Finite Group Databasebeta

← #470973

Subgroups — #470973

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 541 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11368C211yesmaximal, characteristic, complemented
2#11306C211yesmaximal, characteristic, complemented
3#11300C211yesmaximal, characteristic, complemented
4#11300C211yesmaximal, characteristic, complemented
5#11260C211yesmaximal, characteristic, complemented
6#11250C211yesmaximal, characteristic, complemented
7#11177C211yesmaximal, characteristic
8#11165C211yesmaximal, characteristic
9#11150C211yesmaximal, characteristic
10#11146C211yesmaximal, characteristic
11#11104C211yesmaximal, characteristic
12#11099C211yesmaximal, characteristic
13#10930C211yesmaximal, characteristic
14#10860C211yesmaximal, characteristic, complemented
15#10823C211yesmaximal, characteristic, complemented
16C2×#430C2211yescharacteristic
17C22×#108C2211yescharacteristic
18C2×#427C2211yescharacteristic
19C2×#421C2211yescharacteristic
20C2×#421C2211yescharacteristic
21C22×#104C2211yescharacteristic
22C2×#416C2211yescharacteristic
23C2×#413C2211yescharacteristic
24C2×#413C2211yescharacteristic
25C2×#408C2211yescharacteristic
26C22×#97C2211yescharacteristic
27C2×#344C2211yescharacteristic
28C2×#344C2211yescharacteristic
29C2×#344C2211yescharacteristic
30C2×#344C2211yescharacteristic
31C2×#344C2211yescharacteristic
32C2×#344C2211yescharacteristic
33#1738C2211yescharacteristic
34C2×#341C2211yescharacteristic
35C2×#341C2211yescharacteristic
36C2×#341C2211yescharacteristic
37C2×#341C2211yescharacteristic
38C2×#341C2211yescharacteristic
39C2×#339C2211yescharacteristic
40C2×#339C2211yescharacteristic
41C2×#339C2211yescharacteristic
42C2×#338C2211yescharacteristic
43C2×#338C2211yescharacteristic
44C2×H(3,ℤ/4)C2211yescharacteristic
45C2×H(3,ℤ/4)C2211yescharacteristic
46#1729C2211yescharacteristic
47#1729C2211yescharacteristic
48#1729C2211yescharacteristic
49#1728C2211yescharacteristic
50#1728C2211yescharacteristic
51C22×#41C2311yescharacteristic
52C22×M16C2311yescharacteristic
53C22×M16C2311yescharacteristic
54C22×M16C2311yescharacteristic
55C22×M16C2311yescharacteristic
56C22×M16C2311yescharacteristic
57C22×M16C2311yescharacteristic
58C22×M16C2311yescharacteristic
59C22×M16C2311yescharacteristic
60C23×C8C2311yescharacteristic
61C23×C8C2311yescharacteristic
62C2×#117C2311yescharacteristic
63C2×#117C2311yescharacteristic
64C2×#117C2311yescharacteristic
65C22×C42C2311yescharacteristic
66#430—22no—
67#430—41no—
68C2×#108—22no—
69C2×#108—22no—
70C2×#108—41no—
71C2×#108—41no—
72#427—41no—
73#427—41no—
74C2×#107—21no—
75C2×#106—21no—
76#421—41no—
77#421—41no—
78#421—41no—
79#421—41no—
80C2×#104—22no—
81C2×#104—41no—
82C2×#104—41no—
83C2×#104—41no—
84C2×#104—41no—
85#416—22no—
86#416—41no—
87C2×#103—21no—
88C2×#102—21no—
89#413—22no—
90#413—41no—
91#413—41no—
92#413—41no—
93C2×#101—21no—
94C2×#101—21no—
95C2×#100—21no—
96C2×#100—21no—
97#408—21no—
98#408—21no—
99#408—22no—
100C2×C22⋊C8—22no—
101C2×#97—22no—
102C2×#97—22no—
103C2×#97—41no—
104#344—41no—
105#344—41no—
106#344—41no—
107#344—41no—
108#344—41no—
109#344—41no—
110#341—22no—
111#341—22no—
112#341—21no—
113#341—21no—
114#341—41no—
115#341—41no—
116#341—41no—
117#339—41no—
118#339—41no—
119#339—41no—
120#338—41no—
121#338—41no—
122H(3,ℤ/4)—41no—
123H(3,ℤ/4)—41no—
124#336—22no—
125#336—22no—
126#336—22no—
127#336—41no—
128#336—41no—
129C2×#41—21no—
130C2×#41—21no—
131C2×#41—21no—
132C2×#41—21no—
133C2×#41—21no—
134C2×#41—21no—
135C2×#41—21no—
136C2×#41—21no—
137C2×#41—21no—
138C2×#41—21no—
139C2×#41—21no—
140C2×#41—21no—
141C22×Q8#4111yescharacteristic
142C22×D4—21no—
143C22×D4#4111yescharacteristic
144C23×C4—21no—
145C23×C4—21no—
146C23×C4C2411yescharacteristic
147C2×M16—21no—
148C2×M16—21no—
149C2×M16—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16—21no—
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#4112yes—
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#4112yes—
214C22×C8#4112yes—
215C22×C8#4111yescharacteristic
216C22×C8#4111yescharacteristic
217C22×C8#4112yes—
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—
230C2×C4⋊C4—21no—
231C2×C4⋊C4—21no—
232C2×C4⋊C4—21no—
233C2×C22⋊C4—21no—
234C2×C22⋊C4—21no—
235C2×C22⋊C4—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42—21no—
239C2×C42—21no—
240C2×C42—21no—
241C2×C42—21no—
242C2×C42—21no—
243C2×C42—21no—
244C2×C42#4112yes—
245#108—44no—
246#108—44no—
247#107—42no—
248#106—42no—
249#104—44no—
250#104—44no—
251#104—44no—
252#103—42no—
253#102—42no—
254#101—42no—
255#101—42no—
256#100—42no—
257#100—42no—
258C22⋊C8—42no—
259C22⋊C8—42no—
260#97—44no—
261C24—21no—
262C242−1+411yescharacteristic
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267#41—41no—
268#41—41no—
269#41—41no—
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—41no—
292C2×D4—41no—
293C2×D4—41no—
294C2×D4—41no—
295C2×D4—41no—
296C2×D4—41no—
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—41no—
310C22×C4—82no—
311C22×C4C2×#4111yescharacteristic
312C22×C4C2×#4111yescharacteristic
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4C2×#4111yescharacteristic
317C22×C4C2×#4111yescharacteristic
318C22×C4H(5,𝔽2)11yescharacteristic
319C22×C4H(5,𝔽2)11yescharacteristic
320C22×C4H(5,𝔽2)11yescharacteristic
321C22×C4H(5,𝔽2)11yescharacteristic
322C22×C4H(5,𝔽2)11yescharacteristic
323C22×C4H(5,𝔽2)11yescharacteristic
324C22×C42−1+411yescharacteristic
325M16—22no—
326M16—22no—
327M16—41no—
328M16—41no—
329M16—41no—
330M16—41no—
331M16—41no—
332M16—41no—
333M16—41no—
334M16—41no—
335M16—41no—
336M16—41no—
337M16—41no—
338M16—41no—
339M16—41no—
340M16—41no—
341M16—41no—
342M16—41no—
343M16—41no—
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—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—22no—
364C2×C8—22no—
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—41no—
372C2×C8—41no—
373C2×C8—41no—
374C2×C8—41no—
375C2×C8—41no—
376C2×C8—41no—
377C2×C8—41no—
378C2×C8—41no—
379C2×C8—41no—
380C2×C8—41no—
381C2×C8—41no—
382C2×C8—41no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—41no—
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#12612yes—
397C2×C8#12612yes—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—41no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C22⋊C4—41no—
410C42—22no—
411C42—22no—
412C42—41no—
413C42—41no—
414C42—41no—
415C42—41no—
416C42—41no—
417C42—41no—
418C42—41no—
419C42—41no—
420C23—42no—
421C23—41no—
422C23—41no—
423C23—41no—
424C23—82no—
425C23#54111yescharacteristic
426C23#54211yescharacteristic
427C23#54211yescharacteristic
428C23#55211yescharacteristic
429C23#55211yescharacteristic
430C23#56211yescharacteristic
431C23#56211yescharacteristic
432Q8—41no—
433Q8—41no—
434Q8—41no—
435Q8—41no—
436D4—41no—
437D4—41no—
438D4—41no—
439D4—41no—
440D4—81no—
441D4—81no—
442D4—81no—
443D4—81no—
444C2×C4—44no—
445C2×C4—41no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4C2×#12611yescharacteristic
462C2×C4#52911yescharacteristic
463C2×C4#52911yescharacteristic
464C2×C4#53811yescharacteristic
465C2×C4#53811yescharacteristic
466C2×C4#53811yescharacteristic
467C2×C4#53811yescharacteristic
468C2×C4#53911yescharacteristic
469C2×C4#53911yescharacteristic
470C2×C4#54011yescharacteristic
471C2×C4#54011yescharacteristic
472C2×C4#54011yescharacteristic
473C2×C4#54211yescharacteristic
474C2×C4#54211yescharacteristic
475C2×C4#54211yescharacteristic
476C2×C4#54211yescharacteristic
477C2×C4#55111yescharacteristic
478C2×C4#55111yescharacteristic
479C2×C4#55111yescharacteristic
480C2×C4#55111yescharacteristic
481C2×C4#55311yescharacteristic
482C2×C4#55311yescharacteristic
483C2×C4#55311yescharacteristic
484C2×C4#55311yescharacteristic
485C2×C4#55511yescharacteristic
486C2×C4#55911yescharacteristic
487C2×C4#56011yescharacteristic
488C2×C4#56011yescharacteristic
489C8—22no—
490C8—22no—
491C8—42no—
492C8—42no—
493C8—81no—
494C8—81no—
495C8—81no—
496C8—81no—
497C8—81no—
498C8—81no—
499C8—81no—
500C22—22no—
501C22—22no—
502C22—22no—
503C22—41no—
504C22—41no—
505C22—41no—
506C22—41no—
507C22—81no—
508C22—82no—
509C22—161no—
510C22—161no—
511C22#249511yescharacteristic
512C22#265011yescharacteristic
513C22#273911yescharacteristic
514C22#274111yescharacteristic
515C22#275411yescharacteristic
516C22#276111yescharacteristic
517C22#282911yescharacteristic
518C4—82no—
519C4—81no—
520C4—161no—
521C4—161no—
522C4—161no—
523C4#251011yescharacteristic
524C4#251611yescharacteristic
525C4#264411yescharacteristic
526C4#265311yescharacteristic
527C4#274611yescharacteristic
528C4#274911yescharacteristic
529C4#279511yescharacteristic
530C4#280311yescharacteristic
531C2—21no—
532C2—21no—
533C2—21no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—161no—
538C2#1672311yescharacteristic
539C2#2783412yes—
subgroupG/H
centerC2×C4#555
commutator, Frattini, cosocleC23×C4C24
socleC22#2495
prime radicalC22×#41C23
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #11368C2semidirect
2. #11306C2semidirect
3. #11300C2semidirect
4. #11300C2semidirect
5. #11260C2semidirect
6. #11250C2semidirect
14. #10860C2semidirect
15. #10823C2semidirect

Maximal subgroups

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

subgroupindex
#113682
#113062
#113002
#112602
#112502
#111772
#111652
#111502
#111462
#111042
#110992
#109302
#108602
#108232