Finite Group Databasebeta

← #117705

Subgroups — #117705

Order 512.

798 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21195C211yesmaximal, characteristic
2C2×#1564C211yesmaximal, characteristic, complemented
3C2×#1502C211yesmaximal, characteristic
4C2×#1480C211yesmaximal, characteristic
5#10834C211yesmaximal, characteristic, complemented
6#8014C211yesmaximal, characteristic, complemented
7#7893C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C2×#416—21no—
13C2×#416—21no—
14C22×C4×C8C2211yescharacteristic
15#2285C2211yescharacteristic
16C2×C4×C4⋊C4C2211yescharacteristic
17#1958—21no—
18#1958—21no—
19#1957C411yescharacteristic, complemented
20#1957C411yescharacteristic, complemented
21#1956C411yescharacteristic, complemented
22#1956C411yescharacteristic, complemented
23C2×#336—21no—
24C2×#336—21no—
25#1564—22no—
26#1564—22no—
27C2×#335—21no—
28C2×#334—21no—
29#1502—22no—
30#1502—22no—
31C2×#327—22no—
32#1480—22no—
33#1480—22no—
34C2×D4⋊C8—22no—
35#1294—22no—
36#1279—22no—
37C22×#41—21no—
38C23×C8—21no—
39C23×C8D411yescharacteristic
40C23×C8D411yescharacteristic
41#532—21no—
42#532—21no—
43#532—21no—
44#532—21no—
45#532—21no—
46#532—21no—
47#532D412yes—
48#532Q812yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×#127C2×C411yescharacteristic
51C2×#124—21no—
52C2×#121—21no—
53C2×#121—21no—
54C2×C4×D4—21no—
55C2×C4×D4—21no—
56C2×#117—21no—
57C22×C4⋊C4—21no—
58C22×C42—21no—
59C22×C42C2311yescharacteristic
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431D412yes—
67#431D412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8D412yes—
76#416—42no—
77#416—42no—
78#416—42no—
79#416—42no—
80C2×#103—21no—
81C2×#103—21no—
82C2×#103—21no—
83C2×#103—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×C22⋊C8—21no—
89C2×C22⋊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×C8C2×C411yescharacteristic
97C2×C4×C8C2×C411yescharacteristic
98C2×C4×C8D411yescharacteristic
99C2×C4×C8D411yescharacteristic
100#384C2×C411yescharacteristic
101#384C2×C411yescharacteristic
102#382—21no—
103#380—21no—
104#380—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4—21no—
107C4×C4⋊C4—21no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4—21no—
111C2×#95D412yes—
112#336—42no—
113#336—42no—
114#335—24no—
115#334—24no—
116#327—28no—
117D4⋊C8—28no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—41no—
121C2×#41—41no—
122C2×#41—41no—
123C2×#41—41no—
124C2×#41—41no—
125C2×#41—41no—
126C2×#41—41no—
127C2×#41—41no—
128C2×#41—41no—
129C2×#41—41no—
130C22×Q8—21no—
131C22×D4—21no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4—41no—
135C23×C4—22no—
136C23×C4—41no—
137C23×C4C2×D411yescharacteristic
138C23×C4C2×D411yescharacteristic
139C23×C4#4111yescharacteristic
140C22×C8—21no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—21no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—41no—
152C22×C8—41no—
153C22×C8—41no—
154C22×C8—41no—
155C22×C8—41no—
156C22×C8—41no—
157C22×C8—41no—
158C22×C8—41no—
159C22×C8C22⋊C411yescharacteristic
160C22×C8C22⋊C411yescharacteristic
161C22×C8C22⋊C411yescharacteristic
162C22×C8C22⋊C411yescharacteristic
163C22×C8D812yes—
164C22×C8SD1612yes—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8C4⋊C412yes—
169#127—21no—
170#127—21no—
171#127—21no—
172#127C4⋊C412yes—
173#124—22no—
174#124—22no—
175#124—41no—
176#124—41no—
177#121—22no—
178#121—22no—
179#121—22no—
180#121—22no—
181#121—41no—
182#121—41no—
183#121—41no—
184#121—41no—
185C4×D4—22no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—22no—
189C4×D4—41no—
190C4×D4—41no—
191C4×D4—41no—
192C4×D4—41no—
193#117—41no—
194#117—41no—
195#117—41no—
196#117—41no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—24no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4—41no—
212C2×C42—41no—
213C2×C42—41no—
214C2×C42—41no—
215C2×C42—41no—
216C2×C42—41no—
217C2×C42—41no—
218C2×C42—41no—
219C2×C42C22×C411yescharacteristic
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D411yescharacteristic
225C2×C42C2×Q811yescharacteristic
226C2×C42#4111yescharacteristic
227C2×C42#4111yescharacteristic
228C2×C42#4111yescharacteristic
229C2×C42#4111yescharacteristic
230C2×C42#4111yescharacteristic
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—22no—
234C4⋊C8—22no—
235C4⋊C8—41no—
236C4⋊C8—41no—
237C4⋊C8—41no—
238C4⋊C8—41no—
239#103—42no—
240#103—42no—
241#103—42no—
242#103—42no—
243#102—42no—
244#102—42no—
245#102—42no—
246#102—42no—
247C22⋊C8—22no—
248C22⋊C8—22no—
249C22⋊C8—41no—
250C22⋊C8—41no—
251C4×C8—21no—
252C4×C8—21no—
253C4×C8—21no—
254C4×C8—21no—
255C4×C8—21no—
256C4×C8—21no—
257C4×C8—22no—
258C4×C8—22no—
259C4×C8—21no—
260C4×C8—21no—
261C4×C8—41no—
262C4×C8—41no—
263#95—22no—
264#95—22no—
265#95—41no—
266#95—41no—
267C24—41no—
268C24#12111yescharacteristic
269#41—42no—
270#41—42no—
271#41—42no—
272#41—42no—
273#41—42no—
274#41—42no—
275#41—42no—
276#41—42no—
277#41—81no—
278#41—81no—
279#41—81no—
280#41—81no—
281#41—81no—
282#41—81no—
283#41—81no—
284#41—81no—
285C2×Q8—22no—
286C2×Q8—41no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—42no—
304C2×D4—42no—
305C2×D4—81no—
306C2×D4—81no—
307C2×D4—81no—
308C2×D4—81no—
309C2×D4—81no—
310C2×D4—81no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—81no—
328C22×C4—81no—
329C22×C4—81no—
330C22×C4—81no—
331C22×C4C2×C22⋊C411yescharacteristic
332C22×C4C4×D411yescharacteristic
333C22×C4C4×D411yescharacteristic
334C22×C4#12011yescharacteristic
335C22×C4#12111yescharacteristic
336C22×C4#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4#12111yescharacteristic
340C22×C4C22⋊Q811yescharacteristic
341C22×C4C22⋊Q811yescharacteristic
342C22×C4C22⋊Q811yescharacteristic
343C22×C4#12311yescharacteristic
344C22×C4#12311yescharacteristic
345C22×C4#12411yescharacteristic
346C22×C4#12411yescharacteristic
347C22×C4#12611yescharacteristic
348C22×C4#12711yescharacteristic
349C22×C4C2×D811yescharacteristic
350C22×C4C2×SD1611yescharacteristic
351C22×C4#13511yescharacteristic
352C22×C4#13511yescharacteristic
353C22×C4AGL(1,ℤ/8)11yescharacteristic
354C22×C4AGL(1,ℤ/8)11yescharacteristic
355C22×C4AGL(1,ℤ/8)11yescharacteristic
356C22×C4#13711yescharacteristic
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—21no—
366C2×C8—21no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
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—
380C2×C8—81no—
381C2×C8—81no—
382C2×C8—81no—
383C2×C8—81no—
384C2×C8—81no—
385C2×C8—81no—
386C2×C8—81no—
387C2×C8—81no—
388C2×C8#10212yes—
389C2×C8#10212yes—
390C4⋊C4—22no—
391C4⋊C4—22no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—41no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—42no—
401C4⋊C4—42no—
402C4⋊C4—81no—
403C4⋊C4—81no—
404C22⋊C4—42no—
405C22⋊C4—42no—
406C22⋊C4—42no—
407C22⋊C4—42no—
408C22⋊C4—81no—
409C22⋊C4—81no—
410C42—81no—
411C42—81no—
412C42—81no—
413C42—81no—
414C42—81no—
415C42—81no—
416C42C2×C4⋊C411yescharacteristic
417C42#11711yescharacteristic
418C42C4×D411yescharacteristic
419C42C4×D411yescharacteristic
420C42#12011yescharacteristic
421C42#12011yescharacteristic
422C42#12111yescharacteristic
423C42#12111yescharacteristic
424C42C22⋊Q811yescharacteristic
425C42C22⋊Q811yescharacteristic
426C42C22⋊Q811yescharacteristic
427C42#12311yescharacteristic
428C42#12311yescharacteristic
429C42#12311yescharacteristic
430C42#12411yescharacteristic
431C42#12611yescharacteristic
432C23—21no—
433C23—21no—
434C23—21no—
435C23—21no—
436C23—42no—
437C23—81no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23#39011yescharacteristic
444C23#39411yescharacteristic
445C23#39611yescharacteristic
446C23#45911yescharacteristic
447C23#46011yescharacteristic
448C23#46311yescharacteristic
449C23#46411yescharacteristic
450Q8—42no—
451Q8—42no—
452Q8—81no—
453Q8—81no—
454D4—42no—
455D4—42no—
456D4—44no—
457D4—82no—
458D4—82no—
459D4—82no—
460D4—82no—
461D4—82no—
462D4—82no—
463D4—81no—
464D4—81no—
465D4—82no—
466C2×C4—21no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—44no—
481C2×C4—44no—
482C2×C4—44no—
483C2×C4—42no—
484C2×C4—82no—
485C2×C4—82no—
486C2×C4—82no—
487C2×C4—82no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—81no—
497C2×C4—81no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4#38011yescharacteristic
501C2×C4#38511yescharacteristic
502C2×C4#38511yescharacteristic
503C2×C4#38611yescharacteristic
504C2×C4#38811yescharacteristic
505C2×C4#38811yescharacteristic
506C2×C4#39211yescharacteristic
507C2×C4#39311yescharacteristic
508C2×C4#39411yescharacteristic
509C2×C4#39611yescharacteristic
510C2×C4#39711yescharacteristic
511C2×C4#39911yescharacteristic
512C2×C4C2×#10211yescharacteristic
513C2×C4#41611yescharacteristic
514C2×C4#41711yescharacteristic
515C2×C4#41811yescharacteristic
516C2×C4#44312yes—
517C2×C4#44412yes—
518C2×C4#45911yescharacteristic
519C2×C4#45911yescharacteristic
520C2×C4#46011yescharacteristic
521C2×C4#46111yescharacteristic
522C2×C4#46311yescharacteristic
523C2×C4#46311yescharacteristic
524C2×C4#46311yescharacteristic
525C2×C4#46411yescharacteristic
526C2×C4#46511yescharacteristic
527C2×C4#46511yescharacteristic
528C2×C4#46611yescharacteristic
529C2×C4#46611yescharacteristic
530C2×C4#46811yescharacteristic
531C2×C4#46811yescharacteristic
532C2×C4#46911yescharacteristic
533C2×C4#46911yescharacteristic
534C2×C4#47411yescharacteristic
535C2×C4#47611yescharacteristic
536C2×C4#47811yescharacteristic
537C2×C4#47811yescharacteristic
538C8—22no—
539C8—22no—
540C8—41no—
541C8—41no—
542C8—82no—
543C8—82no—
544C8—82no—
545C22—21no—
546C22—21no—
547C22—21no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—82no—
558C22—82no—
559C22—82no—
560C22—161no—
561C22—161no—
562C22—161no—
563C22—161no—
564C22#143911yescharacteristic
565C22#197411yescharacteristic
566C22#197511yescharacteristic
567C22#197611yescharacteristic
568C22#197611yescharacteristic
569C22#198211yescharacteristic
570C22#198311yescharacteristic
571C22#205411yescharacteristic
572C22#205511yescharacteristic
573C22#206511yescharacteristic
574C22#206711yescharacteristic
575C4—21no—
576C4—21no—
577C4—21no—
578C4—21no—
579C4—21no—
580C4—21no—
581C4—84no—
582C4—82no—
583C4—161no—
584C4—161no—
585C4#180211yescharacteristic
586C4#180411yescharacteristic
587C4#186612yes—
588C4#189311yescharacteristic
589C4#189511yescharacteristic
590C4#192912yes—
591C4#205411yescharacteristic
592C4#205511yescharacteristic
593C4#206511yescharacteristic
594C4#206711yescharacteristic
595C2—21no—
596C2—21no—
597C2—21no—
598C2—21no—
599C2—162no—
600C2#975011yescharacteristic
601C2#975111yescharacteristic
602C2#975311yescharacteristic
603C2#975411yescharacteristic
604C2#1107611yescharacteristic
605C2#1112511yescharacteristic
606C2#1112611yescharacteristic
subgroupG/H
centerC22×C4#121
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC23#390
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
2. C2×#1564C2semidirect
5. #10834C2semidirect
6. #8014C2semidirect
7. #7893C2semidirect
19. #1957C4semidirect
20. #1957C4semidirect
21. #1956C4semidirect
22. #1956C4semidirect

Maximal subgroups

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

subgroupindex
#211952
C2×#15642
C2×#15022
C2×#14802
#108342
#80142
#78932