Finite Group Databasebeta

← #115839

Subgroups — #115839

Order 512.

880 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21386C211yesmaximal, characteristic
2C2×#1980C211yesmaximal, characteristic
3#21146C211yesmaximal, characteristic, complemented
4C2×#1492C211yesmaximal, characteristic
5C4×#340C211yesmaximal, characteristic, complemented
6C2×#1303C211yesmaximal, characteristic, complemented
7C2×#1278C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#423—21no—
11C2×#423C2211yescharacteristic
12C22×#104—21no—
13C22×#104—21no—
14C2×C4×M16C2211yescharacteristic
15C22×C4×C8C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17C2×C4×C4⋊C4C2211yescharacteristic
18#1988C411yescharacteristic, complemented
19#1988C411yescharacteristic, complemented
20#1982C411yescharacteristic, complemented
21#1982C411yescharacteristic, complemented
22#1980—22no—
23#1980—22no—
24#1940—22no—
25#1939—22no—
26#1874—22no—
27C4×#102—22no—
28C2×#340—22no—
29#1728—21no—
30#1492—22no—
31#1492—22no—
32C2×D4⋊C8—22no—
33#1303—22no—
34#1278—22no—
35C22×#41—21no—
36C22×M16—21no—
37C22×M16D411yescharacteristic
38C22×M16D411yescharacteristic
39C23×C8—21no—
40#532—21no—
41#532—21no—
42#532—21no—
43#532—21no—
44#532—21no—
45#532—21no—
46#532D412yes—
47#532Q812yes—
48C2×C4⋊Q8C2×C411yescharacteristic
49C2×#127C2×C411yescharacteristic
50C2×#124—21no—
51C2×#121—21no—
52C2×#121—21no—
53C2×C4×D4—21no—
54C2×C4×D4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D412yes—
57C22×C42—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#423—21no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423—42no—
69#423—42no—
70C2×C4⋊C8—24no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8C2×C411yescharacteristic
73C2×C4⋊C8C2×C411yescharacteristic
74C2×C4⋊C8D412yes—
75C2×#104—24no—
76C2×#104—24no—
77C2×#104—42no—
78C2×#104—42no—
79C2×#104—42no—
80C2×#104—42no—
81C2×#102—22no—
82C2×#102—22no—
83C2×C22⋊C8—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C2×#97—21no—
89C2×C4×C8—21no—
90C2×C4×C8—21no—
91C2×C4×C8—21no—
92C2×C4×C8—21no—
93C2×C4×C8—21no—
94C2×C4×C8—21no—
95C2×C4×C8—21no—
96C2×C4×C8D412yes—
97C4×C4⋊C4—21no—
98C4×C4⋊C4—21no—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4D412yes—
105C4×C4⋊C4D412yes—
106C2×#95—21no—
107C2×#95—21no—
108#340—28no—
109#340—22no—
110#340—22no—
111#336—24no—
112D4⋊C8—28no—
113C2×#41—22no—
114C2×#41—22no—
115C2×#41—41no—
116C2×#41—41no—
117C2×#41—41no—
118C2×#41—41no—
119C2×#41—41no—
120C2×#41—41no—
121C2×#41—41no—
122C2×#41—41no—
123C2×#41—41no—
124C2×#41—41no—
125C22×Q8—21no—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—41no—
130C23×C4—22no—
131C23×C4—41no—
132C23×C4—41no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4#4111yescharacteristic
136C2×M16—21no—
137C2×M16—21no—
138C2×M16—21no—
139C2×M16—21no—
140C2×M16—21no—
141C2×M16—21no—
142C2×M16—21no—
143C2×M16—21no—
144C2×M16—21no—
145C2×M16—21no—
146C2×M16—41no—
147C2×M16—41no—
148C2×M16—41no—
149C2×M16—41no—
150C2×M16—41no—
151C2×M16—41no—
152C2×M16D812yes—
153C2×M16SD1612yes—
154C22×C8—24no—
155C22×C8—21no—
156C22×C8—21no—
157C22×C8—22no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—22no—
161C22×C8—42no—
162C22×C8—41no—
163C22×C8—41no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8C4⋊C412yes—
168#127—21no—
169#127—21no—
170#127—21no—
171#127C4⋊C412yes—
172#124—22no—
173#124—22no—
174#124—41no—
175#124—41no—
176#121—22no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—41no—
181#121—41no—
182#121—41no—
183#121—41no—
184C4×D4—22no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—41no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C2×C4⋊C4—24no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—24no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4C22⋊C412yes—
202C2×C4⋊C4C22⋊C412yes—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—41no—
210C2×C42—41no—
211C2×C42—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42C22×C411yescharacteristic
219C2×C42C2×D411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×Q811yescharacteristic
225C2×C42#4111yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C4⋊C8—21no—
231C4⋊C8—21no—
232C4⋊C8—21no—
233C4⋊C8—21no—
234C4⋊C8—22no—
235C4⋊C8—22no—
236C4⋊C8—48no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239#104—48no—
240#104—48no—
241#102—44no—
242#102—44no—
243C22⋊C8—41no—
244C22⋊C8—41no—
245#97—41no—
246#97—41no—
247C4×C8—22no—
248C4×C8—22no—
249C4×C8—41no—
250C4×C8—41no—
251C4×C8—41no—
252C4×C8—41no—
253C4×C8—41no—
254C4×C8—41no—
255#95—21no—
256#95—21no—
257#95—22no—
258#95—22no—
259#95—41no—
260C24—41no—
261C24#12111yescharacteristic
262#41—42no—
263#41—42no—
264#41—42no—
265#41—42no—
266#41—42no—
267#41—42no—
268#41—42no—
269#41—42no—
270#41—81no—
271#41—81no—
272#41—81no—
273#41—81no—
274#41—81no—
275#41—81no—
276#41—81no—
277#41—81no—
278C2×Q8—22no—
279C2×Q8—41no—
280C2×Q8—41no—
281C2×Q8—41no—
282C2×Q8—41no—
283C2×Q8—41no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—44no—
288C2×D4—42no—
289C2×D4—42no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—81no—
300C2×D4—81no—
301C2×D4—81no—
302C2×D4—81no—
303C22×C4—22no—
304C22×C4—22no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—41no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—82no—
317C22×C4—82no—
318C22×C4—81no—
319C22×C4—81no—
320C22×C4—81no—
321C22×C4—81no—
322C22×C4—81no—
323C22×C4—81no—
324C22×C4C2×C22⋊C411yescharacteristic
325C22×C4C4×D411yescharacteristic
326C22×C4C4×D411yescharacteristic
327C22×C4#12011yescharacteristic
328C22×C4#12011yescharacteristic
329C22×C4#12111yescharacteristic
330C22×C4#12111yescharacteristic
331C22×C4#12111yescharacteristic
332C22×C4#12111yescharacteristic
333C22×C4#12111yescharacteristic
334C22×C4C22⋊Q811yescharacteristic
335C22×C4C22⋊Q811yescharacteristic
336C22×C4C22⋊Q811yescharacteristic
337C22×C4#12311yescharacteristic
338C22×C4#12311yescharacteristic
339C22×C4#12311yescharacteristic
340C22×C4#12411yescharacteristic
341C22×C4#12611yescharacteristic
342C22×C4C2×D811yescharacteristic
343C22×C4C2×SD1611yescharacteristic
344C22×C4#13511yescharacteristic
345C22×C4#13511yescharacteristic
346C22×C4AGL(1,ℤ/8)11yescharacteristic
347C22×C4AGL(1,ℤ/8)11yescharacteristic
348C22×C4AGL(1,ℤ/8)11yescharacteristic
349C22×C4#13711yescharacteristic
350M16—22no—
351M16—22no—
352M16—22no—
353M16—22no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—81no—
359M16—81no—
360M16—81no—
361M16—81no—
362C2×C8—24no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—44no—
366C2×C8—44no—
367C2×C8—42no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—42no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—42no—
376C2×C8—42no—
377C2×C8—42no—
378C2×C8—42no—
379C2×C8—42no—
380C4⋊C4—22no—
381C4⋊C4—24no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—44no—
387C4⋊C4—44no—
388C4⋊C4—42no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4#10414yescomplemented
392C22⋊C4—42no—
393C22⋊C4—42no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C42—41no—
397C42—41no—
398C42—41no—
399C42—41no—
400C42—42no—
401C42—42no—
402C42—42no—
403C42—42no—
404C42—81no—
405C42—81no—
406C42C2×C4⋊C411yescharacteristic
407C42#11711yescharacteristic
408C42C4×D411yescharacteristic
409C42C4×D411yescharacteristic
410C42#12011yescharacteristic
411C42#12111yescharacteristic
412C42#12111yescharacteristic
413C42C22⋊Q811yescharacteristic
414C42C22⋊Q811yescharacteristic
415C42C22⋊Q811yescharacteristic
416C42#12311yescharacteristic
417C42#12311yescharacteristic
418C42#12411yescharacteristic
419C42#12411yescharacteristic
420C42#12611yescharacteristic
421C42#12711yescharacteristic
422C23—22no—
423C23—22no—
424C23—42no—
425C23—82no—
426C23—81no—
427C23—81no—
428C23—82no—
429C23#38611yescharacteristic
430C23#39411yescharacteristic
431C23#39611yescharacteristic
432C23#45911yescharacteristic
433C23#46011yescharacteristic
434C23#46311yescharacteristic
435C23#46411yescharacteristic
436Q8—42no—
437Q8—42no—
438Q8—81no—
439Q8—81no—
440D4—44no—
441D4—44no—
442D4—42no—
443D4—42no—
444D4—82no—
445D4—82no—
446D4—82no—
447D4—82no—
448D4—82no—
449D4—81no—
450D4—81no—
451D4—81no—
452D4—81no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—44no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—84no—
475C2×C4—84no—
476C2×C4—84no—
477C2×C4—84no—
478C2×C4—82no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—82no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4#38011yescharacteristic
487C2×C4#38511yescharacteristic
488C2×C4#38511yescharacteristic
489C2×C4#38811yescharacteristic
490C2×C4#38811yescharacteristic
491C2×C4#39011yescharacteristic
492C2×C4#39211yescharacteristic
493C2×C4#39311yescharacteristic
494C2×C4#39411yescharacteristic
495C2×C4#39611yescharacteristic
496C2×C4#39711yescharacteristic
497C2×C4#39911yescharacteristic
498C2×C4C2×#10412yes—
499C2×C4#42112yes—
500C2×C4C4×D811yescharacteristic
501C2×C4C4×SD1611yescharacteristic
502C2×C4#44011yescharacteristic
503C2×C4#44211yescharacteristic
504C2×C4#44711yescharacteristic
505C2×C4#44711yescharacteristic
506C2×C4#44811yescharacteristic
507C2×C4#44911yescharacteristic
508C2×C4#44911yescharacteristic
509C2×C4#44911yescharacteristic
510C2×C4#45011yescharacteristic
511C2×C4#45211yescharacteristic
512C2×C4#46511yescharacteristic
513C2×C4#46611yescharacteristic
514C2×C4#46811yescharacteristic
515C2×C4#46911yescharacteristic
516C2×C4#47411yescharacteristic
517C2×C4#47611yescharacteristic
518C2×C4#47811yescharacteristic
519C2×C4#47811yescharacteristic
520C2×C4#48611yescharacteristic
521C2×C4#48811yescharacteristic
522C2×C4#48911yescharacteristic
523C2×C4#49011yescharacteristic
524C8—44no—
525C8—42no—
526C8—42no—
527C8—81no—
528C8—81no—
529C8—82no—
530C22—22no—
531C22—22no—
532C22—22no—
533C22—22no—
534C22—21no—
535C22—21no—
536C22—21no—
537C22—21no—
538C22—82no—
539C22—82no—
540C22—82no—
541C22—162no—
542C22—162no—
543C22#143911yescharacteristic
544C22#187112yes—
545C22#187911yescharacteristic
546C22#188011yescharacteristic
547C22#188112yes—
548C22#204111yescharacteristic
549C22#204211yescharacteristic
550C22#206511yescharacteristic
551C22#206711yescharacteristic
552C4—21no—
553C4—21no—
554C4—21no—
555C4—21no—
556C4—21no—
557C4—21no—
558C4—44no—
559C4—84no—
560C4—82no—
561C4—82no—
562C4#180612yes—
563C4#186211yescharacteristic
564C4#186411yescharacteristic
565C4#193411yescharacteristic
566C4#193611yescharacteristic
567C4#196211yescharacteristic
568C4#196311yescharacteristic
569C4#199911yescharacteristic
570C4#200011yescharacteristic
571C4#202311yescharacteristic
572C4#202511yescharacteristic
573C2—21no—
574C2—21no—
575C2—21no—
576C2—21no—
577C2—162no—
578C2#899112yes—
579C2#899212yes—
580C2#1105311yescharacteristic
581C2#1106511yescharacteristic
582C2#1106611yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#386
prime radicalC2×#532C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21146C2semidirect
5. C4×#340C2semidirect
6. C2×#1303C2semidirect
7. C2×#1278C2semidirect
18. #1988C4semidirect
19. #1988C4semidirect
20. #1982C4semidirect
21. #1982C4semidirect
391. C4⋊C4#104semidirect

Maximal subgroups

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

subgroupindex
#213862
C2×#19802
#211462
C2×#14922
C4×#3402
C2×#13032
C2×#12782