Finite Group Databasebeta

← #361047

Subgroups — #361047

Order 512.

896 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
1C2×#1827C211yesmaximal, characteristic
2C2×#1772C211yesmaximal, characteristic, complemented
3C4×#408C211yesmaximal, characteristic
4#11260C211yesmaximal, characteristic, complemented
5#11259C211yesmaximal, characteristic, complemented
6#11099C211yesmaximal, characteristic
7#11098C211yesmaximal, characteristic
8#10961C211yesmaximal, characteristic
9#10961C211yesmaximal, characteristic
10#10825C211yesmaximal, characteristic, complemented
11#10818C211yesmaximal, characteristic, complemented
12#10805C211yesmaximal, characteristic, complemented
13#10800C211yesmaximal, characteristic, complemented
14#10799C211yesmaximal, characteristic, complemented
15#10758C211yesmaximal, characteristic
16C2×C4×#41C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#413C2211yescharacteristic
20C2×#413C2211yescharacteristic
21C2×#410C2211yescharacteristic
22C2×#410C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C2×#405C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×C4×M16C2211yescharacteristic
28C22×C4×C8C2211yescharacteristic
29#1827—24no—
30#1772—24no—
31#1750C414yescomplemented
32C4×C22⋊C8C414yescomplemented
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#341C2211yescharacteristic
38#1735C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#338C2211yescharacteristic
44C2×#338C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46C2×H(3,ℤ/4)C2211yescharacteristic
47C2×H(3,ℤ/4)C2211yescharacteristic
48C2×H(3,ℤ/4)C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51#1729C2211yescharacteristic
52#1729C2211yescharacteristic
53#1729C2211yescharacteristic
54#1729C2211yescharacteristic
55C22×#41C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C4×#41—22no—
65C4×#41—22no—
66C4×#41—41no—
67C4×#41—41no—
68C2×C4×Q8—21no—
69C2×C4×D4—21no—
70C2×C4×D4—22no—
71C2×#117C2×C412yes—
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2×C412yes—
78C22×C42C2311yescharacteristic
79C2×#108—22no—
80C2×#108—22no—
81C2×#108—22no—
82#427—22no—
83#427—22no—
84C2×#107—21no—
85C2×#106—21no—
86#413—22no—
87#413—22no—
88#413—41no—
89#413—41no—
90C2×#101—21no—
91C2×#101—21no—
92C2×#100—21no—
93C2×#100—21no—
94#410—22no—
95#410—22no—
96#410—41no—
97#410—41no—
98C2×#99—22no—
99C2×#99—22no—
100#408—22no—
101#408—22no—
102#408—21no—
103#408—21no—
104#408—21no—
105#408—21no—
106C2×C22⋊C8C2×C414yes—
107C2×C22⋊C8C2×C414yes—
108#405—22no—
109#405—22no—
110#405—41no—
111#405—41no—
112C4×M16—41no—
113C4×M16—41no—
114C2×#97—21no—
115C2×#97—21no—
116C2×#97C2×C412yes—
117C2×C4×C8—22no—
118C2×C4×C8—22no—
119C2×C4×C8—21no—
120C2×C4×C8—21no—
121C2×C4×C8C2×C412yes—
122C2×C4×C8C2×C412yes—
123C2×C4×C8C2×C412yes—
124#344—22no—
125#344—22no—
126#344—41no—
127#344—41no—
128#341—21no—
129#341—21no—
130#340—22no—
131#339—22no—
132#339—22no—
133#339—22no—
134#339D414yes—
135#338—41no—
136#338—41no—
137H(3,ℤ/4)—22no—
138H(3,ℤ/4)—22no—
139H(3,ℤ/4)—41no—
140H(3,ℤ/4)—41no—
141H(3,ℤ/4)—41no—
142H(3,ℤ/4)—41no—
143#336—22no—
144#336—22no—
145#336C2×C414yes—
146#336C2×C414yes—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C22×Q8#4111yescharacteristic
160C22×D4#4112yes—
161C22×D4#4111yescharacteristic
162C23×C4—21no—
163C23×C4—22no—
164C23×C4C22×C412yes—
165C23×C4C2411yescharacteristic
166C2×M16—21no—
167C2×M16—21no—
168C2×M16—21no—
169C2×M16—21no—
170C2×M16—21no—
171C2×M16—21no—
172C2×M16—21no—
173C2×M16—21no—
174C2×M16—21no—
175C2×M16—21no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—21no—
179C2×M16—21no—
180C2×M16—21no—
181C2×M16—21no—
182C2×M16—21no—
183C2×M16—21no—
184C2×M16—21no—
185C2×M16—21no—
186C2×M16—21no—
187C2×M16—21no—
188C2×M16—21no—
189C2×M16—21no—
190C2×M16—21no—
191C2×M16—21no—
192C2×M16—21no—
193C2×M16—21no—
194C2×M16—21no—
195C2×M16—21no—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16C2×D412yes—
201C2×M16#4112yes—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8—21no—
213C22×C8—21no—
214C22×C8C22×C412yes—
215C22×C8C22×C412yes—
216C22×C8C22×C412yes—
217C22×C8C22×C412yes—
218C22×C8#4112yes—
219C22×C8#4112yes—
220C22×C8#4111yescharacteristic
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224C4×Q8—42no—
225C4×Q8—42no—
226C4×D4—42no—
227C4×D4—42no—
228C4×D4—44no—
229C4×D4—44no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—24no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117—21no—
243#117—21no—
244#117—21no—
245#117—21no—
246#117—21no—
247#117—21no—
248#117—21no—
249#117—42no—
250#117C2×D412yes—
251#117#4112yes—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—22no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—22no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C22⋊C4—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—24no—
270C2×C42—22no—
271C2×C42—21no—
272C2×C42—21no—
273C2×C42—42no—
274C2×C42C22×C412yes—
275C2×C42C22×C411yescharacteristic
276C2×C42C22×C411yescharacteristic
277C2×C42C2×D412yes—
278C2×C42#4112yes—
279#108—28no—
280#107—24no—
281#106—24no—
282#101—42no—
283#101—42no—
284#100—42no—
285#100—42no—
286#99—44no—
287#99—44no—
288C22⋊C8—24no—
289C22⋊C8—24no—
290C22⋊C8—24no—
291C22⋊C8—24no—
292#97—42no—
293#97—42no—
294#97—42no—
295C4×C8—24no—
296C4×C8—24no—
297C4×C8—42no—
298C4×C8—42no—
299C4×C8—42no—
300C24#11712yes—
301C24C2×#4111yescharacteristic
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310#41—41no—
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—21no—
321C2×Q8—21no—
322C2×Q8—21no—
323C2×Q8—21no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—22no—
329C2×D4—22no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—22no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—44no—
348C22×C4—44no—
349C22×C4—48no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4#11712yes—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×D411yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4C2×#4111yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C42−1+411yescharacteristic
369M16—22no—
370M16—22no—
371M16—22no—
372M16—22no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381M16—41no—
382M16—41no—
383M16—41no—
384M16—41no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—21no—
398C2×C8—21no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—22no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—22no—
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—22no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—41no—
418C2×C8—41no—
419C2×C8—41no—
420C2×C8—41no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8#11712yes—
428C2×C8#11712yes—
429C2×C8#11712yes—
430C2×C8#11712yes—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—44no—
434C4⋊C4—41no—
435C4⋊C4—41no—
436C4⋊C4—41no—
437C4⋊C4—41no—
438C4⋊C4—41no—
439C4⋊C4—41no—
440C4⋊C4—41no—
441C4⋊C4—41no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—44no—
445C22⋊C4—41no—
446C22⋊C4—41no—
447C22⋊C4—41no—
448C22⋊C4—41no—
449C22⋊C4—41no—
450C22⋊C4—41no—
451C22⋊C4—41no—
452C22⋊C4—41no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42—22no—
457C42—22no—
458C42—22no—
459C42—22no—
460C42—22no—
461C42—44no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C42—41no—
467C42—41no—
468C42C4×D412yes—
469C42C4×D412yes—
470C23—22no—
471C23—22no—
472C23—22no—
473C23—42no—
474C23—44no—
475C23C2×#11711yescharacteristic
476C23#52911yescharacteristic
477C23#52911yescharacteristic
478C23#54611yescharacteristic
479C23#55111yescharacteristic
480C23#55111yescharacteristic
481C23#55211yescharacteristic
482Q8—41no—
483Q8—41no—
484Q8—41no—
485Q8—41no—
486D4—42no—
487D4—42no—
488D4—42no—
489D4—42no—
490D4—41no—
491D4—41no—
492D4—41no—
493D4—41no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—44no—
501C2×C4—44no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—44no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—88no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4C2×#11711yescharacteristic
519C2×C4C2×#11711yescharacteristic
520C2×C4C2×C4×D411yescharacteristic
521C2×C4C4×#4111yescharacteristic
522C2×C4#51811yescharacteristic
523C2×C4#52011yescharacteristic
524C2×C4#52511yescharacteristic
525C2×C4#52511yescharacteristic
526C2×C4#52911yescharacteristic
527C2×C4#52911yescharacteristic
528C2×C4#52911yescharacteristic
529C2×C4#52911yescharacteristic
530C2×C4#54011yescharacteristic
531C2×C4#54011yescharacteristic
532C2×C4#54211yescharacteristic
533C2×C4#54211yescharacteristic
534C2×C4#54611yescharacteristic
535C2×C4#54611yescharacteristic
536C2×C4#54811yescharacteristic
537C2×C4#55111yescharacteristic
538C2×C4#55111yescharacteristic
539C2×C4#55211yescharacteristic
540C2×C4#55211yescharacteristic
541C2×C4#55311yescharacteristic
542C2×C4#55311yescharacteristic
543C2×C4#55311yescharacteristic
544C2×C4#55911yescharacteristic
545C2×C4#55911yescharacteristic
546C8—22no—
547C8—22no—
548C8—22no—
549C8—22no—
550C8—42no—
551C8—42no—
552C8—42no—
553C8—42no—
554C8—81no—
555C8—81no—
556C22—22no—
557C22—22no—
558C22—22no—
559C22—41no—
560C22—41no—
561C22—41no—
562C22—41no—
563C22—44no—
564C22—42no—
565C22—82no—
566C22—82no—
567C22#228311yescharacteristic
568C22#234211yescharacteristic
569C22#234311yescharacteristic
570C22#245711yescharacteristic
571C22#245811yescharacteristic
572C22#251511yescharacteristic
573C22#251611yescharacteristic
574C4—24no—
575C4—42no—
576C4—42no—
577C4—82no—
578C4—82no—
579C4—82no—
580C4—161no—
581C4—161no—
582C4#229011yescharacteristic
583C4#231911yescharacteristic
584C4#235211yescharacteristic
585C4#235711yescharacteristic
586C4#245711yescharacteristic
587C4#245811yescharacteristic
588C4#251511yescharacteristic
589C4#251611yescharacteristic
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—82no—
597C2#1388711yescharacteristic
598C2#2078712yes—
subgroupG/H
centerC2×C4C2×#117
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22#2283
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ #1729 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#1772C2semidirect
4. #11260C2semidirect
5. #11259C2semidirect
10. #10825C2semidirect
11. #10818C2semidirect
12. #10805C2semidirect
13. #10800C2semidirect
14. #10799C2semidirect
31. #1750C4semidirect
32. C4×C22⋊C8C4semidirect

Maximal subgroups

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

subgroupindex
C2×#18272
C2×#17722
C4×#4082
#112602
#112592
#110992
#110982
#109612
#108252
#108182
#108052
#108002
#107992
#107582