Finite Group Databasebeta

← #390641

Subgroups — #390641

Order 512.

775 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23737C211yesmaximal, characteristic
2#21072C211yesmaximal, characteristic
3#21037C211yesmaximal, characteristic, complemented
4#11259C211yesmaximal, characteristic, complemented
5#11213C211yesmaximal, characteristic, complemented
6#11185C211yesmaximal, characteristic
7#11154C211yesmaximal, characteristic
8#11072C211yesmaximal, characteristic
9#10954C211yesmaximal, characteristic
10#10944C211yesmaximal, characteristic
11#10904C211yesmaximal, characteristic, complemented
12#10889C211yesmaximal, characteristic, complemented
13#10888C211yesmaximal, characteristic, complemented
14#10870C211yesmaximal, characteristic, complemented
15#10799C211yesmaximal, characteristic, complemented
16C22×#108C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C22×#107C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#417C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#410C2211yescharacteristic
24#2879C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#405C2211yescharacteristic
27#2317C2211yescharacteristic
28#2316C2211yescharacteristic
29#2311C2211yescharacteristic
30#2274C2211yescharacteristic
31#2097C412yescomplemented
32#2095C412yescomplemented
33#2093C412yescomplemented
34#2092C412yescomplemented
35#1929—22no—
36#1929—22no—
37#1921—22no—
38#1918—22no—
39C2×#344C2211yescharacteristic
40C2×#344C2211yescharacteristic
41C2×#343C2211yescharacteristic
42C2×#343C2211yescharacteristic
43#1740C2211yescharacteristic
44#1736C2211yescharacteristic
45#1736C2211yescharacteristic
46#1736C2211yescharacteristic
47#1736C2211yescharacteristic
48#1735C2211yescharacteristic
49#1733C2211yescharacteristic
50#1733C2211yescharacteristic
51C2×#338C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54C2×H(3,ℤ/4)C2211yescharacteristic
55C2×H(3,ℤ/4)C2211yescharacteristic
56C2×H(3,ℤ/4)C2211yescharacteristic
57#1729C2211yescharacteristic
58#1729C2211yescharacteristic
59C22×#41C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C22×M16C2311yescharacteristic
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C23×C8C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C2×#117C2311yescharacteristic
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C2×#108—22no—
75C2×#108—21no—
76C2×#108—21no—
77C2×#108—41no—
78#428—41no—
79#428—41no—
80C2×#107—21no—
81C2×#107—21no—
82C2×#107—21no—
83C2×#107—21no—
84C2×#107C2×C412yes—
85C2×#107C2×C412yes—
86C2×#107C2×C412yes—
87C2×#106C2×C412yes—
88#421—22no—
89#421—22no—
90#421—41no—
91#421—41no—
92C2×#104—22no—
93C2×#104—41no—
94#417—41no—
95#417—41no—
96#416—21no—
97#416—21no—
98#416—21no—
99#416—21no—
100C2×#103C2×C411yescharacteristic
101C2×#103C2×C411yescharacteristic
102C2×#103C2×C412yes—
103C2×#102C2×C411yescharacteristic
104C2×#102C2×C411yescharacteristic
105C2×#102C2×C412yes—
106#410—22no—
107#410—22no—
108C2×#99—41no—
109#408—22no—
110#408—41no—
111#407—41no—
112#405—22no—
113#405—22no—
114#405—41no—
115#405—41no—
116C4×M16—41no—
117C2×#97—21no—
118C2×C4×C8—21no—
119#391C2×C412yes—
120#390C2×C412yes—
121#389—22no—
122#389—41no—
123#389—41no—
124#386—41no—
125#385—41no—
126#384C2×C412yes—
127#382C2×C412yes—
128#380—41no—
129#380—41no—
130C4×C4⋊C4—22no—
131C4×C4⋊C4—41no—
132C4×C22⋊C4—41no—
133#344—22no—
134#344—41no—
135#343—41no—
136#343—41no—
137#342—41no—
138#340—21no—
139#340—22no—
140#340—22no—
141#340—21no—
142#340—41no—
143#340—41no—
144#339—22no—
145#339—41no—
146#338—22no—
147H(3,ℤ/4)—22no—
148H(3,ℤ/4)—22no—
149H(3,ℤ/4)—22no—
150H(3,ℤ/4)—22no—
151H(3,ℤ/4)—41no—
152#336C2×C412yes—
153#336C2×C412yes—
154#336D412yes—
155#336D412yes—
156C2×#41—21no—
157C2×#41—21no—
158C2×#41—21no—
159C2×#41—21no—
160C2×#41—21no—
161C2×#41—21no—
162C2×#41—21no—
163C2×#41—21no—
164C2×#41—21no—
165C2×#41—21no—
166C2×#41—21no—
167C2×#41—21no—
168C22×Q8C22×C411yescharacteristic
169C22×D4—21no—
170C22×D4C22×C411yescharacteristic
171C23×C4—21no—
172C23×C4—21no—
173C23×C4—21no—
174C23×C4C2411yescharacteristic
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×M16—21no—
201C2×M16—21no—
202C2×M16—21no—
203C2×M16#4112yes—
204C2×M16#4112yes—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—21no—
212C22×C8C22×C412yes—
213C22×C8C22×C411yescharacteristic
214C22×C8C22×C411yescharacteristic
215C22×C8C2×D412yes—
216C22×C8C2×D411yescharacteristic
217C22×C8C2×D411yescharacteristic
218C22×C8#4111yescharacteristic
219C22×C8#4111yescharacteristic
220#117—21no—
221#117—21no—
222#117—21no—
223#117—21no—
224#117—21no—
225#117—21no—
226#117—21no—
227#117—21no—
228#117—21no—
229#117—21no—
230#117—21no—
231#117—21no—
232#117—21no—
233#117—21no—
234#117—21no—
235#117—21no—
236#117—21no—
237#117—21no—
238#117—21no—
239#117—21no—
240#117—21no—
241#117—21no—
242#117#4112yes—
243#117#4112yes—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—42no—
260C2×C4⋊C4—42no—
261C2×C4⋊C4C22×C412yes—
262C2×C4⋊C4C22×C412yes—
263C2×C4⋊C4C22×C411yescharacteristic
264C2×C4⋊C4C22×C411yescharacteristic
265C2×C22⋊C4—21no—
266C2×C22⋊C4—21no—
267C2×C22⋊C4—21no—
268C2×C22⋊C4—21no—
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—42no—
272C2×C42—21no—
273C2×C42—21no—
274C2×C42—21no—
275C2×C42—21no—
276C2×C42—42no—
277C2×C42C22×C412yes—
278C2×C42C2×D412yes—
279#108—42no—
280#108—42no—
281#107—22no—
282#107—22no—
283#107—22no—
284#107—22no—
285#107—42no—
286#106—42no—
287#104—44no—
288#104—44no—
289#103—21no—
290#103—21no—
291#103—21no—
292#103—21no—
293#103—42no—
294#102—21no—
295#102—21no—
296#102—21no—
297#102—21no—
298#102—42no—
299#99—44no—
300C22⋊C8—44no—
301#97—42no—
302#97—42no—
303C4×C8—42no—
304C4×C8—42no—
305#95—42no—
306#95—42no—
307#95—42no—
308#95—42no—
309C24—21no—
310C24H(5,𝔽2)11yescharacteristic
311#41—41no—
312#41—41no—
313#41—41no—
314#41—41no—
315#41—41no—
316#41—41no—
317#41—41no—
318#41—41no—
319#41—41no—
320#41—41no—
321#41—41no—
322#41—41no—
323#41—41no—
324#41—41no—
325#41—41no—
326#41—41no—
327C2×Q8—21no—
328C2×Q8—21no—
329C2×Q8—21no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—21no—
333C2×D4—21no—
334C2×D4—21no—
335C2×D4—21no—
336C2×D4—21no—
337C2×D4—21no—
338C2×D4—21no—
339C2×D4—41no—
340C2×D4—41no—
341C2×D4—41no—
342C2×D4—41no—
343C2×D4—41no—
344C2×D4—41no—
345C22×C4—22no—
346C22×C4—21no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—41no—
357C22×C4—41no—
358C22×C4—41no—
359C22×C4—41no—
360C22×C4—41no—
361C22×C4—41no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—161no—
366C22×C4—161no—
367C22×C4C23×C411yescharacteristic
368C22×C4C22×D411yescharacteristic
369C22×C4C2×#4111yescharacteristic
370C22×C4C2×#4111yescharacteristic
371C22×C4C2×#4111yescharacteristic
372C22×C4C2×#4111yescharacteristic
373C22×C4C2×#4111yescharacteristic
374C22×C4H(5,𝔽2)11yescharacteristic
375C22×C4H(5,𝔽2)11yescharacteristic
376C22×C4H(5,𝔽2)11yescharacteristic
377C22×C4H(5,𝔽2)11yescharacteristic
378C22×C42−1+411yescharacteristic
379C22×C42−1+411yescharacteristic
380C22×C42−1+411yescharacteristic
381M16—22no—
382M16—22no—
383M16—22no—
384M16—22no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390M16—41no—
391M16—41no—
392M16—41no—
393M16—41no—
394M16—41no—
395M16—41no—
396M16—41no—
397M16—41no—
398M16—41no—
399M16—41no—
400M16—41no—
401C2×C8—21no—
402C2×C8—21no—
403C2×C8—21no—
404C2×C8—21no—
405C2×C8—21no—
406C2×C8—21no—
407C2×C8—21no—
408C2×C8—21no—
409C2×C8—21no—
410C2×C8—21no—
411C2×C8—21no—
412C2×C8—21no—
413C2×C8—22no—
414C2×C8—22no—
415C2×C8—22no—
416C2×C8—22no—
417C2×C8—22no—
418C2×C8—22no—
419C2×C8—22no—
420C2×C8—22no—
421C2×C8—41no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C2×C8—41no—
426C2×C8—41no—
427C2×C8—41no—
428C2×C8C4×D411yescharacteristic
429C2×C8C4×D411yescharacteristic
430C2×C8C4×D411yescharacteristic
431C2×C8C4×D411yescharacteristic
432C4⋊C4—21no—
433C4⋊C4—21no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—22no—
437C4⋊C4—21no—
438C4⋊C4—22no—
439C4⋊C4—21no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—41no—
443C4⋊C4—41no—
444C4⋊C4—41no—
445C4⋊C4—41no—
446C4⋊C4—41no—
447C4⋊C4—41no—
448C4⋊C4—41no—
449C4⋊C4—41no—
450C4⋊C4—41no—
451C4⋊C4—41no—
452C4⋊C4—41no—
453C4⋊C4—41no—
454C4⋊C4—162no—
455C4⋊C4—162no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—41no—
462C22⋊C4—41no—
463C22⋊C4—41no—
464C22⋊C4—41no—
465C22⋊C4—41no—
466C22⋊C4—41no—
467C22⋊C4—41no—
468C22⋊C4—162no—
469C42—22no—
470C42—22no—
471C42—22no—
472C42—22no—
473C42—22no—
474C42—22no—
475C42—41no—
476C42—41no—
477C42—41no—
478C42—41no—
479C42—41no—
480C42—41no—
481C42—162no—
482C23—41no—
483C23—41no—
484C23—41no—
485C23—41no—
486C23—41no—
487C23—81no—
488C23—81no—
489C23#51811yescharacteristic
490C23#53511yescharacteristic
491C23#54011yescharacteristic
492C23#55111yescharacteristic
493C23#55911yescharacteristic
494C23#56211yescharacteristic
495C23#56311yescharacteristic
496Q8—41no—
497Q8—41no—
498Q8—41no—
499Q8—41no—
500D4—41no—
501D4—41no—
502D4—41no—
503D4—41no—
504D4—81no—
505D4—81no—
506D4—81no—
507D4—81no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—41no—
517C2×C4—42no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—42no—
522C2×C4—42no—
523C2×C4—81no—
524C2×C4—82no—
525C2×C4—82no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—81no—
529C2×C4—81no—
530C2×C4—81no—
531C2×C4—81no—
532C2×C4—81no—
533C2×C4—82no—
534C2×C4—81no—
535C2×C4—82no—
536C2×C4—81no—
537C2×C4—81no—
538C2×C4—81no—
539C2×C4—81no—
540C2×C4—161no—
541C2×C4—161no—
542C2×C4C2×C4×D411yescharacteristic
543C2×C4C4×#4111yescharacteristic
544C2×C4C4×#4111yescharacteristic
545C2×C4#51811yescharacteristic
546C2×C4#51911yescharacteristic
547C2×C4#52011yescharacteristic
548C2×C4#53511yescharacteristic
549C2×C4#53811yescharacteristic
550C2×C4#53911yescharacteristic
551C2×C4#53911yescharacteristic
552C2×C4#53911yescharacteristic
553C2×C4#54011yescharacteristic
554C2×C4#54011yescharacteristic
555C2×C4#54211yescharacteristic
556C2×C4#54211yescharacteristic
557C2×C4#54211yescharacteristic
558C2×C4#54611yescharacteristic
559C2×C4#54711yescharacteristic
560C2×C4#54711yescharacteristic
561C2×C4#54811yescharacteristic
562C2×C4#55111yescharacteristic
563C2×C4#55211yescharacteristic
564C2×C4#55211yescharacteristic
565C2×C4#55311yescharacteristic
566C2×C4#55311yescharacteristic
567C2×C4#55611yescharacteristic
568C2×C4#56211yescharacteristic
569C2×C4#56311yescharacteristic
570C8—21no—
571C8—21no—
572C8—21no—
573C8—21no—
574C8—42no—
575C8—42no—
576C8—42no—
577C8—42no—
578C8—81no—
579C22—21no—
580C22—21no—
581C22—21no—
582C22—21no—
583C22—21no—
584C22—21no—
585C22—41no—
586C22—41no—
587C22—41no—
588C22—41no—
589C22—82no—
590C22—81no—
591C22—81no—
592C22—81no—
593C22—161no—
594C22#232811yescharacteristic
595C22#237411yescharacteristic
596C22#237811yescharacteristic
597C22#271011yescharacteristic
598C22#272311yescharacteristic
599C22#274011yescharacteristic
600C22#277311yescharacteristic
601C4—82no—
602C4—82no—
603C4—82no—
604C4—82no—
605C4—81no—
606C4—81no—
607C4—82no—
608C4—82no—
609C4—81no—
610C4—161no—
611C4#233211yescharacteristic
612C4#233511yescharacteristic
613C4#235911yescharacteristic
614C4#236211yescharacteristic
615C4#271111yescharacteristic
616C4#271811yescharacteristic
617C4#274111yescharacteristic
618C4#277711yescharacteristic
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—21no—
624C2—21no—
625C2—161no—
626C2#1457811yescharacteristic
627C2#2228511yescharacteristic
628C2#2228611yescharacteristic
subgroupG/H
center, socleC22#2328
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21037C2semidirect
4. #11259C2semidirect
5. #11213C2semidirect
11. #10904C2semidirect
12. #10889C2semidirect
13. #10888C2semidirect
14. #10870C2semidirect
15. #10799C2semidirect
31. #2097C4semidirect
32. #2095C4semidirect
33. #2093C4semidirect
34. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237372
#210722
#210372
#112592
#112132
#111852
#111542
#110722
#109542
#109442
#109042
#108892
#108882
#108702
#107992