Finite Group Databasebeta

← #365563

Subgroups — #365563

Order 512.

1678 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2326C211yesmaximal, characteristic
2C2×#1959C211yesmaximal, characteristic
3C2×#1958C211yesmaximal, characteristic
4C2×#1863C212yesmaximal, complemented
5C2×#1862C212yesmaximal
6C2×#1845C211yesmaximal, characteristic, complemented
7#14277C211yesmaximal, characteristic, complemented
8#11124C212yesmaximal, complemented
9#11123C212yesmaximal
10#10843C211yesmaximal, characteristic, complemented
11#10694C211yesmaximal, characteristic, complemented
12C22×#124C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#102C2212yes—
20C22×#102C2212yes—
21C22×#97C2211yescharacteristic
22C22×C4×C8C2211yescharacteristic
23#2326—24no—
24#2326—24no—
25C2×#391C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#388—22no—
29C2×#385—22no—
30C2×#384C2211yescharacteristic
31C2×#384C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#376C2211yescharacteristic
34#1959—24no—
35#1958C418yescomplemented
36#1863—28no—
37#1862—28no—
38#1845—24no—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1444—22no—
44#1442—22no—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×D4C2311yescharacteristic
48C24×C4—21no—
49C23×C8C2312yes—
50C23×C8C2312yes—
51C2×#124—22no—
52C2×#124—22no—
53C2×#124—22no—
54C2×#124C2×C414yes—
55C2×#124D414yes—
56C2×#124Q814yes—
57C2×C4×Q8—22no—
58C2×C4×Q8—22no—
59C2×C4×Q8—24no—
60C2×C4×Q8—42no—
61C2×C4×D4—22no—
62C2×C4×D4—22no—
63C2×C4×D4—24no—
64C2×C4×D4—42no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C22⋊C4—22no—
74C22×C22⋊C4—21no—
75C22×C22⋊C4—22no—
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—42no—
91C2×#103C2×C418yescomplemented
92C2×#102—24no—
93C2×#102—24no—
94C2×#102—24no—
95C2×#102—24no—
96C2×#102—42no—
97C2×#102C2×C418yes—
98C2×#97—22no—
99C2×#97—22no—
100C2×#97—22no—
101C2×C4×C8—22no—
102C2×C4×C8—22no—
103C2×C4×C8C2×C414yescomplemented
104#391—22no—
105#391—22no—
106#390—22no—
107#390—22no—
108#389—24no—
109#389—24no—
110#388—216no—
111#385—216no—
112#384—22no—
113#384—22no—
114#384—22no—
115#384D414yes—
116#382—22no—
117#382C2×C414yescomplemented
118#376—22no—
119#376—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—22no—
123C2×#95—24no—
124C2×#95D412yes—
125C2×#95D412yes—
126#340—24no—
127#340—24no—
128#336—24no—
129#336D418yes—
130C25—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8C22×C412yes—
139C22×Q8C2×D412yes—
140C22×Q8C2×Q812yes—
141C22×D4—22no—
142C22×D4—21no—
143C22×D4—22no—
144C22×D4—21no—
145C22×D4—21no—
146C22×D4—22no—
147C22×D4—22no—
148C22×D4C22×C412yes—
149C22×D4C2×D412yes—
150C22×D4C2×Q812yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—24no—
156C23×C4—22no—
157C23×C4—42no—
158C23×C4—42no—
159C23×C4—84no—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2×D411yescharacteristic
163C23×C4C2×D412yes—
164C23×C4C2411yescharacteristic
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8C22×C414yes—
176C22×C8C2×D414yes—
177C22×C8C2×D414yes—
178C22×C8#4114yes—
179#124—28no—
180#124—28no—
181#124—28no—
182#124C4⋊C4116yescomplemented
183C4×Q8—48no—
184C4×Q8—48no—
185C4×D4—48no—
186C4×D4—48no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—28no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—44no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—84no—
211C2×C4⋊C4—84no—
212C2×C4⋊C4—84no—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D412yes—
216C2×C22⋊C4—28no—
217C2×C22⋊C4—28no—
218C2×C22⋊C4—28no—
219C2×C22⋊C4—24no—
220C2×C22⋊C4—28no—
221C2×C22⋊C4—42no—
222C2×C22⋊C4—42no—
223C2×C22⋊C4—84no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—24no—
228C2×C42—21no—
229C2×C42—21no—
230C2×C42—22no—
231C2×C42—22no—
232C2×C42—42no—
233C2×C42—42no—
234C2×C42C22×C412yes—
235C2×C42C2×D414yes—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×Q812yes—
239C2×C42#4112yes—
240C2×C42#4112yes—
241#107—48no—
242#107—48no—
243#103—216no—
244#103—48no—
245#102—216no—
246#102—48no—
247#97—44no—
248C4×C8—28no—
249#95—28no—
250#95—28no—
251#95—24no—
252#95—24no—
253#95—44no—
254#95—84no—
255C24—22no—
256C24—22no—
257C24—22no—
258C24—41no—
259C24—41no—
260C24—41no—
261C24—41no—
262C24—44no—
263C24—44no—
264C24C22×D411yescharacteristic
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—24no—
269C2×Q8—24no—
270C2×Q8—24no—
271C2×Q8—24no—
272C2×Q8—24no—
273C2×Q8—24no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8C2×C4⋊C414yes—
281C2×D4—24no—
282C2×D4—22no—
283C2×D4—24no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4—24no—
287C2×D4—24no—
288C2×D4—24no—
289C2×D4—24no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—42no—
293C2×D4—42no—
294C2×D4—42no—
295C2×D4—42no—
296C2×D4C2×C4⋊C414yes—
297C22×C4—24no—
298C22×C4—24no—
299C22×C4—22no—
300C22×C4—21no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—24no—
307C22×C4—24no—
308C22×C4—24no—
309C22×C4—22no—
310C22×C4—22no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—22no—
314C22×C4—24no—
315C22×C4—24no—
316C22×C4—48no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—41no—
323C22×C4—41no—
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—44no—
328C22×C4—44no—
329C22×C4—44no—
330C22×C4—44no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—88no—
337C22×C4—88no—
338C22×C4—88no—
339C22×C4C4×D414yes—
340C22×C4#12014yes—
341C22×C4C22⋊Q814yes—
342C22×C4#12712yes—
343C22×C4#12712yes—
344C22×C4C23×C411yescharacteristic
345C22×C4C22×D411yescharacteristic
346C22×C4C22×D411yescharacteristic
347C22×C4C22×D412yes—
348C22×C4C22×Q811yescharacteristic
349C22×C4C2×#4111yescharacteristic
350C22×C4C2×#4112yes—
351C22×C4C2×#4111yescharacteristic
352C22×C4H(5,𝔽2)12yes—
353C22×C42−1+412yes—
354C2×C8—24no—
355C2×C8—24no—
356C2×C8—24no—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8C4×D418yes—
368C4⋊C4—24no—
369C4⋊C4—24no—
370C4⋊C4—24no—
371C4⋊C4—24no—
372C4⋊C4—48no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—42no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—88no—
382C4⋊C4—88no—
383C4⋊C4—88no—
384C22⋊C4—232no—
385C22⋊C4—48no—
386C22⋊C4—88no—
387C42—24no—
388C42—24no—
389C42—48no—
390C42—42no—
391C42—42no—
392C42C2×C4⋊C414yes—
393C42C4×D414yes—
394C42C22⋊Q814yes—
395C42#12314yes—
396C23—24no—
397C23—42no—
398C23—42no—
399C23—48no—
400C23—42no—
401C23—42no—
402C23—42no—
403C23—48no—
404C23—44no—
405C23—44no—
406C23—42no—
407C23—82no—
408C23—82no—
409C23—84no—
410C23—82no—
411C23—82no—
412C23—84no—
413C23C2×C4×D411yescharacteristic
414C23C2×#12011yescharacteristic
415C23C2×C22⋊Q811yescharacteristic
416C23C2×#12711yescharacteristic
417C23#53211yescharacteristic
418C23#53511yescharacteristic
419C23#53611yescharacteristic
420C23#57614yes—
421C23#57814yes—
422Q8—28no—
423Q8—44no—
424Q8—44no—
425Q8—44no—
426D4—28no—
427D4—44no—
428D4—44no—
429D4—44no—
430C2×C4—28no—
431C2×C4—28no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—24no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—44no—
450C2×C4—44no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—48no—
455C2×C4—44no—
456C2×C4—42no—
457C2×C4—44no—
458C2×C4—44no—
459C2×C4—42no—
460C2×C4—816no—
461C2×C4—82no—
462C2×C4—82no—
463C2×C4—82no—
464C2×C4—82no—
465C2×C4—84no—
466C2×C4—82no—
467C2×C4—82no—
468C2×C4—82no—
469C2×C4—82no—
470C2×C4—82no—
471C2×C4—84no—
472C2×C4—84no—
473C2×C4#38518yes—
474C2×C4C22×C4⋊C411yescharacteristic
475C2×C4C2×C4×D412yes—
476C2×C4C2×C4×D411yescharacteristic
477C2×C4#52012yes—
478C2×C4C2×C22⋊Q811yescharacteristic
479C2×C4C2×#12311yescharacteristic
480C2×C4#53212yes—
481C2×C4#54212yes—
482C2×C4D4212yes—
483C2×C4D4212yes—
484C2×C4#54712yes—
485C2×C4#54712yes—
486C2×C4#54812yes—
487C2×C4D4×Q812yes—
488C2×C4#55312yes—
489C2×C4#55412yes—
490C8—28no—
491C8—44no—
492C22—48no—
493C22—44no—
494C22—84no—
495C22—84no—
496C22—84no—
497C22—82no—
498C22—82no—
499C22—84no—
500C22—84no—
501C22C2×#38511yescharacteristic
502C22C4×#12711yescharacteristic
503C22#232611yescharacteristic
504C22#243211yescharacteristic
505C22#243311yescharacteristic
506C22#254211yescharacteristic
507C22#256611yescharacteristic
508C22#295114yes—
509C22#300812yes—
510C22#301112yes—
511C22#308314yes—
512C22#315512yes—
513C22#315612yes—
514C22#317514yes—
515C22#322012yes—
516C22#322312yes—
517C22#324314yes—
518C4—28no—
519C4—42no—
520C4—42no—
521C4—44no—
522C4—82no—
523C4—82no—
524C4—82no—
525C4—84no—
526C4—162no—
527C4D4×C4⋊C412yes—
528C4#236112yes—
529C4#242212yes—
530C4#245212yes—
531C2—84no—
532C2#1397711yescharacteristic
533C2#2088112yes—
534C2#2169612yes—
535C2#2222812yes—
536C2#2568112yes—
537C2#2573411yescharacteristic
538C2#2573511yescharacteristic
539C2#2852312yes—
540C2#2920012yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1863C2semidirect
6. C2×#1845C2semidirect
7. #14277C2semidirect
8. #11124C2semidirect
10. #10843C2semidirect
11. #10694C2semidirect
35. #1958C4semidirect
91. C2×#103C2×C4semidirect
103. C2×C4×C8C2×C4semidirect
117. #382C2×C4semidirect
182. #124C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23262
C2×#19592
C2×#19582
C2×#18632
C2×#18622
C2×#18452
#142772
#111242
#111232
#108432
#106942