Finite Group Databasebeta

← #382227

Subgroups — #382227

Order 512.

865 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#23451C211yesmaximal, characteristic
2#21301C211yesmaximal, characteristic, complemented
3#21047C211yesmaximal, characteristic
4#11386C211yesmaximal, characteristic, complemented
5#11317C211yesmaximal, characteristic, complemented
6#11317C211yesmaximal, characteristic, complemented
7#11269C211yesmaximal, characteristic, complemented
8#11191C211yesmaximal, characteristic
9#11191C211yesmaximal, characteristic
10#11053C211yesmaximal, characteristic
11#10930C211yesmaximal, characteristic
12#10929C211yesmaximal, characteristic
13#10883C211yesmaximal, characteristic, complemented
14#10882C211yesmaximal, characteristic, complemented
15#10696C211yesmaximal, characteristic, complemented
16C2×#533C2211yescharacteristic
17C2×#424C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C22×#104C2211yescharacteristic
20C22×#104C2211yescharacteristic
21C2×#416C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#408C2211yescharacteristic
24C2×#408C2211yescharacteristic
25C22×#97C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C22×C4×C8C2211yescharacteristic
28#2297C2211yescharacteristic
29#2026C412yescomplemented
30#2025C412yescomplemented
31#2024C412yescomplemented
32#2023C412yescomplemented
33#1985—24no—
34#1924—24no—
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39#1735C2211yescharacteristic
40#1735C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#339C2211yescharacteristic
43C2×#339C2211yescharacteristic
44C2×#339C2211yescharacteristic
45C2×#339C2211yescharacteristic
46C2×#339C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1729C2211yescharacteristic
50#1729C2211yescharacteristic
51#1729C2211yescharacteristic
52#1728C2211yescharacteristic
53C2×#336C2211yescharacteristic
54C2×#336C2211yescharacteristic
55C2×#336C2211yescharacteristic
56C4×#95C2211yescharacteristic
57C22×#41C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C23×C8C2311yescharacteristic
64C23×C8C2311yescharacteristic
65C23×C8C2311yescharacteristic
66#533—41no—
67#533—41no—
68#533—41no—
69#533—41no—
70C2×C4⋊Q8C2×C412yes—
71C2×#125C2×C412yes—
72C2×C22⋊Q8—22no—
73C2×C4×Q8—22no—
74C2×#117C2311yescharacteristic
75C2×#117C2311yescharacteristic
76C2×#117C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#424—22no—
81#424—22no—
82#423—41no—
83#423—41no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×#104—22no—
87C2×#104—22no—
88C2×#104—41no—
89C2×#104—41no—
90C2×#104—41no—
91C2×#104—41no—
92#416—41no—
93#416—41no—
94#416—41no—
95#416—41no—
96C2×#103C2×C412yes—
97C2×#103C2×C412yes—
98C2×#102C2×C412yes—
99C2×#102C2×C412yes—
100#408—22no—
101#408—22no—
102#408—41no—
103#408—41no—
104C2×C22⋊C8—22no—
105C2×C22⋊C8—22no—
106C2×#97—22no—
107C2×#97—41no—
108C2×#97—41no—
109C2×#97—41no—
110C2×#97—41no—
111C2×#97D414yes—
112C2×C4×C8—22no—
113C2×C4×C8—41no—
114C2×C4×C8—41no—
115#391C2×C411yescharacteristic
116#391C2×C411yescharacteristic
117#390C2×C411yescharacteristic
118#390C2×C411yescharacteristic
119#386—22no—
120C4×C22⋊C4—22no—
121C2×#95—22no—
122#341—41no—
123#341—41no—
124#341—41no—
125#341—41no—
126#340C2×C412yes—
127#340C2×C412yes—
128#340C2×C412yes—
129#340C2×C412yes—
130#339—22no—
131#339—22no—
132#339—41no—
133#339—41no—
134#339—41no—
135#339—41no—
136#336—22no—
137#336—22no—
138#336—22no—
139#336—22no—
140#336—22no—
141#336—22no—
142#336—22no—
143#336—22no—
144#336C2×C412yes—
145#336C2×C412yes—
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×#41—21no—
155C2×#41—21no—
156C2×#41—21no—
157C2×#41—21no—
158C22×Q8—22no—
159C22×Q8C22×C411yescharacteristic
160C22×D4—21no—
161C22×D4C22×C411yescharacteristic
162C23×C4—21no—
163C23×C4—21no—
164C23×C4—21no—
165C23×C4#4112yes—
166C23×C4C2411yescharacteristic
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—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8—21no—
195C22×C8—21no—
196C22×C8—21no—
197C22×C8—21no—
198C22×C8—21no—
199C22×C8—21no—
200C22×C8—21no—
201C22×C8—21no—
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×C8—21no—
215C22×C8—21no—
216C22×C8—21no—
217C22×C8C22×C412yes—
218C22×C8C22×C412yes—
219C22×C8C2×D412yes—
220C22×C8C2×D412yes—
221C22×C8#4112yes—
222C22×C8#4112yes—
223C4⋊Q8—42no—
224C4⋊Q8—42no—
225#125—42no—
226#125—42no—
227C22⋊Q8—44no—
228C22⋊Q8—44no—
229C4×Q8—44no—
230C4×Q8—44no—
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—21no—
243C2×C4⋊C4—24no—
244C2×C4⋊C4—22no—
245C2×C4⋊C4C22×C412yes—
246C2×C4⋊C4C22×C412yes—
247C2×C4⋊C4C22×C411yescharacteristic
248C2×C4⋊C4C22×C411yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—22no—
253C2×C22⋊C4—22no—
254C2×C42—21no—
255C2×C42—21no—
256C2×C42—21no—
257C2×C42—21no—
258C2×C42—21no—
259C2×C42—21no—
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—21no—
271C2×C42C22×C411yescharacteristic
272C2×C42C22×C411yescharacteristic
273C2×C42C2×D412yes—
274C2×C42#4112yes—
275C2×C42#4112yes—
276C4⋊C8—44no—
277C4⋊C8—44no—
278#104—44no—
279#104—44no—
280#103—42no—
281#103—42no—
282#102—42no—
283#102—42no—
284C22⋊C8—44no—
285C22⋊C8—44no—
286#97—44no—
287#97—44no—
288C4×C8—44no—
289#95—22no—
290#95—22no—
291#95—44no—
292C24—21no—
293C24C2×#4111yescharacteristic
294#41—41no—
295#41—41no—
296#41—41no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
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×Q8—21no—
316C2×Q8—44no—
317C2×Q8—82no—
318C2×Q8—82no—
319C2×D4—21no—
320C2×D4—21no—
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—21no—
324C2×D4—21no—
325C2×D4—41no—
326C2×D4—41no—
327C2×D4—41no—
328C2×D4—41no—
329C2×D4—41no—
330C2×D4—41no—
331C22×C4—24no—
332C22×C4—22no—
333C22×C4—22no—
334C22×C4—22no—
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—22no—
341C22×C4—42no—
342C22×C4—42no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4—42no—
347C22×C4—42no—
348C22×C4—41no—
349C22×C4—41no—
350C22×C4—41no—
351C22×C4—41no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4C4×D412yes—
357C22×C4C4×D412yes—
358C22×C4C23×C411yescharacteristic
359C22×C4C22×D411yescharacteristic
360C22×C4C2×#4111yescharacteristic
361C22×C4C2×#4111yescharacteristic
362C22×C4C2×#4111yescharacteristic
363C22×C4C2×#4111yescharacteristic
364C22×C4H(5,𝔽2)11yescharacteristic
365C22×C4H(5,𝔽2)11yescharacteristic
366C22×C4H(5,𝔽2)11yescharacteristic
367C22×C4H(5,𝔽2)11yescharacteristic
368C22×C4H(5,𝔽2)11yescharacteristic
369C22×C42−1+411yescharacteristic
370C22×C42−1+411yescharacteristic
371C22×C42−1+411yescharacteristic
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—
388C2×C8—22no—
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—22no—
398C2×C8—22no—
399C2×C8—22no—
400C2×C8—22no—
401C2×C8—22no—
402C2×C8—22no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C2×C8—41no—
411C2×C8—41no—
412C2×C8—41no—
413C2×C8—41no—
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
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#12414yes—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—21no—
431C4⋊C4—21no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4—48no—
437C4⋊C4—44no—
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—22no—
453C42—22no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—41no—
459C42—41no—
460C42—41no—
461C42—41no—
462C42—41no—
463C42—41no—
464C42—41no—
465C42—41no—
466C23—22no—
467C23—22no—
468C23—41no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23C2×C4×D411yescharacteristic
474C23#52911yescharacteristic
475C23#52911yescharacteristic
476C23#53811yescharacteristic
477C23#53811yescharacteristic
478C23#54111yescharacteristic
479C23#54211yescharacteristic
480Q8—41no—
481Q8—41no—
482Q8—41no—
483Q8—41no—
484Q8—88no—
485D4—41no—
486D4—41no—
487D4—41no—
488D4—41no—
489D4—81no—
490D4—81no—
491D4—81no—
492D4—81no—
493C2×C4—24no—
494C2×C4—22no—
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—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—41no—
518C2×C4—41no—
519C2×C4—41no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—82no—
524C2×C4—81no—
525C2×C4—81no—
526C2×C4—81no—
527C2×C4—81no—
528C2×C4C4×#4111yescharacteristic
529C2×C4C4×#4111yescharacteristic
530C2×C4#51811yescharacteristic
531C2×C4#51811yescharacteristic
532C2×C4#51911yescharacteristic
533C2×C4#52011yescharacteristic
534C2×C4C2×#12411yescharacteristic
535C2×C4#53211yescharacteristic
536C2×C4#53411yescharacteristic
537C2×C4#53511yescharacteristic
538C2×C4#53611yescharacteristic
539C2×C4#53711yescharacteristic
540C2×C4#53811yescharacteristic
541C2×C4#53811yescharacteristic
542C2×C4#53811yescharacteristic
543C2×C4#53811yescharacteristic
544C2×C4#53911yescharacteristic
545C2×C4#53911yescharacteristic
546C2×C4#53911yescharacteristic
547C2×C4#53911yescharacteristic
548C2×C4#54211yescharacteristic
549C2×C4#54211yescharacteristic
550C2×C4#54211yescharacteristic
551C2×C4#54211yescharacteristic
552C2×C4#54211yescharacteristic
553C2×C4#54211yescharacteristic
554C2×C4#54211yescharacteristic
555C2×C4#54211yescharacteristic
556C8—24no—
557C8—42no—
558C8—42no—
559C8—42no—
560C8—81no—
561C8—81no—
562C8—81no—
563C8—81no—
564C22—22no—
565C22—22no—
566C22—22no—
567C22—22no—
568C22—21no—
569C22—21no—
570C22—21no—
571C22—21no—
572C22—82no—
573C22—82no—
574C22—81no—
575C22—81no—
576C22—81no—
577C22C4×#12411yescharacteristic
578C22#232111yescharacteristic
579C22#232311yescharacteristic
580C22#265011yescharacteristic
581C22#265011yescharacteristic
582C22#265311yescharacteristic
583C22#265311yescharacteristic
584C22#295612yes—
585C22#295712yes—
586C4—44no—
587C4—42no—
588C4—42no—
589C4—82no—
590C4—82no—
591C4—82no—
592C4—81no—
593C4—81no—
594C4—81no—
595C4#233211yescharacteristic
596C4#233211yescharacteristic
597C4#233711yescharacteristic
598C4#233711yescharacteristic
599C4#263911yescharacteristic
600C4#264011yescharacteristic
601C4#266211yescharacteristic
602C4#266311yescharacteristic
603C2—21no—
604C2—21no—
605C2—21no—
606C2—21no—
607C2—161no—
608C2#1432311yescharacteristic
609C2#2165911yescharacteristic
610C2#2166211yescharacteristic
611C2#2206212yes—
612C2#2213212yes—
subgroupG/H
centerC22×C4C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC23C2×C4×D4
prime radicalC22×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C22×#97 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21301C2semidirect
4. #11386C2semidirect
5. #11317C2semidirect
6. #11317C2semidirect
7. #11269C2semidirect
13. #10883C2semidirect
14. #10882C2semidirect
15. #10696C2semidirect
29. #2026C4semidirect
30. #2025C4semidirect
31. #2024C4semidirect
32. #2023C4semidirect

Maximal subgroups

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

subgroupindex
#234512
#213012
#210472
#113862
#113172
#112692
#111912
#110532
#109302
#109292
#108832
#108822
#106962