Finite Group Databasebeta

← #392061

Subgroups — #392061

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1797C211yesmaximal, characteristic
2#14685C211yesmaximal, characteristic
3#14682C211yesmaximal, characteristic
4#11325C212yesmaximal, complemented
5#11268C212yesmaximal, complemented
6#11232C212yesmaximal, complemented
7#11231C212yesmaximal, complemented
8#11096C212yesmaximal
9#11095C212yesmaximal
10C23×C42C2211yescharacteristic
11C22×#107C2212yes—
12C22×#106C2212yes—
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C22⋊C8C2212yes—
16C2×#382C2212yes—
17C2×#381—21no—
18C2×#380C2211yescharacteristic
19C2×#380C2212yes—
20C2×#380C2211yescharacteristic
21#1797C418yescomplemented
22C2×#340C2212yes—
23C2×#340C2212yes—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#340C2212yes—
28C2×#336C2212yes—
29C4⋊#95C2211yescharacteristic
30C4⋊#95C2211yescharacteristic
31C4⋊#95C2212yes—
32C4⋊#95C2212yes—
33#1434—21no—
34#1434—21no—
35#1434—21no—
36#1434—21no—
37C22⋊#95—21no—
38C22⋊#95—21no—
39C22⋊#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—21no—
50C22×C22⋊C4—21no—
51C22×C22⋊C4—22no—
52C22×C42—42no—
53C22×C42—42no—
54C22×C42C2×C414yes—
55C22×C42D411yescharacteristic
56C22×C42D411yescharacteristic
57C22×C42Q814yes—
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#107—42no—
61C2×#107—42no—
62C2×#107—42no—
63C2×#106—42no—
64C2×#106—42no—
65C2×#106—42no—
66C2×C4⋊C8—42no—
67C2×C4⋊C8—42no—
68C2×C4⋊C8C2×C414yes—
69C2×C4⋊C8C2×C414yes—
70C2×C22⋊C8—42no—
71C2×C22⋊C8—42no—
72C2×C22⋊C8C2×C418yes—
73#382—42no—
74#382—42no—
75#381—44no—
76#380—41no—
77#380—41no—
78#380—42no—
79#380—42no—
80#380—41no—
81#380—41no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95—21no—
94C2×#95—22no—
95#340—42no—
96#340—42no—
97#340—42no—
98#340—42no—
99#340—42no—
100#340—42no—
101#336C2×C418yes—
102C25C2×D411yescharacteristic
103C23×C4—22no—
104C23×C4—22no—
105C23×C4—24no—
106C23×C4—21no—
107C23×C4—21no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—41no—
119C23×C4—41no—
120C23×C4—42no—
121C23×C4—42no—
122C23×C4—42no—
123C23×C4C4⋊C414yes—
124C23×C4C4⋊C414yes—
125C23×C4C22×C412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×Q812yes—
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C22×C414yes—
146C22×C8C22×C414yes—
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—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—44no—
175C2×C22⋊C4—42no—
176C2×C22⋊C4—42no—
177C2×C22⋊C4—42no—
178C2×C22⋊C4—44no—
179C2×C22⋊C4—42no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42—48no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42C4⋊C414yes—
190C2×C42C4⋊C414yes—
191C2×C42C22×C412yes—
192C2×C42C22×C412yes—
193C2×C42C2×Q812yes—
194C2×C42C2×Q812yes—
195#107—48no—
196#106—48no—
197C4⋊C8—44no—
198C4⋊C8—44no—
199C22⋊C8—48no—
200#95—42no—
201#95—42no—
202#95—44no—
203#95—44no—
204#95—42no—
205#95—42no—
206#95—42no—
207#95—44no—
208#95—42no—
209#95—44no—
210#95—44no—
211#95—44no—
212#95—44no—
213C24—22no—
214C24—21no—
215C24—21no—
216C24—22no—
217C24—22no—
218C24—22no—
219C24—21no—
220C24—21no—
221C24C2×C4⋊C412yes—
222C24AGL(1,ℤ/8)12yes—
223C24#13712yes—
224C24C22×D411yescharacteristic
225C22×C4—22no—
226C22×C4—21no—
227C22×C4—21no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—28no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—28no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—48no—
261C22×C4—48no—
262C22×C4—48no—
263C22×C4—48no—
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×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—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4C2×C4⋊C412yes—
301C22×C4C2×C4⋊C412yes—
302C22×C4C2×C4⋊C412yes—
303C22×C4C2×C4⋊C412yes—
304C22×C4C2×C4⋊C412yes—
305C22×C4#13514yes—
306C22×C4C23×C411yescharacteristic
307C22×C4C22×Q811yescharacteristic
308C22×C4H(5,𝔽2)12yes—
309C22×C4H(5,𝔽2)12yes—
310C22×C4H(5,𝔽2)12yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C4H(5,𝔽2)11yescharacteristic
313C22×C42−1+412yes—
314C22×C42−1+411yescharacteristic
315C22×C42−1+411yescharacteristic
316C2×C8—24no—
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—24no—
321C2×C8—24no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C4⋊C4—42no—
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—42no—
346C22⋊C4—88no—
347C22⋊C4—84no—
348C22⋊C4—84no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—24no—
354C42—24no—
355C42—44no—
356C42—44no—
357C23—22no—
358C23—22no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—22no—
363C23—24no—
364C23—24no—
365C23—22no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—24no—
370C23—24no—
371C23—42no—
372C23—42no—
373C23—42no—
374C23—42no—
375C23—42no—
376C23—42no—
377C23—42no—
378C23—42no—
379C23#42814yes—
380C23C22×C4⋊C411yescharacteristic
381C23#53412yes—
382C23#53511yescharacteristic
383C23#53611yescharacteristic
384C23#53712yes—
385C23C2×#13512yes—
386C23C2×AGL(1,ℤ/8)11yescharacteristic
387C23C2×#13711yescharacteristic
388C23#57511yescharacteristic
389C23#57511yescharacteristic
390C23#57512yes—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—416no—
408C2×C4—416no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
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—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4#42714yes—
452C2×C4#51811yescharacteristic
453C2×C4#51811yescharacteristic
454C2×C4#51811yescharacteristic
455C2×C4#51911yescharacteristic
456C2×C4#52011yescharacteristic
457C2×C4#52011yescharacteristic
458C2×C4#54312yes—
459C2×C4#54412yes—
460C2×C4#54411yescharacteristic
461C2×C4#54411yescharacteristic
462C2×C4#56012yes—
463C2×C4#56012yes—
464C2×C4#56212yes—
465C2×C4#56212yes—
466C2×C4#56212yes—
467C2×C4#56212yes—
468C2×C4#56312yes—
469C2×C4#56312yes—
470C8—44no—
471C8—44no—
472C22—24no—
473C22—24no—
474C22—24no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—42no—
481C22—44no—
482C22—44no—
483C22#231511yescharacteristic
484C22#231611yescharacteristic
485C22#231711yescharacteristic
486C22#281312yes—
487C22#281412yes—
488C22C2×#42711yescharacteristic
489C22C2×#42811yescharacteristic
490C22#291111yescharacteristic
491C22#291111yescharacteristic
492C22#319112yes—
493C22#319112yes—
494C22#319412yes—
495C22#319612yes—
496C22#321212yes—
497C22#321311yescharacteristic
498C22#321511yescharacteristic
499C22#323312yes—
500C22#323411yescharacteristic
501C22#323611yescharacteristic
502C22#324712yes—
503C22#324812yes—
504C22#325612yes—
505C22#325612yes—
506C4—24no—
507C4—42no—
508C4—42no—
509C4—42no—
510C4—42no—
511C4—162no—
512C4—162no—
513C4—162no—
514C4—162no—
515C4#237711yescharacteristic
516C4#237711yescharacteristic
517C4#237811yescharacteristic
518C4#237811yescharacteristic
519C4#284112yes—
520C4#284112yes—
521C2—44no—
522C2#1467511yescharacteristic
523C2#2193911yescharacteristic
524C2#2194411yescharacteristic
525C2#2196211yescharacteristic
526C2#2196511yescharacteristic
527C2#2198511yescharacteristic
528C2#2198611yescharacteristic
529C2#3272712yes—
530C2#3272912yes—
531C2#3274712yes—
532C2#3274912yes—
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. #11325C2semidirect
5. #11268C2semidirect
6. #11232C2semidirect
7. #11231C2semidirect
21. #1797C4semidirect

Maximal subgroups

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

subgroupindex
C2×#17972
#146852
#146822
#113252
#112682
#112322
#112312
#110962
#110952