Finite Group Databasebeta

← #381391

Subgroups — #381391

Order 512.

1298 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic
3C4×#380C211yesmaximal, characteristic
4#11232C212yesmaximal, complemented
5#11231C212yesmaximal, complemented
6#11096C212yesmaximal
7#11095C212yesmaximal
8#10908C212yesmaximal, complemented
9#10907C212yesmaximal, complemented
10C23×C42C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#381—21no—
17C2×#380C2212yes—
18C2×#380C2211yescharacteristic
19C2×#380C2211yescharacteristic
20C2×C4×C4⋊C4C2211yescharacteristic
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C22⋊C4—21no—
23#1797C418yescomplemented
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#336C2212yes—
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33C4⋊#95C2212yes—
34#1434—21no—
35#1434—21no—
36#1434—21no—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C4×#95—21no—
40C24×C4D411yescharacteristic
41C24×C4D411yescharacteristic
42C24×C4C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4C2312yes—
46C22×C4⋊C4C2312yes—
47C22×C4⋊C4C2312yes—
48C22×C4⋊C4C2312yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—21no—
52C22×C42—21no—
53C22×C42—21no—
54C22×C42—41no—
55C22×C42—41no—
56C22×C42—41no—
57C22×C42—41no—
58C22×C42C2×C414yes—
59C22×C42D411yescharacteristic
60C22×C42D411yescharacteristic
61C22×C42Q814yes—
62C22×C42C2311yescharacteristic
63C22×C42C2311yescharacteristic
64C2×#107—24no—
65C2×#107—42no—
66C2×#107—42no—
67C2×#106—24no—
68C2×#106—42no—
69C2×#106—42no—
70C2×C4⋊C8—41no—
71C2×C4⋊C8—41no—
72C2×C4⋊C8—41no—
73C2×C4⋊C8—41no—
74C2×C4⋊C8C2×C414yes—
75C2×C4⋊C8C2×C414yes—
76C2×C22⋊C8—42no—
77C2×C22⋊C8—42no—
78C2×C22⋊C8C2×C418yes—
79#381—44no—
80#380—24no—
81#380—24no—
82#380—41no—
83#380—41no—
84#380—41no—
85#380—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4—41no—
88C4×C4⋊C4—41no—
89C4×C4⋊C4—41no—
90C4×C22⋊C4—44no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—21no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102#340—24no—
103#340—24no—
104#340—42no—
105#340—42no—
106#340—42no—
107#340—42no—
108#336C2×C418yes—
109C25C2×D411yescharacteristic
110C23×C4—22no—
111C23×C4—22no—
112C23×C4—24no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—42no—
128C23×C4—41no—
129C23×C4—41no—
130C23×C4—41no—
131C23×C4—41no—
132C23×C4C4⋊C414yes—
133C23×C4C4⋊C414yes—
134C23×C4C22×C412yes—
135C23×C4C2×D411yescharacteristic
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×Q812yes—
141C23×C4C2411yescharacteristic
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8C22×C414yes—
155C22×C8C22×C414yes—
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—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4#4114yes—
179C2×C4⋊C4#4114yes—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—44no—
182C2×C22⋊C4—44no—
183C2×C22⋊C4—42no—
184C2×C22⋊C4—42no—
185C2×C22⋊C4—42no—
186C2×C22⋊C4—42no—
187C2×C22⋊C4—42no—
188C2×C22⋊C4—42no—
189C2×C42—21no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—21no—
195C2×C42—21no—
196C2×C42—21no—
197C2×C42—44no—
198C2×C42—44no—
199C2×C42—44no—
200C2×C42—44no—
201C2×C42—41no—
202C2×C42—41no—
203C2×C42—41no—
204C2×C42—41no—
205C2×C42—42no—
206C2×C42—42no—
207C2×C42—42no—
208C2×C42—42no—
209C2×C42—42no—
210C2×C42—42no—
211C2×C42C4⋊C414yes—
212C2×C42C4⋊C414yes—
213C2×C42C22×C412yes—
214C2×C42C22×C412yes—
215C2×C42C2×Q812yes—
216C2×C42C2×Q812yes—
217#107—48no—
218#106—48no—
219C4⋊C8—44no—
220C4⋊C8—44no—
221C22⋊C8—48no—
222#95—28no—
223#95—44no—
224#95—44no—
225#95—42no—
226#95—42no—
227#95—42no—
228#95—42no—
229#95—44no—
230#95—44no—
231#95—44no—
232#95—44no—
233C24—22no—
234C24—21no—
235C24—21no—
236C24—22no—
237C24—22no—
238C24—21no—
239C24—21no—
240C24—22no—
241C24C2×C4⋊C412yes—
242C24AGL(1,ℤ/8)12yes—
243C24#13712yes—
244C24C22×D411yescharacteristic
245C22×C4—21no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—28no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—44no—
282C22×C4—44no—
283C22×C4—44no—
284C22×C4—44no—
285C22×C4—44no—
286C22×C4—44no—
287C22×C4—44no—
288C22×C4—48no—
289C22×C4—48no—
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×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4C2×C4⋊C412yes—
325C22×C4C2×C4⋊C412yes—
326C22×C4C2×C4⋊C412yes—
327C22×C4C2×C4⋊C412yes—
328C22×C4C2×C4⋊C412yes—
329C22×C4#13514yes—
330C22×C4AGL(1,ℤ/8)12yes—
331C22×C4#13712yes—
332C22×C4C23×C411yescharacteristic
333C22×C4C22×Q811yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4H(5,𝔽2)12yes—
337C22×C4H(5,𝔽2)11yescharacteristic
338C22×C4H(5,𝔽2)11yescharacteristic
339C22×C4H(5,𝔽2)11yescharacteristic
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C42−1+412yes—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C4⋊C4—24no—
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—42no—
372C22⋊C4—416no—
373C22⋊C4—84no—
374C22⋊C4—84no—
375C42—22no—
376C42—22no—
377C42—22no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—44no—
384C42—44no—
385C42—84no—
386C42—84no—
387C23—22no—
388C23—22no—
389C23—24no—
390C23—24no—
391C23—22no—
392C23—22no—
393C23—22no—
394C23—22no—
395C23—22no—
396C23—22no—
397C23—22no—
398C23—24no—
399C23—24no—
400C23—22no—
401C23—42no—
402C23—42no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23#42814yes—
410C23C22×C4⋊C411yescharacteristic
411C23#52512yes—
412C23#53411yescharacteristic
413C23#53411yescharacteristic
414C23#53712yes—
415C23C2×#13512yes—
416C23C2×AGL(1,ℤ/8)11yescharacteristic
417C23C2×AGL(1,ℤ/8)11yescharacteristic
418C23C2×#13711yescharacteristic
419C23C2×#13711yescharacteristic
420C23#57512yes—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—24no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—416no—
441C2×C4—416no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—44no—
455C2×C4—44no—
456C2×C4—44no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—44no—
460C2×C4—44no—
461C2×C4—44no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4#42714yes—
485C2×C4#42814yes—
486C2×C4C4×#4111yescharacteristic
487C2×C4C4×#4111yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#51811yescharacteristic
490C2×C4#51811yescharacteristic
491C2×C4#51911yescharacteristic
492C2×C4#53312yes—
493C2×C4#53812yes—
494C2×C4#53812yes—
495C2×C4#53912yes—
496C2×C4#53912yes—
497C2×C4#54012yes—
498C2×C4#54012yes—
499C2×C4#54212yes—
500C2×C4#54212yes—
501C2×C4#54311yescharacteristic
502C2×C4#54311yescharacteristic
503C2×C4#54412yes—
504C8—44no—
505C8—44no—
506C22—24no—
507C22—24no—
508C22—24no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—42no—
517C22#229611yescharacteristic
518C22#231511yescharacteristic
519C22#231611yescharacteristic
520C22#263012yes—
521C22#263012yes—
522C22C2×#42711yescharacteristic
523C22C2×#42811yescharacteristic
524C22C2×#42811yescharacteristic
525C22#291111yescharacteristic
526C22#310612yes—
527C22#310712yes—
528C22#311412yes—
529C22#311712yes—
530C22#319112yes—
531C22#319112yes—
532C22#319211yescharacteristic
533C22#319311yescharacteristic
534C22#319311yescharacteristic
535C22#319511yescharacteristic
536C22#324712yes—
537C22#324812yes—
538C22#325112yes—
539C22#325212yes—
540C4—24no—
541C4—24no—
542C4—42no—
543C4—42no—
544C4—42no—
545C4—84no—
546C4—84no—
547C4—162no—
548C4—162no—
549C4#233211yescharacteristic
550C4#233211yescharacteristic
551C4#233311yescharacteristic
552C4#233311yescharacteristic
553C4#266512yes—
554C4#266712yes—
555C2—44no—
556C2#1429711yescharacteristic
557C2#2136211yescharacteristic
558C2#2136311yescharacteristic
559C2#2193911yescharacteristic
560C2#2194211yescharacteristic
561C2#2196211yescharacteristic
562C2#2196311yescharacteristic
563C2#3001512yes—
564C2#3001712yes—
565C2#3001912yes—
566C2#3002112yes—
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
4. #11232C2semidirect
5. #11231C2semidirect
8. #10908C2semidirect
9. #10907C2semidirect
23. #1797C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
#146852
C4×#3802
#112322
#112312
#110962
#110952
#109082
#109072