Finite Group Databasebeta

← #380163

Subgroups — #380163

Order 512.

1268 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1958C211yesmaximal, characteristic
2C2×#1958C211yesmaximal, characteristic
3C2×#1744C211yesmaximal, characteristic, complemented
4#14708C211yesmaximal, characteristic
5#14707C211yesmaximal, characteristic, complemented
6#11190C212yesmaximal, complemented
7#11189C212yesmaximal
8#11067C212yesmaximal, complemented
9#11066C212yesmaximal
10#10842C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#102C2212yes—
16C22×#102C2212yes—
17C22×C4×C8C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2211yescharacteristic
21C2×#390C2211yescharacteristic
22C2×#390C2211yescharacteristic
23C2×#389C2212yes—
24C2×#382C2211yescharacteristic
25C2×#382C2211yescharacteristic
26C2×#382C2211yescharacteristic
27C2×C43C2211yescharacteristic
28#1958—24no—
29#1958C418yescomplemented
30#1744—24no—
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37#1437—22no—
38#1437—22no—
39#1436—22no—
40#1436—22no—
41C4⋊#95C2212yes—
42C23×Q8C2311yescharacteristic
43C23×D4C2311yescharacteristic
44C23×C8C2312yes—
45C23×C8C2312yes—
46C2×#124—22no—
47C2×#124—22no—
48C2×#124—22no—
49C2×#124—22no—
50C2×#124—22no—
51C2×#124C2×C414yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—21no—
58C22×C22⋊C4—21no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C42D412yes—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#103—24no—
67C2×#103—42no—
68C2×#103—42no—
69C2×#103—42no—
70C2×#103—42no—
71C2×#103C2×C418yes—
72C2×#102—24no—
73C2×#102—42no—
74C2×#102—42no—
75C2×#102—42no—
76C2×#102—42no—
77C2×#102C2×C418yes—
78C2×C4×C8—22no—
79C2×C4×C8—22no—
80C2×C4×C8—22no—
81C2×C4×C8—22no—
82C2×C4×C8C2×C414yes—
83C2×C4×C8D414yes—
84#391—42no—
85#391—42no—
86#390—42no—
87#390—42no—
88#389—42no—
89#389—42no—
90#382—22no—
91#382—22no—
92#382—22no—
93#382—22no—
94#382—22no—
95#382C2×C414yes—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107C43—22no—
108C43—22no—
109#340—42no—
110#340—42no—
111#340—42no—
112#340—42no—
113#336—24no—
114#336—24no—
115C25—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8C22×C412yes—
126C22×D4—21no—
127C22×D4—21no—
128C22×D4—21no—
129C22×D4—21no—
130C22×D4—21no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4—22no—
134C22×D4—22no—
135C22×D4C22×C412yes—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—21no—
139C23×C4—21no—
140C23×C4—22no—
141C23×C4C2×D412yes—
142C23×C4C2×D412yes—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2411yescharacteristic
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C414yes—
159C22×C8C2×D414yes—
160#124—28no—
161#124—28no—
162#124—44no—
163#124—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4C22×C414yes—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—42no—
190C2×C22⋊C4—42no—
191C2×C22⋊C4—42no—
192C2×C22⋊C4—42no—
193C2×C22⋊C4—44no—
194C2×C22⋊C4—44no—
195C2×C22⋊C4—44no—
196C2×C22⋊C4—44no—
197C2×C22⋊C4—44no—
198C2×C22⋊C4—44no—
199C2×C42—24no—
200C2×C42—24no—
201C2×C42—24no—
202C2×C42—24no—
203C2×C42—21no—
204C2×C42—21no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—22no—
210C2×C42—22no—
211C2×C42—42no—
212C2×C42—42no—
213C2×C42C22×C412yes—
214C2×C42C2×D412yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217C2×C42#4112yes—
218C2×C42#4112yes—
219#103—48no—
220#103—48no—
221#102—48no—
222#102—48no—
223C4×C8—28no—
224C4×C8—28no—
225#95—24no—
226#95—24no—
227#95—44no—
228#95—44no—
229#95—44no—
230#95—44no—
231#95—42no—
232#95—42no—
233#95—42no—
234#95—42no—
235#95—44no—
236C24—22no—
237C24—41no—
238C24—41no—
239C24—41no—
240C24—41no—
241C24—41no—
242C24—41no—
243C24—42no—
244C24—42no—
245C24—42no—
246C24—42no—
247C24C22×D411yescharacteristic
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—41no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—24no—
267C2×D4—24no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—42no—
271C2×D4—42no—
272C2×D4—42no—
273C2×D4—42no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4—42no—
279C2×D4—42no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—24no—
297C22×C4—22no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—41no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4—42no—
326C22×C4—42no—
327C22×C4C4×D412yes—
328C22×C4C4×D412yes—
329C22×C4#12112yes—
330C22×C4#12112yes—
331C22×C4#12412yes—
332C22×C4#12412yes—
333C22×C4#13514yes—
334C22×C4#13514yes—
335C22×C4C23×C411yescharacteristic
336C22×C4C22×D411yescharacteristic
337C22×C4C2×#4111yescharacteristic
338C22×C4C2×#4111yescharacteristic
339C22×C4C2×#4111yescharacteristic
340C22×C4C2×#4111yescharacteristic
341C22×C4H(5,𝔽2)12yes—
342C22×C4H(5,𝔽2)12yes—
343C22×C4H(5,𝔽2)12yes—
344C22×C42−1+412yes—
345C2×C8—24no—
346C2×C8—24no—
347C2×C8—24no—
348C2×C8—24no—
349C2×C8—24no—
350C2×C8—24no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8—42no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—42no—
371C4⋊C4—88no—
372C22⋊C4—48no—
373C22⋊C4—48no—
374C22⋊C4—88no—
375C22⋊C4—88no—
376C42—216no—
377C42—48no—
378C42—42no—
379C42—42no—
380C42—42no—
381C42—42no—
382C42#11714yes—
383C42C4×D414yes—
384C42#12314yes—
385C42#12614yes—
386C23—42no—
387C23—42no—
388C23—44no—
389C23—44no—
390C23—42no—
391C23—81no—
392C23—81no—
393C23—81no—
394C23—81no—
395C23—81no—
396C23—81no—
397C23—81no—
398C23—81no—
399C23—82no—
400C23—82no—
401C23—82no—
402C23—82no—
403C23—82no—
404C23—82no—
405C23—82no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23C2×C4×D411yescharacteristic
410C23C2×#12111yescharacteristic
411C23C2×#12411yescharacteristic
412C23#53511yescharacteristic
413C23#53511yescharacteristic
414C23#53511yescharacteristic
415C23#53611yescharacteristic
416C23C2×#13512yes—
417C23C2×#13512yes—
418C23#57512yes—
419C23#57512yes—
420Q8—44no—
421Q8—44no—
422Q8—82no—
423Q8—82no—
424D4—44no—
425D4—44no—
426D4—82no—
427D4—82no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—24no—
437C2×C4—24no—
438C2×C4—24no—
439C2×C4—24no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—42no—
453C2×C4—44no—
454C2×C4—44no—
455C2×C4—42no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—82no—
472C2×C4—82no—
473C2×C4—82no—
474C2×C4—82no—
475C2×C4—82no—
476C2×C4—82no—
477C2×C4—82no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4#38814yes—
482C2×C4#38814yes—
483C2×C4C2×#11711yescharacteristic
484C2×C4C2×C4×D411yescharacteristic
485C2×C4#51812yes—
486C2×C4#52012yes—
487C2×C4C2×#12311yescharacteristic
488C2×C4C2×#12611yescharacteristic
489C2×C4#53512yes—
490C2×C4#53712yes—
491C2×C4#54012yes—
492C2×C4#54012yes—
493C2×C4#54012yes—
494C2×C4#54012yes—
495C2×C4#54212yes—
496C2×C4#54212yes—
497C2×C4#54212yes—
498C2×C4#54212yes—
499C8—44no—
500C8—44no—
501C22—82no—
502C22—82no—
503C22—82no—
504C22—84no—
505C22—84no—
506C22—161no—
507C22—161no—
508C22—161no—
509C22—162no—
510C22—162no—
511C22—161no—
512C22C2×#38811yescharacteristic
513C22#232611yescharacteristic
514C22#232811yescharacteristic
515C22#261911yescharacteristic
516C22#262011yescharacteristic
517C22#264011yescharacteristic
518C22#264111yescharacteristic
519C22C4×#13512yes—
520C22#294312yes—
521C22#305012yes—
522C22#305412yes—
523C22#313111yescharacteristic
524C22#313111yescharacteristic
525C22#313511yescharacteristic
526C22#313511yescharacteristic
527C22#321112yes—
528C22#321112yes—
529C22#321112yes—
530C22#321212yes—
531C22#321212yes—
532C22#321212yes—
533C22#323212yes—
534C22#323312yes—
535C4—24no—
536C4—24no—
537C4—44no—
538C4—44no—
539C4—82no—
540C4—84no—
541C4—161no—
542C4—161no—
543C4—162no—
544C4#231912yes—
545C4#232412yes—
546C4#262412yes—
547C4#264412yes—
548C2—162no—
549C2#1426011yescharacteristic
550C2#2108311yescharacteristic
551C2#2108611yescharacteristic
552C2#2222212yes—
553C2#2227112yes—
554C2#2980912yes—
555C2#2984612yes—
556C2#3020711yescharacteristic
557C2#3020811yescharacteristic
558C2#3022111yescharacteristic
559C2#3022111yescharacteristic
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#1744C2semidirect
5. #14707C2semidirect
6. #11190C2semidirect
8. #11067C2semidirect
10. #10842C2semidirect
11. #10842C2semidirect
29. #1958C4semidirect

Maximal subgroups

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

subgroupindex
C2×#19582
C2×#17442
#147082
#147072
#111902
#111892
#110672
#110662
#108422