complemented
7#10582C211yesmaximal, characteristic, complemented
8#10580C211yesmaximal, characteristic, complemented
9#9619C211yesmaximal, characteristic, complemented
10#9596C211yesmaximal, characteristic, complemented
11#8890C211yesmaximal, characteristic
12#8886C211yesmaximal, characteristic
13#8821C211yesmaximal, characteristic, complemented
14#8648C211yesmaximal, characteristic
15#8202C211yesmaximal, characteristic
16C2×#478—21no—
17C2×C8×D4C2211yescharacteristic
18#2552C2211yescharacteristic
19#2552C2211yescharacteristic
20D4×C4⋊C4C2211yescharacteristic
21D4×C4⋊C4C2211yescharacteristic
22#2099—21no—
23#2087—21no—
24#2086—21no—
25#2084—21no—
26#2059—21no—
27#2059—21no—
28#1992—21no—
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1948C2211yescharacteristic
32#1948C2211yescharacteristic
33#1947C2211yescharacteristic
34#1947C2211yescharacteristic
35#1944—21no—
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1862—21no—
39#1862—21no—
40#1853—21no—
41#1853—21no—
42#1845C2211yescharacteristic
43#1844C2211yescharacteristic
44#1834C2211yescharacteristic
45#1834C2211yescharacteristic
46#1834C2211yescharacteristic
47#1834C2211yescharacteristic
48#1807—21no—
49#1802—21no—
50#1802C2211yescharacteristic
51#1802C2211yescharacteristic
52C4×#106C2211yescharacteristic
53#1770C2211yescharacteristic
54C4×#102C2211yescharacteristic
55C2×#340—21no—
56C2×#335C2211yescharacteristic
57C2×D4⋊C8C2211yescharacteristic
58C2×#322C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1279C2211yescharacteristic
64#1279C2211yescharacteristic
65#1279C2211yescharacteristic
66#1277C2211yescharacteristic
67C23×C8—21no—
68C2×#125—21no—
69C2×#125—21no—
70C2×C4×D4—21no—
71C2×C4×D4—21no—
72C2×C4×D4C2311yescharacteristic
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75#478—44no—
76C8×D4—22no—
77C8×D4Q814yes—
78C2×#107—21no—
79C2×#107—21no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84C2×#106—21no—
85C2×#106—21no—
86C2×#106—41no—
87C2×#106D411yescharacteristic
88C2×#106D411yescharacteristic
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8C2311yescharacteristic
92C2×C4⋊C8C2311yescharacteristic
93C2×C4⋊C8C2311yescharacteristic
94C2×C4⋊C8C2311yescharacteristic
95C2×#102—21no—
96C2×#102—21no—
97C2×#102D411yescharacteristic
98C2×#102D411yescharacteristic
99C2×C22⋊C8—21no—
100C2×#97—21no—
101C2×C4×C8C2311yescharacteristic
102C2×C4×C8C2311yescharacteristic
103C2×C4×C8C2311yescharacteristic
104#398—41no—
105#398—41no—
106#389—21no—
107#389—21no—
108#389—21no—
109#389—21no—
110#389—21no—
111#389—21no—
112#385—41no—
113#385—41no—
114#384—21no—
115#384—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#384C2311yescharacteristic
119#384C2311yescharacteristic
120#380—21no—
121#380—21no—
122C4×C4⋊C4C2311yescharacteristic
123C4×C4⋊C4C2311yescharacteristic
124C4×C4⋊C4C2311yescharacteristic
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#340—41no—
140#340D411yescharacteristic
141#340D411yescharacteristic
142#340D411yescharacteristic
143#340D411yescharacteristic
144#336—21no—
145#336—21no—
146#336—21no—
147#335—22no—
148D4⋊C8Q814yes—
149#322—22no—
150C22×D4C2×D411yescharacteristic
151C23×C4—21no—
152C23×C4—41no—
153C23×C4—41no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—24no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—81no—
161C22×C8—81no—
162C22×C8C2×D411yescharacteristic
163C22×C8C2×D411yescharacteristic
164C22×C8C2×D411yescharacteristic
165C22×C8C2×D411yescharacteristic
166#125—42no—
167#125—42no—
168#125—81no—
169#125—81no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—42no—
173C4×D4—42no—
174C4×D4—81no—
175C4×D4—81no—
176C4×D4C2×Q812yes—
177C4×D4C2×Q812yes—
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—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4—41no—
195C2×C4⋊C4—41no—
196C2×C4⋊C4—41no—
197C2×C4⋊C4—41no—
198C2×C4⋊C4—41no—
199C2×C4⋊C4—41no—
200C2×C4⋊C4—81no—
201C2×C4⋊C4—81no—
202C2×C4⋊C4C2×D411yescharacteristic
203C2×C4⋊C4C2×D411yescharacteristic
204C2×C4⋊C4C2×D411yescharacteristic
205C2×C4⋊C4C2×D411yescharacteristic
206C2×C4⋊C4C2×D411yescharacteristic
207C2×C4⋊C4C2×D411yescharacteristic
208C2×C4⋊C4C2×D411yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—21no—
213C2×C42—21no—
214C2×C42—21no—
215C2×C42C2411yescharacteristic
216#107—42no—
217#107—81no—
218#107—81no—
219#107—81no—
220#106—22no—
221#106—41no—
222#106—42no—
223#106—42no—
224#106—81no—
225#106—81no—
226#106—82no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8—21no—
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—42no—
234C4⋊C8—81no—
235C4⋊C8C2×Q812yes—
236C4⋊C8C2×Q812yes—
237#102—22no—
238#102—22no—
239#102—42no—
240#102—42no—
241C22⋊C8—24no—
242#97—81no—
243C4×C8—21no—
244C4×C8—21no—
245C4×C8—21no—
246C4×C8C2×Q812yes—
247C4×C8C2×Q812yes—
248C4×C8#4112yes—
249#95—21no—
250#95—21no—
251#95—21no—
252#95—41no—
253#95—41no—
254#95—41no—
255#95—41no—
256C24—21no—
257C2×D4—21no—
258C2×D4—21no—
259C2×D4—21no—
260C2×D4C4⋊Q812yes—
261C2×D4AGL(1,ℤ/8)12yes—
262C2×D4#13712yes—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
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—82no—
281C22×C4—82no—
282C22×C4—82no—
283C22×C4—82no—
284C22×C4—161no—
285C22×C4—161no—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4H(5,𝔽2)11yescharacteristic
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—22no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—22no—
297C2×C8—22no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C2×C8—21no—
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—41no—
313C2×C8—41no—
314C2×C8—84no—
315C2×C8C22⋊Q812yes—
316C2×C8C22⋊Q812yes—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
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—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—82no—
346C4⋊C4—82no—
347C4⋊C4—82no—
348C4⋊C4—81no—
349C4⋊C4—81no—
350C4⋊C4—81no—
351C4⋊C4—81no—
352C4⋊C4—82no—
353C4⋊C4—82no—
354C4⋊C4—82no—
355C4⋊C4—161no—
356C4⋊C4—161no—
357C4⋊C4C4⋊Q812yes—
358C4⋊C4#13512yes—
359C22⋊C4—22no—
360C22⋊C4—22no—
361C22⋊C4—82no—
362C22⋊C4—82no—
363C42—41no—
364C42—41no—
365C42—41no—
366C42—41no—
367C42—41no—
368C42—41no—
369C42C22×Q811yescharacteristic
370C42C22×Q811yescharacteristic
371C42C2×#4111yescharacteristic
372C422−1+411yescharacteristic
373C23—22no—
374C23—22no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—41no—
379C23—41no—
380C23D4211yescharacteristic
381D4—22no—
382D4—22no—
383D4—22no—
384D4#50114yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—41no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—41no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
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—164no—
425C2×C4—164no—
426C2×C4C2×C22⋊Q811yescharacteristic
427C2×C4C2×C4⋊Q811yescharacteristic
428C2×C4#54411yescharacteristic
429C2×C4#54711yescharacteristic
430C2×C4D4×Q811yescharacteristic
431C2×C4#55411yescharacteristic
432C2×C4C2×#13511yescharacteristic
433C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
434C2×C4C2×#13711yescharacteristic
435C2×C4#57511yescharacteristic
436C2×C4#57611yescharacteristic
437C2×C4#57711yescharacteristic
438C2×C4#57711yescharacteristic
439C2×C4#57711yescharacteristic
440C8—22no—
441C8—22no—
442C8—42no—
443C8—42no—
444C8—41no—
445C8—41no—
446C8—42no—
447C8—81no—
448C8—81no—
449C22—24no—
450C22—42no—
451C22—42no—
452C22—42no—
453C22—42no—
454C22—42no—
455C22—42no—
456C22#244011yescharacteristic
457C22#326511yescharacteristic
458C22#326611yescharacteristic
459C22#327511yescharacteristic
460C22#328011yescharacteristic
461C22#328811yescharacteristic
462C22#328911yescharacteristic
463C4—21no—
464C4—21no—
465C4—42no—
466C4—42no—
467C4—81no—
468C4—81no—
469C4—81no—
470C4—81no—
471C4—81no—
472C4—161no—
473C4—161no—
474C4—161no—
475C4—161no—
476C4#307611yescharacteristic
477C4#308111yescharacteristic
478C4C2×#50111yescharacteristic
479C4#316311yescharacteristic
480C4#334411yescharacteristic
481C4#334511yescharacteristic
482C4#338111yescharacteristic
483C4#338911yescharacteristic
484C2—44no—
485C2#1977411yescharacteristic
486C2#1977911yescharacteristic
487C2#2018011yescharacteristic
488C2#2019311yescharacteristic
489C2#2604911yescharacteristic
490C2#2606811yescharacteristic
491C2#2608511yescharacteristic
subgroupG/H
center, socleC23D42
commutator, Frattini, cosocleC2×C42C24
prime radicalC22×D4C2×D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. #28800C2semidirect
5. #10655C2semidirect
6. #10652C2semidirect
7. #10582C2semidirect
8. #10580C2semidirect
9. #9619C2semidirect
10. #9596C2semidirect
13. #8821C2semidirect

Maximal subgroups

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

subgroupindex
#288002
#261062
#260412
#227032
#106552
#106522
#105822
#105802
#96192
#95962
#88902
#88862
#88212
#86482
#82022