Finite Group Databasebeta

← #391613

Subgroups — #391613

Order 512.

698 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
1#23733C211yesmaximal, characteristic
2#11300C211yesmaximal, characteristic, complemented
3#11299C211yesmaximal, characteristic, complemented
4#11214C211yesmaximal, characteristic, complemented
5#11214C211yesmaximal, characteristic, complemented
6#11213C211yesmaximal, characteristic, complemented
7#11211C211yesmaximal, characteristic, complemented
8#11185C211yesmaximal, characteristic
9#11185C211yesmaximal, characteristic
10#11160C211yesmaximal, characteristic
11#11057C211yesmaximal, characteristic
12#11056C211yesmaximal, characteristic
13#10954C211yesmaximal, characteristic
14#10903C211yesmaximal, characteristic, complemented
15#10892C211yesmaximal, characteristic, complemented
16C2×#428C2211yescharacteristic
17C2×#428C2211yescharacteristic
18#2905C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C2×#417C2211yescharacteristic
22C2×#417C2211yescharacteristic
23C2×#410C2211yescharacteristic
24#2879C2211yescharacteristic
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
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C2×#117C2311yescharacteristic
67C22×C4⋊C4C2311yescharacteristic
68C22×C4⋊C4C2311yescharacteristic
69C22×C42C2311yescharacteristic
70#428—41no—
71#428—41no—
72#428—41no—
73#428—41no—
74C2×#107C2×C412yes—
75C2×#107C2×C412yes—
76C2×#106C2×C412yes—
77C2×#106C2×C412yes—
78#424—41no—
79#424—41no—
80#423—41no—
81#421—22no—
82#421—41no—
83#421—41no—
84#421—41no—
85C2×#104—41no—
86C2×#104—41no—
87#417—41no—
88#417—41no—
89#417—41no—
90#417—41no—
91C2×#103C2×C412yes—
92C2×#103C2×C412yes—
93C2×#102C2×C412yes—
94C2×#102C2×C412yes—
95#410—22no—
96#410—41no—
97C2×#99—41no—
98#408—41no—
99#408—41no—
100#407—41no—
101#391C2×C411yescharacteristic
102#391C2×C411yescharacteristic
103#390C2×C411yescharacteristic
104#390C2×C411yescharacteristic
105#389—41no—
106#389—41no—
107#386—41no—
108#382—41no—
109#382C2×C412yes—
110#382C2×C412yes—
111#380—41no—
112C4×C4⋊C4—41no—
113C4×C4⋊C4—41no—
114C4×C22⋊C4—41no—
115C4×C22⋊C4—41no—
116C2×#95—41no—
117#344—22no—
118#344—41no—
119#344—41no—
120#344—41no—
121#343—41no—
122#343—41no—
123#342—41no—
124#340—41no—
125#340—41no—
126#340—41no—
127#340—41no—
128#339—22no—
129#339—22no—
130#339—41no—
131#339—41no—
132#339—41no—
133#339—41no—
134#338—41no—
135#338—41no—
136H(3,ℤ/4)—22no—
137H(3,ℤ/4)—41no—
138#336C2×C412yes—
139#336C2×C412yes—
140C2×#41—21no—
141C2×#41—21no—
142C2×#41—21no—
143C2×#41—21no—
144C2×#41—21no—
145C2×#41—21no—
146C2×#41—21no—
147C2×#41—21no—
148C2×#41—21no—
149C2×#41—21no—
150C2×#41—21no—
151C2×#41—21no—
152C22×Q8C22×C411yescharacteristic
153C22×D4—21no—
154C22×D4C22×C411yescharacteristic
155C23×C4—21no—
156C23×C4—21no—
157C23×C4—21no—
158C23×C4—21no—
159C23×C4C2411yescharacteristic
160C2×M16—21no—
161C2×M16—21no—
162C2×M16—21no—
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—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8C22×C412yes—
201C22×C8C22×C412yes—
202#117—21no—
203#117—21no—
204#117—21no—
205#117—21no—
206#117—21no—
207#117—21no—
208#117—21no—
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#4112yes—
221#117#4112yes—
222C2×C4⋊C4—21no—
223C2×C4⋊C4—21no—
224C2×C4⋊C4—21no—
225C2×C4⋊C4—21no—
226C2×C4⋊C4—21no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4—21no—
229C2×C4⋊C4—21no—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—42no—
237C2×C4⋊C4C22×C412yes—
238C2×C4⋊C4C22×C412yes—
239C2×C4⋊C4C22×C411yescharacteristic
240C2×C4⋊C4C22×C411yescharacteristic
241C2×C22⋊C4—21no—
242C2×C22⋊C4—21no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—41no—
247C2×C22⋊C4—41no—
248C2×C42—21no—
249C2×C42—21no—
250C2×C42—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—42no—
259C2×C42C22×C411yescharacteristic
260C2×C42C22×C411yescharacteristic
261#107—42no—
262#107—42no—
263#106—42no—
264#106—42no—
265C4⋊C8—44no—
266#104—44no—
267#104—44no—
268#103—42no—
269#103—42no—
270#102—42no—
271#102—42no—
272#99—44no—
273C22⋊C8—44no—
274#95—42no—
275#95—42no—
276#95—41no—
277#95—41no—
278#95—42no—
279#95—42no—
280C24—21no—
281C24H(5,𝔽2)11yescharacteristic
282#41—41no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41—41no—
288#41—41no—
289#41—41no—
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298C2×Q8—21no—
299C2×Q8—21no—
300C2×Q8—21no—
301C2×Q8—21no—
302C2×Q8—21no—
303C2×Q8—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—21no—
309C2×D4—21no—
310C2×D4—41no—
311C2×D4—41no—
312C2×D4—41no—
313C2×D4—41no—
314C2×D4—41no—
315C2×D4—41no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4—41no—
334C22×C4—41no—
335C22×C4—41no—
336C22×C4—41no—
337C22×C4—41no—
338C22×C4—41no—
339C22×C4—81no—
340C22×C4—81no—
341C22×C4—82no—
342C22×C4—82no—
343C22×C4C23×C411yescharacteristic
344C22×C4C2×#4111yescharacteristic
345C22×C4C2×#4111yescharacteristic
346C22×C4H(5,𝔽2)11yescharacteristic
347C22×C4H(5,𝔽2)11yescharacteristic
348C22×C4H(5,𝔽2)11yescharacteristic
349C22×C4H(5,𝔽2)11yescharacteristic
350C22×C4H(5,𝔽2)11yescharacteristic
351C22×C4H(5,𝔽2)11yescharacteristic
352C22×C4H(5,𝔽2)11yescharacteristic
353C22×C42−1+411yescharacteristic
354C22×C42−1+411yescharacteristic
355C22×C42−1+411yescharacteristic
356C22×C42−1+411yescharacteristic
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368M16—41no—
369M16—41no—
370M16—41no—
371M16—41no—
372M16—41no—
373M16—41no—
374M16—41no—
375M16—41no—
376M16—41no—
377M16—41no—
378M16—41no—
379M16—41no—
380M16—41no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—41no—
388C2×C8—41no—
389C2×C8—41no—
390C2×C8—41no—
391C2×C8—41no—
392C2×C8—41no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C2×C8—41no—
398C2×C8—41no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—21no—
403C4⋊C4—22no—
404C4⋊C4—22no—
405C4⋊C4—21no—
406C4⋊C4—21no—
407C4⋊C4—22no—
408C4⋊C4—22no—
409C4⋊C4—41no—
410C4⋊C4—41no—
411C4⋊C4—41no—
412C4⋊C4—41no—
413C4⋊C4—41no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—162no—
420C22⋊C4—22no—
421C22⋊C4—22no—
422C22⋊C4—41no—
423C22⋊C4—41no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C22⋊C4—41no—
428C22⋊C4—41no—
429C22⋊C4—41no—
430C22⋊C4—82no—
431C22⋊C4—82no—
432C42—21no—
433C42—21no—
434C42—21no—
435C42—21no—
436C42—22no—
437C42—22no—
438C42—41no—
439C42—41no—
440C42—41no—
441C42—41no—
442C42—41no—
443C42—41no—
444C42—41no—
445C42—41no—
446C42—41no—
447C42—41no—
448C42—162no—
449C23—41no—
450C23—41no—
451C23—41no—
452C23—41no—
453C23—41no—
454C23—81no—
455C23—81no—
456C23#51811yescharacteristic
457C23#53811yescharacteristic
458C23#53911yescharacteristic
459C23#56011yescharacteristic
460C23#56011yescharacteristic
461C23#56211yescharacteristic
462C23#56311yescharacteristic
463Q8—41no—
464Q8—41no—
465Q8—41no—
466Q8—41no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—41no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—41no—
482C2×C4—41no—
483C2×C4—41no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—41no—
490C2×C4—81no—
491C2×C4—81no—
492C2×C4—82no—
493C2×C4—82no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4—82no—
497C2×C4—82no—
498C2×C4—81no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4—81no—
502C2×C4—81no—
503C2×C4—81no—
504C2×C4—82no—
505C2×C4—82no—
506C2×C4—81no—
507C2×C4—81no—
508C2×C4—81no—
509C2×C4—81no—
510C2×C4—81no—
511C2×C4—81no—
512C2×C4—161no—
513C2×C4—161no—
514C2×C4C4×#4111yescharacteristic
515C2×C4#51811yescharacteristic
516C2×C4#51811yescharacteristic
517C2×C4#51911yescharacteristic
518C2×C4#52011yescharacteristic
519C2×C4#52011yescharacteristic
520C2×C4#53811yescharacteristic
521C2×C4#53911yescharacteristic
522C2×C4#53911yescharacteristic
523C2×C4#53911yescharacteristic
524C2×C4#54011yescharacteristic
525C2×C4#54011yescharacteristic
526C2×C4#54211yescharacteristic
527C2×C4#54211yescharacteristic
528C2×C4#54211yescharacteristic
529C2×C4#54211yescharacteristic
530C2×C4#56011yescharacteristic
531C2×C4#56011yescharacteristic
532C2×C4#56011yescharacteristic
533C2×C4#56211yescharacteristic
534C2×C4#56211yescharacteristic
535C2×C4#56211yescharacteristic
536C2×C4#56211yescharacteristic
537C2×C4#56211yescharacteristic
538C2×C4#56311yescharacteristic
539C2×C4#56311yescharacteristic
540C2×C4#56311yescharacteristic
541C2×C4#56311yescharacteristic
542C8—42no—
543C8—42no—
544C8—81no—
545C8—81no—
546C8—81no—
547C8—81no—
548C22—21no—
549C22—21no—
550C22—21no—
551C22—21no—
552C22—21no—
553C22—21no—
554C22—41no—
555C22—41no—
556C22—41no—
557C22—41no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—161no—
563C22#233511yescharacteristic
564C22#237811yescharacteristic
565C22#238011yescharacteristic
566C22#282711yescharacteristic
567C22#282711yescharacteristic
568C22#282911yescharacteristic
569C22#283811yescharacteristic
570C4—81no—
571C4—81no—
572C4—82no—
573C4—82no—
574C4—81no—
575C4—81no—
576C4—81no—
577C4—82no—
578C4—82no—
579C4—161no—
580C4—161no—
581C4#233211yescharacteristic
582C4#233511yescharacteristic
583C4#237611yescharacteristic
584C4#237811yescharacteristic
585C4#282911yescharacteristic
586C4#283711yescharacteristic
587C4#283711yescharacteristic
588C4#283811yescharacteristic
589C2—21no—
590C2—21no—
591C2—21no—
592C2—21no—
593C2—21no—
594C2—21no—
595C2—161no—
596C2#1464611yescharacteristic
597C2#2336211yescharacteristic
598C2#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. #11214C2semidirect
5. #11214C2semidirect
6. #11213C2semidirect
7. #11211C2semidirect
14. #10903C2semidirect
15. #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
#112142
#112132
#112112
#111852
#111602
#110572
#110562
#109542
#109032
#108922