Finite Group Databasebeta

← #418103

Subgroups — #418103

Order 512.

1966 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2487C211yesmaximal, characteristic
3C2×#2043C212yesmaximal, complemented
4C2×#1872C212yesmaximal, complemented
5#10970C212yesmaximal, complemented
6#10970C214yesmaximal, complemented
7#10964C211yesmaximal, characteristic
8#10885C212yesmaximal, complemented
9C22×C4⋊Q8C2211yescharacteristic
10C22×C22⋊Q8C2212yes—
11C22×C22⋊Q8C2212yes—
12C22×C4×D4C2211yescharacteristic
13C23×C4⋊C4C2211yescharacteristic
14C22×#103C2214yes—
15C22×#103C2214yes—
16C22×C22⋊C8C2212yes—
17C22×C22⋊C8C2212yes—
18#2542—28no—
19#2487—28no—
20C2×#400—21no—
21C2×#399—21no—
22C2×#399—21no—
23C2×#399—21no—
24C2×#390—21no—
25C2×#389C2212yes—
26C2×#386—21no—
27C2×#385—21no—
28C2×#384—21no—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×C4×C4⋊C4C2211yescharacteristic
32#2043—28no—
33#1872—28no—
34C2×#340C2214yes—
35C2×#340C2214yes—
36C2×#336C2212yes—
37C23×Q8C2312yes—
38C23×D4D411yescharacteristic
39C24×C4—21no—
40C24×C4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C23×C8C2314yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C22⋊Q8—24no—
47C2×C22⋊Q8—24no—
48C2×C22⋊Q8—24no—
49C2×C22⋊Q8—28no—
50C2×C22⋊Q8—28no—
51C2×C22⋊Q8—44no—
52C2×C22⋊Q8—44no—
53C2×C22⋊Q8D418yes—
54C2×C4×D4—24no—
55C2×C4×D4—24no—
56C2×C4×D4—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—22no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—24no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—42no—
66C22×C4⋊C4D411yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2312yes—
69C22×C4⋊C4C2312yes—
70C22×C22⋊C4—22no—
71C22×C22⋊C4—22no—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4D411yescharacteristic
75C22×C22⋊C4D411yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C2×#103—28no—
79C2×#103—28no—
80C2×#103—28no—
81C2×#103—44no—
82C2×#103—44no—
83C2×#103—44no—
84C2×C22⋊C8—24no—
85C2×C22⋊C8—44no—
86C2×C22⋊C8—44no—
87C2×C22⋊C8—42no—
88#400—28no—
89#399—28no—
90#399—28no—
91#399—28no—
92#390—28no—
93#389—24no—
94#389D418yes—
95#386—28no—
96#385—28no—
97#384—28no—
98#380—24no—
99#380—24no—
100#380—24no—
101C4×C4⋊C4—24no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106#340—28no—
107#340—28no—
108#336—24no—
109C25C2×D411yescharacteristic
110C25C2×D411yescharacteristic
111C22×Q8—22no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—24no—
116C22×Q8—24no—
117C22×Q8—24no—
118C22×Q8C2×D414yes—
119C22×D4—22no—
120C22×D4—22no—
121C22×D4—22no—
122C22×D4—22no—
123C22×D4—41no—
124C22×D4—41no—
125C22×D4Q1614yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4—22no—
130C23×C4—24no—
131C23×C4—22no—
132C23×C4—22no—
133C23×C4—22no—
134C23×C4—24no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—24no—
138C23×C4—22no—
139C23×C4—24no—
140C23×C4—42no—
141C23×C4—42no—
142C23×C4—84no—
143C23×C4C2×D411yescharacteristic
144C23×C4C2×D411yescharacteristic
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D414yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2411yescharacteristic
149C22×C8—24no—
150C22×C8—24no—
151C22×C8—24no—
152C22×C8—24no—
153C22×C8—24no—
154C22×C8—24no—
155C22×C8—24no—
156C4⋊Q8—416no—
157C22⋊Q8—216no—
158C22⋊Q8—48no—
159C22⋊Q8—416no—
160C22⋊Q8—416no—
161C22⋊Q8—48no—
162C22⋊Q8—48no—
163C4×D4—416no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—28no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—28no—
180C2×C4⋊C4—416no—
181C2×C4⋊C4—41no—
182C2×C4⋊C4—48no—
183C2×C4⋊C4—48no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—84no—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4#4118yes—
195C2×C22⋊C4—28no—
196C2×C22⋊C4—24no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—28no—
203C2×C22⋊C4—41no—
204C2×C22⋊C4—41no—
205C2×C22⋊C4—48no—
206C2×C22⋊C4—48no—
207C2×C22⋊C4—42no—
208C2×C22⋊C4—42no—
209C2×C22⋊C4—84no—
210C2×C42—22no—
211C2×C42—22no—
212C2×C42—24no—
213C2×C42—42no—
214C2×C42—42no—
215C2×C42C2×D414yes—
216#103—416no—
217#103—416no—
218C22⋊C8—48no—
219C22⋊C8—48no—
220#95—24no—
221#95—28no—
222#95—24no—
223#95—28no—
224C24—21no—
225C24—21no—
226C24—24no—
227C24—21no—
228C24—21no—
229C24—24no—
230C24—21no—
231C24—21no—
232C24—21no—
233C24—24no—
234C24—21no—
235C24—21no—
236C24—21no—
237C24—24no—
238C24#12012yes—
239C24C2×Q1612yes—
240C24C2×Q1612yes—
241C24#13712yes—
242C24C22×D411yescharacteristic
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—28no—
246C2×Q8—28no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—44no—
250C2×Q8—44no—
251C2×Q8—44no—
252C2×Q8—44no—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—44no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×D4—28no—
260C2×D4—28no—
261C2×D4—44no—
262C2×D4—44no—
263C2×D4—44no—
264C2×D4—44no—
265C2×D4—44no—
266C22×C4—21no—
267C22×C4—21no—
268C22×C4—21no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—24no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—28no—
276C22×C4—24no—
277C22×C4—24no—
278C22×C4—24no—
279C22×C4—24no—
280C22×C4—28no—
281C22×C4—28no—
282C22×C4—21no—
283C22×C4—24no—
284C22×C4—24no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—44no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—44no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—48no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—44no—
305C22×C4—44no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—44no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—44no—
316C22×C4—44no—
317C22×C4—44no—
318C22×C4—44no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4—88no—
322C22×C4—88no—
323C22×C4—88no—
324C22×C4#12012yes—
325C22×C4#12012yes—
326C22×C4#12012yes—
327C22×C4C2×Q1612yes—
328C22×C4#13712yes—
329C22×C4C22×D412yes—
330C22×C4C22×D412yes—
331C22×C4C2×#4114yes—
332C22×C4H(5,𝔽2)14yes—
333C22×C4H(5,𝔽2)12yes—
334C2×C8—44no—
335C2×C8—44no—
336C2×C8—44no—
337C2×C8—44no—
338C2×C8—44no—
339C2×C8—44no—
340C2×C8—44no—
341C4⋊C4—24no—
342C4⋊C4—28no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—28no—
346C4⋊C4—44no—
347C4⋊C4—416no—
348C4⋊C4—48no—
349C4⋊C4—416no—
350C4⋊C4—48no—
351C4⋊C4—48no—
352C4⋊C4—44no—
353C4⋊C4—44no—
354C4⋊C4—88no—
355C4⋊C4#12318yes—
356C22⋊C4—44no—
357C22⋊C4—44no—
358C22⋊C4—416no—
359C22⋊C4—416no—
360C22⋊C4—48no—
361C22⋊C4—88no—
362C42—28no—
363C42—48no—
364C23—22no—
365C23—24no—
366C23—22no—
367C23—22no—
368C23—22no—
369C23—22no—
370C23—22no—
371C23—24no—
372C23—28no—
373C23—28no—
374C23—22no—
375C23—22no—
376C23—22no—
377C23—28no—
378C23—44no—
379C23—42no—
380C23—42no—
381C23—42no—
382C23—42no—
383C23—44no—
384C23—44no—
385C23—41no—
386C23—42no—
387C23—41no—
388C23—42no—
389C23—42no—
390C23—42no—
391C23—41no—
392C23—41no—
393C23—42no—
394C23—42no—
395C23—44no—
396C23—44no—
397C23#45114yes—
398C23C2×#12011yescharacteristic
399C23C2×#12011yescharacteristic
400C23#53411yescharacteristic
401C23#54614yes—
402C23C22×Q1611yescharacteristic
403C23C2×#13711yescharacteristic
404C23C2×#13711yescharacteristic
405C23#57511yescharacteristic
406C23#57714yes—
407Q8—48no—
408Q8—48no—
409Q8—84no—
410Q8—84no—
411D4—416no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—28no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—48no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—44no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—48no—
431C2×C4—48no—
432C2×C4—48no—
433C2×C4—41no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—44no—
437C2×C4—44no—
438C2×C4—42no—
439C2×C4—44no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—44no—
443C2×C4—44no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—816no—
447C2×C4—84no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4#45114yes—
456C2×C4C2×#12312yes—
457C2×C4#53212yes—
458C2×C4#53412yes—
459C2×C4#53512yes—
460C2×C4#54014yes—
461C2×C4D4214yes—
462C2×C4#54614yes—
463C2×C4#55014yes—
464C2×C4#55912yes—
465C2×C4#56012yes—
466C8—84no—
467C22—24no—
468C22—28no—
469C22—44no—
470C22—44no—
471C22—42no—
472C22—42no—
473C22—44no—
474C22—44no—
475C22—44no—
476C22—44no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—82no—
481C22—81no—
482C22—81no—
483C22—82no—
484C22—82no—
485C22#240311yescharacteristic
486C22#240512yes—
487C22#243112yes—
488C22#270912yes—
489C22C2×#45111yescharacteristic
490C22C2×#45111yescharacteristic
491C22#300711yescharacteristic
492C22#300711yescharacteristic
493C22#300912yes—
494C22#301012yes—
495C22#318911yescharacteristic
496C22#319511yescharacteristic
497C22#320012yes—
498C22#329914yes—
499C22#330714yes—
500C22#332014yes—
501C22#333114yes—
502C4—24no—
503C4—42no—
504C4—42no—
505C4—48no—
506C4—84no—
507C4—81no—
508C4—84no—
509C4—84no—
510C4#244514yes—
511C4#267314yes—
512C2—44no—
513C2—82no—
514C2#1526111yescharacteristic
515C2#2392111yescharacteristic
516C2#2392611yescharacteristic
517C2#2404212yes—
518C2#2405512yes—
519C2#2557312yes—
520C2#2558212yes—
521C2#3202912yes—
522C2#3203712yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×D4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. C2×#2043C2semidirect
4. C2×#1872C2semidirect
5. #10970C2semidirect
6. #10970C2semidirect
8. #10885C2semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#24872
C2×#20432
C2×#18722
#109702
#109642
#108852