Finite Group Databasebeta

← #360362

Subgroups — #360362

Order 512.

1180 conjugacy classes of subgroups.

The subgroup lattice has 549 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20700C211yesmaximal, characteristic
2C2×#1824C211yesmaximal, 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—21no—
22#1824—21no—
23#1824—22no—
24#1824—21no—
25#1824—21no—
26#1823—22no—
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—84no—
144C23×C4C22×C412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4#4111yescharacteristic
150C23×C4#4112yes—
151C23×C4#4112yes—
152C23×C4#4112yes—
153C23×C4C2411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—24no—
159C2×M16—24no—
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—22no—
163C2×M16—22no—
164C2×M16—22no—
165C2×M16—22no—
166C2×M16—28no—
167C2×M16—28no—
168C2×M16—22no—
169C2×M16—22no—
170C2×M16—22no—
171C2×M16—22no—
172C2×M16—22no—
173C2×M16—22no—
174C2×M16—44no—
175C2×M16—44no—
176C2×M16—44no—
177C2×M16—44no—
178C2×M16—48no—
179C2×M16—48no—
180C2×M16C22×C412yes—
181C2×M16C22×C412yes—
182C2×M16C22×C412yes—
183C2×M16C22×C412yes—
184C2×M16#4114yes—
185C2×M16#4114yes—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—24no—
189C22×C8—24no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—22no—
193C22×C8—22no—
194C22×C8—22no—
195C22×C8—22no—
196C22×C8—22no—
197C22×C8—22no—
198C22×C8—22no—
199C22×C8—44no—
200C22×C8—44no—
201C22×C8C2×D412yes—
202C22×C8C2×D412yes—
203#117—22no—
204#117—22no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—24no—
208C2×C4⋊C4—24no—
209C2×C4⋊C4—24no—
210C2×C4⋊C4—24no—
211C2×C4⋊C4—21no—
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—24no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—21no—
222C2×C4⋊C4—22no—
223C2×C4⋊C4—22no—
224C2×C4⋊C4—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—22no—
228C2×C4⋊C4—41no—
229C2×C4⋊C4—41no—
230C2×C4⋊C4—42no—
231C2×C4⋊C4—42no—
232C2×C4⋊C4—42no—
233C2×C4⋊C4—42no—
234C2×C4⋊C4—42no—
235C2×C4⋊C4C22×C412yes—
236C2×C4⋊C4C22×C412yes—
237C2×C4⋊C4C2×D412yes—
238C2×C4⋊C4C2×D412yes—
239C2×C22⋊C4—21no—
240C2×C22⋊C4—21no—
241C2×C22⋊C4—22no—
242C2×C22⋊C4—22no—
243C2×C22⋊C4—41no—
244C2×C22⋊C4—41no—
245C2×C22⋊C4—82no—
246C2×C42—22no—
247#107—44no—
248#106—44no—
249C22⋊C8—22no—
250C22⋊C8—22no—
251C22⋊C8—24no—
252C22⋊C8—42no—
253C22⋊C8—42no—
254#97—44no—
255C4×C8—44no—
256#95—22no—
257#95—21no—
258#95—21no—
259#95—41no—
260#95—41no—
261C24—22no—
262C24—22no—
263C24—22no—
264C24—22no—
265C24—42no—
266C24#11712yes—
267C24#12111yescharacteristic
268C24#12111yescharacteristic
269C24#12612yes—
270C24C2×#4111yescharacteristic
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—84no—
305C22×C4—84no—
306C22×C4—88no—
307C22×C4—82no—
308C22×C4—82no—
309C22×C4—88no—
310C22×C4—82no—
311C22×C4—82no—
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×#4112yes—
328C22×C4C2×#4112yes—
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)12yes—
333C22×C42−1+412yes—
334M16—22no—
335M16—22no—
336M16—24no—
337M16—24no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—48no—
343M16—48no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349M16—42no—
350M16#11714yes—
351M16#11714yes—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—48no—
365C2×C8—48no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C2×C8—42no—
372C2×C8—42no—
373C2×C8—42no—
374C2×C8—42no—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—22no—
379C4⋊C4—22no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—44no—
388C4⋊C4—44no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—42no—
392C4⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—42no—
397C22⋊C4—82no—
398C22⋊C4—82no—
399C22⋊C4—82no—
400C42—42no—
401C42—42no—
402C23—22no—
403C23—21no—
404C23—21no—
405C23—22no—
406C23—22no—
407C23—22no—
408C23—22no—
409C23—24no—
410C23—24no—
411C23—24no—
412C23—22no—
413C23—21no—
414C23—21no—
415C23—22no—
416C23—42no—
417C23—42no—
418C23—44no—
419C23—44no—
420C23—44no—
421C23—44no—
422C23#38812yes—
423C23#38812yes—
424C23#47112yes—
425C23#47312yes—
426C23C2×#11711yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×#12611yescharacteristic
429C23#52912yes—
430C23#53912yes—
431C2×C4—22no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—24no—
435C2×C4—22no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—88no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—88no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—82no—
466C2×C4—82no—
467C2×C4—162no—
468C2×C4—162no—
469C2×C4#38812yes—
470C2×C4#38812yes—
471C2×C4C2×#11711yescharacteristic
472C2×C4C2×#11711yescharacteristic
473C2×C4C2×C4×D411yescharacteristic
474C2×C4C2×C4×D411yescharacteristic
475C2×C4#51811yescharacteristic
476C2×C4#51911yescharacteristic
477C2×C4C2×#12311yescharacteristic
478C2×C4C2×#12411yescharacteristic
479C2×C4#52912yes—
480C2×C4#53912yes—
481C2×C4#54612yes—
482C2×C4#54612yes—
483C2×C4#54712yes—
484C2×C4#54712yes—
485C2×C4#55112yes—
486C2×C4#55112yes—
487C2×C4#55212yes—
488C2×C4#55212yes—
489C8—24no—
490C8—24no—
491C8—42no—
492C8—42no—
493C8—42no—
494C8—42no—
495C8—82no—
496C22—22no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—24no—
505C22—22no—
506C22—22no—
507C22—24no—
508C22—22no—
509C22—22no—
510C22—22no—
511C22—44no—
512C22—48no—
513C22—44no—
514C22—42no—
515C22—42no—
516C22#193014yes—
517C22#228311yescharacteristic
518C22C2×#38811yescharacteristic
519C22#231311yescharacteristic
520C22#242512yes—
521C22#249312yes—
522C22C2×#47111yescharacteristic
523C22C2×#47311yescharacteristic
524C22#306812yes—
525C4—22no—
526C4—22no—
527C4—82no—
528C4—82no—
529C4—82no—
530C4—82no—
531C4—162no—
532C4#234211yescharacteristic
533C4#234211yescharacteristic
534C4#234411yescharacteristic
535C4#234411yescharacteristic
536C4#241912yes—
537C4#248812yes—
538C2—21no—
539C2—24no—
540C2—22no—
541C2—21no—
542C2—42no—
543C2#1387411yescharacteristic
544C2C2×#193011yescharacteristic
545C2#2107811yescharacteristic
546C2#2526612yes—
547C2#2527012yes—
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×#18242
#206612
#203892
C22×#3432
#113032
#112902
#108962
#108272
#107392
#107342