Finite Group Databasebeta

← #224267

Subgroups — #224267

Order 512.

829 conjugacy classes of subgroups.

The subgroup lattice has 600 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36573C211yesmaximal, characteristic
2#22825C211yesmaximal, characteristic, complemented
3#22722C211yesmaximal, characteristic, complemented
4#9907C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9904C211yesmaximal, characteristic, complemented
7#9902C211yesmaximal, characteristic, complemented
8#9901C211yesmaximal, characteristic, complemented
9#9900C211yesmaximal, characteristic, complemented
10#8588C211yesmaximal, characteristic
11#8588C211yesmaximal, characteristic
12#8585C211yesmaximal, characteristic
13#8234C211yesmaximal, characteristic
14#8234C211yesmaximal, characteristic
15#8231C211yesmaximal, characteristic
16C2×#446—21no—
17#2859C2211yescharacteristic
18#2859C2211yescharacteristic
19#2859C2211yescharacteristic
20#2580—21no—
21#2580—21no—
22#2580C412yes—
23#2454—21no—
24#2454—21no—
25#2454—21no—
26#2454—21no—
27#2454C412yes—
28#2454C412yes—
29#2349—21no—
30#2349—21no—
31#2349C412yes—
32C4×#127C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C42×Q8C2211yescharacteristic
36#1984—21no—
37#1940—21no—
38#1940—21no—
39#1939—21no—
40#1917—21no—
41#1917—21no—
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1770—21no—
49#1767C2211yescharacteristic
50#1767C2211yescharacteristic
51#1767C2211yescharacteristic
52#1767C2211yescharacteristic
53C4×C4⋊C8C2211yescharacteristic
54C4×C4⋊C8C2211yescharacteristic
55C2×#331C2211yescharacteristic
56C2×#331C2211yescharacteristic
57C2×#331C2211yescharacteristic
58C2×#331C2211yescharacteristic
59#1296C2211yescharacteristic
60#1296C2211yescharacteristic
61#1296C2211yescharacteristic
62#1296C2211yescharacteristic
63#1296C2211yescharacteristic
64#1296C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69#1296C2211yescharacteristic
70#1296C2211yescharacteristic
71C2×#127C2311yescharacteristic
72C2×#125—21no—
73C2×#125—21no—
74C2×#125C2×C412yes—
75C2×C4×Q8C2×C411yescharacteristic
76C2×C4×Q8D411yescharacteristic
77C2×C4×Q8D411yescharacteristic
78C2×C4×D4—21no—
79C2×C4×D4—21no—
80C2×C4×D4—21no—
81C2×C4×D4—21no—
82C2×C4×D4—21no—
83C2×C4×D4—21no—
84C2×C4×D4C2×C412yes—
85C2×C4×D4C2×C412yes—
86C2×C4×D4C2×C412yes—
87#446—44no—
88C2×C4⋊C8—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⋊C8C2311yescharacteristic
97C2×C4⋊C8C2311yescharacteristic
98C2×C4⋊C8C2311yescharacteristic
99C2×C4⋊C8C2311yescharacteristic
100C2×C4⋊C8C2311yescharacteristic
101C2×C4⋊C8C2311yescharacteristic
102C2×C4⋊C8C2311yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—21no—
107C2×#97—21no—
108C2×C4×C8—21no—
109C2×C4×C8—21no—
110#399—22no—
111#399—22no—
112#399—22no—
113#396—22no—
114#396—22no—
115#393—22no—
116#390—21no—
117#390—21no—
118#390—21no—
119#390—21no—
120#390—21no—
121#390—21no—
122#390C2×C412yes—
123#390C2×C412yes—
124#390C2×C412yes—
125#389—21no—
126#389—21no—
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389—21no—
132#389—21no—
133#389C2×C412yes—
134#389C2×C412yes—
135#389C2×C412yes—
136#389C2×C412yes—
137#385—22no—
138#385—22no—
139#385—22no—
140#385—22no—
141#385—22no—
142#385—22no—
143#384C2311yescharacteristic
144#384C2311yescharacteristic
145#384C2311yescharacteristic
146#382—21no—
147#382—21no—
148#382C2×C412yes—
149#380—22no—
150#380—22no—
151#380—22no—
152C4×C4⋊C4C2×C411yescharacteristic
153C4×C4⋊C4C2×C411yescharacteristic
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4D411yescharacteristic
156C4×C4⋊C4D411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C4⋊C4D411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C2×#95—22no—
162C43C2311yescharacteristic
163C43C2311yescharacteristic
164C43C2311yescharacteristic
165#336—21no—
166#336—21no—
167#336—21no—
168#336—21no—
169#336—21no—
170#336—21no—
171#336—21no—
172#336—21no—
173#336—21no—
174#336—21no—
175#331—22no—
176#331—22no—
177#331—22no—
178#331Q814yescomplemented
179C22×Q8C22⋊C411yescharacteristic
180C22×D4—21no—
181C22×D4—21no—
182C22×D4C22×C412yes—
183C23×C4—22no—
184C23×C4—22no—
185C23×C4—22no—
186C22×C8—21no—
187C22×C8—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194#127—21no—
195#127—21no—
196#127C2×Q812yes—
197#127#4112yes—
198#125—81no—
199#125—81no—
200#125—81no—
201#125—81no—
202#125—82no—
203C4×Q8—22no—
204C4×Q8—22no—
205C4×Q8—41no—
206C4×Q8—41no—
207C4×Q8—42no—
208C4×D4—42no—
209C4×D4—42no—
210C4×D4—44no—
211C4×D4—42no—
212C4×D4—42no—
213C4×D4—81no—
214C4×D4—81no—
215C4×D4—81no—
216C4×D4—81no—
217C4×D4—81no—
218C4×D4—81no—
219C4×D4—81no—
220C4×D4—81no—
221C4×D4—82no—
222C4×D4—82no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—22no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4C22⋊C411yescharacteristic
236C2×C4⋊C4C22⋊C411yescharacteristic
237C2×C4⋊C4C22⋊C411yescharacteristic
238C2×C4⋊C4C22⋊C411yescharacteristic
239C2×C4⋊C4C22⋊C411yescharacteristic
240C2×C4⋊C4C22⋊C411yescharacteristic
241C2×C4⋊C4C22⋊C411yescharacteristic
242C2×C4⋊C4C22⋊C411yescharacteristic
243C2×C4⋊C4C22⋊C411yescharacteristic
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C22⋊C4—22no—
248C2×C22⋊C4—22no—
249C2×C22⋊C4—22no—
250C2×C22⋊C4—22no—
251C2×C22⋊C4—22no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—22no—
256C2×C42C22×C411yescharacteristic
257C2×C42C22×C411yescharacteristic
258C2×C42C22×C411yescharacteristic
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261C2×C42C22×C411yescharacteristic
262C2×C42C2×D411yescharacteristic
263C2×C42C2×D411yescharacteristic
264C2×C42C2×D411yescharacteristic
265C2×C42C2×D411yescharacteristic
266C2×C42C2×D411yescharacteristic
267C2×C42C2×D411yescharacteristic
268C2×C42C2×D411yescharacteristic
269C2×C42C2×D411yescharacteristic
270C2×C42C2×D411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2411yescharacteristic
275C4⋊C8—21no—
276C4⋊C8—21no—
277C4⋊C8—21no—
278C4⋊C8—21no—
279C4⋊C8—21no—
280C4⋊C8—21no—
281C4⋊C8—21no—
282C4⋊C8—21no—
283C4⋊C8—21no—
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—42no—
290C4⋊C8—42no—
291C4⋊C8—42no—
292C4⋊C8—42no—
293C4⋊C8—81no—
294C4⋊C8—81no—
295C4⋊C8—81no—
296C4⋊C8—81no—
297C4⋊C8C2×Q812yes—
298C4⋊C8C2×Q812yes—
299#97—42no—
300#97—42no—
301#97—81no—
302#97—81no—
303C4×C8—42no—
304C4×C8—42no—
305#95—22no—
306#95—22no—
307#95—22no—
308#95—22no—
309#95—22no—
310#95—22no—
311#95C22⋊C411yescharacteristic
312#95C22⋊C411yescharacteristic
313#95C22⋊C411yescharacteristic
314#95C22⋊C411yescharacteristic
315#95C22⋊C411yescharacteristic
316#95C22⋊C411yescharacteristic
317C24—22no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8—41no—
321C2×D4—22no—
322C2×D4—22no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—42no—
327C2×D4—42no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C2×D4—41no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—41no—
336C2×D4C4×Q814yes—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4—44no—
343C22×C4—82no—
344C22×C4—82no—
345C22×C4—82no—
346C22×C4—82no—
347C22×C4—82no—
348C22×C4—82no—
349C22×C4C2×C22⋊C411yescharacteristic
350C22×C4C2×C22⋊C411yescharacteristic
351C22×C4C2×C22⋊C411yescharacteristic
352C22×C4C2×C22⋊C411yescharacteristic
353C22×C4C2×C22⋊C411yescharacteristic
354C22×C4C2×C22⋊C411yescharacteristic
355C22×C4C2×C22⋊C411yescharacteristic
356C22×C4C2×C22⋊C411yescharacteristic
357C22×C4C2×C22⋊C411yescharacteristic
358C22×C4C2×C22⋊C411yescharacteristic
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C23×C411yescharacteristic
362C22×C4C22×D411yescharacteristic
363C22×C4C22×D411yescharacteristic
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
386C2×C8—41no—
387C2×C8—41no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—42no—
398C4⋊C4—42no—
399C4⋊C4—42no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—41no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—82no—
413C4⋊C4—82no—
414C4⋊C4—82no—
415C4⋊C4—82no—
416C4⋊C4—82no—
417C4⋊C4—82no—
418C22⋊C4—44no—
419C22⋊C4—44no—
420C22⋊C4—44no—
421C22⋊C4—82no—
422C22⋊C4—82no—
423C22⋊C4—82no—
424C22⋊C4—82no—
425C22⋊C4—82no—
426C22⋊C4—82no—
427C42—21no—
428C42—21no—
429C42—21no—
430C42—21no—
431C42—21no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—21no—
437C42—21no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—22no—
450C42—41no—
451C42—41no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42AGL(1,ℤ/8)12yes—
456C42#13712yes—
457C42C22×Q811yescharacteristic
458C42C2×#4111yescharacteristic
459C422−1+411yescharacteristic
460C422−1+411yescharacteristic
461C23—24no—
462C23—42no—
463C23—42no—
464C23—42no—
465C23—42no—
466C23—42no—
467C23—42no—
468C23C22×C22⋊C411yescharacteristic
469Q8—44no—
470D4—44no—
471D4—44no—
472D4—42no—
473D4—42no—
474D4—42no—
475D4—42no—
476D4—42no—
477D4—42no—
478D4—42no—
479D4—42no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—21no—
487C2×C4—21no—
488C2×C4—21no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—21no—
499C2×C4—21no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—42no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—164no—
525C2×C4—164no—
526C2×C4—164no—
527C2×C4#41312yes—
528C2×C4#41712yes—
529C2×C4#41712yes—
530C2×C4C2×C4×Q811yescharacteristic
531C2×C4#51911yescharacteristic
532C2×C4#54811yescharacteristic
533C2×C4#54811yescharacteristic
534C2×C4D4×Q811yescharacteristic
535C2×C4D4×Q811yescharacteristic
536C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
537C2×C4C2×#13711yescharacteristic
538C2×C4#57511yescharacteristic
539C2×C4#57511yescharacteristic
540C2×C4#57611yescharacteristic
541C2×C4#57611yescharacteristic
542C2×C4#57711yescharacteristic
543C2×C4#57711yescharacteristic
544C8—42no—
545C8—42no—
546C8—81no—
547C8—81no—
548C8—81no—
549C8—81no—
550C8—81no—
551C8—81no—
552C22—44no—
553C22—44no—
554C22—44no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22Q8×C22⋊C411yescharacteristic
560C22C2×#41311yescharacteristic
561C22#288811yescharacteristic
562C22C2×#41711yescharacteristic
563C22#289611yescharacteristic
564C22#289711yescharacteristic
565C22#289711yescharacteristic
566C4—22no—
567C4—22no—
568C4—22no—
569C4—21no—
570C4—21no—
571C4—41no—
572C4—41no—
573C4—41no—
574C4—41no—
575C4—41no—
576C4—41no—
577C4—41no—
578C4—41no—
579C4—41no—
580C4—82no—
581C4—82no—
582C4—82no—
583C4#336611yescharacteristic
584C4#337011yescharacteristic
585C4#337211yescharacteristic
586C4#337611yescharacteristic
587C4#338411yescharacteristic
588C4#338511yescharacteristic
589C4#338811yescharacteristic
590C4#338911yescharacteristic
591C2—84no—
592C2#1775311yescharacteristic
593C2#1776211yescharacteristic
594C2#1776611yescharacteristic
595C2#1777611yescharacteristic
596C2#2251511yescharacteristic
597C2#2252211yescharacteristic
598C2#2252611yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC2×#127C23
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22825C2semidirect
3. #22722C2semidirect
4. #9907C2semidirect
5. #9906C2semidirect
6. #9904C2semidirect
7. #9902C2semidirect
8. #9901C2semidirect
9. #9900C2semidirect
178. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#365732
#228252
#227222
#99072
#99062
#99042
#99022
#99012
#99002
#85882
#85852
#82342
#82312