Finite Group Databasebeta

← #390965

Subgroups — #390965

Order 512.

872 conjugacy classes of subgroups.

The subgroup lattice has 620 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#23731C211yesmaximal, characteristic
2#11300C211yesmaximal, characteristic, complemented
3#11299C211yesmaximal, characteristic, complemented
4#11214C211yesmaximal, characteristic, complemented
5#11212C211yesmaximal, characteristic, complemented
6#11058C211yesmaximal, characteristic
7#10958C211yesmaximal, characteristic
8#10953C211yesmaximal, characteristic
9#10904C211yesmaximal, characteristic, complemented
10#10903C211yesmaximal, characteristic, complemented
11#10903C211yesmaximal, characteristic, complemented
12#10893C211yesmaximal, characteristic, complemented
13#10785C211yesmaximal, characteristic
14#10785C211yesmaximal, characteristic
15#10749C211yesmaximal, characteristic
16C2×#518C2211yescharacteristic
17C2×#428C2211yescharacteristic
18C2×#428C2211yescharacteristic
19#2905C2211yescharacteristic
20C2×#418C2211yescharacteristic
21C2×#418C2211yescharacteristic
22C2×#418C2211yescharacteristic
23C2×#418C2211yescharacteristic
24C22×#100C2211yescharacteristic
25#2873C2211yescharacteristic
26C2×#390C2211yescharacteristic
27#2285C2211yescharacteristic
28#2274C2211yescharacteristic
29#2094C412yescomplemented
30#2093C412yescomplemented
31#2092C412yescomplemented
32#2092C412yescomplemented
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#344C2211yescharacteristic
37C2×#343C2211yescharacteristic
38#1740C2211yescharacteristic
39#1740C2211yescharacteristic
40#1736C2211yescharacteristic
41#1736C2211yescharacteristic
42#1733C2211yescharacteristic
43#1733C2211yescharacteristic
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
55C23×D4C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62#518—22no—
63#518—22no—
64#518—22no—
65#518—41no—
66C2×C4×D4—41no—
67C2×C4×D4—41no—
68C2×#117C2311yescharacteristic
69C2×#117C2311yescharacteristic
70C2×#117C2311yescharacteristic
71C2×#117C2311yescharacteristic
72C2×#117C2311yescharacteristic
73C2×#117C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—41no—
76C22×C22⋊C4—41no—
77C22×C42C2311yescharacteristic
78#428—41no—
79#428—41no—
80#428—41no—
81#428—41no—
82C2×#107C2×C412yescomplemented
83C2×#107C2×C412yescomplemented
84C2×#106C2×C412yescomplemented
85C2×#106C2×C412yescomplemented
86#424—41no—
87#424—41no—
88#423—41no—
89#418—22no—
90#418—22no—
91#418—41no—
92#418—41no—
93#418—41no—
94#418—41no—
95#418—41no—
96#418Q814yes—
97C2×#102—41no—
98C2×#102—41no—
99C2×#102C2×C412yes—
100C2×#102C2×C412yes—
101C2×#102C2×C412yes—
102C2×#102C2×C412yes—
103C2×#100—22no—
104C2×#100—22no—
105C2×#100—41no—
106#405—41no—
107#405—41no—
108C4×M16—41no—
109#390—41no—
110#390C2×C412yes—
111#390C2×C412yes—
112#384C2×C411yescharacteristic, complemented
113#384C2×C411yescharacteristic, complemented
114#382C2×C411yescharacteristic, complemented
115#382C2×C411yescharacteristic, complemented
116#380—41no—
117C4×C4⋊C4—41no—
118C4×C4⋊C4—41no—
119C4×C4⋊C4—41no—
120C4×C22⋊C4—41no—
121#344—22no—
122#344—22no—
123#344—22no—
124#344—41no—
125#343—41no—
126#342—41no—
127#342—41no—
128#340—41no—
129#340—41no—
130#339—22no—
131#339—22no—
132#339—41no—
133#339—41no—
134#339—41no—
135#339—41no—
136#339—41no—
137#339—41no—
138#338—22no—
139#338—41no—
140H(3,ℤ/4)—22no—
141H(3,ℤ/4)—22no—
142#336C2×C412yescomplemented
143#336C2×C412yescomplemented
144C25—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4—21no—
152C22×D4—21no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156C22×D4C22×C412yes—
157C22×D4C2×Q812yes—
158C22×D4#4112yes—
159C23×C4—21no—
160C23×C4—21no—
161C23×C4—42no—
162C23×C4C2411yescharacteristic
163C2×M16—21no—
164C2×M16—21no—
165C2×M16—21no—
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×M16C2×Q812yes—
198C2×M16#4112yes—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8C22×C412yes—
204C22×C8C22×C412yes—
205C4×D4—44no—
206C4×D4—44no—
207C4×D4—44no—
208C4×D4—44no—
209#117—21no—
210#117—21no—
211#117—21no—
212#117—21no—
213#117—21no—
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#117C2×Q812yes—
230#117#4112yes—
231#117#4112yes—
232#117#4112yes—
233C2×C4⋊C4—21no—
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—41no—
243C2×C4⋊C4—41no—
244C2×C4⋊C4C22×C412yes—
245C2×C4⋊C4C22×C411yescharacteristic
246C2×C4⋊C4C22×C411yescharacteristic
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4—21no—
255C2×C22⋊C4—44no—
256C2×C22⋊C4—44no—
257C2×C22⋊C4—42no—
258C2×C22⋊C4—42no—
259C2×C22⋊C4—42no—
260C2×C22⋊C4—84no—
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—21no—
271C2×C42—21no—
272C2×C42—42no—
273C2×C42C22×C411yescharacteristic
274C2×C42C22×C411yescharacteristic
275#107—42no—
276#107—42no—
277#106—42no—
278#106—42no—
279C4⋊C8—44no—
280#102—44no—
281#102—42no—
282#102—42no—
283#102—42no—
284#102—42no—
285#102—44no—
286#100—44no—
287#97—42no—
288C4×C8—42no—
289#95—41no—
290#95—41no—
291#95—42no—
292C24—22no—
293C24—22no—
294C24—22no—
295C24—41no—
296C24—41no—
297C24—41no—
298C24—41no—
299C24—82no—
300C24—82no—
301C24H(5,𝔽2)11yescharacteristic
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—22no—
305C2×D4—22no—
306C2×D4—22no—
307C2×D4—22no—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—22no—
313C2×D4—22no—
314C2×D4—22no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—41no—
324C2×D4—41no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4C4×Q814yes—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—21no—
335C22×C4—21no—
336C22×C4—22no—
337C22×C4—42no—
338C22×C4—41no—
339C22×C4—41no—
340C22×C4—41no—
341C22×C4—41no—
342C22×C4—41no—
343C22×C4—41no—
344C22×C4—41no—
345C22×C4—48no—
346C22×C4—41no—
347C22×C4—81no—
348C22×C4—82no—
349C22×C4—162no—
350C22×C4—162no—
351C22×C4C23×C411yescharacteristic
352C22×C4C22×Q811yescharacteristic
353C22×C4C2×#4111yescharacteristic
354C22×C4C2×#4111yescharacteristic
355C22×C4C2×#4111yescharacteristic
356C22×C4C2×#4111yescharacteristic
357C22×C4C2×#4111yescharacteristic
358C22×C4H(5,𝔽2)11yescharacteristic
359C22×C4H(5,𝔽2)11yescharacteristic
360C22×C4H(5,𝔽2)11yescharacteristic
361C22×C4H(5,𝔽2)11yescharacteristic
362C22×C4H(5,𝔽2)11yescharacteristic
363C22×C4H(5,𝔽2)11yescharacteristic
364C22×C42−1+411yescharacteristic
365M16—22no—
366M16—22no—
367M16—22no—
368M16—22no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
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—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—22no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—22no—
396C2×C8—22no—
397C2×C8—41no—
398C2×C8—41no—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C4⋊C4—21no—
408C4⋊C4—21no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C4⋊C4—21no—
412C4⋊C4—21no—
413C4⋊C4—22no—
414C4⋊C4—22no—
415C4⋊C4—21no—
416C4⋊C4—21no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—41no—
422C4⋊C4—41no—
423C4⋊C4—41no—
424C4⋊C4—41no—
425C4⋊C4—41no—
426C4⋊C4—41no—
427C4⋊C4—82no—
428C4⋊C4—82no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—22no—
432C22⋊C4—22no—
433C22⋊C4—48no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—41no—
440C22⋊C4—41no—
441C22⋊C4—41no—
442C22⋊C4—84no—
443C22⋊C4—84no—
444C42—21no—
445C42—21no—
446C42—21no—
447C42—21no—
448C42—22no—
449C42—22no—
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—24no—
464C23—41no—
465C23—41no—
466C23—42no—
467C23—42no—
468C23—42no—
469C23—42no—
470C23—42no—
471C23—42no—
472C23—82no—
473C23—82no—
474C23—82no—
475C23—82no—
476C23—84no—
477C23—84no—
478C23—161no—
479C23—161no—
480C23—162no—
481C23—162no—
482C23#51811yescharacteristic
483C23#53811yescharacteristic
484C23#53911yescharacteristic
485C23#55111yescharacteristic
486C23#55411yescharacteristic
487C23#56011yescharacteristic
488C23#56011yescharacteristic
489D4—24no—
490D4—24no—
491D4—42no—
492D4—42no—
493D4—42no—
494D4—42no—
495D4—42no—
496D4—42no—
497C2×C4—22no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—82no—
513C2×C4—82no—
514C2×C4—81no—
515C2×C4—81no—
516C2×C4—82no—
517C2×C4—81no—
518C2×C4—81no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4—81no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4—161no—
529C2×C4—168no—
530C2×C4C2×C4×Q811yescharacteristic
531C2×C4C4×#4111yescharacteristic
532C2×C4C4×#4111yescharacteristic
533C2×C4#51811yescharacteristic
534C2×C4#51811yescharacteristic
535C2×C4#52011yescharacteristic
536C2×C4#53811yescharacteristic
537C2×C4#53811yescharacteristic
538C2×C4#53911yescharacteristic
539C2×C4#53911yescharacteristic
540C2×C4#54011yescharacteristic
541C2×C4#54011yescharacteristic
542C2×C4#54211yescharacteristic
543C2×C4#54211yescharacteristic
544C2×C4#54311yescharacteristic
545C2×C4#54411yescharacteristic
546C2×C4#55111yescharacteristic
547C2×C4#55111yescharacteristic
548C2×C4#55111yescharacteristic
549C2×C4#55111yescharacteristic
550C2×C4#55311yescharacteristic
551C2×C4#55311yescharacteristic
552C2×C4#55411yescharacteristic
553C2×C4#55411yescharacteristic
554C2×C4#55411yescharacteristic
555C2×C4#55911yescharacteristic
556C2×C4#56011yescharacteristic
557C2×C4#56011yescharacteristic
558C8—42no—
559C8—42no—
560C8—42no—
561C8—42no—
562C8—81no—
563C8—81no—
564C22—21no—
565C22—21no—
566C22—21no—
567C22—21no—
568C22—21no—
569C22—21no—
570C22—41no—
571C22—41no—
572C22—41no—
573C22—41no—
574C22—44no—
575C22—84no—
576C22—84no—
577C22—84no—
578C22—82no—
579C22—82no—
580C22—162no—
581C22—162no—
582C22—162no—
583C22#233511yescharacteristic
584C22#236811yescharacteristic
585C22#238011yescharacteristic
586C22#273511yescharacteristic
587C22#274611yescharacteristic
588C22#278011yescharacteristic
589C22#280711yescharacteristic
590C4—42no—
591C4—81no—
592C4—82no—
593C4—82no—
594C4—82no—
595C4—82no—
596C4—81no—
597C4—81no—
598C4—82no—
599C4—81no—
600C4—81no—
601C4#232911yescharacteristic
602C4#233511yescharacteristic
603C4#236811yescharacteristic
604C4#237411yescharacteristic
605C4#274211yescharacteristic
606C4#274611yescharacteristic
607C4#275211yescharacteristic
608C4#278011yescharacteristic
609C2—21no—
610C2—21no—
611C2—21no—
612C2—21no—
613C2—21no—
614C2—21no—
615C2—84no—
616C2#1459211yescharacteristic
617C2#2336111yescharacteristic
618C2#2336211yescharacteristic
subgroupG/H
center, socleC22#2380
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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. #11214C2semidirect
5. #11212C2semidirect
9. #10904C2semidirect
10. #10903C2semidirect
11. #10903C2semidirect
12. #10893C2semidirect
29. #2094C4semidirect
30. #2093C4semidirect
31. #2092C4semidirect
32. #2092C4semidirect
82. C2×#107C2×C4semidirect
83. C2×#107C2×C4semidirect
84. C2×#106C2×C4semidirect
85. C2×#106C2×C4semidirect
112. #384C2×C4semidirect
113. #384C2×C4semidirect
114. #382C2×C4semidirect
115. #382C2×C4semidirect
142. #336C2×C4semidirect
143. #336C2×C4semidirect

Maximal subgroups

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

subgroupindex
#237312
#113002
#112992
#112142
#112122
#110582
#109582
#109532
#109042
#109032
#108932
#107852
#107492