Finite Group Databasebeta

← #360343

Subgroups — #360343

Order 512.

1144 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1826C211yesmaximal, characteristic
2#20685C211yesmaximal, characteristic
3C2×#1811C211yesmaximal, characteristic
4#20379C211yesmaximal, characteristic
5#20347C211yesmaximal, characteristic
6#11317C212yesmaximal, complemented
7#11258C212yesmaximal, complemented
8#10910C212yesmaximal, complemented
9#10809C212yesmaximal, complemented
10#10733C211yesmaximal, characteristic
11#10715C211yesmaximal, characteristic
12C23×M16C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C22×#117C2211yescharacteristic
15C22×#108C2212yes—
16C2×#407C2211yescharacteristic
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×#405C2212yes—
20C2×C4×C22⋊C4C2211yescharacteristic
21#1826—24no—
22#1818—21no—
23#1818—21no—
24#1817—22no—
25#1811—24no—
26#1750—22no—
27#1749—21no—
28C4×C22⋊C8—21no—
29C2×#341C2212yes—
30#1736C412yescomplemented
31#1736C412yescomplemented
32#1736C412yescomplemented
33#1736C412yescomplemented
34#1735C2211yescharacteristic
35#1735C2211yescharacteristic
36#1735C2211yescharacteristic
37#1735C2211yescharacteristic
38C2×#339C2212yes—
39C2×#339C2212yes—
40C2×#339C2212yes—
41C2×#339C2212yes—
42#1729C2212yes—
43#1729C2212yes—
44#1729C2212yes—
45C2×#336C2212yes—
46C2×#336C2212yes—
47C24×C4C2311yescharacteristic
48C22×M16—22no—
49C22×M16—22no—
50C22×M16—24no—
51C22×M16—42no—
52C22×M16C2×C412yes—
53C22×M16C2×C412yes—
54C22×M16C2312yes—
55C22×M16C2312yes—
56C23×C8C2311yescharacteristic
57C23×C8C2312yes—
58C23×C8C2311yescharacteristic
59C2×#117—21no—
60C2×#117—21no—
61C2×#117—21no—
62C2×#117—21no—
63C2×#117—22no—
64C2×#117—22no—
65C2×#117—22no—
66C2×#117—22no—
67C2×#117—24no—
68C2×#117—42no—
69C2×#117C2×C411yescharacteristic
70C2×#117C2×C411yescharacteristic
71C2×#117C2×C412yes—
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C22×C4⋊C4C2×C411yescharacteristic
77C22×C4⋊C4C2×C411yescharacteristic
78C22×C4⋊C4C2×C412yes—
79C22×C22⋊C4—21no—
80C22×C22⋊C4—21no—
81C22×C22⋊C4—21no—
82C22×C42—21no—
83C22×C42—21no—
84C22×C42C2312yes—
85C2×#108—24no—
86C2×#108—24no—
87C2×#108—42no—
88#407—22no—
89#407—22no—
90#407—22no—
91#407D414yes—
92C2×C22⋊C8—21no—
93C2×C22⋊C8—21no—
94C2×C22⋊C8—22no—
95C2×C22⋊C8—22no—
96C2×C22⋊C8—21no—
97C2×C22⋊C8—21no—
98C2×C22⋊C8D412yes—
99C2×C22⋊C8D411yescharacteristic
100C2×C22⋊C8D411yescharacteristic
101#405—24no—
102#405—42no—
103C2×#97—22no—
104C2×C4×C8—22no—
105C4×C22⋊C4—22no—
106C4×C22⋊C4—22no—
107C2×#95—21no—
108#341—24no—
109#340C2×C412yes—
110#340C2×C412yes—
111#340C2×C412yes—
112#340C2×C412yes—
113#340C2×C412yes—
114#340C2×C412yes—
115#340C2×C412yes—
116#340C2×C412yes—
117#339—24no—
118#339—24no—
119#339—24no—
120#339—42no—
121#336—22no—
122#336—22no—
123#336—22no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—24no—
128#336—42no—
129C25#4111yescharacteristic
130C23×C4—22no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4C22×C411yescharacteristic
139C23×C4C22×C411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4#4112yes—
144C23×C4#4112yes—
145C23×C4#4112yes—
146C23×C4#4111yescharacteristic
147C23×C4C2411yescharacteristic
148C2×M16—24no—
149C2×M16—24no—
150C2×M16—22no—
151C2×M16—24no—
152C2×M16—24no—
153C2×M16—22no—
154C2×M16—22no—
155C2×M16—22no—
156C2×M16—22no—
157C2×M16—22no—
158C2×M16—22no—
159C2×M16—22no—
160C2×M16—22no—
161C2×M16—22no—
162C2×M16—22no—
163C2×M16—42no—
164C2×M16—42no—
165C2×M16—48no—
166C2×M16—48no—
167C2×M16C2×D414yes—
168C22×C8—21no—
169C22×C8—21no—
170C22×C8—22no—
171C22×C8—21no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—21no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8—22no—
183C22×C8C22×C412yes—
184C22×C8C22×C412yes—
185C22×C8C2×D412yes—
186C22×C8C2×D412yes—
187C22×C8#4114yes—
188#117—22no—
189#117—22no—
190#117—22no—
191#117—22no—
192#117—22no—
193#117—21no—
194#117—21no—
195#117—21no—
196#117—21no—
197#117—21no—
198#117—21no—
199#117—21no—
200#117—21no—
201#117—22no—
202#117—22no—
203#117—24no—
204#117—44no—
205#117—44no—
206#117—42no—
207#117—42no—
208#117—42no—
209#117—42no—
210#117—42no—
211#117#4114yes—
212C2×C4⋊C4—22no—
213C2×C4⋊C4—22no—
214C2×C4⋊C4—22no—
215C2×C4⋊C4—22no—
216C2×C4⋊C4—24no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—42no—
221C2×C4⋊C4C22×C411yescharacteristic
222C2×C4⋊C4C22×C411yescharacteristic
223C2×C4⋊C4C22×C411yescharacteristic
224C2×C4⋊C4C22×C411yescharacteristic
225C2×C4⋊C4C22×C412yes—
226C2×C4⋊C4C22×C412yes—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—22no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—44no—
235C2×C22⋊C4—44no—
236C2×C22⋊C4—42no—
237C2×C22⋊C4—42no—
238C2×C22⋊C4—42no—
239C2×C42—21no—
240C2×C42—22no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42—21no—
245C2×C42—21no—
246C2×C42—22no—
247C2×C42—22no—
248C2×C42—22no—
249C2×C42—22no—
250C2×C42—22no—
251C2×C42—22no—
252C2×C42—44no—
253C2×C42—42no—
254C2×C42—42no—
255#108—48no—
256C22⋊C8—24no—
257C22⋊C8—24no—
258C22⋊C8—24no—
259C22⋊C8—22no—
260C22⋊C8—22no—
261C22⋊C8—42no—
262C22⋊C8—42no—
263#97—44no—
264C4×C8—44no—
265#95—42no—
266C24—22no—
267C24—22no—
268C24—42no—
269C24—42no—
270C24C4×D411yescharacteristic
271C24C4×D411yescharacteristic
272C24#12412yes—
273C24#12612yes—
274C24C2×#4111yescharacteristic
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—24no—
295C22×C4—24no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—44no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4#11712yes—
318C22×C4#11712yes—
319C22×C4C4×D412yes—
320C22×C4C4×D412yes—
321C22×C4C4×D411yescharacteristic
322C22×C4C4×D411yescharacteristic
323C22×C4#12112yes—
324C22×C4#12112yes—
325C22×C4#12312yes—
326C22×C4#12312yes—
327C22×C4#12412yes—
328C22×C4#12612yes—
329C22×C4C23×C411yescharacteristic
330C22×C4C22×D411yescharacteristic
331C22×C4C22×D411yescharacteristic
332C22×C4C2×#4111yescharacteristic
333C22×C4C2×#4111yescharacteristic
334C22×C4C2×#4112yes—
335C22×C4C2×#4112yes—
336C22×C4C2×#4111yescharacteristic
337C22×C4H(5,𝔽2)12yes—
338C22×C42−1+412yes—
339M16—24no—
340M16—24no—
341M16—48no—
342M16—48no—
343M16—42no—
344M16—42no—
345M16—42no—
346M16—42no—
347M16—42no—
348M16—42no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—24no—
352C2×C8—24no—
353C2×C8—22no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—24no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—22no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
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#11714yes—
375C4⋊C4—22no—
376C4⋊C4—22no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—21no—
381C4⋊C4—21no—
382C4⋊C4—21no—
383C4⋊C4—21no—
384C4⋊C4—21no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—42no—
388C4⋊C4—42no—
389C4⋊C4#11714yes—
390C22⋊C4—24no—
391C22⋊C4—44no—
392C22⋊C4—48no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C22⋊C4—41no—
396C22⋊C4—41no—
397C22⋊C4—42no—
398C22⋊C4—42no—
399C22⋊C4—42no—
400C42—24no—
401C42—44no—
402C42—41no—
403C42—41no—
404C42—41no—
405C42—41no—
406C42—42no—
407C42—42no—
408C42—42no—
409C42—42no—
410C42—42no—
411C42—42no—
412C42—42no—
413C23—22no—
414C23—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—22no—
420C23—22no—
421C23—22no—
422C23—22no—
423C23—24no—
424C23—24no—
425C23—42no—
426C23—42no—
427C23—42no—
428C23—42no—
429C23—82no—
430C23—82no—
431C23—82no—
432C23—82no—
433C23#38812yes—
434C23#38812yes—
435C23C2×C4×D411yescharacteristic
436C23C2×#12411yescharacteristic
437C23C2×#12611yescharacteristic
438C23#55112yes—
439C23#55212yes—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—24no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—44no—
462C2×C4—44no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—88no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—84no—
480C2×C4—84no—
481C2×C4—82no—
482C2×C4#38812yes—
483C2×C4#38812yes—
484C2×C4C2×#11711yescharacteristic
485C2×C4C2×#11711yescharacteristic
486C2×C4C2×#11711yescharacteristic
487C2×C4C2×C4×D411yescharacteristic
488C2×C4#51811yescharacteristic
489C2×C4#51911yescharacteristic
490C2×C4C2×#12111yescharacteristic
491C2×C4C2×#12311yescharacteristic
492C2×C4#52912yes—
493C2×C4#52912yes—
494C2×C4#53912yes—
495C2×C4#53912yes—
496C2×C4#54612yes—
497C2×C4#54612yes—
498C2×C4#54712yes—
499C2×C4#54712yes—
500C2×C4#55112yes—
501C2×C4#55212yes—
502C8—24no—
503C8—42no—
504C8—42no—
505C8—42no—
506C8—44no—
507C8—82no—
508C22—22no—
509C22—22no—
510C22—21no—
511C22—21no—
512C22—21no—
513C22—21no—
514C22—22no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—22no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—84no—
527C22—84no—
528C22C2×#38811yescharacteristic
529C22#234211yescharacteristic
530C22#234411yescharacteristic
531C22#248812yes—
532C22#249312yes—
533C22#295512yes—
534C22#295612yes—
535C4—22no—
536C4—22no—
537C4—44no—
538C4—84no—
539C4—82no—
540C4—81no—
541C4—81no—
542C4—81no—
543C4—81no—
544C4#228311yescharacteristic
545C4#231311yescharacteristic
546C4#234211yescharacteristic
547C4#234411yescharacteristic
548C4#241912yes—
549C4#242512yes—
550C2—22no—
551C2—22no—
552C2—42no—
553C2—42no—
554C2#1387411yescharacteristic
555C2#2107011yescharacteristic
556C2#2107111yescharacteristic
557C2#2258812yes—
558C2#2260812yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC25#41
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C22⋊C8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11317C2semidirect
7. #11258C2semidirect
8. #10910C2semidirect
9. #10809C2semidirect
30. #1736C4semidirect
31. #1736C4semidirect
32. #1736C4semidirect
33. #1736C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18262
#206852
C2×#18112
#203792
#203472
#113172
#112582
#109102
#108092
#107332
#107152