Finite Group Databasebeta

← #432141

Subgroups — #432141

Order 512.

1586 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×#2567C211yesmaximal, characteristic
2C2×#2024C212yesmaximal
3C2×#2016C211yesmaximal, characteristic
4C2×#1956C211yesmaximal, characteristic
5C2×#1878C212yesmaximal
6#15180C211yesmaximal, characteristic
7#11371C211yesmaximal, characteristic
8#11050C212yesmaximal
9#11024C212yesmaximal
10#10926C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C23×Q16C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2567—24no—
22#2567—24no—
23C2×#395—22no—
24C2×#391C2211yescharacteristic
25C2×#391C2212yes—
26C2×#391C2212yes—
27C2×#391C2211yescharacteristic
28C2×#389—22no—
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2024—28no—
33#2016—24no—
34#1956—24no—
35#1878—28no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1442—22no—
43#1441—22no—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2312yes—
47C23×Q8C2311yescharacteristic
48C22×Q16—24no—
49C22×Q16—24no—
50C22×Q16—24no—
51C22×Q16—24no—
52C22×Q16—24no—
53C22×Q16—42no—
54C22×Q16—42no—
55C23×C8C2312yes—
56C23×C8C2312yes—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—24no—
60C2×C4⋊Q8—24no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—42no—
64C2×C4⋊Q8—42no—
65C2×C4×Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—24no—
68C2×C4×Q8—42no—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4—22no—
74C22×C4⋊C4D412yes—
75C22×C4⋊C4C2312yes—
76C22×C4⋊C4C2311yescharacteristic
77C22×C4⋊C4C2311yescharacteristic
78C22×C4⋊C4C2311yescharacteristic
79C22×C42—21no—
80C22×C42C2311yescharacteristic
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—42no—
92C2×#103—42no—
93C2×#103—42no—
94C2×#103—42no—
95C2×C4×C8—22no—
96C2×C4×C8—42no—
97#395—216no—
98#391—22no—
99#391—22no—
100#391—22no—
101#391—24no—
102#391—24no—
103#391—24no—
104#391—42no—
105#391D414yes—
106#389—216no—
107#384—22no—
108#384—22no—
109#384—22no—
110#384D414yes—
111C4×C4⋊C4—22no—
112C4×C4⋊C4—22no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95—22no—
116C2×#95—22no—
117C2×#95—24no—
118C2×#95—22no—
119C2×#95D412yes—
120#340—24no—
121#340—24no—
122#340—24no—
123#336—22no—
124#336—42no—
125#336D418yes—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—42no—
147C22×Q8—42no—
148C22×Q8—84no—
149C22×Q8C2×D412yes—
150C22×Q8#4114yes—
151C22×Q8#4114yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4—22no—
158C23×C4C2×D411yescharacteristic
159C23×C4C2×D412yes—
160C23×C4C2×D412yes—
161C23×C4C2×D411yescharacteristic
162C23×C4C2411yescharacteristic
163C2×Q16—216no—
164C2×Q16—48no—
165C2×Q16—48no—
166C2×Q16—48no—
167C2×Q16—48no—
168C2×Q16—48no—
169C2×Q16—48no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8—22no—
175C22×C8—22no—
176C22×C8—22no—
177C22×C8—22no—
178C22×C8—22no—
179C22×C8—22no—
180C22×C8—22no—
181C22×C8—22no—
182C22×C8C2×D414yes—
183C22×C8C2×D414yes—
184C4⋊Q8—28no—
185C4⋊Q8—48no—
186C4⋊Q8—48no—
187C4⋊Q8—48no—
188C4⋊Q8—44no—
189C4×Q8—48no—
190C4×Q8—48no—
191C2×C4⋊C4—24no—
192C2×C4⋊C4—24no—
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—24no—
200C2×C4⋊C4—24no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—22no—
209C2×C4⋊C4—22no—
210C2×C4⋊C4—28no—
211C2×C4⋊C4—28no—
212C2×C4⋊C4—48no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—42no—
217C2×C4⋊C4—42no—
218C2×C4⋊C4—84no—
219C2×C4⋊C4—84no—
220C2×C4⋊C4Q1618yescomplemented
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4C2×D412yes—
223C2×C4⋊C4C2×D412yes—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—22no—
229C2×C42—24no—
230C2×C42—21no—
231C2×C42—21no—
232C2×C42—22no—
233C2×C42—22no—
234C2×C42—42no—
235C2×C42—42no—
236C2×C42C2×D412yes—
237C2×C42C2×D412yes—
238C2×C42C2×D414yes—
239#103—48no—
240#103—48no—
241#103—48no—
242#103—48no—
243C4×C8—44no—
244#95—28no—
245#95—28no—
246#95—24no—
247#95—48no—
248#95—44no—
249#95—44no—
250#95—44no—
251C24C22×D411yescharacteristic
252C2×Q8—22no—
253C2×Q8—24no—
254C2×Q8—24no—
255C2×Q8—24no—
256C2×Q8—24no—
257C2×Q8—24no—
258C2×Q8—24no—
259C2×Q8—24no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—48no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8—42no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—42no—
274C2×Q8—42no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—88no—
297C2×Q8—88no—
298C2×Q8—88no—
299C2×Q8#12414yes—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—24no—
303C22×C4—22no—
304C22×C4—24no—
305C22×C4—22no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—24no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—24no—
313C22×C4—24no—
314C22×C4—22no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—24no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—44no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—42no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—44no—
340C22×C4—44no—
341C22×C4#12014yes—
342C22×C4#12712yes—
343C22×C4#12712yes—
344C22×C4C2×Q1614yes—
345C22×C4C2×Q1614yes—
346C22×C4C2×Q1614yes—
347C22×C4#13714yes—
348C22×C4C22×D411yescharacteristic
349C22×C4C22×D412yes—
350C22×C4C22×D411yescharacteristic
351C22×C4C2×#4112yes—
352C22×C4C2×#4112yes—
353C22×C4H(5,𝔽2)12yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C42−1+412yes—
356Q16—432no—
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×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—216no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—48no—
376C4⋊C4—48no—
377C4⋊C4—416no—
378C4⋊C4—42no—
379C4⋊C4—42no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—88no—
386C4⋊C4—88no—
387C42—22no—
388C42—22no—
389C42—24no—
390C42—24no—
391C42—24no—
392C42—48no—
393C42—41no—
394C42—41no—
395C42—42no—
396C23C2×#12011yescharacteristic
397C23C2×#12711yescharacteristic
398C23#53511yescharacteristic
399C23#54612yes—
400C23#54812yes—
401C23C22×Q1612yes—
402C23C2×#13712yes—
403C23#57714yes—
404Q8—28no—
405Q8—44no—
406Q8—44no—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—816no—
413Q8—82no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—24no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—28no—
424C2×C4—28no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—24no—
428C2×C4—24no—
429C2×C4—28no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—44no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—44no—
447C2×C4—44no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—82no—
457C2×C4—82no—
458C2×C4—84no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4#45118yes—
464C2×C4#49414yes—
465C2×C4#52511yescharacteristic
466C2×C4#52512yes—
467C2×C4C2×#12411yescharacteristic
468C2×C4#53511yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53712yes—
471C2×C4#53812yes—
472C2×C4#53912yes—
473C2×C4D4212yes—
474C2×C4D4212yes—
475C2×C4#54612yes—
476C2×C4#54612yes—
477C2×C4#54812yes—
478C2×C4#54812yes—
479C2×C4#55112yes—
480C2×C4#56212yes—
481C8—44no—
482C8—44no—
483C22#240411yescharacteristic
484C22#243311yescharacteristic
485C22#243611yescharacteristic
486C22#254012yes—
487C22#269612yes—
488C22C2×#45112yes—
489C22#300912yes—
490C22C2×#49411yescharacteristic
491C22#315211yescharacteristic
492C22#315412yes—
493C22#320912yes—
494C22#322212yes—
495C22#329914yes—
496C22#331614yes—
497C22#336314yes—
498C22#336914yes—
499C4—24no—
500C4—28no—
501C4—44no—
502C4—42no—
503C4—44no—
504C4—84no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—162no—
509C4#246312yes—
510C4#247212yes—
511C4#267612yes—
512C4#268112yes—
513C2#1545311yescharacteristic
514C2#2397012yes—
515C2#2572211yescharacteristic
516C2#2573211yescharacteristic
517C2#2593612yes—
518C2#2840112yes—
519C2#2843212yes—
520C2#3164014yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
220. C2×C4⋊C4Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#25672
C2×#20242
C2×#20162
C2×#19562
C2×#18782
#151802
#113712
#110502
#110242
#109262
#108742