Finite Group Databasebeta

← #329920

Subgroups — #329920

Order 512.

609 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#33148C211yesmaximal, characteristic
2#33146C211yesmaximal, characteristic
3#33049C211yesmaximal, characteristic
4#21146C211yesmaximal, characteristic
5#10658C211yesmaximal, characteristic, complemented
6#10657C211yesmaximal, characteristic, complemented
7#10600C211yesmaximal, characteristic, complemented
8#10596C211yesmaximal, characteristic, complemented
9#10593C211yesmaximal, characteristic, complemented
10#10589C211yesmaximal, characteristic, complemented
11#10541C211yesmaximal, characteristic, complemented
12#10541C211yesmaximal, characteristic, complemented
13#9139C211yesmaximal, characteristic
14#9135C211yesmaximal, characteristic
15#8597C211yesmaximal, characteristic
16C2×#487C2211yescharacteristic
17C2×#423—21no—
18C2×#423C2211yescharacteristic
19C2×C4×M16C2211yescharacteristic
20#2857C2211yescharacteristic
21#2857C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1990C2211yescharacteristic
24#1988C2211yescharacteristic
25#1988C2211yescharacteristic
26#1987C2211yescharacteristic
27#1987C2211yescharacteristic
28#1956C2211yescharacteristic
29#1948C2211yescharacteristic
30#1948C2211yescharacteristic
31#1947C2211yescharacteristic
32#1947C2211yescharacteristic
33#1940—21no—
34#1940—21no—
35#1939C2211yescharacteristic
36#1939C2211yescharacteristic
37#1939C2211yescharacteristic
38#1939C2211yescharacteristic
39#1935—21no—
40#1935—21no—
41#1920—21no—
42#1920—21no—
43#1918—21no—
44#1918—21no—
45#1894—21no—
46#1894—21no—
47#1853—21no—
48#1852—21no—
49#1834C2211yescharacteristic
50#1823—21no—
51#1823—21no—
52#1797C2211yescharacteristic
53#1797C2211yescharacteristic
54#1728—21no—
55C2×#335C2211yescharacteristic
56C2×#334C2211yescharacteristic
57C2×#333C2211yescharacteristic
58C2×#333C2211yescharacteristic
59C2×#332C2211yescharacteristic
60C2×#332C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1277C2211yescharacteristic
66#1277C2211yescharacteristic
67C22×M16—21no—
68C22×M16D411yescharacteristic
69C22×M16D411yescharacteristic
70C2×C4⋊Q8C2311yescharacteristic
71C2×C4⋊Q8C2311yescharacteristic
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8—21no—
74C22×C4⋊C4D411yescharacteristic
75C22×C4⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78#487—41no—
79#487—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#106—21no—
83C2×#106—21no—
84#423—41no—
85#423—41no—
86#423—44no—
87C2×C4⋊C8—21no—
88C2×C4⋊C8—21no—
89C2×C4⋊C8—21no—
90C2×C4⋊C8—21no—
91C2×C4⋊C8—21no—
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×C22⋊C8—21no—
103C2×C22⋊C8—21no—
104C4×M16—41no—
105C4×M16—41no—
106C2×#97D411yescharacteristic
107C2×#97D411yescharacteristic
108C2×C4×C8C2311yescharacteristic
109C2×C4×C8C2311yescharacteristic
110#391—21no—
111#391—21no—
112#389—21no—
113#389—21no—
114#389—21no—
115#389—21no—
116#384C2311yescharacteristic
117#384C2311yescharacteristic
118#380—21no—
119#380—21no—
120#380—21no—
121#380—21no—
122C4×C4⋊C4—21no—
123C4×C4⋊C4—21no—
124C4×C4⋊C4D411yescharacteristic
125C4×C4⋊C4D411yescharacteristic
126C4×C4⋊C4C2311yescharacteristic
127C4×C4⋊C4C2311yescharacteristic
128C2×#95—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#340—21no—
141#340—21no—
142#340—21no—
143#340—21no—
144#340—21no—
145#336—21no—
146#336—21no—
147#336—21no—
148#336—21no—
149#336—21no—
150#336—21no—
151#335—41no—
152#334—41no—
153#333—41no—
154#333—41no—
155#332—41no—
156#332—41no—
157C22×Q8—21no—
158C22×Q8—21no—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4#4111yescharacteristic
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—41no—
168C2×M16—41no—
169C2×M16—41no—
170C2×M16—42no—
171C2×M16—42no—
172C2×M16—42no—
173C22×C8—21no—
174C22×C8—21no—
175C22×C8—21no—
176C22×C8—21no—
177C22×C8—21no—
178C22×C8—21no—
179C22×C8C2×D411yescharacteristic
180C22×C8C2×D411yescharacteristic
181C22×C8C2×D411yescharacteristic
182C22×C8C2×D411yescharacteristic
183C4⋊Q8—21no—
184C4⋊Q8—21no—
185C4⋊Q8—21no—
186C4⋊Q8—21no—
187C4⋊Q8—21no—
188C4⋊Q8—21no—
189C4⋊Q8—21no—
190C4⋊Q8—21no—
191C22⋊Q8—42no—
192C22⋊Q8—42no—
193C22⋊Q8—82no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—21no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—41no—
210C2×C4⋊C4—41no—
211C2×C4⋊C4—41no—
212C2×C4⋊C4C2×D411yescharacteristic
213C2×C4⋊C4C2×D411yescharacteristic
214C2×C4⋊C4C2×D411yescharacteristic
215C2×C4⋊C4C2×D411yescharacteristic
216C2×C22⋊C4—41no—
217C2×C22⋊C4—41no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—21no—
222C2×C42—21no—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—42no—
227C2×C42—42no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230C2×C42#4111yescharacteristic
231C2×C42C2411yescharacteristic
232#107—42no—
233#107—42no—
234#106—42no—
235#106—42no—
236C4⋊C8—21no—
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—21no—
241C4⋊C8—21no—
242C4⋊C8—21no—
243C4⋊C8—21no—
244C4⋊C8—21no—
245C4⋊C8—21no—
246C4⋊C8—21no—
247C4⋊C8—21no—
248C4⋊C8—42no—
249C4⋊C8—42no—
250C4⋊C8—42no—
251C4⋊C8—42no—
252C4⋊C8—42no—
253#103—42no—
254#103—42no—
255#103—42no—
256#103—42no—
257C22⋊C8—42no—
258C22⋊C8—42no—
259#97—41no—
260#97—41no—
261C4×C8—21no—
262C4×C8—21no—
263C4×C8—21no—
264C4×C8—21no—
265#95—21no—
266#95—21no—
267#95—21no—
268#95—21no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273C24#12111yescharacteristic
274C2×Q8—41no—
275C2×Q8—41no—
276C2×Q8—41no—
277C2×Q8—41no—
278C2×Q8—41no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—81no—
283C2×Q8—81no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4#12111yescharacteristic
313C22×C4#12111yescharacteristic
314C22×C4#12111yescharacteristic
315C22×C4#13512yes—
316C22×C4AGL(1,ℤ/8)12yes—
317C22×C4C22×D411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320M16—44no—
321M16—44no—
322M16—84no—
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—41no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8—41no—
351C2×C8—41no—
352C2×C8#13712yes—
353C4⋊C4—21no—
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—41no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—81no—
378C4⋊C4—81no—
379C4⋊C4—82no—
380C4⋊C4—82no—
381C4⋊C4#13712yes—
382C4⋊C4#13712yes—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C42—21no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—21no—
391C42—41no—
392C42—41no—
393C42—41no—
394C42—41no—
395C42—42no—
396C42—42no—
397C42H(5,𝔽2)11yescharacteristic
398C42H(5,𝔽2)11yescharacteristic
399C422−1+411yescharacteristic
400C422−1+411yescharacteristic
401C23—21no—
402C23—21no—
403C23—21no—
404C23—21no—
405C23—21no—
406C23—21no—
407C23#46312yes—
408C23C2×#12111yescharacteristic
409Q8—81no—
410Q8—81no—
411Q8—81no—
412Q8—81no—
413Q8—161no—
414Q8—161no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—41no—
436C2×C4—41no—
437C2×C4—41no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4#46312yes—
462C2×C4#53411yescharacteristic
463C2×C4#53611yescharacteristic
464C2×C4#53711yescharacteristic
465C2×C4#53711yescharacteristic
466C2×C4#53911yescharacteristic
467C2×C4#53911yescharacteristic
468C2×C4C2×#13511yescharacteristic
469C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
470C2×C4C2×#13711yescharacteristic
471C2×C4C2×#13711yescharacteristic
472C2×C4C2×#13711yescharacteristic
473C2×C4#57511yescharacteristic
474C2×C4#57511yescharacteristic
475C2×C4#57511yescharacteristic
476C8—22no—
477C8—41no—
478C8—41no—
479C8—41no—
480C8—81no—
481C8—81no—
482C8—81no—
483C8—81no—
484C8—81no—
485C8—81no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22#261811yescharacteristic
494C22C2×#46311yescharacteristic
495C22#303711yescharacteristic
496C22#305611yescharacteristic
497C22#305611yescharacteristic
498C22#306811yescharacteristic
499C22#306911yescharacteristic
500C4—22no—
501C4—21no—
502C4—21no—
503C4—41no—
504C4—41no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—81no—
509C4—161no—
510C4—161no—
511C4—161no—
512C4—161no—
513C4#319011yescharacteristic
514C4#319411yescharacteristic
515C4#323211yescharacteristic
516C4#323811yescharacteristic
517C4#325211yescharacteristic
518C4#325211yescharacteristic
519C4#325511yescharacteristic
520C4#325611yescharacteristic
521C2—42no—
522C2#1916711yescharacteristic
523C2#1916811yescharacteristic
524C2#1930911yescharacteristic
525C2#1931111yescharacteristic
526C2#2976011yescharacteristic
527C2#2978311yescharacteristic
528C2#2979311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
5. #10658C2semidirect
6. #10657C2semidirect
7. #10600C2semidirect
8. #10596C2semidirect
9. #10593C2semidirect
10. #10589C2semidirect
11. #10541C2semidirect
12. #10541C2semidirect

Maximal subgroups

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

subgroupindex
#331482
#331462
#330492
#211462
#106582
#106572
#106002
#105962
#105932
#105892
#105412
#91392
#91352
#85972