Finite Group Databasebeta

← #414227

Subgroups — #414227

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 543 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1942C211yesmaximal, characteristic, complemented
2C2×C8×C22⋊C4C211yesmaximal, characteristic
3#15131C212yesmaximal
4#11328C211yesmaximal, characteristic, complemented
5#11256C211yesmaximal, characteristic, complemented
6#11207C212yesmaximal, complemented
7#11096C211yesmaximal, characteristic
8#11096C211yesmaximal, characteristic
9#10880C212yesmaximal, complemented
10#10841C211yesmaximal, characteristic, complemented
11#10721C212yesmaximal
12C24×C8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C4×C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#396—22no—
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#384C2212yes—
22C2×#380C2212yes—
23C2×#380C2212yes—
24C2×C4×C22⋊C4C2211yescharacteristic
25#1942—24no—
26C8×C22⋊C4—24no—
27C2×#340C2211yescharacteristic
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1441—22no—
38#1441—22no—
39#1439—22no—
40C4⋊#95C2212yes—
41C24×C4C2311yescharacteristic
42C23×C8—22no—
43C23×C8—22no—
44C23×C8—42no—
45C23×C8—42no—
46C23×C8Q814yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C22×C4⋊C4—22no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4C2312yes—
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4D411yescharacteristic
62C22×C22⋊C4D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—24no—
67C2×#107—22no—
68C2×#107—42no—
69C2×C22⋊C8—22no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8Q814yes—
72C2×C4×C8—22no—
73C2×C4×C8—42no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#396—48no—
77#389—24no—
78#389—24no—
79#389—24no—
80#389—42no—
81#384—24no—
82#384—42no—
83#380—24no—
84#380—24no—
85#380—42no—
86#380—42no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95D411yescharacteristic
96C2×#95D411yescharacteristic
97#340—24no—
98#340—22no—
99#340—24no—
100#340—22no—
101#340—24no—
102#340—42no—
103#340—42no—
104#340—42no—
105#336—22no—
106#336—22no—
107C25C2×D411yescharacteristic
108C23×C4—21no—
109C23×C4—21no—
110C23×C4—21no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×Q812yes—
126C23×C4C2×Q812yes—
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—28no—
141C22×C8—48no—
142C22×C8—44no—
143C22×C8—44no—
144C22×C8—48no—
145C22×C8C2×D412yes—
146C22×C8C2×D412yes—
147C22×C8C2×Q812yes—
148C22×C8C2×Q812yes—
149C22×C8C2×Q812yes—
150C22×C8C2×Q812yes—
151C22×C8#4112yes—
152C22×C8#4112yes—
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—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—82no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4#4114yes—
188C2×C4⋊C4#4114yes—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—22no—
191C2×C22⋊C4—22no—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—82no—
198C2×C22⋊C4—82no—
199C2×C22⋊C4D814yes—
200C2×C22⋊C4Q1614yes—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42C2×D412yes—
208C2×C42#4112yes—
209#107—28no—
210#107—44no—
211C22⋊C8—28no—
212C4×C8—44no—
213C4×C8—44no—
214#95—22no—
215#95—22no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—82no—
222C24—21no—
223C24—21no—
224C24—22no—
225C24—22no—
226C24—21no—
227C24—21no—
228C24—22no—
229C24—22no—
230C24C4⋊Q812yes—
231C24C2×D812yes—
232C24C2×Q1612yes—
233C24C22×D411yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—41no—
261C22×C4—42no—
262C22×C4—41no—
263C22×C4—41no—
264C22×C4—41no—
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×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4#12112yes—
297C22×C4#12112yes—
298C22×C4C4⋊Q812yes—
299C22×C4C2×D812yes—
300C22×C4C2×D812yes—
301C22×C4C2×Q1612yes—
302C22×C4C2×Q1612yes—
303C22×C4#13512yes—
304C22×C4#13512yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×Q811yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4C2×#4111yescharacteristic
309C22×C4C2×#4111yescharacteristic
310C22×C4C2×#4112yes—
311C22×C4C2×#4112yes—
312C22×C4C2×#4111yescharacteristic
313C22×C4H(5,𝔽2)12yes—
314C22×C4H(5,𝔽2)12yes—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—24no—
326C2×C8—24no—
327C2×C8—416no—
328C2×C8—41no—
329C2×C8—41no—
330C2×C8—41no—
331C2×C8—41no—
332C2×C8—42no—
333C2×C8—42no—
334C2×C8C22⋊Q814yes—
335C2×C8C22⋊Q814yes—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—24no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—88no—
353C4⋊C4—88no—
354C4⋊C4—88no—
355C4⋊C4—88no—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—88no—
359C22⋊C4—88no—
360C42—22no—
361C42—22no—
362C42—24no—
363C42—41no—
364C42—41no—
365C42—42no—
366C23—22no—
367C23—22no—
368C23—24no—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—24no—
373C23—24no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—42no—
379C23—42no—
380C23—42no—
381C23—41no—
382C23—42no—
383C23—42no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—42no—
388C23#50014yes—
389C23C2×#12111yescharacteristic
390C23C2×C4⋊Q811yescharacteristic
391C23#53211yescharacteristic
392C23#54612yes—
393C23#54612yes—
394C23C22×D811yescharacteristic
395C23C22×Q1611yescharacteristic
396C23C2×#13511yescharacteristic
397C23C2×#13511yescharacteristic
398C23#57712yes—
399C23#57712yes—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—44no—
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—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4#46614yes—
458C2×C4#46714yes—
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4C2×C22⋊Q811yescharacteristic
461C2×C4#52511yescharacteristic
462C2×C4#52912yes—
463C2×C4#53211yescharacteristic
464C2×C4#53311yescharacteristic
465C2×C4#53311yescharacteristic
466C2×C4#53312yes—
467C2×C4#53411yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#53812yes—
470C2×C4#54312yes—
471C2×C4#55312yes—
472C2×C4#55312yes—
473C2×C4#55312yes—
474C2×C4#55312yes—
475C2×C4#55412yes—
476C2×C4#55412yes—
477C8—24no—
478C8—24no—
479C8—42no—
480C8—42no—
481C22—28no—
482C22—24no—
483C22—42no—
484C22—44no—
485C22—44no—
486C22—44no—
487C22—42no—
488C22—42no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22#253511yescharacteristic
493C22#254311yescharacteristic
494C22#254912yes—
495C22#255912yes—
496C22#285711yescharacteristic
497C22C2×#46611yescharacteristic
498C22C2×#46711yescharacteristic
499C22#306011yescharacteristic
500C22#306011yescharacteristic
501C22C2×#50011yescharacteristic
502C22#316011yescharacteristic
503C22#316312yes—
504C22#316911yescharacteristic
505C22#317011yescharacteristic
506C22#317412yes—
507C22#329412yes—
508C22#329412yes—
509C22#329612yes—
510C22#329912yes—
511C22#329912yes—
512C22#330112yes—
513C22#330312yes—
514C22#330412yes—
515C4—24no—
516C4—24no—
517C4—42no—
518C4—42no—
519C4—42no—
520C4—84no—
521C4—84no—
522C4—162no—
523C4—162no—
524C4#239611yescharacteristic
525C4#256611yescharacteristic
526C4#257911yescharacteristic
527C4#258712yes—
528C4#260812yes—
529C4#262611yescharacteristic
530C2—44no—
531C2#1516511yescharacteristic
532C2#2814011yescharacteristic
533C2#2814211yescharacteristic
534C2#2855111yescharacteristic
535C2#2855311yescharacteristic
536C2#2876912yes—
537C2#2877412yes—
538C2#2903912yes—
539C2#2904112yes—
540C2#3320911yescharacteristic
541C2#3321411yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#1942C2semidirect
4. #11328C2semidirect
5. #11256C2semidirect
6. #11207C2semidirect
9. #10880C2semidirect
10. #10841C2semidirect

Maximal subgroups

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

subgroupindex
C2×#19422
C2×C8×C22⋊C42
#151312
#113282
#112562
#112072
#110962
#108802
#108412
#107212