Finite Group Databasebeta

← #365695

Subgroups — #365695

Order 512.

1238 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2355C211yesmaximal, characteristic
2C2×#1985C211yesmaximal, characteristic
3C2×#1949C211yesmaximal, characteristic
4C2×#1836C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic
6#11268C212yesmaximal
7#11268C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2355—24no—
18#2355—24no—
19C2×#391—21no—
20C2×#387—22no—
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×#382C2211yescharacteristic
24C2×C4×C4⋊C4—21no—
25C2×C4×C4⋊C4C2212yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1985C418yescomplemented
29#1949—24no—
30#1836—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1446—21no—
42#1445—21no—
43#1436—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8D411yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
50C2×C4×Q8—22no—
51C2×C4×Q8—22no—
52C2×C4×Q8—22no—
53C2×C4×Q8—22no—
54C2×C4×Q8C2×C414yes—
55C2×C4×Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4D412yes—
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—22no—
68C22×C42—21no—
69C22×C42C2312yes—
70C22×C42C2311yescharacteristic
71C2×#107—22no—
72C2×#107—24no—
73C2×#106—22no—
74C2×#106—24no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8C2×C414yes—
78C2×#97—22no—
79C2×#97—22no—
80C2×#97C2×C414yes—
81#391—28no—
82#387—216no—
83#384—22no—
84#384D414yes—
85#382—24no—
86#382—24no—
87#382—22no—
88#382—22no—
89C4×C4⋊C4—28no—
90C4×C4⋊C4—24no—
91C4×C4⋊C4—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4D414yes—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95—22no—
106C2×#95—22no—
107#340—24no—
108#340—24no—
109#340—22no—
110#340—22no—
111#340—24no—
112#340—24no—
113#340—24no—
114#340—24no—
115#336C2×C418yes—
116#336C2×C418yes—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—42no—
120C22×Q8—42no—
121C22×Q8C4⋊C414yes—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4—22no—
126C23×C4—21no—
127C23×C4—21no—
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—22no—
134C22×C8—21no—
135C22×C8—21no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8C22×C414yes—
148C22×C8C22×C412yes—
149C22×C8C22×C412yes—
150C4×Q8—28no—
151C4×Q8—28no—
152C4×Q8—28no—
153C4×Q8—28no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
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—24no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—81no—
185C2×C4⋊C4—81no—
186C2×C4⋊C4C4⋊C414yes—
187C2×C4⋊C4C4⋊C418yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D412yes—
191C2×C4⋊C4#4114yes—
192C2×C4⋊C4#4114yes—
193C2×C42—22no—
194C2×C42—22no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—28no—
199C2×C42—44no—
200C2×C42—42no—
201C2×C42—81no—
202C2×C42—81no—
203C2×C42—82no—
204C2×C42—82no—
205C2×C42C22×C414yes—
206C2×C42C22×C412yes—
207C2×C42C2×D412yes—
208C2×C42C2×D412yes—
209C2×C42C2×Q814yes—
210C2×C42C2×Q812yes—
211C2×C42#4112yes—
212C2×C42#4112yes—
213#107—44no—
214#106—44no—
215C4⋊C8—28no—
216#97—28no—
217#95—28no—
218#95—28no—
219#95—28no—
220#95—24no—
221#95—44no—
222#95—44no—
223#95—42no—
224#95—42no—
225#95—42no—
226#95—82no—
227#95—82no—
228C24C22×D411yescharacteristic
229C2×Q8—28no—
230C2×Q8—48no—
231C2×Q8—48no—
232C2×Q8—44no—
233C2×Q8—44no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—22no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—42no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4C2×C4⋊C414yes—
287C22×C4C2×C4⋊C414yes—
288C22×C4C2×C4⋊C412yes—
289C22×C4C2×C4⋊C412yes—
290C22×C4#12012yes—
291C22×C4#12012yes—
292C22×C4#12312yes—
293C22×C4#12312yes—
294C22×C4C23×C411yescharacteristic
295C22×C4C22×D411yescharacteristic
296C22×C4C22×D411yescharacteristic
297C22×C4C22×Q811yescharacteristic
298C22×C4C2×#4111yescharacteristic
299C22×C4C2×#4111yescharacteristic
300C22×C4C2×#4112yes—
301C22×C4C2×#4112yes—
302C22×C42−1+412yes—
303C22×C42−1+412yes—
304C2×C8—24no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—24no—
308C2×C8—24no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—41no—
312C2×C8—41no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—42no—
320C2×C8—42no—
321C4⋊C4—22no—
322C4⋊C4—24no—
323C4⋊C4—28no—
324C4⋊C4—216no—
325C4⋊C4—24no—
326C4⋊C4—22no—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—48no—
332C4⋊C4—42no—
333C4⋊C4—42no—
334C4⋊C4—42no—
335C4⋊C4—42no—
336C4⋊C4—42no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—84no—
344C4⋊C4#12414yes—
345C42—24no—
346C42—24no—
347C42—24no—
348C42—24no—
349C42—48no—
350C42—84no—
351C42—88no—
352C42C4×D414yes—
353C42C4×D414yes—
354C42C22⋊Q814yes—
355C42C22⋊Q814yes—
356C23C22×C4⋊C411yescharacteristic
357C23C2×#12011yescharacteristic
358C23C2×#12311yescharacteristic
359C23#54812yes—
360C23#54812yes—
361C23#57512yes—
362C23#57512yes—
363C23#57611yescharacteristic
364C23#57711yescharacteristic
365C23#57711yescharacteristic
366C23#57811yescharacteristic
367Q8—416no—
368C2×C4—24no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—24no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—44no—
390C2×C4—44no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—44no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—82no—
407C2×C4—82no—
408C2×C4—82no—
409C2×C4—82no—
410C2×C4—82no—
411C2×C4—82no—
412C2×C4—82no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4#38514yes—
427C2×C4#38514yes—
428C2×C4C2×C4×D411yescharacteristic
429C2×C4C2×C4×D411yescharacteristic
430C2×C4C4×#4112yes—
431C2×C4#51912yes—
432C2×C4C2×C22⋊Q811yescharacteristic
433C2×C4C2×C22⋊Q811yescharacteristic
434C2×C4C2×#12411yescharacteristic
435C2×C4#52912yes—
436C2×C4#53211yescharacteristic
437C2×C4#53611yescharacteristic
438C2×C4#53611yescharacteristic
439C2×C4#54112yes—
440C2×C4#54712yes—
441C2×C4#54812yes—
442C2×C4#55212yes—
443C2×C4#55612yes—
444C2×C4#55712yes—
445C2×C4#55712yes—
446C8—44no—
447C8—42no—
448C8—42no—
449C22C2×#38511yescharacteristic
450C22#235112yes—
451C22#242112yes—
452C22#243211yescharacteristic
453C22#243611yescharacteristic
454C22#291112yes—
455C22#291211yescharacteristic
456C22#291311yescharacteristic
457C22#300511yescharacteristic
458C22#300511yescharacteristic
459C22#301211yescharacteristic
460C22#301311yescharacteristic
461C22#309312yes—
462C22#309611yescharacteristic
463C22#309711yescharacteristic
464C22#337012yes—
465C22#337012yes—
466C22#337112yes—
467C22#337112yes—
468C22#337612yes—
469C22#337612yes—
470C22#337712yes—
471C22#337712yes—
472C4—24no—
473C4—24no—
474C4—44no—
475C4—44no—
476C4—44no—
477C4—82no—
478C4—84no—
479C4—84no—
480C4—161no—
481C4—161no—
482C4C4×#12411yescharacteristic
483C4#232711yescharacteristic
484C4#236212yes—
485C4#245312yes—
486C4#256511yescharacteristic
487C4#256811yescharacteristic
488C2#1397811yescharacteristic
489C2#2084511yescharacteristic
490C2#2084911yescharacteristic
491C2#2274412yes—
492C2#2274512yes—
493C2#2500312yes—
494C2#2500512yes—
495C2#2565611yescharacteristic
496C2#2565811yescharacteristic
497C2#2594011yescharacteristic
498C2#2594111yescharacteristic
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1985C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23552
C2×#19852
C2×#19492
C2×#18362
#139252
#112682
#109082
#109072
#108492
#107032