Finite Group Databasebeta

← #370171

Subgroups — #370171

Order 512.

1610 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1853C211yesmaximal, characteristic
2C2×#1850C211yesmaximal, characteristic
3C22×#340C211yesmaximal, characteristic
4#13880C211yesmaximal, characteristic
5C22⋊C4×C4⋊C4C211yesmaximal, characteristic
6#11267C212yesmaximal, complemented
7#11231C212yesmaximal, complemented
8#10908C212yesmaximal, complemented
9#10849C212yesmaximal, complemented
10#10730C211yesmaximal, characteristic
11#10721C211yesmaximal, characteristic
12C24×C8C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#106C2211yescharacteristic
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2212yes—
18C22×C22⋊C8C2211yescharacteristic
19C2×#385—21no—
20C2×#385—21no—
21C2×#384C2212yes—
22C2×#380C2211yescharacteristic
23C2×#380C2211yescharacteristic
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C22⋊C4—21no—
26C2×C4×C22⋊C4C2211yescharacteristic
27#1853—24no—
28#1850C418yescomplemented
29C2×#340—24no—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39#1444—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C22⋊#95—21no—
44C24×C4—21no—
45C24×C4—21no—
46C24×C4C2311yescharacteristic
47C23×C8—22no—
48C23×C8—22no—
49C23×C8—42no—
50C23×C8—42no—
51C23×C8C2×C414yes—
52C23×C8C2311yescharacteristic
53C23×C8C2311yescharacteristic
54C23×C8C2312yes—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—24no—
57C22×C4⋊C4—24no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—42no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D414yes—
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—21no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4D411yescharacteristic
76C22×C22⋊C4D411yescharacteristic
77C22×C42—22no—
78C22×C42C2312yes—
79C2×#106—22no—
80C2×#106—22no—
81C2×#106—24no—
82C2×#106—42no—
83C2×C4⋊C8—42no—
84C2×C4⋊C8—42no—
85C2×C4⋊C8C2×C418yescomplemented
86C2×C22⋊C8—22no—
87C2×C22⋊C8—22no—
88C2×C22⋊C8C2×C414yes—
89#385—44no—
90#385—44no—
91#384—24no—
92#384—42no—
93#380—22no—
94#380—22no—
95#380—22no—
96#380Q814yes—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—42no—
99C4×C22⋊C4—44no—
100C4×C22⋊C4C2×C414yes—
101C4×C22⋊C4Q814yes—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—22no—
109C2×#95D412yes—
110#340—22no—
111#340—22no—
112#340—24no—
113#340—24no—
114#340—24no—
115#340—42no—
116#340—42no—
117#340—42no—
118#336C2×C418yescomplemented
119C25C2×D411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—21no—
128C23×C4—24no—
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—44no—
132C23×C4—44no—
133C23×C4—42no—
134C23×C4—42no—
135C23×C4—42no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—82no—
139C23×C4—82no—
140C23×C4C22×C412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×Q812yes—
146C23×C4C2×Q812yes—
147C23×C4#4112yes—
148C23×C4#4112yes—
149C23×C4C2411yescharacteristic
150C22×C8—28no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—48no—
166C22×C8—44no—
167C22×C8—44no—
168C22×C8—48no—
169C22×C8C22×C414yes—
170C22×C8C22×C412yes—
171C22×C8C22×C412yes—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—48no—
200C2×C4⋊C4—48no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—44no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—82no—
210C2×C4⋊C4—82no—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×Q814yes—
214C2×C4⋊C4#4114yes—
215C2×C4⋊C4#4114yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—24no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—42no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—42no—
224C2×C22⋊C4C4⋊C414yes—
225C2×C22⋊C4C4⋊C414yes—
226C2×C22⋊C4SD1614yes—
227C2×C22⋊C4SD1614yes—
228C2×C42—22no—
229C2×C42—22no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—44no—
233C2×C42—44no—
234C2×C42—82no—
235C2×C42C22×C414yes—
236C2×C42C2×Q814yes—
237#106—44no—
238#106—44no—
239C4⋊C8—48no—
240C22⋊C8—28no—
241#95—24no—
242#95—44no—
243#95—44no—
244#95—42no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95C4⋊C418yes—
249C24—22no—
250C24—22no—
251C24—22no—
252C24—22no—
253C24C2×C4⋊C412yes—
254C24#12112yes—
255C24C22⋊Q812yes—
256C24C2×SD1612yes—
257C24C2×SD1612yes—
258C24AGL(1,ℤ/8)12yes—
259C24#13712yes—
260C24C22×D411yescharacteristic
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—24no—
291C22×C4—24no—
292C22×C4—48no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—48no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—41no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—48no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—41no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—84no—
325C22×C4—88no—
326C22×C4—88no—
327C22×C4—88no—
328C22×C4—88no—
329C22×C4—84no—
330C22×C4—84no—
331C22×C4C2×C4⋊C414yes—
332C22×C4C2×C4⋊C412yes—
333C22×C4C2×C4⋊C414yes—
334C22×C4C4×D414yes—
335C22×C4C4×Q814yes—
336C22×C4#12112yes—
337C22×C4C22⋊Q812yes—
338C22×C4#12514yes—
339C22×C4C4⋊Q814yes—
340C22×C4C2×SD1614yes—
341C22×C4C2×SD1614yes—
342C22×C4C23×C411yescharacteristic
343C22×C4C22×D411yescharacteristic
344C22×C4C22×Q811yescharacteristic
345C22×C4C22×Q811yescharacteristic
346C22×C4C2×#4111yescharacteristic
347C22×C4C2×#4111yescharacteristic
348C22×C4C2×#4112yes—
349C22×C4C2×#4112yes—
350C22×C4H(5,𝔽2)12yes—
351C22×C4H(5,𝔽2)12yes—
352C2×C8—22no—
353C2×C8—22no—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—416no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—42no—
363C2×C8—41no—
364C2×C8—42no—
365C2×C8—41no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C4⋊C4—22no—
371C4⋊C4—24no—
372C4⋊C4—22no—
373C4⋊C4—24no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—24no—
377C4⋊C4—24no—
378C4⋊C4—48no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—42no—
390C4⋊C4—42no—
391C4⋊C4—88no—
392C4⋊C4—88no—
393C4⋊C4#12314yes—
394C22⋊C4—28no—
395C22⋊C4—48no—
396C22⋊C4—84no—
397C22⋊C4#106116yescomplemented
398C42—24no—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—88no—
403C23—24no—
404C23—24no—
405C23—24no—
406C23—22no—
407C23—22no—
408C23—24no—
409C23—24no—
410C23—24no—
411C23—22no—
412C23—24no—
413C23—24no—
414C23—22no—
415C23—24no—
416C23—24no—
417C23—24no—
418C23—24no—
419C23#38914yes—
420C23C2×#10614yes—
421C23#42814yes—
422C23#46014yes—
423C23#46114yes—
424C23#47514yes—
425C23#47614yes—
426C23C22×C4⋊C411yescharacteristic
427C23C2×#12111yescharacteristic
428C23C2×C22⋊Q811yescharacteristic
429C23#54612yes—
430C23#54612yes—
431C23C22×SD1611yescharacteristic
432C23C22×SD1611yescharacteristic
433C23C2×AGL(1,ℤ/8)11yescharacteristic
434C23C2×#13711yescharacteristic
435C23#57612yes—
436C23#57812yes—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—22no—
441C2×C4—22no—
442C2×C4—24no—
443C2×C4—24no—
444C2×C4—24no—
445C2×C4—24no—
446C2×C4—24no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—28no—
450C2×C4—24no—
451C2×C4—28no—
452C2×C4—24no—
453C2×C4—24no—
454C2×C4—24no—
455C2×C4—24no—
456C2×C4—24no—
457C2×C4—42no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—44no—
466C2×C4—44no—
467C2×C4—42no—
468C2×C4—44no—
469C2×C4—42no—
470C2×C4—44no—
471C2×C4—42no—
472C2×C4—44no—
473C2×C4—44no—
474C2×C4—816no—
475C2×C4—816no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—82no—
495C2×C4—82no—
496C2×C4#38914yes—
497C2×C4C2×#10618yes—
498C2×C4C2×C4×D411yescharacteristic
499C2×C4C2×C4×Q811yescharacteristic
500C2×C4C4×#4112yes—
501C2×C4#51812yes—
502C2×C4C2×#12311yescharacteristic
503C2×C4#52511yescharacteristic
504C2×C4C2×#12511yescharacteristic
505C2×C4#52912yes—
506C2×C4C2×C4⋊Q811yescharacteristic
507C2×C4#53411yescharacteristic
508C2×C4#53511yescharacteristic
509C2×C4#53812yes—
510C2×C4#55112yes—
511C2×C4#55112yes—
512C2×C4#55412yes—
513C2×C4#55412yes—
514C2×C4#55412yes—
515C2×C4#55412yes—
516C8—42no—
517C8—44no—
518C8—42no—
519C22—28no—
520C22—28no—
521C22—24no—
522C22—24no—
523C22—28no—
524C22—24no—
525C22—28no—
526C22—24no—
527C22—24no—
528C22—24no—
529C22#194818yes—
530C22C2×#38911yescharacteristic
531C22#234912yes—
532C22#241111yescharacteristic
533C22#241211yescharacteristic
534C22#245012yes—
535C22C22×#10611yescharacteristic
536C22C2×#42811yescharacteristic
537C22#291312yes—
538C22C2×#46111yescharacteristic
539C22C2×#46011yescharacteristic
540C22#304111yescharacteristic
541C22#304411yescharacteristic
542C22C2×#47611yescharacteristic
543C22C2×#47511yescharacteristic
544C22#308312yes—
545C22#329512yes—
546C22#329512yes—
547C22#329812yes—
548C22#329812yes—
549C22#330812yes—
550C22#331212yes—
551C22#331412yes—
552C22#331812yes—
553C4—28no—
554C4—24no—
555C4—44no—
556C4—44no—
557C4—84no—
558C4—84no—
559C4—82no—
560C4—161no—
561C4—161no—
562C4C4×#12311yescharacteristic
563C4#232111yescharacteristic
564C4#236812yes—
565C4#250211yescharacteristic
566C4#250512yes—
567C4#258311yescharacteristic
568C2—28no—
569C2#1404411yescharacteristic
570C2C2×#194811yescharacteristic
571C2#2117311yescharacteristic
572C2#2270312yes—
573C2#2270912yes—
574C2#2447811yescharacteristic
575C2#2448011yescharacteristic
576C2#2452811yescharacteristic
577C2#2453011yescharacteristic
578C2#2641012yes—
579C2#2641212yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #11267C2semidirect
7. #11231C2semidirect
8. #10908C2semidirect
9. #10849C2semidirect
28. #1850C4semidirect
85. C2×C4⋊C8C2×C4semidirect
118. #336C2×C4semidirect
397. C22⋊C4#106semidirect

Maximal subgroups

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

subgroupindex
C2×#18532
C2×#18502
C22×#3402
#138802
C22⋊C4×C4⋊C42
#112672
#112312
#109082
#108492
#107302
#107212