Finite Group Databasebeta

← #360360

Subgroups — #360360

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20700C211yesmaximal, characteristic
2C2×#1823C211yesmaximal, characteristic
3#20661C211yesmaximal, characteristic
4#20389C211yesmaximal, characteristic
5C22×#343C211yesmaximal, characteristic
6#11303C212yesmaximal, complemented
7#11290C212yesmaximal, complemented
8#10896C212yesmaximal, complemented
9#10827C212yesmaximal, complemented
10#10739C211yesmaximal, characteristic
11#10734C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C23×M16C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C2×#428C2212yes—
16C2×#407C2211yescharacteristic
17C22×C22⋊C8C2211yescharacteristic
18C2×#405C2212yes—
19#2280C2211yescharacteristic
20C2×#380C2211yescharacteristic
21#1824—22no—
22#1823—21no—
23#1823—21no—
24#1823—22no—
25#1823—21no—
26#1823—21no—
27#1809—22no—
28#1808—21no—
29#1807—21no—
30#1753—22no—
31#1752—21no—
32C8×C22⋊C4—21no—
33C2×#343C418yescomplemented
34C2×#343C2212yes—
35C2×#343C2212yes—
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39C2×#343C2211yescharacteristic
40C2×#341C2212yes—
41C2×#341C2212yes—
42#1736C2212yes—
43#1736C2212yes—
44#1736C2212yes—
45C2×#340C2212yes—
46C2×#340C2212yes—
47#1729C2212yes—
48C24×C4—21no—
49C24×C4C2311yescharacteristic
50C22×M16—21no—
51C22×M16—21no—
52C22×M16—22no—
53C22×M16—22no—
54C22×M16—24no—
55C22×M16—24no—
56C22×M16—42no—
57C22×M16—42no—
58C22×M16C2×C414yes—
59C22×M16C2×C414yes—
60C22×M16C2312yes—
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C23×C8—21no—
66C23×C8—21no—
67C23×C8C2312yes—
68C2×#117C2312yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—21no—
76C22×C4⋊C4—22no—
77C22×C4⋊C4C2×C414yes—
78C22×C4⋊C4C2312yes—
79C22×C4⋊C4C2311yescharacteristic
80C22×C4⋊C4C2311yescharacteristic
81C22×C22⋊C4—21no—
82#428—24no—
83#428—42no—
84C2×#107—22no—
85C2×#106—22no—
86#407—22no—
87#407—22no—
88C2×C22⋊C8—22no—
89C2×C22⋊C8—22no—
90C2×C22⋊C8—21no—
91C2×C22⋊C8—21no—
92C2×C22⋊C8D411yescharacteristic
93C2×C22⋊C8D411yescharacteristic
94C2×C22⋊C8D412yes—
95#405—24no—
96#405—42no—
97C2×#97—22no—
98C2×C4×C8—22no—
99#381—21no—
100#380—21no—
101#380—21no—
102#380—21no—
103#380—21no—
104#380D411yescharacteristic
105#380D411yescharacteristic
106#380D412yes—
107#379—21no—
108C4×C22⋊C4—21no—
109C4×C22⋊C4—21no—
110C2×#95—21no—
111#343—24no—
112#343—42no—
113#343C2×C414yes—
114#343C2×C414yes—
115#343C2×C414yes—
116#343C2×C414yes—
117#341—24no—
118#341—42no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340—22no—
124#340—22no—
125#340—22no—
126#340—22no—
127#340—22no—
128#340—22no—
129#336—22no—
130#336—22no—
131C25#4111yescharacteristic
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—22no—
136C23×C4—24no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—41no—
141C23×C4—41no—
142C23×C4—42no—
143C23×C4—82no—
144C23×C4—82no—
145C23×C4C22×C412yes—
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4#4111yescharacteristic
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4#4112yes—
154C23×C4C2411yescharacteristic
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—24no—
160C2×M16—24no—
161C2×M16—21no—
162C2×M16—21no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—22no—
167C2×M16—28no—
168C2×M16—28no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—22no—
175C2×M16—44no—
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—44no—
179C2×M16—48no—
180C2×M16—48no—
181C2×M16C22×C412yes—
182C2×M16C22×C412yes—
183C2×M16C22×C412yes—
184C2×M16C22×C412yes—
185C2×M16#4114yes—
186C2×M16#4114yes—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—24no—
190C22×C8—24no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—22no—
200C22×C8—44no—
201C22×C8—44no—
202C22×C8C2×D412yes—
203C22×C8C2×D412yes—
204#117—22no—
205#117—22no—
206#117—22no—
207#117—22no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—24no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—24no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—22no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—41no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4—42no—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C4⋊C4C2×D412yes—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—21no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—22no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—41no—
246C2×C22⋊C4—82no—
247C2×C42—22no—
248#107—44no—
249#106—44no—
250C22⋊C8—22no—
251C22⋊C8—22no—
252C22⋊C8—24no—
253C22⋊C8—42no—
254C22⋊C8—42no—
255#97—44no—
256C4×C8—44no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—41no—
261#95—41no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—22no—
266C24—42no—
267C24#11712yes—
268C24#12111yescharacteristic
269C24#12111yescharacteristic
270C24#12612yes—
271C24C2×#4111yescharacteristic
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—44no—
301C22×C4—44no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—88no—
306C22×C4—82no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—82no—
310C22×C4—88no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#11712yes—
318C22×C4C4×D412yes—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4C4×D412yes—
322C22×C4#12111yescharacteristic
323C22×C4#12111yescharacteristic
324C22×C4#12312yes—
325C22×C4#12312yes—
326C22×C4#12412yes—
327C22×C4#12412yes—
328C22×C4#12612yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4112yes—
333C22×C4C2×#4112yes—
334C22×C4C2×#4111yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339M16—22no—
340M16—22no—
341M16—24no—
342M16—24no—
343M16—22no—
344M16—22no—
345M16—22no—
346M16—22no—
347M16—48no—
348M16—48no—
349M16—42no—
350M16—42no—
351M16—42no—
352M16—42no—
353M16—42no—
354M16—42no—
355M16#11714yes—
356M16#11714yes—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—48no—
370C2×C8—48no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—42no—
395C4⋊C4—42no—
396C4⋊C4—42no—
397C4⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—42no—
402C22⋊C4—82no—
403C22⋊C4—82no—
404C22⋊C4—82no—
405C42—42no—
406C42—42no—
407C23—22no—
408C23—21no—
409C23—21no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—22no—
414C23—24no—
415C23—24no—
416C23—24no—
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—42no—
422C23—42no—
423C23—44no—
424C23—44no—
425C23—44no—
426C23—44no—
427C23#38812yes—
428C23#38812yes—
429C23#47112yes—
430C23#47212yes—
431C23C2×#11711yescharacteristic
432C23C2×#12111yescharacteristic
433C23C2×#12611yescharacteristic
434C23#52912yes—
435C23#53912yes—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—24no—
450C2×C4—24no—
451C2×C4—24no—
452C2×C4—24no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—88no—
464C2×C4—88no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—162no—
475C2×C4—162no—
476C2×C4#38812yes—
477C2×C4#38812yes—
478C2×C4C2×#11711yescharacteristic
479C2×C4C2×#11711yescharacteristic
480C2×C4C2×C4×D411yescharacteristic
481C2×C4C2×C4×D411yescharacteristic
482C2×C4#51811yescharacteristic
483C2×C4#51911yescharacteristic
484C2×C4C2×#12311yescharacteristic
485C2×C4C2×#12411yescharacteristic
486C2×C4#52912yes—
487C2×C4#53912yes—
488C2×C4#54612yes—
489C2×C4#54612yes—
490C2×C4#54712yes—
491C2×C4#54712yes—
492C2×C4#55112yes—
493C2×C4#55112yes—
494C2×C4#55212yes—
495C2×C4#55212yes—
496C8—24no—
497C8—24no—
498C8—42no—
499C8—42no—
500C8—42no—
501C8—42no—
502C8—82no—
503C22—22no—
504C22—21no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—24no—
512C22—22no—
513C22—22no—
514C22—24no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—44no—
519C22—48no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22#193114yes—
524C22#228311yescharacteristic
525C22C2×#38811yescharacteristic
526C22#231311yescharacteristic
527C22#242512yes—
528C22#249312yes—
529C22C2×#47111yescharacteristic
530C22C2×#47211yescharacteristic
531C22#306912yes—
532C4—22no—
533C4—22no—
534C4—82no—
535C4—82no—
536C4—82no—
537C4—82no—
538C4—162no—
539C4#234211yescharacteristic
540C4#234211yescharacteristic
541C4#234411yescharacteristic
542C4#234411yescharacteristic
543C4#241912yes—
544C4#248812yes—
545C2—21no—
546C2—24no—
547C2—22no—
548C2—21no—
549C2—42no—
550C2#1387411yescharacteristic
551C2C2×#193111yescharacteristic
552C2#2107711yescharacteristic
553C2#2526412yes—
554C2#2526812yes—
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
33. C2×#343C4semidirect

Maximal subgroups

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

subgroupindex
#207002
C2×#18232
#206612
#203892
C22×#3432
#113032
#112902
#108962
#108272
#107392
#107342