Finite Group Databasebeta

← #382357

Subgroups — #382357

Order 512.

715 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2081C211yesmaximal, characteristic
2#11308C211yesmaximal, characteristic, complemented
3#11301C211yesmaximal, characteristic, complemented
4#11283C211yesmaximal, characteristic, complemented
5#11252C211yesmaximal, characteristic, complemented
6#11181C211yesmaximal, characteristic
7#11139C211yesmaximal, characteristic
8#11133C211yesmaximal, characteristic
9#10943C211yesmaximal, characteristic
10#10943C211yesmaximal, characteristic
11#10943C211yesmaximal, characteristic
12#10898C211yesmaximal, characteristic, complemented
13#10898C211yesmaximal, characteristic, complemented
14#10898C211yesmaximal, characteristic, complemented
15#10699C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C2×#430C2211yescharacteristic
18C2×#430C2211yescharacteristic
19C2×#418C2211yescharacteristic
20C2×#418C2211yescharacteristic
21C2×#418C2211yescharacteristic
22C22×#102C2211yescharacteristic
23C22×#100C2211yescharacteristic
24C22×#100C2211yescharacteristic
25C22×#100C2211yescharacteristic
26C2×#405C2211yescharacteristic
27C2×#405C2211yescharacteristic
28C2×#405C2211yescharacteristic
29#2081C418yescomplemented
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33C2×#344C2211yescharacteristic
34C2×#344C2211yescharacteristic
35C2×#344C2211yescharacteristic
36C2×#343C2211yescharacteristic
37C2×#343C2211yescharacteristic
38C2×#343C2211yescharacteristic
39#1738C2211yescharacteristic
40#1738C2211yescharacteristic
41#1738C2211yescharacteristic
42C2×#341C2211yescharacteristic
43C2×#341C2211yescharacteristic
44C2×#341C2211yescharacteristic
45#1736C2211yescharacteristic
46#1736C2211yescharacteristic
47#1736C2211yescharacteristic
48C2×#338C2211yescharacteristic
49C2×H(3,ℤ/4)C2211yescharacteristic
50C2×H(3,ℤ/4)C2211yescharacteristic
51C2×H(3,ℤ/4)C2211yescharacteristic
52C23×D4C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C22×M16C2311yescharacteristic
61C22×M16C2311yescharacteristic
62C23×C8C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C2×#117C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67#430—22no—
68#430—22no—
69#430—41no—
70#430—41no—
71#430—41no—
72#430C2×C414yes—
73C2×#108—41no—
74C2×#108—41no—
75C2×#108—41no—
76#418—22no—
77#418—41no—
78#418—41no—
79#418—41no—
80#418C2×C414yes—
81#418C2×C414yes—
82C2×#102—21no—
83C2×#102—21no—
84C2×#102—21no—
85C2×#102—21no—
86C2×#102—21no—
87C2×#102—21no—
88C2×#102—41no—
89C2×#102—41no—
90C2×#102—41no—
91C2×#100—22no—
92C2×#100—22no—
93C2×#100—41no—
94C2×#100—41no—
95C2×#100—41no—
96C2×#100—41no—
97C2×#100—41no—
98C2×#100—41no—
99C2×#100C2×C414yes—
100#405—41no—
101#405—41no—
102#405—41no—
103#405—41no—
104#405—41no—
105#405—41no—
106C2×#97—21no—
107C2×#97—21no—
108C2×#97—21no—
109C2×C4×C8—21no—
110C2×C4×C8—21no—
111C2×C4×C8—21no—
112#344—22no—
113#344—41no—
114#344—41no—
115#344—41no—
116#344C2×C414yes—
117#344C2×C414yes—
118#343—41no—
119#343—41no—
120#343—41no—
121#341—21no—
122#341—21no—
123#341—21no—
124#341—21no—
125#341—21no—
126#341—21no—
127#341—41no—
128#341—41no—
129#341—41no—
130#340—41no—
131#340—41no—
132#340—41no—
133#338—22no—
134H(3,ℤ/4)—22no—
135H(3,ℤ/4)—22no—
136H(3,ℤ/4)C2×C414yes—
137C25—21no—
138C22×D4—21no—
139C22×D4—21no—
140C22×D4—21no—
141C22×D4—21no—
142C22×D4—21no—
143C22×D4—21no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—21no—
147C22×D4—21no—
148C22×D4—21no—
149C22×D4—21no—
150C22×D4—21no—
151C22×D4C22×C412yes—
152C23×C4—21no—
153C23×C4C2411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16—21no—
159C2×M16—21no—
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—
196C2×M16—21no—
197C2×M16—21no—
198C2×M16—21no—
199C2×M16—21no—
200C2×M16—21no—
201C2×M16—21no—
202C2×M16C22×C412yes—
203C2×M16C22×C412yes—
204C2×M16C22×C412yes—
205C2×M16C22×C412yes—
206C2×M16#4112yes—
207C2×M16#4112yes—
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×C8—21no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8#4111yescharacteristic
222C22×C8#4111yescharacteristic
223C22×C8#4111yescharacteristic
224C22×C8#4111yescharacteristic
225C22×C8#4111yescharacteristic
226C22×C8#4111yescharacteristic
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#117C22×C412yes—
237#117C22×C412yes—
238#117#4112yes—
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×C22⋊C4—21no—
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C42—21no—
252C2×C42—21no—
253C2×C42—21no—
254#108—44no—
255#108—44no—
256#108—44no—
257#102—44no—
258#102—44no—
259#102—44no—
260#102—41no—
261#102—41no—
262#102—41no—
263#102—41no—
264#100—44no—
265#100—44no—
266#100—44no—
267#97—42no—
268#97—42no—
269#97—42no—
270C4×C8—42no—
271C4×C8—42no—
272C4×C8—42no—
273C24—22no—
274C24—41no—
275C24—41no—
276C24—41no—
277C24—41no—
278C24—41no—
279C24—41no—
280C24—161no—
281C24—161no—
282C242−1+411yescharacteristic
283C2×D4—22no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—22no—
287C2×D4—22no—
288C2×D4—22no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C2×D4—41no—
298C2×D4—41no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—161no—
318C22×C4C23×C411yescharacteristic
319C22×C4C2×#4111yescharacteristic
320C22×C4C2×#4111yescharacteristic
321C22×C4C2×#4111yescharacteristic
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4111yescharacteristic
325C22×C4H(5,𝔽2)11yescharacteristic
326C22×C4H(5,𝔽2)11yescharacteristic
327C22×C4H(5,𝔽2)11yescharacteristic
328C22×C4H(5,𝔽2)11yescharacteristic
329C22×C4H(5,𝔽2)11yescharacteristic
330C22×C4H(5,𝔽2)11yescharacteristic
331C22×C42−1+411yescharacteristic
332M16—22no—
333M16—22no—
334M16—22no—
335M16—22no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—22no—
340M16—22no—
341M16—22no—
342M16—22no—
343M16—22no—
344M16—41no—
345M16—41no—
346M16—41no—
347M16—41no—
348M16—41no—
349M16—41no—
350M16—41no—
351M16—41no—
352M16—41no—
353M16—41no—
354M16—41no—
355M16—41no—
356M16—41no—
357M16—41no—
358M16—41no—
359M16—41no—
360M16—41no—
361M16—41no—
362M16—41no—
363M16—41no—
364M16—41no—
365M16—41no—
366M16—41no—
367M16—41no—
368C2×C8—21no—
369C2×C8—21no—
370C2×C8—21no—
371C2×C8—21no—
372C2×C8—21no—
373C2×C8—21no—
374C2×C8—21no—
375C2×C8—21no—
376C2×C8—21no—
377C2×C8—21no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—41no—
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—
399C2×C8—41no—
400C2×C8—41no—
401C2×C8—41no—
402C2×C8—41no—
403C2×C8—41no—
404C2×C8—41no—
405C2×C8—41no—
406C2×C8—41no—
407C2×C8#12611yescharacteristic
408C2×C8#12611yescharacteristic
409C2×C8#12611yescharacteristic
410C2×C8#12611yescharacteristic
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—41no—
415C4⋊C4—41no—
416C4⋊C4—41no—
417C4⋊C4—41no—
418C4⋊C4—41no—
419C4⋊C4—41no—
420C4⋊C4—41no—
421C22⋊C4—22no—
422C22⋊C4—22no—
423C22⋊C4—22no—
424C22⋊C4—41no—
425C22⋊C4—41no—
426C22⋊C4—41no—
427C42—22no—
428C42—22no—
429C42—22no—
430C42—41no—
431C42—41no—
432C42—41no—
433C23—41no—
434C23—41no—
435C23—42no—
436C23—42no—
437C23—42no—
438C23—81no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—161no—
443C23—161no—
444C23—161no—
445C23—161no—
446C23—161no—
447C23—161no—
448C23—162no—
449C23—162no—
450C23—162no—
451C23—162no—
452C23#52011yescharacteristic
453C23#53911yescharacteristic
454C23#54111yescharacteristic
455C23#54211yescharacteristic
456C23#54211yescharacteristic
457C23#54211yescharacteristic
458C23#54211yescharacteristic
459D4—42no—
460D4—42no—
461D4—42no—
462D4—42no—
463D4—81no—
464D4—81no—
465D4—81no—
466D4—81no—
467C2×C4—42no—
468C2×C4—42no—
469C2×C4—42no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—161no—
478C2×C4—161no—
479C2×C4—161no—
480C2×C4C4×#4111yescharacteristic
481C2×C4C4×#4111yescharacteristic
482C2×C4C4×#4111yescharacteristic
483C2×C4#51811yescharacteristic
484C2×C4#51811yescharacteristic
485C2×C4#52011yescharacteristic
486C2×C4C2×#12611yescharacteristic
487C2×C4#52911yescharacteristic
488C2×C4#52911yescharacteristic
489C2×C4#52911yescharacteristic
490C2×C4#53811yescharacteristic
491C2×C4#53811yescharacteristic
492C2×C4#53811yescharacteristic
493C2×C4#53811yescharacteristic
494C2×C4#53811yescharacteristic
495C2×C4#53811yescharacteristic
496C2×C4#53811yescharacteristic
497C2×C4#53811yescharacteristic
498C2×C4#53911yescharacteristic
499C2×C4#53911yescharacteristic
500C2×C4#53911yescharacteristic
501C2×C4#54011yescharacteristic
502C2×C4#54011yescharacteristic
503C2×C4#54011yescharacteristic
504C2×C4#54011yescharacteristic
505C2×C4#54011yescharacteristic
506C2×C4#54211yescharacteristic
507C2×C4#54211yescharacteristic
508C8—21no—
509C8—21no—
510C8—21no—
511C8—21no—
512C8—42no—
513C8—42no—
514C8—42no—
515C8—42no—
516C8—42no—
517C8—42no—
518C8—81no—
519C8—81no—
520C8—81no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—21no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—82no—
532C22—162no—
533C22—162no—
534C22—162no—
535C22—162no—
536C22—162no—
537C22—162no—
538C22—161no—
539C22—161no—
540C22#233411yescharacteristic
541C22#233511yescharacteristic
542C22#233711yescharacteristic
543C22#264411yescharacteristic
544C22#265011yescharacteristic
545C22#265311yescharacteristic
546C22#265411yescharacteristic
547C4—82no—
548C4—82no—
549C4—82no—
550C4—161no—
551C4C4×#12611yescharacteristic
552C4#233311yescharacteristic
553C4#233511yescharacteristic
554C4#233511yescharacteristic
555C4#264311yescharacteristic
556C4#264811yescharacteristic
557C4#264811yescharacteristic
558C4#265111yescharacteristic
559C2—21no—
560C2—21no—
561C2—21no—
562C2—21no—
563C2—21no—
564C2—21no—
565C2—162no—
566C2#1433211yescharacteristic
567C2#3023611yescharacteristic
568C2#3023711yescharacteristic
subgroupG/H
center, socleC22#2644
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. #11308C2semidirect
3. #11301C2semidirect
4. #11283C2semidirect
5. #11252C2semidirect
12. #10898C2semidirect
13. #10898C2semidirect
14. #10898C2semidirect
15. #10699C2semidirect
29. #2081C4semidirect

Maximal subgroups

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

subgroupindex
C2×#20812
#113082
#113012
#112832
#112522
#111812
#111392
#111332
#109432
#108982
#106992