Finite Group Databasebeta

← #390964

Subgroups — #390964

Order 512.

807 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23733C211yesmaximal, characteristic
2#11300C211yesmaximal, characteristic, complemented
3#11299C211yesmaximal, characteristic, complemented
4#11213C211yesmaximal, characteristic, complemented
5#11211C211yesmaximal, characteristic, complemented
6#11057C211yesmaximal, characteristic
7#10959C211yesmaximal, characteristic
8#10954C211yesmaximal, characteristic
9#10904C211yesmaximal, characteristic, complemented
10#10904C211yesmaximal, characteristic, complemented
11#10903C211yesmaximal, characteristic, complemented
12#10892C211yesmaximal, characteristic, complemented
13#10784C211yesmaximal, characteristic
14#10784C211yesmaximal, characteristic
15#10751C211yesmaximal, characteristic
16C2×#568C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C2×#428C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#417C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#410C2211yescharacteristic
24#2873C2211yescharacteristic
25#2316C2211yescharacteristic
26#2297C2211yescharacteristic
27C2×#382C2211yescharacteristic
28#2274C2211yescharacteristic
29#2097C412yescomplemented
30#2097C412yescomplemented
31#2092C412yescomplemented
32#2092C412yescomplemented
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39#1740C2211yescharacteristic
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1736C2211yescharacteristic
43#1736C2211yescharacteristic
44#1733C2211yescharacteristic
45#1733C2211yescharacteristic
46C2×#339C2211yescharacteristic
47C2×#339C2211yescharacteristic
48C2×#339C2211yescharacteristic
49C2×#339C2211yescharacteristic
50C2×#338C2211yescharacteristic
51C2×#338C2211yescharacteristic
52C2×H(3,ℤ/4)C2211yescharacteristic
53C2×H(3,ℤ/4)C2211yescharacteristic
54#1728C2211yescharacteristic
55C22×#41C2311yescharacteristic
56#568—22no—
57#568—22no—
58#568—22no—
59#568—41no—
60C2×#131—41no—
61C2×#131—41no—
62C22×M16—41no—
63C22×M16C2311yescharacteristic
64C22×M16C2311yescharacteristic
65C22×M16C2311yescharacteristic
66C22×M16C2311yescharacteristic
67C22×M16C2311yescharacteristic
68C22×M16C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42C2311yescharacteristic
77#428—41no—
78#428—41no—
79#428—41no—
80#428—41no—
81C2×#107C2×C412yes—
82C2×#107C2×C412yes—
83C2×#106C2×C412yes—
84C2×#106C2×C412yes—
85#421—22no—
86#421—22no—
87#421—41no—
88#421Q814yes—
89C2×#104—41no—
90C2×#104—41no—
91#417—41no—
92#417—41no—
93#417—41no—
94#417—41no—
95C2×#103C2×C412yes—
96C2×#103C2×C412yes—
97C2×#102C2×C412yes—
98C2×#102C2×C412yes—
99#410—22no—
100#410—22no—
101C2×#99—41no—
102#405—41no—
103#405—41no—
104C4×M16—41no—
105#391C2×C411yescharacteristic
106#391C2×C411yescharacteristic
107#390C2×C411yescharacteristic
108#390C2×C411yescharacteristic
109#389—41no—
110#389—41no—
111#386—41no—
112#382—41no—
113#382C2×C412yes—
114#382C2×C412yes—
115#380—41no—
116C4×C4⋊C4—41no—
117C4×C4⋊C4—41no—
118C4×C22⋊C4—41no—
119C4×C22⋊C4—41no—
120C2×#95—41no—
121#344—22no—
122#344—22no—
123#344—22no—
124#344—41no—
125#343—41no—
126#343—41no—
127#342—41no—
128#340—41no—
129#340—41no—
130#340—41no—
131#340—41no—
132#339—22no—
133#339—22no—
134#339—41no—
135#339—41no—
136#339—41no—
137#339—41no—
138#338—22no—
139#338—22no—
140H(3,ℤ/4)—22no—
141H(3,ℤ/4)—41no—
142#336C2×C412yes—
143#336C2×C412yes—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C2×#41—21no—
153C2×#41—21no—
154C2×#41C2×Q812yes—
155C2×#41#4112yes—
156C22×Q8C22×C411yescharacteristic
157C22×D4—21no—
158C22×D4C22×C411yescharacteristic
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—21no—
162C23×C4—21no—
163C23×C4C2411yescharacteristic
164#131—44no—
165#131—44no—
166#131—44no—
167#131—44no—
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×M16—44no—
201C2×M16—44no—
202C2×M16—42no—
203C2×M16C2×Q812yes—
204C2×M16#4112yes—
205C2×M16#4112yes—
206C2×M16#4112yes—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8—21no—
211C22×C8—42no—
212C22×C8C22×C412yes—
213C22×C8C22×C412yes—
214#117—21no—
215#117—21no—
216#117—21no—
217#117—21no—
218#117—21no—
219#117—21no—
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#117C2×Q812yes—
233#117#4112yes—
234C2×C4⋊C4—21no—
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—21no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—21no—
247C2×C4⋊C4—21no—
248C2×C4⋊C4—42no—
249C2×C4⋊C4C22×C412yes—
250C2×C4⋊C4C22×C412yes—
251C2×C4⋊C4C22×C411yescharacteristic
252C2×C4⋊C4C22×C411yescharacteristic
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—41no—
259C2×C22⋊C4—41no—
260C2×C42—21no—
261C2×C42—21no—
262C2×C42—21no—
263C2×C42—21no—
264C2×C42—21no—
265C2×C42—21no—
266C2×C42—21no—
267C2×C42—21no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—42no—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273#107—42no—
274#107—42no—
275#106—42no—
276#106—42no—
277#104—44no—
278#104—44no—
279#103—42no—
280#103—42no—
281#102—42no—
282#102—42no—
283#99—44no—
284#97—42no—
285C4×C8—42no—
286#95—42no—
287#95—42no—
288#95—41no—
289#95—41no—
290#95—42no—
291#95—42no—
292C24—21no—
293C24H(5,𝔽2)11yescharacteristic
294#41—22no—
295#41—22no—
296#41—22no—
297#41—22no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—41no—
303#41—41no—
304#41—41no—
305#41—41no—
306#41—41no—
307#41—41no—
308#41—41no—
309#41—41no—
310C2×Q8—21no—
311C2×Q8—21no—
312C2×Q8—21no—
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8C4×Q812yes—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—21no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4C4×Q812yes—
328C22×C4—21no—
329C22×C4—22no—
330C22×C4—22no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—41no—
347C22×C4—41no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—81no—
352C22×C4—81no—
353C22×C4—82no—
354C22×C4—82no—
355C22×C4C23×C411yescharacteristic
356C22×C4C22×Q811yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4C2×#4111yescharacteristic
359C22×C4C2×#4111yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
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—22no—
374M16—22no—
375M16—22no—
376M16—22no—
377M16—48no—
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—
393M16—41no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—22no—
404C2×C8—48no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—22no—
422C4⋊C4—22no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—41no—
428C4⋊C4—41no—
429C4⋊C4—41no—
430C4⋊C4—41no—
431C4⋊C4—41no—
432C4⋊C4—41no—
433C4⋊C4—162no—
434C22⋊C4—22no—
435C22⋊C4—22no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—41no—
443C22⋊C4—41no—
444C22⋊C4—82no—
445C22⋊C4—82no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—21no—
450C42—22no—
451C42—22no—
452C42—41no—
453C42—41no—
454C42—41no—
455C42—41no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—162no—
463C23—22no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—81no—
469C23—81no—
470C23#51811yescharacteristic
471C23#54011yescharacteristic
472C23#54211yescharacteristic
473C23#55111yescharacteristic
474C23#55411yescharacteristic
475C23#56011yescharacteristic
476C23#56011yescharacteristic
477Q8—22no—
478Q8—22no—
479Q8—41no—
480Q8—41no—
481D4—22no—
482D4—22no—
483D4—42no—
484D4—42no—
485D4—42no—
486D4—42no—
487D4—41no—
488D4—41no—
489C2×C4—22no—
490C2×C4—41no—
491C2×C4—41no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—42no—
507C2×C4—42no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—82no—
511C2×C4—82no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4—82no—
515C2×C4—82no—
516C2×C4—81no—
517C2×C4—81no—
518C2×C4—82no—
519C2×C4—82no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—81no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—161no—
527C2×C4—161no—
528C2×C4C2×C4×Q811yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4C4×#4111yescharacteristic
531C2×C4#51811yescharacteristic
532C2×C4#51811yescharacteristic
533C2×C4#52011yescharacteristic
534C2×C4#53811yescharacteristic
535C2×C4#53811yescharacteristic
536C2×C4#53811yescharacteristic
537C2×C4#53911yescharacteristic
538C2×C4#53911yescharacteristic
539C2×C4#53911yescharacteristic
540C2×C4#54011yescharacteristic
541C2×C4#54211yescharacteristic
542C2×C4#54311yescharacteristic
543C2×C4#54411yescharacteristic
544C2×C4#55111yescharacteristic
545C2×C4#55111yescharacteristic
546C2×C4#55111yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4#55311yescharacteristic
549C2×C4#55311yescharacteristic
550C2×C4#55411yescharacteristic
551C2×C4#55411yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55911yescharacteristic
554C2×C4#56011yescharacteristic
555C2×C4#56011yescharacteristic
556C8—42no—
557C8—42no—
558C8—42no—
559C8—42no—
560C8—42no—
561C8—42no—
562C22—21no—
563C22—21no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—41no—
569C22—41no—
570C22—41no—
571C22—41no—
572C22—42no—
573C22—82no—
574C22—81no—
575C22—81no—
576C22—161no—
577C22#233511yescharacteristic
578C22#236811yescharacteristic
579C22#238011yescharacteristic
580C22#274211yescharacteristic
581C22#274611yescharacteristic
582C22#275211yescharacteristic
583C22#278011yescharacteristic
584C4—42no—
585C4—81no—
586C4—81no—
587C4—82no—
588C4—82no—
589C4—81no—
590C4—81no—
591C4—82no—
592C4—82no—
593C4—161no—
594C4—161no—
595C4#232911yescharacteristic
596C4#233511yescharacteristic
597C4#236811yescharacteristic
598C4#237411yescharacteristic
599C4#273511yescharacteristic
600C4#274611yescharacteristic
601C4#278011yescharacteristic
602C4#280711yescharacteristic
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—21no—
608C2—21no—
609C2—82no—
610C2#1459211yescharacteristic
611C2#2336211yescharacteristic
612C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
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
2. #11300C2semidirect
3. #11299C2semidirect
4. #11213C2semidirect
5. #11211C2semidirect
9. #10904C2semidirect
10. #10904C2semidirect
11. #10903C2semidirect
12. #10892C2semidirect
29. #2097C4semidirect
30. #2097C4semidirect
31. #2092C4semidirect
32. #2092C4semidirect

Maximal subgroups

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

subgroupindex
#237332
#113002
#112992
#112132
#112112
#110572
#109592
#109542
#109042
#109032
#108922
#107842
#107512