Finite Group Databasebeta

← #366401

Subgroups — #366401

Order 512.

1162 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20845C211yesmaximal, characteristic
2#20700C211yesmaximal, characteristic
3C2×#1821C211yesmaximal, characteristic
4#20386C211yesmaximal, characteristic
5C2×#1736C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10728C211yesmaximal, characteristic
11#10724C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C2×#407C2211yescharacteristic
19C2×#405C2211yescharacteristic
20C2×#405C2211yescharacteristic
21#2280C2211yescharacteristic
22#1854—22no—
23#1854—22no—
24#1824—22no—
25#1823—22no—
26#1821C418yescomplemented
27#1753—21no—
28#1753—21no—
29#1753—21no—
30#1753—21no—
31C2×#343C2212yes—
32C2×#343C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#341C2212yes—
36C2×#341C2212yes—
37#1736—21no—
38#1736—21no—
39#1736—21no—
40#1736—21no—
41#1736C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44C2×#340C2212yes—
45C2×#340C2212yes—
46#1729C2211yescharacteristic
47#1729C2211yescharacteristic
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—22no—
51C22×M16—22no—
52C22×M16—24no—
53C22×M16—42no—
54C22×M16D411yescharacteristic
55C22×M16D411yescharacteristic
56C22×M16D411yescharacteristic
57C22×M16D411yescharacteristic
58C22×M16C2312yes—
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2312yes—
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C2×#117—21no—
65C2×#117—21no—
66C2×#117—21no—
67C2×#117—21no—
68C2×#117—24no—
69C2×#117—42no—
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4—22no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—22no—
80C22×C4⋊C4C2×C414yes—
81C22×C4⋊C4C2312yes—
82C22×C4⋊C4C2312yes—
83C22×C22⋊C4—21no—
84C22×C42—21no—
85#428—24no—
86#428—42no—
87C2×#107—22no—
88C2×#106—22no—
89#407—22no—
90#407—22no—
91#407—22no—
92#407—22no—
93#407C2×C414yes—
94#407C2×C414yes—
95C2×C22⋊C8—21no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8—21no—
99C2×C22⋊C8—21no—
100C2×C22⋊C8—21no—
101#405—24no—
102#405—42no—
103C2×#97—21no—
104C2×#97—21no—
105C2×C4×C8—21no—
106C2×C4×C8—21no—
107#381—21no—
108#380D412yes—
109#380D412yes—
110#379—21no—
111C4×C22⋊C4—21no—
112C4×C22⋊C4—21no—
113#343—24no—
114#343—42no—
115#343C2×C418yes—
116#343C2×C418yes—
117#341—24no—
118#341—42no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—24no—
124#340—24no—
125#340—24no—
126#336—21no—
127#336—21no—
128#336—21no—
129#336—21no—
130C25#4111yescharacteristic
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—24no—
136C23×C4—22no—
137C23×C4—22no—
138C23×C4—41no—
139C23×C4—41no—
140C23×C4—42no—
141C23×C4—84no—
142C23×C4C22×C412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4#4111yescharacteristic
148C23×C4#4112yes—
149C23×C4#4112yes—
150C23×C4#4112yes—
151C23×C4C2411yescharacteristic
152C2×M16—22no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—22no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—22no—
176C2×M16—22no—
177C2×M16—41no—
178C2×M16—41no—
179C2×M16—41no—
180C2×M16—41no—
181C2×M16—48no—
182C2×M16—48no—
183C2×M16C22×C412yes—
184C2×M16C22×C414yes—
185C2×M16C22×C412yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—22no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—22no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—21no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8C2×D411yescharacteristic
198C22×C8C2×D411yescharacteristic
199C22×C8C2×D411yescharacteristic
200C22×C8C2×D411yescharacteristic
201C22×C8#4114yes—
202#117—24no—
203#117—24no—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208#117—44no—
209#117—44no—
210#117—42no—
211#117—42no—
212C2×C4⋊C4—24no—
213C2×C4⋊C4—24no—
214C2×C4⋊C4—24no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—22no—
220C2×C4⋊C4—22no—
221C2×C4⋊C4—22no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—24no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—41no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—42no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4C22×C414yes—
235C2×C4⋊C4C2×D412yes—
236C2×C4⋊C4C2×D412yes—
237C2×C4⋊C4#4114yes—
238C2×C22⋊C4—21no—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—22no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—44no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C42—21no—
246C2×C42—21no—
247C2×C42—44no—
248C2×C42—41no—
249C2×C42—41no—
250#107—44no—
251#106—44no—
252C22⋊C8—24no—
253C22⋊C8—24no—
254C22⋊C8—24no—
255C22⋊C8—24no—
256C22⋊C8—42no—
257C22⋊C8—42no—
258#97—42no—
259#97—42no—
260C4×C8—42no—
261C4×C8—42no—
262#95—22no—
263#95—22no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24—42no—
268C24#11712yes—
269C24#12111yescharacteristic
270C24#12111yescharacteristic
271C24#12612yes—
272C24C2×#4111yescharacteristic
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—24no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—24no—
294C22×C4—24no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—84no—
306C22×C4—84no—
307C22×C4—88no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4#11712yes—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4C4×D412yes—
316C22×C4C4×D412yes—
317C22×C4#12111yescharacteristic
318C22×C4#12111yescharacteristic
319C22×C4#12312yes—
320C22×C4#12312yes—
321C22×C4#12412yes—
322C22×C4#12412yes—
323C22×C4#12612yes—
324C22×C4C23×C411yescharacteristic
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C2×#4111yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4112yes—
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334M16—24no—
335M16—24no—
336M16—24no—
337M16—24no—
338M16—44no—
339M16—44no—
340M16—44no—
341M16—44no—
342M16—42no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349M16—42no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—24no—
358C2×C8—22no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—21no—
364C2×C8—21no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8#12312yes—
377C2×C8#12312yes—
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—24no—
383C4⋊C4—22no—
384C4⋊C4—24no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4—44no—
390C4⋊C4—44no—
391C22⋊C4—44no—
392C22⋊C4—44no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—82no—
396C22⋊C4—82no—
397C42—44no—
398C42—44no—
399C42—41no—
400C42—41no—
401C42—41no—
402C42—41no—
403C23—22no—
404C23—22no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—24no—
409C23—21no—
410C23—21no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—22no—
415C23—42no—
416C23—42no—
417C23—42no—
418C23—42no—
419C23—48no—
420C23—44no—
421C23—44no—
422C23#38812yes—
423C23#38812yes—
424C23C2×#11711yescharacteristic
425C23C2×#12111yescharacteristic
426C23C2×#12611yescharacteristic
427C23#55112yes—
428C23#55212yes—
429C2×C4—24no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—24no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—42no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—84no—
456C2×C4—88no—
457C2×C4—84no—
458C2×C4—88no—
459C2×C4—82no—
460C2×C4—84no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4#38812yes—
465C2×C4#38812yes—
466C2×C4C2×C4×D411yescharacteristic
467C2×C4C2×C4×D411yescharacteristic
468C2×C4C4×#4112yes—
469C2×C4#52012yes—
470C2×C4C2×#12311yescharacteristic
471C2×C4C2×#12311yescharacteristic
472C2×C4#52511yescharacteristic
473C2×C4C2×#12411yescharacteristic
474C2×C4#52912yes—
475C2×C4#53411yescharacteristic
476C2×C4#53611yescharacteristic
477C2×C4#53912yes—
478C2×C4#54612yes—
479C2×C4#54712yes—
480C2×C4#55112yes—
481C2×C4#55112yes—
482C2×C4#55212yes—
483C2×C4#55212yes—
484C8—22no—
485C8—22no—
486C8—42no—
487C8—44no—
488C8—41no—
489C8—41no—
490C8—42no—
491C8—82no—
492C22—24no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—48no—
506C22—48no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22C2×#38811yescharacteristic
512C22#234312yes—
513C22#248611yescharacteristic
514C22#249312yes—
515C22#253411yescharacteristic
516C22#306812yes—
517C22#306912yes—
518C4—22no—
519C4—22no—
520C4—82no—
521C4—84no—
522C4—82no—
523C4—84no—
524C4C4×#12311yescharacteristic
525C4#232211yescharacteristic
526C4#235912yes—
527C4#248311yescharacteristic
528C4#248411yescharacteristic
529C4#248812yes—
530C2—22no—
531C2—22no—
532C2—42no—
533C2—42no—
534C2#1399211yescharacteristic
535C2#2107711yescharacteristic
536C2#2107811yescharacteristic
537C2#2767612yes—
538C2#2813912yes—
subgroupG/H
center, socleC23C2×#121
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11303C2semidirect
7. #11290C2semidirect
8. #10896C2semidirect
9. #10827C2semidirect
26. #1821C4semidirect

Maximal subgroups

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

subgroupindex
#208452
#207002
C2×#18212
#203862
C2×#17362
#113032
#112902
#108962
#108272
#107282
#107242