Finite Group Databasebeta

← #360716

Subgroups — #360716

Order 512.

1654 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2317C211yesmaximal, characteristic
2C2×#1854C212yesmaximal
3C2×#1852C212yesmaximal
4C2×#1847C211yesmaximal, characteristic, complemented
5C2×#1845C211yesmaximal, characteristic, complemented
6C2×#1750C211yesmaximal, characteristic
7#14340C211yesmaximal, characteristic
8#11325C212yesmaximal, complemented
9#11324C212yesmaximal, complemented
10#10843C211yesmaximal, characteristic, complemented
11#10842C211yesmaximal, characteristic, complemented
12C22×#117C2211yescharacteristic
13C22×#117C2211yescharacteristic
14C23×C4⋊C4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C22⋊C8C2212yes—
20C22×#97C2211yescharacteristic
21C22×C4×C8C2211yescharacteristic
22#2317—24no—
23#2317—24no—
24C2×#389—22no—
25C2×#385—22no—
26C2×#384C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2211yescharacteristic
29C2×#382C2211yescharacteristic
30C2×#382C2211yescharacteristic
31C2×#382C2211yescharacteristic
32C2×#380C2212yes—
33#1854—28no—
34#1852—28no—
35#1847—24no—
36#1845—24no—
37#1750C418yescomplemented
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#336C2212yes—
43#1441—22no—
44#1439—22no—
45C4⋊#95C2212yes—
46C24×C4—21no—
47C24×C4C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×#117—22no—
51C2×#117—22no—
52C2×#117—22no—
53C2×#117—22no—
54C2×#117—22no—
55C2×#117—22no—
56C2×#117—24no—
57C2×#117—42no—
58C2×#117C2×C414yes—
59C2×#117Q814yes—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—24no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4—42no—
68C22×C4⋊C4D412yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C22⋊C4—21no—
76C22×C22⋊C4—22no—
77C22×C22⋊C4D412yes—
78C22×C42—21no—
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—24no—
85C2×#107—24no—
86C2×#107—24no—
87C2×#107—42no—
88C2×#107—42no—
89C2×#107—42no—
90C2×#106—24no—
91C2×#106—24no—
92C2×#106—42no—
93C2×C22⋊C8C2×C418yes—
94C2×C22⋊C8D418yes—
95C2×C22⋊C8Q818yes—
96C2×#97—22no—
97C2×#97—22no—
98C2×#97C2×C414yescomplemented
99C2×C4×C8—22no—
100C2×C4×C8—22no—
101C2×C4×C8C2×C414yescomplemented
102#389—216no—
103#385—216no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384—22no—
109#384D414yes—
110#382—22no—
111#382—22no—
112#382—22no—
113#382—22no—
114#382—22no—
115#382D414yes—
116#380—24no—
117#380—24no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—22no—
121C2×#95—22no—
122C2×#95—24no—
123#340—24no—
124#340—24no—
125#340—24no—
126#340—24no—
127#336C2×C418yescomplemented
128C25C2×D411yescharacteristic
129C23×C4—21no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—24no—
140C23×C4—22no—
141C23×C4—22no—
142C23×C4—42no—
143C23×C4—42no—
144C23×C4—84no—
145C23×C4C22×C412yes—
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D414yes—
150C23×C4C2×Q814yes—
151C23×C4C2×Q812yes—
152C23×C4C2411yescharacteristic
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8C22×C414yes—
162C22×C8C22×C414yes—
163C22×C8C2×D414yes—
164C22×C8C2×D414yes—
165C22×C8C2×Q814yes—
166C22×C8C2×Q814yes—
167#117—28no—
168#117—28no—
169#117—28no—
170#117—28no—
171#117—48no—
172#117—48no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
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—24no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
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—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—28no—
202C2×C4⋊C4—48no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—84no—
208C2×C4⋊C4—84no—
209C2×C4⋊C4—84no—
210C2×C4⋊C4C4⋊C418yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D412yes—
214C2×C4⋊C4C2×D412yes—
215C2×C22⋊C4—24no—
216C2×C22⋊C4—24no—
217C2×C22⋊C4—24no—
218C2×C22⋊C4—42no—
219C2×C22⋊C4—42no—
220C2×C22⋊C4—44no—
221C2×C22⋊C4—84no—
222C2×C22⋊C4C4⋊C418yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—21no—
227C2×C42—24no—
228C2×C42—42no—
229C2×C42—42no—
230C2×C42C22×C412yes—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×Q812yes—
235C2×C42C2×Q812yes—
236C2×C42#4112yes—
237C2×C42#4112yes—
238#107—48no—
239#107—48no—
240#107—48no—
241#106—48no—
242C22⋊C8C4⋊C4132yescomplemented
243#97—28no—
244C4×C8—28no—
245#95—28no—
246#95—28no—
247#95—44no—
248#95—44no—
249#95—84no—
250C24—21no—
251C24—21no—
252C24—21no—
253C24—21no—
254C24—24no—
255C24—24no—
256C24C2×C4⋊C412yes—
257C24#12712yes—
258C24C4⋊Q812yes—
259C24C22×D411yescharacteristic
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—24no—
274C22×C4—24no—
275C22×C4—24no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—48no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—44no—
298C22×C4—44no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—44no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—88no—
312C22×C4—88no—
313C22×C4—88no—
314C22×C4C2×C4⋊C414yes—
315C22×C4C2×C4⋊C418yes—
316C22×C4C2×C4⋊C412yes—
317C22×C4C2×C4⋊C414yes—
318C22×C4#12014yes—
319C22×C4#12314yes—
320C22×C4#12712yes—
321C22×C4C4⋊Q818yes—
322C22×C4C4⋊Q812yes—
323C22×C4C23×C411yescharacteristic
324C22×C4C22×D412yes—
325C22×C4C22×D411yescharacteristic
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q812yes—
328C22×C4C22×Q811yescharacteristic
329C22×C4C2×#4111yescharacteristic
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C42−1+412yes—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—24no—
344C2×C8—24no—
345C2×C8C2×C4⋊C418yes—
346C2×C8C2×C4⋊C418yes—
347C4⋊C4—216no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—24no—
352C4⋊C4—48no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—88no—
366C4⋊C4—88no—
367C4⋊C4—88no—
368C22⋊C4—216no—
369C22⋊C4—48no—
370C22⋊C4—88no—
371C42—22no—
372C42—22no—
373C42—22no—
374C42—22no—
375C42—48no—
376C42C4×D414yes—
377C42C4×D414yes—
378C42C22⋊Q814yes—
379C42C22⋊Q814yes—
380C23—22no—
381C23—24no—
382C23—24no—
383C23—28no—
384C23—28no—
385C23—22no—
386C23—22no—
387C23—22no—
388C23—22no—
389C23—22no—
390C23—44no—
391C23—42no—
392C23—42no—
393C23—44no—
394C23—42no—
395C23—42no—
396C23#38414yes—
397C23C22×C4⋊C411yescharacteristic
398C23C2×#12011yescharacteristic
399C23C2×#12311yescharacteristic
400C23C2×#12711yescharacteristic
401C23C2×C4⋊Q811yescharacteristic
402C23#53511yescharacteristic
403C23#53611yescharacteristic
404C23#57614yes—
405C23#57814yes—
406C2×C4—22no—
407C2×C4—28no—
408C2×C4—28no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—44no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—816no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—84no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—82no—
452C2×C4#38414yes—
453C2×C4#38518yes—
454C2×C4C22×C4⋊C412yes—
455C2×C4C2×C4×D411yescharacteristic
456C2×C4C2×C4×D411yescharacteristic
457C2×C4#52012yes—
458C2×C4C2×C22⋊Q811yescharacteristic
459C2×C4C2×C22⋊Q811yescharacteristic
460C2×C4C2×C4⋊Q812yes—
461C2×C4#54412yes—
462C2×C4D4212yes—
463C2×C4D4212yes—
464C2×C4#54712yes—
465C2×C4#54712yes—
466C2×C4D4×Q812yes—
467C2×C4D4×Q812yes—
468C2×C4#55412yes—
469C2×C4#55412yes—
470C8—28no—
471C8—28no—
472C22—24no—
473C22—28no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22—42no—
479C22—44no—
480C22—44no—
481C22C2×#38411yescharacteristic
482C22C2×#38511yescharacteristic
483C22#231711yescharacteristic
484C22#243311yescharacteristic
485C22#243411yescharacteristic
486C22#250111yescharacteristic
487C22#250211yescharacteristic
488C22#291314yes—
489C22#300812yes—
490C22#301112yes—
491C22#309814yes—
492C22#315512yes—
493C22#315612yes—
494C22#316414yes—
495C22#322012yes—
496C22#322312yes—
497C22#324314yes—
498C4—24no—
499C4—28no—
500C4—44no—
501C4—82no—
502C4—82no—
503C4—82no—
504C4—82no—
505C4—162no—
506C4—162no—
507C4D4×C4⋊C412yes—
508C4D4×C4⋊C412yes—
509C4#244012yes—
510C4#244012yes—
511C2—44no—
512C2#1388011yescharacteristic
513C2#2080312yes—
514C2#2084712yes—
515C2#2199112yes—
516C2#2573411yescharacteristic
517C2#2573511yescharacteristic
518C2#2582112yes—
519C2#2792112yes—
520C2#2795812yes—
subgroupG/H
center, socleC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×#117 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. C2×#1847C2semidirect
5. C2×#1845C2semidirect
8. #11325C2semidirect
9. #11324C2semidirect
10. #10843C2semidirect
11. #10842C2semidirect
37. #1750C4semidirect
98. C2×#97C2×C4semidirect
101. C2×C4×C8C2×C4semidirect
127. #336C2×C4semidirect
242. C22⋊C8C4⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23172
C2×#18542
C2×#18522
C2×#18472
C2×#18452
C2×#17502
#143402
#113252
#113242
#108432
#108422