Finite Group Databasebeta

← #224282

Subgroups — #224282

Order 512.

931 conjugacy classes of subgroups.

The subgroup lattice has 612 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36559C211yesmaximal, characteristic
2#22823C211yesmaximal, characteristic, complemented
3#22722C211yesmaximal, characteristic, complemented
4#9906C211yesmaximal, characteristic, complemented
5#9906C211yesmaximal, characteristic, complemented
6#9906C211yesmaximal, characteristic, complemented
7#9903C211yesmaximal, characteristic, complemented
8#9901C211yesmaximal, characteristic, complemented
9#9898C211yesmaximal, characteristic, complemented
10#8588C211yesmaximal, characteristic
11#8588C211yesmaximal, characteristic
12#8585C211yesmaximal, characteristic
13#8234C211yesmaximal, characteristic
14#8234C211yesmaximal, characteristic
15#8231C211yesmaximal, characteristic
16C2×#554C412yes—
17C2×#446—21no—
18#2868C2211yescharacteristic
19#2859C2211yescharacteristic
20#2859C2211yescharacteristic
21#2566C412yes—
22#2566C412yes—
23#2563C412yes—
24#2538—21no—
25#2538—21no—
26#2452—21no—
27#2452—21no—
28#2452—21no—
29#2452—21no—
30D4×C4⋊C4—21no—
31D4×C4⋊C4—21no—
32C4×#127C2211yescharacteristic
33C4×#127C2211yescharacteristic
34C4×#127C2211yescharacteristic
35C42×Q8C2211yescharacteristic
36#1985—21no—
37#1940—21no—
38#1940—21no—
39#1939—21no—
40#1917—21no—
41#1916—21no—
42#1842C2211yescharacteristic
43#1842C2211yescharacteristic
44#1842C2211yescharacteristic
45#1842C2211yescharacteristic
46#1831C2211yescharacteristic
47#1831C2211yescharacteristic
48#1771—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
71#554—48no—
72C2×C4⋊Q8—21no—
73C2×C4⋊Q8—21no—
74C2×C4⋊Q8C2×C412yes—
75C2×#127C2311yescharacteristic
76C2×#125C2×C412yes—
77C2×#125C2×C412yes—
78C2×C22⋊Q8—22no—
79C2×C22⋊Q8—22no—
80C2×C22⋊Q8—22no—
81C2×C4×Q8C2×C411yescharacteristic
82C2×C4×Q8D411yescharacteristic
83C2×C4×Q8D411yescharacteristic
84C2×C4×D4—21no—
85C2×C4×D4—21no—
86C2×C4×D4—21no—
87C2×C4×D4—21no—
88C2×C4×D4—21no—
89C2×C4×D4—21no—
90C2×C4×D4C2×C412yes—
91C2×C4×D4C2×C412yes—
92C2×C4×D4C2×C412yes—
93C22×C4⋊C4—22no—
94#446—44no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8—21no—
97C2×C4⋊C8—21no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—21no—
101C2×C4⋊C8—21no—
102C2×C4⋊C8—21no—
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×C4⋊C8C2311yescharacteristic
110C2×C4⋊C8C2311yescharacteristic
111C2×#97—21no—
112C2×#97—21no—
113C2×#97—21no—
114C2×#97—21no—
115C2×C4×C8—21no—
116C2×C4×C8—21no—
117#396—22no—
118#396—22no—
119#396—22no—
120#396—22no—
121#393—22no—
122#393—22no—
123#390—21no—
124#390—21no—
125#390—21no—
126#390—21no—
127#390—21no—
128#390—21no—
129#390C2×C412yes—
130#390C2×C412yes—
131#390C2×C412yes—
132#389—21no—
133#389—21no—
134#389—21no—
135#389—21no—
136#389—21no—
137#389—21no—
138#389—21no—
139#389—21no—
140#385—22no—
141#385—22no—
142#385—22no—
143#385—22no—
144#384—21no—
145#384—21no—
146#384C2×C412yes—
147#384C2311yescharacteristic
148#384C2311yescharacteristic
149#384C2311yescharacteristic
150#382C2×C412yes—
151#382C2×C412yes—
152#380—22no—
153#380—22no—
154C4×C4⋊C4C2×C411yescharacteristic
155C4×C4⋊C4C2×C411yescharacteristic
156C4×C4⋊C4C2×C411yescharacteristic
157C4×C4⋊C4D411yescharacteristic
158C4×C4⋊C4D411yescharacteristic
159C4×C4⋊C4D411yescharacteristic
160C4×C4⋊C4D411yescharacteristic
161C4×C4⋊C4D411yescharacteristic
162C4×C4⋊C4D411yescharacteristic
163C43C2311yescharacteristic
164C43C2311yescharacteristic
165C43C2311yescharacteristic
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#336—21no—
176#331—22no—
177#331—22no—
178#331—22no—
179#331Q814yescomplemented
180C22×Q8—22no—
181C22×Q8C22⋊C411yescharacteristic
182C22×D4—21no—
183C22×D4—21no—
184C22×D4C22×C412yes—
185C23×C4—22no—
186C23×C4—22no—
187C23×C4—22no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C4⋊Q8—44no—
197C4⋊Q8—81no—
198C4⋊Q8—81no—
199C4⋊Q8—81no—
200C4⋊Q8—81no—
201#127—21no—
202#127—21no—
203#127C2×Q812yes—
204#127#4112yes—
205#125—44no—
206#125—44no—
207C22⋊Q8—48no—
208C22⋊Q8—48no—
209C22⋊Q8—48no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—41no—
214C4×Q8—41no—
215C4×D4—42no—
216C4×D4—42no—
217C4×D4—44no—
218C4×D4—44no—
219C4×D4—44no—
220C4×D4—42no—
221C4×D4—42no—
222C4×D4—81no—
223C4×D4—81no—
224C4×D4—81no—
225C4×D4—81no—
226C4×D4—81no—
227C4×D4—81no—
228C4×D4—81no—
229C4×D4—81no—
230C2×C4⋊C4—22no—
231C2×C4⋊C4—22no—
232C2×C4⋊C4—22no—
233C2×C4⋊C4—22no—
234C2×C4⋊C4—22no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—22no—
242C2×C4⋊C4—22no—
243C2×C4⋊C4—48no—
244C2×C4⋊C4C22⋊C411yescharacteristic
245C2×C4⋊C4C22⋊C411yescharacteristic
246C2×C4⋊C4C22⋊C411yescharacteristic
247C2×C4⋊C4C22⋊C411yescharacteristic
248C2×C4⋊C4C22⋊C411yescharacteristic
249C2×C4⋊C4C22⋊C411yescharacteristic
250C2×C4⋊C4C22⋊C411yescharacteristic
251C2×C4⋊C4C22⋊C411yescharacteristic
252C2×C4⋊C4C22⋊C411yescharacteristic
253C2×C4⋊C4C22×C412yes—
254C2×C4⋊C4C22×C412yes—
255C2×C4⋊C4C22×C412yes—
256C2×C22⋊C4—22no—
257C2×C22⋊C4—22no—
258C2×C22⋊C4—22no—
259C2×C22⋊C4—22no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—22no—
262C2×C22⋊C4—22no—
263C2×C22⋊C4—22no—
264C2×C22⋊C4—22no—
265C2×C42C22×C411yescharacteristic
266C2×C42C22×C411yescharacteristic
267C2×C42C22×C411yescharacteristic
268C2×C42C22×C411yescharacteristic
269C2×C42C22×C411yescharacteristic
270C2×C42C22×C411yescharacteristic
271C2×C42C2×D411yescharacteristic
272C2×C42C2×D411yescharacteristic
273C2×C42C2×D411yescharacteristic
274C2×C42C2×D411yescharacteristic
275C2×C42C2×D411yescharacteristic
276C2×C42C2×D411yescharacteristic
277C2×C42C2×D411yescharacteristic
278C2×C42C2×D411yescharacteristic
279C2×C42C2×D411yescharacteristic
280C2×C42C2×D411yescharacteristic
281C2×C42C2×D411yescharacteristic
282C2×C42C2×D411yescharacteristic
283C2×C42C2411yescharacteristic
284C4⋊C8—21no—
285C4⋊C8—21no—
286C4⋊C8—21no—
287C4⋊C8—21no—
288C4⋊C8—21no—
289C4⋊C8—21no—
290C4⋊C8—21no—
291C4⋊C8—21no—
292C4⋊C8—21no—
293C4⋊C8—21no—
294C4⋊C8—21no—
295C4⋊C8—21no—
296C4⋊C8—21no—
297C4⋊C8—21no—
298C4⋊C8—42no—
299C4⋊C8—42no—
300C4⋊C8—42no—
301C4⋊C8—42no—
302C4⋊C8—81no—
303C4⋊C8—81no—
304C4⋊C8—81no—
305C4⋊C8—81no—
306C4⋊C8C2×Q812yes—
307C4⋊C8C2×Q812yes—
308#97—42no—
309#97—42no—
310#97—42no—
311#97—81no—
312C4×C8—42no—
313C4×C8—81no—
314#95—22no—
315#95—22no—
316#95—22no—
317#95—22no—
318#95C22⋊C411yescharacteristic
319#95C22⋊C411yescharacteristic
320#95C22⋊C411yescharacteristic
321#95C22⋊C411yescharacteristic
322#95C22⋊C411yescharacteristic
323#95C22⋊C411yescharacteristic
324C24—22no—
325C2×Q8—22no—
326C2×Q8—22no—
327C2×Q8—41no—
328C2×Q8—44no—
329C2×Q8—82no—
330C2×Q8—82no—
331C2×D4—22no—
332C2×D4—22no—
333C2×D4—22no—
334C2×D4—22no—
335C2×D4—22no—
336C2×D4—42no—
337C2×D4—42no—
338C2×D4—41no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C2×D4—41no—
346C2×D4C4×Q814yes—
347C22×C4—22no—
348C22×C4—22no—
349C22×C4—22no—
350C22×C4—44no—
351C22×C4—44no—
352C22×C4—44no—
353C22×C4—44no—
354C22×C4—44no—
355C22×C4—82no—
356C22×C4—82no—
357C22×C4—82no—
358C22×C4—82no—
359C22×C4C2×C22⋊C411yescharacteristic
360C22×C4C2×C22⋊C411yescharacteristic
361C22×C4C2×C22⋊C411yescharacteristic
362C22×C4C2×C22⋊C411yescharacteristic
363C22×C4C2×C22⋊C411yescharacteristic
364C22×C4C2×C22⋊C411yescharacteristic
365C22×C4C2×C22⋊C411yescharacteristic
366C22×C4C2×C22⋊C411yescharacteristic
367C22×C4C2×C22⋊C411yescharacteristic
368C22×C4C2×C22⋊C411yescharacteristic
369C22×C4C2×C22⋊C411yescharacteristic
370C22×C4C2×C22⋊C411yescharacteristic
371C22×C4C23×C411yescharacteristic
372C22×C4C22×D411yescharacteristic
373C22×C4C22×D411yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
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—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C4⋊C4—22no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—22no—
403C4⋊C4—41no—
404C4⋊C4—41no—
405C4⋊C4—41no—
406C4⋊C4—42no—
407C4⋊C4—42no—
408C4⋊C4—42no—
409C4⋊C4—44no—
410C4⋊C4—44no—
411C4⋊C4—44no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—44no—
425C4⋊C4—44no—
426C4⋊C4—82no—
427C4⋊C4—82no—
428C4⋊C4#10014yes—
429C22⋊C4—44no—
430C22⋊C4—44no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—44no—
434C22⋊C4—82no—
435C22⋊C4—82no—
436C22⋊C4—82no—
437C22⋊C4—82no—
438C42—21no—
439C42—21no—
440C42—21no—
441C42—21no—
442C42—21no—
443C42—21no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—21no—
451C42—21no—
452C42—21no—
453C42—21no—
454C42—21no—
455C42—21no—
456C42—21no—
457C42—21no—
458C42—21no—
459C42—21no—
460C42—22no—
461C42—22no—
462C42—22no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42AGL(1,ℤ/8)12yes—
467C42#13712yes—
468C42C22×Q811yescharacteristic
469C42C2×#4111yescharacteristic
470C422−1+411yescharacteristic
471C422−1+411yescharacteristic
472C23—24no—
473C23—42no—
474C23—42no—
475C23—42no—
476C23—42no—
477C23—42no—
478C23—42no—
479C23C22×C22⋊C411yescharacteristic
480Q8—44no—
481Q8—164no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—42no—
491D4—42no—
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—21no—
516C2×C4—21no—
517C2×C4—21no—
518C2×C4—21no—
519C2×C4—21no—
520C2×C4—21no—
521C2×C4—21no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—21no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—42no—
533C2×C4—42no—
534C2×C4—88no—
535C2×C4—164no—
536C2×C4—164no—
537C2×C4C2×#10012yes—
538C2×C4C2×#10012yes—
539C2×C4C2×#10012yes—
540C2×C4#41712yes—
541C2×C4#41712yes—
542C2×C4C2×C4×Q811yescharacteristic
543C2×C4#51911yescharacteristic
544C2×C4#54811yescharacteristic
545C2×C4#54811yescharacteristic
546C2×C4D4×Q811yescharacteristic
547C2×C4D4×Q811yescharacteristic
548C2×C4C2×AGL(1,ℤ/8)11yescharacteristic
549C2×C4C2×#13711yescharacteristic
550C2×C4#57511yescharacteristic
551C2×C4#57511yescharacteristic
552C2×C4#57611yescharacteristic
553C2×C4#57611yescharacteristic
554C2×C4#57711yescharacteristic
555C2×C4#57711yescharacteristic
556C8—42no—
557C8—42no—
558C8—81no—
559C8—81no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8—81no—
564C22—44no—
565C22—44no—
566C22—44no—
567C22—82no—
568C22—82no—
569C22—82no—
570C22—82no—
571C22Q8×C22⋊C411yescharacteristic
572C22C22×#10011yescharacteristic
573C22#288911yescharacteristic
574C22C2×#41711yescharacteristic
575C22#289611yescharacteristic
576C22#289711yescharacteristic
577C22#289711yescharacteristic
578C4—22no—
579C4—22no—
580C4—22no—
581C4—22no—
582C4—22no—
583C4—21no—
584C4—21no—
585C4—41no—
586C4—41no—
587C4—41no—
588C4—41no—
589C4—41no—
590C4—41no—
591C4—41no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4#336611yescharacteristic
596C4#337011yescharacteristic
597C4#337211yescharacteristic
598C4#337611yescharacteristic
599C4#338411yescharacteristic
600C4#338511yescharacteristic
601C4#338811yescharacteristic
602C4#338911yescharacteristic
603C2—84no—
604C2#1775011yescharacteristic
605C2#1776511yescharacteristic
606C2#1776911yescharacteristic
607C2#1777011yescharacteristic
608C2Q8×#10011yescharacteristic
609C2#2252211yescharacteristic
610C2#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. #22823C2semidirect
3. #22722C2semidirect
4. #9906C2semidirect
5. #9906C2semidirect
6. #9906C2semidirect
7. #9903C2semidirect
8. #9901C2semidirect
9. #9898C2semidirect
179. #331Q8semidirect

Maximal subgroups

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

subgroupindex
#365592
#228232
#227222
#99062
#99032
#99012
#98982
#85882
#85852
#82342
#82312