Finite Group Databasebeta

← #381959

Subgroups — #381959

Order 512.

1332 conjugacy classes of subgroups.

The subgroup lattice has 553 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2075C211yesmaximal, characteristic
2C2×#1942C211yesmaximal, characteristic, complemented
3C2×C4×C22⋊C8C211yesmaximal, characteristic
4#14697C212yesmaximal
5#11379C211yesmaximal, characteristic, complemented
6#11328C211yesmaximal, characteristic, complemented
7#11292C212yesmaximal, complemented
8#11199C212yesmaximal
9#11096C211yesmaximal, characteristic
10#10880C212yesmaximal, complemented
11#10820C211yesmaximal, characteristic, complemented
12C23×C42C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#107C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C22×C22⋊C8C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#389C2212yes—
20C2×#389C2212yes—
21C2×#385—22no—
22C2×#380C2212yes—
23C2×#380C2212yes—
24#2075C418yescomplemented
25#1942—24no—
26C4×C22⋊C8—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#336C2211yescharacteristic
36C2×#336C2211yescharacteristic
37#1436—22no—
38#1436—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C22⋊#95—22no—
42C24×C4D411yescharacteristic
43C24×C4D411yescharacteristic
44C24×C4C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C22×C4⋊C4—22no—
50C22×C4⋊C4—22no—
51C22×C4⋊C4C2312yes—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C42—22no—
58C22×C42—22no—
59C22×C42—42no—
60C22×C42—42no—
61C22×C42D411yescharacteristic
62C22×C42D411yescharacteristic
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C2×#107—22no—
66C2×#107—22no—
67C2×#107—22no—
68C2×#107—42no—
69C2×#107C2×C414yescomplemented
70C2×C22⋊C8—22no—
71C2×C22⋊C8—22no—
72C2×C22⋊C8—22no—
73C2×C22⋊C8—22no—
74C2×C22⋊C8—22no—
75C2×C22⋊C8C2×C414yes—
76C2×C4×C8—22no—
77C2×C4×C8—42no—
78C2×C4×C8—42no—
79C2×C4×C8D414yes—
80#389—24no—
81#389—24no—
82#389—42no—
83#389—42no—
84#385—48no—
85#380—24no—
86#380—42no—
87#380—42no—
88#380C2×C418yes—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97#340—24no—
98#340—24no—
99#340—22no—
100#340—42no—
101#340—42no—
102#340—42no—
103#340C2×C414yescomplemented
104#340C2×C418yescomplemented
105#336—22no—
106#336—22no—
107C25C2×D411yescharacteristic
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—22no—
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4—22no—
118C23×C4—22no—
119C23×C4—22no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—24no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—42no—
127C23×C4—42no—
128C23×C4—42no—
129C23×C4D814yes—
130C23×C4C22×C412yes—
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4#4112yes—
137C23×C4C2411yescharacteristic
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8C22×C412yes—
153C22×C8C22×C412yes—
154C22×C8C2×D412yes—
155C22×C8C2×D412yes—
156C22×C8#4112yes—
157C22×C8#4112yes—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—44no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22×C414yes—
187C2×C4⋊C4C22×C414yes—
188C2×C22⋊C4—28no—
189C2×C22⋊C4—44no—
190C2×C22⋊C4—44no—
191C2×C22⋊C4—44no—
192C2×C22⋊C4—44no—
193C2×C22⋊C4—44no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—21no—
199C2×C42—21no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—44no—
203C2×C42—44no—
204C2×C42—48no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42C2×D412yes—
208C2×C42#4112yes—
209#107—28no—
210#107—44no—
211C22⋊C8—28no—
212C22⋊C8—28no—
213C4×C8—44no—
214C4×C8—44no—
215#95—28no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—44no—
222#95—44no—
223C24—21no—
224C24—21no—
225C24—22no—
226C24—22no—
227C24—21no—
228C24—21no—
229C24—22no—
230C24—22no—
231C24C4×D412yes—
232C24C2×D812yes—
233C24#13512yes—
234C24C22×D411yescharacteristic
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—22no—
240C22×C4—28no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—28no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—48no—
265C22×C4—48no—
266C22×C4—48no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—48no—
270C22×C4—44no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—44no—
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—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4C4×D412yes—
309C22×C4#12112yes—
310C22×C4#12112yes—
311C22×C4C2×D812yes—
312C22×C4C2×D812yes—
313C22×C4#13512yes—
314C22×C4#13512yes—
315C22×C4#13714yes—
316C22×C4C23×C411yescharacteristic
317C22×C4C22×D411yescharacteristic
318C22×C4C2×#4111yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4H(5,𝔽2)12yes—
323C22×C4H(5,𝔽2)12yes—
324C22×C42−1+412yes—
325C22×C42−1+412yes—
326C2×C8—24no—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—22no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—22no—
336C2×C8—24no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8C4×D414yes—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—88no—
363C4⋊C4—88no—
364C22⋊C4—416no—
365C22⋊C4—88no—
366C42—22no—
367C42—22no—
368C42—24no—
369C42—44no—
370C42—44no—
371C42—41no—
372C42—41no—
373C42—42no—
374C23—24no—
375C23—22no—
376C23—22no—
377C23—24no—
378C23—24no—
379C23—22no—
380C23—24no—
381C23—24no—
382C23—24no—
383C23—22no—
384C23—22no—
385C23—22no—
386C23—41no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—42no—
392C23—42no—
393C23—42no—
394C23—41no—
395C23—41no—
396C23C4×D814yes—
397C23C2×C4×D411yescharacteristic
398C23C2×#12111yescharacteristic
399C23#53211yescharacteristic
400C23#53412yes—
401C23#53612yes—
402C23C22×D811yescharacteristic
403C23C2×#13511yescharacteristic
404C23C2×#13511yescharacteristic
405C23C2×#13511yescharacteristic
406C23C2×#13712yes—
407C23#57512yes—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—416no—
423C2×C4—416no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—44no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—44no—
441C2×C4—44no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
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—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4#46614yes—
468C2×C4#47014yes—
469C2×C4C2×C4×D411yescharacteristic
470C2×C4C4×#4111yescharacteristic
471C2×C4#52012yes—
472C2×C4#52012yes—
473C2×C4#52511yescharacteristic
474C2×C4#52911yescharacteristic
475C2×C4#53411yescharacteristic
476C2×C4#53511yescharacteristic
477C2×C4#53611yescharacteristic
478C2×C4#53611yescharacteristic
479C2×C4#53812yes—
480C2×C4#53912yes—
481C2×C4#53912yes—
482C2×C4#53912yes—
483C2×C4#53912yes—
484C2×C4#54112yes—
485C2×C4#54212yes—
486C2×C4#54212yes—
487C8—24no—
488C8—42no—
489C8—42no—
490C8—42no—
491C22—24no—
492C22—28no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—44no—
497C22—44no—
498C22—44no—
499C22—42no—
500C22—42no—
501C22—42no—
502C22C4×#12111yescharacteristic
503C22#232212yes—
504C22#261711yescharacteristic
505C22#262111yescharacteristic
506C22#266312yes—
507C22C2×C4×D811yescharacteristic
508C22C4×#13511yescharacteristic
509C22C4×#13712yes—
510C22C2×#46611yescharacteristic
511C22#305011yescharacteristic
512C22C2×#47011yescharacteristic
513C22#305411yescharacteristic
514C22#316811yescharacteristic
515C22#316911yescharacteristic
516C22#317212yes—
517C22#318612yes—
518C22#319012yes—
519C22#319012yes—
520C22#319112yes—
521C22#322612yes—
522C22#323112yes—
523C22#323212yes—
524C22#323312yes—
525C4—24no—
526C4—24no—
527C4—42no—
528C4—42no—
529C4—42no—
530C4—84no—
531C4—84no—
532C4—162no—
533C4—162no—
534C4#232211yescharacteristic
535C4#232611yescharacteristic
536C4#233412yes—
537C4#263311yescharacteristic
538C4#263811yescharacteristic
539C4#265212yes—
540C2—44no—
541C2#1431311yescharacteristic
542C2C4×#46611yescharacteristic
543C2C4×#47011yescharacteristic
544C2#2207512yes—
545C2#2207712yes—
546C2#2973011yescharacteristic
547C2#2973311yescharacteristic
548C2#2987611yescharacteristic
549C2#2987911yescharacteristic
550C2#3035612yes—
551C2#3036112yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C23×C42 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1942C2semidirect
5. #11379C2semidirect
6. #11328C2semidirect
7. #11292C2semidirect
10. #10880C2semidirect
11. #10820C2semidirect
24. #2075C4semidirect
69. C2×#107C2×C4semidirect
103. #340C2×C4semidirect
104. #340C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20752
C2×#19422
C2×C4×C22⋊C82
#146972
#113792
#113282
#112922
#111992
#110962
#108802
#108202