Finite Group Databasebeta

← #279167

Subgroups — #279167

Order 512.

652 conjugacy classes of subgroups.

The subgroup lattice has 566 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#29876C211yesmaximal, characteristic, complemented
2#24836C211yesmaximal, characteristic
3#24708C211yesmaximal, characteristic
4#22137C211yesmaximal, characteristic
5C2×#1695C211yesmaximal, characteristic, complemented
6C2×#1573C211yesmaximal, characteristic, complemented
7#10601C211yesmaximal, characteristic, complemented
8#10589C211yesmaximal, characteristic, complemented
9#10481C211yesmaximal, characteristic, complemented
10#10480C211yesmaximal, characteristic, complemented
11#9730C211yesmaximal, characteristic, complemented
12#9613C211yesmaximal, characteristic, complemented
13#9136C211yesmaximal, characteristic, complemented
14#9127C211yesmaximal, characteristic, complemented
15#8654C211yesmaximal, characteristic, complemented
16C2×#500C2211yescharacteristic
17C2×#476C2211yescharacteristic
18C2×#476C2211yescharacteristic
19C2×#436C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22C22×C4⋊C8C2211yescharacteristic
23#2857C2211yescharacteristic
24#2857C2211yescharacteristic
25C4×#121C2211yescharacteristic
26#2088—21no—
27#2086—21no—
28#2086—21no—
29#2077—21no—
30#2075—21no—
31#2075—21no—
32#2065—21no—
33#2033—21no—
34#1990C2211yescharacteristic
35#1990C2211yescharacteristic
36#1987C2211yescharacteristic
37#1987C2211yescharacteristic
38#1954—21no—
39#1954C2211yescharacteristic
40#1948C2211yescharacteristic
41#1947C2211yescharacteristic
42#1943—21no—
43#1939C2211yescharacteristic
44#1936—21no—
45#1923—21no—
46#1893—21no—
47#1870—21no—
48#1851—21no—
49#1834C2211yescharacteristic
50#1807—21no—
51#1802C2211yescharacteristic
52#1802C2211yescharacteristic
53#1797C2211yescharacteristic
54C8×C4⋊C4C2211yescharacteristic
55#1695—24no—
56#1573—24no—
57C2×#335C2211yescharacteristic
58C2×#335C2211yescharacteristic
59C2×#334C2211yescharacteristic
60C2×#334C2211yescharacteristic
61C2×#333C2211yescharacteristic
62C2×#333C2211yescharacteristic
63C2×#333C2211yescharacteristic
64C2×#333C2211yescharacteristic
65#1296C2211yescharacteristic
66#1296C2211yescharacteristic
67#1296C2211yescharacteristic
68#1296C2211yescharacteristic
69C22×M16—21no—
70C23×C8—21no—
71C2×C4⋊Q8C2311yescharacteristic
72C2×C4⋊Q8C2311yescharacteristic
73C2×C22⋊Q8—21no—
74C2×C22⋊Q8—21no—
75C2×#121D411yescharacteristic
76C2×C4×D4—21no—
77C2×C4×D4C2311yescharacteristic
78C22×C42C2311yescharacteristic
79#500—22no—
80#500—22no—
81#476—22no—
82#476—22no—
83#476—41no—
84#476—41no—
85#436—22no—
86#436—22no—
87#432—22no—
88#432—41no—
89#431—22no—
90#431—22no—
91C2×#107—21no—
92C2×#107—21no—
93C2×#107—21no—
94C2×#106—21no—
95C2×#106—21no—
96C2×#106—21no—
97C2×C4⋊C8—22no—
98C2×C4⋊C8—21no—
99C2×C4⋊C8—21no—
100C2×C4⋊C8—41no—
101C2×C4⋊C8D411yescharacteristic
102C2×C4⋊C8D411yescharacteristic
103C2×C4⋊C8C2311yescharacteristic
104C2×C4⋊C8C2311yescharacteristic
105C2×C4⋊C8C2311yescharacteristic
106C2×C4⋊C8C2311yescharacteristic
107C2×C4⋊C8C2311yescharacteristic
108C2×C4⋊C8C2311yescharacteristic
109C2×#102—21no—
110C2×#102—21no—
111C2×#102—21no—
112C2×#102—21no—
113C2×C22⋊C8—21no—
114C2×C22⋊C8—21no—
115C2×C22⋊C8—21no—
116C2×C22⋊C8—21no—
117C2×C22⋊C8—21no—
118C2×C22⋊C8D411yescharacteristic
119C2×C22⋊C8D411yescharacteristic
120C2×#97—21no—
121C2×C4×C8—21no—
122C2×C4×C8C2311yescharacteristic
123C2×C4×C8C2311yescharacteristic
124#391—21no—
125#391—21no—
126#390D411yescharacteristic
127#389—21no—
128#389—21no—
129#389—21no—
130#389—21no—
131#389D411yescharacteristic
132#388—21no—
133#386—21no—
134#384C2311yescharacteristic
135#384C2311yescharacteristic
136#380—21no—
137#380—21no—
138#380—21no—
139#380—21no—
140#380D411yescharacteristic
141C4×C4⋊C4C2311yescharacteristic
142C4×C22⋊C4—21no—
143#340—21no—
144#340—21no—
145#340—21no—
146#340—21no—
147#340—21no—
148#340—21no—
149#340—21no—
150#340—21no—
151#340—21no—
152#340—21no—
153#340—21no—
154#340—21no—
155#340—21no—
156#340—21no—
157#336—21no—
158#336—21no—
159#336—21no—
160#335—22no—
161#335—22no—
162#334—22no—
163#334D414yescomplemented
164#333—22no—
165#333—22no—
166#333—41no—
167#333—41no—
168C22×Q8—21no—
169C22×Q8—21no—
170C22×D4—21no—
171C22×D4C2×D411yescharacteristic
172C23×C4—21no—
173C23×C4C2×D411yescharacteristic
174C23×C4C2×D411yescharacteristic
175C23×C4#4111yescharacteristic
176C2×M16—42no—
177C2×M16—81no—
178C2×M16—81no—
179C22×C8—21no—
180C22×C8—21no—
181C22×C8—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185C22×C8—42no—
186C22×C8—42no—
187C22×C8—81no—
188C22×C8C2×D411yescharacteristic
189C22×C8C2×D411yescharacteristic
190C22×C8C2×D411yescharacteristic
191C22×C8C2×D411yescharacteristic
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—21no—
195C4⋊Q8—21no—
196C4⋊Q8—21no—
197C4⋊Q8—21no—
198C4⋊Q8—21no—
199C4⋊Q8—21no—
200C22⋊Q8—82no—
201C22⋊Q8—82no—
202#121—22no—
203#121—41no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4—21no—
208C4×D4—81no—
209C4×D4—81no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C4⋊C4—21no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C4⋊C4—41no—
222C2×C4⋊C4—41no—
223C2×C4⋊C4C2×D411yescharacteristic
224C2×C4⋊C4C2×D411yescharacteristic
225C2×C4⋊C4C2×D411yescharacteristic
226C2×C22⋊C4—21no—
227C2×C22⋊C4—21no—
228C2×C22⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C22⋊C4—21no—
231C2×C22⋊C4—41no—
232C2×C22⋊C4—41no—
233C2×C42—21no—
234C2×C42—21no—
235C2×C42—21no—
236C2×C42—21no—
237C2×C42—21no—
238C2×C42C2×D411yescharacteristic
239C2×C42C2×D411yescharacteristic
240C2×C42#4111yescharacteristic
241C2×C42C2411yescharacteristic
242#107—42no—
243#107—42no—
244#107—42no—
245#106—42no—
246#106—42no—
247#106—81no—
248C4⋊C8—22no—
249C4⋊C8—22no—
250C4⋊C8—21no—
251C4⋊C8—21no—
252C4⋊C8—21no—
253C4⋊C8—21no—
254C4⋊C8—21no—
255C4⋊C8—21no—
256C4⋊C8—21no—
257C4⋊C8—21no—
258C4⋊C8—21no—
259C4⋊C8—21no—
260C4⋊C8—42no—
261C4⋊C8—81no—
262C4⋊C8C2×D412yes—
263C4⋊C8C2×D412yes—
264#102—42no—
265#102—42no—
266#102—42no—
267#102—81no—
268C22⋊C8—22no—
269C22⋊C8—22no—
270C22⋊C8—42no—
271C22⋊C8—42no—
272C22⋊C8—81no—
273C22⋊C8—81no—
274C22⋊C8—81no—
275#97—42no—
276C4×C8—21no—
277C4×C8—21no—
278C4×C8—81no—
279C4×C8C2×D412yes—
280C4×C8#4112yes—
281#95—21no—
282#95—21no—
283#95—21no—
284#95—41no—
285#95—41no—
286#95—41no—
287#95—41no—
288C24—21no—
289C24#12111yescharacteristic
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—81no—
295C2×Q8—81no—
296C2×Q8—81no—
297C2×Q8—81no—
298C2×D4—21no—
299C2×D4—21no—
300C2×D4—21no—
301C2×D4—21no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4—42no—
305C2×D4—42no—
306C2×D4—81no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—21no—
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—21no—
324C22×C4—21no—
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#12111yescharacteristic
337C22×C4#12111yescharacteristic
338C22×C4#12111yescharacteristic
339C22×C4C22×D411yescharacteristic
340C22×C4C22×D411yescharacteristic
341C22×C4C2×#4111yescharacteristic
342M16—84no—
343C2×C8—21no—
344C2×C8—21no—
345C2×C8—21no—
346C2×C8—21no—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—41no—
356C2×C8—41no—
357C2×C8—41no—
358C2×C8—41no—
359C2×C8—41no—
360C2×C8—41no—
361C2×C8—41no—
362C2×C8—41no—
363C2×C8—41no—
364C2×C8—41no—
365C2×C8—41no—
366C2×C8—41no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—84no—
372C2×C8#12112yes—
373C2×C8#12112yes—
374C4⋊C4—21no—
375C4⋊C4—21no—
376C4⋊C4—21no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—41no—
380C4⋊C4—41no—
381C4⋊C4—41no—
382C4⋊C4—41no—
383C4⋊C4—41no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C4⋊C4—41no—
390C4⋊C4—41no—
391C4⋊C4—41no—
392C4⋊C4—41no—
393C4⋊C4—41no—
394C4⋊C4—41no—
395C4⋊C4—81no—
396C4⋊C4—81no—
397C4⋊C4—81no—
398C4⋊C4—82no—
399C4⋊C4—82no—
400C4⋊C4#13712yes—
401C22⋊C4—42no—
402C22⋊C4—42no—
403C22⋊C4—41no—
404C22⋊C4—41no—
405C22⋊C4—81no—
406C22⋊C4—81no—
407C22⋊C4—82no—
408C22⋊C4—82no—
409C42—21no—
410C42—21no—
411C42—21no—
412C42—21no—
413C42—21no—
414C42—21no—
415C42—41no—
416C42—41no—
417C42C22×D411yescharacteristic
418C42C2×#4111yescharacteristic
419C42H(5,𝔽2)11yescharacteristic
420C422−1+411yescharacteristic
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23—41no—
428C23—41no—
429C23—41no—
430C23—41no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23#47212yes—
435C23C2×#12111yescharacteristic
436Q8—161no—
437Q8—161no—
438Q8—161no—
439Q8—161no—
440D4—41no—
441D4—41no—
442D4—41no—
443D4—41no—
444D4—84no—
445C2×C4—21no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—21no—
455C2×C4—21no—
456C2×C4—21no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—41no—
467C2×C4—41no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—81no—
479C2×C4—81no—
480C2×C4—81no—
481C2×C4—81no—
482C2×C4—81no—
483C2×C4—81no—
484C2×C4—81no—
485C2×C4—81no—
486C2×C4—81no—
487C2×C4—81no—
488C2×C4—81no—
489C2×C4—81no—
490C2×C4—81no—
491C2×C4C2×#12111yescharacteristic
492C2×C4#53711yescharacteristic
493C2×C4D4211yescharacteristic
494C2×C4#54711yescharacteristic
495C2×C4#54811yescharacteristic
496C2×C4#55311yescharacteristic
497C2×C4C2×#13711yescharacteristic
498C2×C4#57511yescharacteristic
499C2×C4#57711yescharacteristic
500C2×C4#57711yescharacteristic
501C2×C4#57711yescharacteristic
502C2×C4#57711yescharacteristic
503C2×C4#57811yescharacteristic
504C2×C4#57811yescharacteristic
505C8—22no—
506C8—22no—
507C8—42no—
508C8—42no—
509C8—41no—
510C8—41no—
511C8—81no—
512C8—81no—
513C8—81no—
514C8—81no—
515C22—22no—
516C22—22no—
517C22—22no—
518C22—41no—
519C22—41no—
520C22—41no—
521C22—41no—
522C22—81no—
523C22—81no—
524C22—81no—
525C22—81no—
526C22—81no—
527C22—81no—
528C22—81no—
529C22#241611yescharacteristic
530C22#304211yescharacteristic
531C22#304611yescharacteristic
532C22#305611yescharacteristic
533C22#306311yescharacteristic
534C22C2×#47211yescharacteristic
535C22#306711yescharacteristic
536C4—21no—
537C4—21no—
538C4—42no—
539C4—41no—
540C4—41no—
541C4—41no—
542C4—81no—
543C4—81no—
544C4—161no—
545C4—161no—
546C4—161no—
547C4—161no—
548C4#305711yescharacteristic
549C4#306311yescharacteristic
550C4#325811yescharacteristic
551C4#326011yescharacteristic
552C4#327511yescharacteristic
553C4#327611yescharacteristic
554C4#335211yescharacteristic
555C4#335611yescharacteristic
556C2—42no—
557C2—161no—
558C2#1946511yescharacteristic
559C2#1949911yescharacteristic
560C2#1956211yescharacteristic
561C2#1959111yescharacteristic
562C2#2476611yescharacteristic
563C2#2479311yescharacteristic
564C2#2480311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#121D4
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #29876C2semidirect
5. C2×#1695C2semidirect
6. C2×#1573C2semidirect
7. #10601C2semidirect
8. #10589C2semidirect
9. #10481C2semidirect
10. #10480C2semidirect
11. #9730C2semidirect
12. #9613C2semidirect
13. #9136C2semidirect
14. #9127C2semidirect
15. #8654C2semidirect
163. #334D4semidirect

Maximal subgroups

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

subgroupindex
#298762
#248362
#247082
#221372
C2×#16952
C2×#15732
#106012
#105892
#104812
#104802
#97302
#96132
#91362
#91272
#86542