Finite Group Databasebeta

← #413389

Subgroups — #413389

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic
2C2×#1845C211yesmaximal, characteristic
3#15146C212yesmaximal
4#11267C212yesmaximal
5#11256C212yesmaximal
6#11219C212yesmaximal
7#10862C211yesmaximal, characteristic
8#10862C211yesmaximal, characteristic
9#10862C211yesmaximal, characteristic
10#10862C211yesmaximal, characteristic
11C8×#95C211yesmaximal, characteristic
12C22×#107C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C22×C4×C8C2211yescharacteristic
20C2×#400—22no—
21C2×#389C2212yes—
22C2×#384C2212yes—
23#1942—24no—
24#1845—24no—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2211yescharacteristic
31C2×#336C2211yescharacteristic
32C2×#336C2211yescharacteristic
33#1442—22no—
34#1442—22no—
35#1441—22no—
36C4⋊#95C2212yes—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2212yes—
40C4×#95C2211yescharacteristic
41C23×C8C2311yescharacteristic
42C23×C8C2311yescharacteristic
43C23×C8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2312yes—
51C22×C4⋊C4C2312yes—
52C22×C42C2311yescharacteristic
53C22×C42C2311yescharacteristic
54C22×C42C2311yescharacteristic
55C2×#107—22no—
56C2×#107—22no—
57C2×#107—22no—
58C2×#107—22no—
59C2×#107—24no—
60C2×#107—22no—
61C2×#107—22no—
62C2×#107—22no—
63C2×#107—22no—
64C2×#107—22no—
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—24no—
68C2×#107—42no—
69C2×#107D414yes—
70C2×C4×C8—22no—
71C2×C4×C8—22no—
72C2×C4×C8—22no—
73C2×C4×C8—22no—
74C2×C4×C8—22no—
75C2×C4×C8—22no—
76C2×C4×C8—22no—
77C2×C4×C8—22no—
78C2×C4×C8Q814yes—
79#400—48no—
80#389—24no—
81#389—42no—
82#384—24no—
83#384—42no—
84C2×#95—22no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95D411yescharacteristic
94C2×#95D411yescharacteristic
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97#340—24no—
98#340—24no—
99#340—42no—
100#340—42no—
101#340—42no—
102#336—22no—
103#336—22no—
104#336Q814yes—
105C23×C4—22no—
106C23×C4—22no—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4C2×D411yescharacteristic
110C23×C4C2×D411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D411yescharacteristic
115C23×C4C2411yescharacteristic
116C22×C8—21no—
117C22×C8—21no—
118C22×C8—21no—
119C22×C8—21no—
120C22×C8—21no—
121C22×C8—21no—
122C22×C8—21no—
123C22×C8—21no—
124C22×C8—21no—
125C22×C8—21no—
126C22×C8—21no—
127C22×C8—21no—
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—21no—
132C22×C8C2×D412yes—
133C22×C8C2×Q812yes—
134C22×C8C2×Q812yes—
135C22×C8C2×Q812yes—
136C22×C8C2×Q812yes—
137C22×C8#4112yes—
138C22×C8#4112yes—
139C22×C8#4112yes—
140C22×C8#4112yes—
141C22×C8#4112yes—
142C22×C8#4112yes—
143C22×C8#4112yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4C2×D414yes—
177C2×C42—21no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—21no—
187C2×C42—21no—
188C2×C42—21no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42C2×Q812yes—
194C2×C42C2×Q812yes—
195#107—28no—
196#107—48no—
197#107—44no—
198#107—44no—
199#107—44no—
200C4×C8—28no—
201C4×C8—44no—
202C4×C8—44no—
203#95—22no—
204#95—22no—
205#95—44no—
206#95—44no—
207#95—44no—
208#95—44no—
209#95—44no—
210#95—44no—
211#95—82no—
212#95—82no—
213#95—82no—
214#95D814yes—
215#95Q1614yes—
216C24C22×D411yescharacteristic
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—24no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4C4⋊Q812yes—
278C22×C4C4⋊Q812yes—
279C22×C4C2×D812yes—
280C22×C4C2×D812yes—
281C22×C4C2×D812yes—
282C22×C4C2×Q1612yes—
283C22×C4C2×Q1612yes—
284C22×C4C2×Q1612yes—
285C22×C4#13512yes—
286C22×C4#13512yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×Q811yescharacteristic
289C22×C4C22×Q811yescharacteristic
290C22×C4C2×#4111yescharacteristic
291C22×C4C2×#4111yescharacteristic
292C22×C4C2×#4111yescharacteristic
293C22×C4C2×#4111yescharacteristic
294C22×C4C2×#4111yescharacteristic
295C22×C4C2×#4111yescharacteristic
296C22×C4C2×#4111yescharacteristic
297C22×C4H(5,𝔽2)11yescharacteristic
298C22×C4H(5,𝔽2)11yescharacteristic
299C22×C42−1+411yescharacteristic
300C22×C42−1+411yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
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×C8C22⋊Q814yes—
322C2×C8#12314yes—
323C2×C8#12414yes—
324C2×C8#12514yes—
325C2×C8#12514yes—
326C2×C8#12514yes—
327C2×C8#12614yes—
328C2×C8#12614yes—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—42no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C4⋊C4—88no—
347C4⋊C4—88no—
348C42—22no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—41no—
353C42—41no—
354C42—41no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—41no—
360C23C2×C4⋊Q811yescharacteristic
361C23#53211yescharacteristic
362C23#53211yescharacteristic
363C23D4211yescharacteristic
364C23#54711yescharacteristic
365C23#54711yescharacteristic
366C23#55011yescharacteristic
367C23C22×D811yescharacteristic
368C23C22×Q1611yescharacteristic
369C23C2×#13511yescharacteristic
370C23C2×#13511yescharacteristic
371C23#57611yescharacteristic
372C23#57611yescharacteristic
373C23#57811yescharacteristic
374C23#57811yescharacteristic
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—22no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
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—41no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4#49914yes—
444C2×C4#50014yes—
445C2×C4C2×C22⋊Q811yescharacteristic
446C2×C4C2×#12311yescharacteristic
447C2×C4C2×#12411yescharacteristic
448C2×C4C2×#12511yescharacteristic
449C2×C4C2×#12511yescharacteristic
450C2×C4C2×#12511yescharacteristic
451C2×C4C2×#12611yescharacteristic
452C2×C4C2×#12611yescharacteristic
453C2×C4#53311yescharacteristic
454C2×C4#53311yescharacteristic
455C2×C4#54011yescharacteristic
456C2×C4#54111yescharacteristic
457C2×C4#54211yescharacteristic
458C2×C4#54211yescharacteristic
459C2×C4#54411yescharacteristic
460C2×C4#54411yescharacteristic
461C2×C4#54611yescharacteristic
462C2×C4#54811yescharacteristic
463C2×C4D4×Q811yescharacteristic
464C2×C4#55211yescharacteristic
465C2×C4#55311yescharacteristic
466C2×C4#55411yescharacteristic
467C2×C4#55411yescharacteristic
468C2×C4#55511yescharacteristic
469C2×C4#55611yescharacteristic
470C2×C4#55611yescharacteristic
471C2×C4#55711yescharacteristic
472C2×C4#55911yescharacteristic
473C8—24no—
474C8—42no—
475C8—42no—
476C8#40018yescomplemented
477C22#244011yescharacteristic
478C22#244511yescharacteristic
479C22#249111yescharacteristic
480C22#249611yescharacteristic
481C22#251211yescharacteristic
482C22#255211yescharacteristic
483C22#285911yescharacteristic
484C22C2×#49911yescharacteristic
485C22C2×#50011yescharacteristic
486C22#316411yescharacteristic
487C22#316411yescharacteristic
488C22#316911yescharacteristic
489C22#316911yescharacteristic
490C22#317011yescharacteristic
491C22#317011yescharacteristic
492C22#317511yescharacteristic
493C22#317511yescharacteristic
494C22#317611yescharacteristic
495C22#317711yescharacteristic
496C22D4×D811yescharacteristic
497C22#328011yescharacteristic
498C22#328511yescharacteristic
499C22D4×Q1611yescharacteristic
500C22#333411yescharacteristic
501C22#333411yescharacteristic
502C22#333611yescharacteristic
503C22#333811yescharacteristic
504C22#333811yescharacteristic
505C22#333911yescharacteristic
506C22#334111yescharacteristic
507C22#334211yescharacteristic
508C22#339111yescharacteristic
509C22#339311yescharacteristic
510C22#339511yescharacteristic
511C22#339611yescharacteristic
512C4—24no—
513C4—24no—
514C4—42no—
515C4—42no—
516C4—42no—
517C4—42no—
518C4—84no—
519C4—162no—
520C4—162no—
521C4—162no—
522C4C2×#40011yescharacteristic
523C4#244211yescharacteristic
524C4#244611yescharacteristic
525C4#249211yescharacteristic
526C4#249711yescharacteristic
527C4#251111yescharacteristic
528C4#257111yescharacteristic
529C4#264711yescharacteristic
530C2#1514611yescharacteristic
531C2#2606511yescharacteristic
532C2#2606711yescharacteristic
533C2#2625411yescharacteristic
534C2#2625511yescharacteristic
535C2#2779111yescharacteristic
536C2#2779211yescharacteristic
537C2#2783911yescharacteristic
538C2#2784011yescharacteristic
539C2#2801411yescharacteristic
540C2#2801511yescharacteristic
541C2#2882711yescharacteristic
542C2#2882811yescharacteristic
543C2#3325811yescharacteristic
544C2#3326111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
476. C8#400semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×#18452
#151462
#112672
#112562
#112192
#108622
C8×#952