Finite Group Databasebeta

← #365651

Subgroups — #365651

Order 512.

1260 conjugacy classes of subgroups.

The subgroup lattice has 503 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2369C211yesmaximal, characteristic
2C2×#1993C211yesmaximal, characteristic
3C2×#1988C211yesmaximal, characteristic
4C2×#1949C211yesmaximal, characteristic
5#13955C211yesmaximal, characteristic
6#11293C212yesmaximal
7#11292C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#107C2211yescharacteristic
14C22×#106C2211yescharacteristic
15C22×C4⋊C8C2211yescharacteristic
16C22×#97C2211yescharacteristic
17#2369—24no—
18#2369—24no—
19C2×#391—22no—
20C2×#389—21no—
21C2×#389C2212yes—
22C2×#382C2211yescharacteristic
23C2×C4×C4⋊C4—21no—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2211yescharacteristic
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C4⋊C4C2211yescharacteristic
28#1993—24no—
29#1988C418yescomplemented
30#1949—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2212yes—
41#1442—21no—
42#1442—21no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45#1434—22no—
46C23×Q8—21no—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8C2×C414yes—
54C2×C4⋊Q8D414yes—
55C2×C4⋊Q8Q814yes—
56C22×C4⋊C4—21no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4D411yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42—21no—
68C22×C42—22no—
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×#97—22no—
81#391—216no—
82#389—28no—
83#389—24no—
84#389—24no—
85#382—22no—
86#382—22no—
87C4×C4⋊C4—28no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C4×C4⋊C4—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4C2×C414yes—
95C4×C4⋊C4D414yes—
96C2×#95—21no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—21no—
100C2×#95—21no—
101C2×#95—22no—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95D412yes—
107#340—24no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340C2×C418yes—
114#340C2×C418yes—
115#336—24no—
116#336—24no—
117C22×Q8—24no—
118C22×Q8—24no—
119C22×Q8—24no—
120C22×Q8—81no—
121C22×Q8—81no—
122C22×Q8—81no—
123C22×Q8—81no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—21no—
130C23×C4C2×D412yes—
131C23×C4C2×D412yes—
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—22no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—21no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8C22×C412yes—
150C22×C8C22×C412yes—
151C22×C8#4114yes—
152C4⋊Q8—28no—
153C4⋊Q8—28no—
154C4⋊Q8—28no—
155C4⋊Q8C4⋊C4116yes—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—24no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—24no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—42no—
185C2×C4⋊C4—82no—
186C2×C4⋊C4—82no—
187C2×C4⋊C4C22×C414yes—
188C2×C4⋊C4C22×C414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D412yes—
192C2×C4⋊C4C2×Q814yes—
193C2×C4⋊C4#4114yes—
194C2×C42—24no—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—28no—
202C2×C42—24no—
203C2×C42—44no—
204C2×C42—81no—
205C2×C42—81no—
206C2×C42—82no—
207C2×C42—82no—
208C2×C42C22×C412yes—
209C2×C42C2×D412yes—
210C2×C42C2×D412yes—
211C2×C42C2×Q812yes—
212C2×C42#4112yes—
213C2×C42#4112yes—
214#107—44no—
215#106—44no—
216C4⋊C8—28no—
217#97—44no—
218#95—28no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—44no—
224#95—42no—
225#95—42no—
226#95—44no—
227#95—82no—
228#95—82no—
229C24C22×D411yescharacteristic
230C2×Q8—216no—
231C2×Q8—84no—
232C2×Q8—84no—
233C2×Q8—84no—
234C2×Q8—84no—
235C2×Q8—84no—
236C2×Q8—84no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—41no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—42no—
273C22×C4—42no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4C4×D412yes—
291C22×C4C4×D412yes—
292C22×C4#12012yes—
293C22×C4#12012yes—
294C22×C4C22⋊Q812yes—
295C22×C4C22⋊Q812yes—
296C22×C4#13714yes—
297C22×C4C23×C411yescharacteristic
298C22×C4C22×D411yescharacteristic
299C22×C4C22×D411yescharacteristic
300C22×C4C22×Q811yescharacteristic
301C22×C4C2×#4111yescharacteristic
302C22×C4C2×#4112yes—
303C22×C4C2×#4112yes—
304C22×C4C2×#4111yescharacteristic
305C22×C42−1+412yes—
306C22×C42−1+412yes—
307C2×C8—22no—
308C2×C8—22no—
309C2×C8—24no—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—42no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—42no—
318C2×C8—42no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C4⋊C4—28no—
325C4⋊C4—24no—
326C4⋊C4—24no—
327C4⋊C4—24no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—216no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—44no—
336C4⋊C4—42no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—88no—
346C4⋊C4C2×C4⋊C418yes—
347C4⋊C4#12414yes—
348C42—48no—
349C42—42no—
350C42—42no—
351C42—42no—
352C42—42no—
353C42—84no—
354C42—88no—
355C42C2×C4⋊C414yes—
356C42C4×D414yes—
357C42C22⋊Q814yes—
358C42#12314yes—
359C23C2×C4×D411yescharacteristic
360C23C2×#12011yescharacteristic
361C23C2×C22⋊Q811yescharacteristic
362C23#54712yes—
363C23#54812yes—
364C23C2×#13712yes—
365C23#57512yes—
366C23#57611yescharacteristic
367C23#57611yescharacteristic
368C23#57711yescharacteristic
369C23#57711yescharacteristic
370Q8—816no—
371C2×C4—24no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—24no—
382C2×C4—24no—
383C2×C4—24no—
384C2×C4—24no—
385C2×C4—28no—
386C2×C4—44no—
387C2×C4—44no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—42no—
393C2×C4—44no—
394C2×C4—44no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—82no—
414C2×C4—82no—
415C2×C4—82no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—81no—
424C2×C4—81no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4#38514yes—
430C2×C4#38514yes—
431C2×C4C22×C4⋊C411yescharacteristic
432C2×C4C2×C4×D411yescharacteristic
433C2×C4C4×#4112yes—
434C2×C4#51912yes—
435C2×C4C2×C22⋊Q811yescharacteristic
436C2×C4C2×#12311yescharacteristic
437C2×C4C2×#12411yescharacteristic
438C2×C4#52912yes—
439C2×C4#53211yescharacteristic
440C2×C4#53611yescharacteristic
441C2×C4#53611yescharacteristic
442C2×C4#54112yes—
443C2×C4#54812yes—
444C2×C4#54812yes—
445C2×C4#55212yes—
446C2×C4#55612yes—
447C2×C4#55712yes—
448C2×C4#55712yes—
449C8—44no—
450C8—42no—
451C8—42no—
452C22C2×#38511yescharacteristic
453C22#236212yes—
454C22#243211yescharacteristic
455C22#243611yescharacteristic
456C22#245312yes—
457C22C4×#13712yes—
458C22#294811yescharacteristic
459C22#294911yescharacteristic
460C22#300411yescharacteristic
461C22#300511yescharacteristic
462C22#301211yescharacteristic
463C22#301211yescharacteristic
464C22#307812yes—
465C22#308111yescharacteristic
466C22#308111yescharacteristic
467C22#334412yes—
468C22#334612yes—
469C22#334712yes—
470C22#334812yes—
471C22#336912yes—
472C22#337012yes—
473C22#337512yes—
474C22#337612yes—
475C4—24no—
476C4—24no—
477C4—44no—
478C4—44no—
479C4—48no—
480C4—84no—
481C4—82no—
482C4—82no—
483C4—161no—
484C4—161no—
485C4C4×#12411yescharacteristic
486C4#232711yescharacteristic
487C4#235112yes—
488C4#242112yes—
489C4#256511yescharacteristic
490C4#256811yescharacteristic
491C2#1397811yescharacteristic
492C2#2087411yescharacteristic
493C2#2087911yescharacteristic
494C2#2307712yes—
495C2#2307912yes—
496C2#2571011yescharacteristic
497C2#2571211yescharacteristic
498C2#2594011yescharacteristic
499C2#2594211yescharacteristic
500C2#2651912yes—
501C2#2652012yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×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
29. #1988C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23692
C2×#19932
C2×#19882
C2×#19492
#139552
#112932
#112922
#109082
#109072
#108352
#107032