Finite Group Databasebeta

← #368925

Subgroups — #368925

Order 512.

1460 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C8×Q8C211yesmaximal, characteristic
2C2×#2584C211yesmaximal, characteristic
3C2×#1991C211yesmaximal, characteristic
4C2×#1989C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic
6#11228C212yesmaximal
7#11227C212yesmaximal
8#10880C212yesmaximal
9#10880C212yesmaximal
10#10850C211yesmaximal, characteristic
11C4×#340C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×C8×Q8—24no—
15C2×C8×Q8—24no—
16C2×C8×Q8—24no—
17C22×C4⋊C8C2212yes—
18C22×C4⋊C8C2211yescharacteristic
19C22×C4×C8C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2584—24no—
22#2584C418yescomplemented
23C2×#398—21no—
24C2×#395—21no—
25C2×#389—22no—
26C2×#389C2212yes—
27C2×#389C2212yes—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1991—24no—
32#1989—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#340C2212yes—
42C2×#340C2212yes—
43#1446—21no—
44#1445—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8D411yescharacteristic
49C23×C8C2312yes—
50C23×C8C2311yescharacteristic
51C23×C8C2311yescharacteristic
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125C2×C414yes—
58C2×C4×Q8—22no—
59C2×C4×Q8—22no—
60C2×C4×Q8—22no—
61C2×C4×Q8—22no—
62C2×C4×Q8C2×C414yes—
63C2×C4×Q8D414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D412yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2312yes—
76C22×C42—21no—
77C22×C42—21no—
78C22×C42C2312yes—
79C22×C42C2311yescharacteristic
80C8×Q8—232no—
81C2×C4⋊C8—24no—
82C2×C4⋊C8—24no—
83C2×C4⋊C8—24no—
84C2×C4⋊C8—22no—
85C2×C4⋊C8—22no—
86C2×C4⋊C8D414yes—
87C2×C4×C8—24no—
88C2×C4×C8—24no—
89C2×C4×C8—24no—
90C2×C4×C8—22no—
91C2×C4×C8—22no—
92C2×C4×C8D414yes—
93#398—28no—
94#395—28no—
95#389—216no—
96#389—24no—
97#389—24no—
98#389C2×C418yes—
99#389C2×C418yes—
100#382—22no—
101#382—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4C2×C414yes—
106C2×#95—21no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—22no—
115C2×#95—22no—
116#340—24no—
117#340—24no—
118#340—22no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—24no—
123#340—24no—
124#340—24no—
125#340Q818yescomplemented
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8C22⋊C414yes—
131C23×C4—22no—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—22no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4C2×D412yes—
138C23×C4C2×D411yescharacteristic
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D412yes—
141C23×C4C2411yescharacteristic
142C22×C8—21no—
143C22×C8—21no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8C2×D414yes—
156C22×C8C2×D412yes—
157C22×C8C2×D412yes—
158C22×C8C2×Q814yes—
159#125—28no—
160#125—28no—
161#125—44no—
162#125—44no—
163C4×Q8—28no—
164C4×Q8—28no—
165C4×Q8—28no—
166C4×Q8—28no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
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—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—28no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—28no—
201C2×C4⋊C4—42no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—44no—
206C2×C4⋊C4—44no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4—82no—
209C2×C4⋊C4C22⋊C418yes—
210C2×C4⋊C4C22⋊C414yes—
211C2×C4⋊C4C22×C412yes—
212C2×C4⋊C4C22×C412yes—
213C2×C4⋊C4C22×C414yes—
214C2×C4⋊C4C2×Q814yes—
215C2×C4⋊C4C2×Q814yes—
216C2×C4⋊C4#4114yes—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—24no—
222C2×C42—42no—
223C2×C42—81no—
224C2×C42—81no—
225C2×C42—81no—
226C2×C42—81no—
227C2×C42C22×C414yes—
228C2×C42C22×C412yes—
229C2×C42C2×D414yes—
230C2×C42C2×D412yes—
231C2×C42C2×D412yes—
232C2×C42C2×D412yes—
233C2×C42#4112yes—
234C2×C42#4112yes—
235C4⋊C8—216no—
236C4⋊C8—28no—
237C4×C8—216no—
238C4×C8—28no—
239#95—24no—
240#95—28no—
241#95—28no—
242#95—42no—
243#95—42no—
244#95—42no—
245#95—44no—
246#95—44no—
247#95—82no—
248#95—82no—
249C24C22×D411yescharacteristic
250C2×Q8—28no—
251C2×Q8—28no—
252C2×Q8—216no—
253C2×Q8—28no—
254C2×Q8—216no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—24no—
277C22×C4—42no—
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×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4C2×C22⋊C414yes—
308C22×C4C2×C22⋊C412yes—
309C22×C4C2×C22⋊C412yes—
310C22×C4C2×C22⋊C414yes—
311C22×C4#12112yes—
312C22×C4#12112yes—
313C22×C4#12112yes—
314C22×C4#12112yes—
315C22×C4#13514yes—
316C22×C4#13514yes—
317C22×C4C23×C411yescharacteristic
318C22×C4C22×D411yescharacteristic
319C22×C4C22×D411yescharacteristic
320C22×C4C22×D411yescharacteristic
321C22×C4C22×Q812yes—
322C22×C4C2×#4111yescharacteristic
323C22×C4C2×#4111yescharacteristic
324C22×C4C2×#4112yes—
325C22×C42−1+412yes—
326C22×C42−1+412yes—
327C2×C8—22no—
328C2×C8—22no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—22no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—42no—
342C2×C8—42no—
343C2×C8C4⋊Q814yes—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—216no—
348C4⋊C4—28no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—22no—
352C4⋊C4—22no—
353C4⋊C4—24no—
354C4⋊C4—24no—
355C4⋊C4—41no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—41no—
359C4⋊C4—42no—
360C4⋊C4—42no—
361C4⋊C4—48no—
362C4⋊C4—48no—
363C4⋊C4—416no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—41no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—88no—
371C4⋊C4C4×Q818yes—
372C42—24no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—84no—
377C42—84no—
378C42C4×D414yes—
379C42C4×D414yes—
380C42#12014yes—
381C42#12314yes—
382C23C22×C22⋊C411yescharacteristic
383C23C2×#12111yescharacteristic
384C23C2×#12111yescharacteristic
385C23#54812yes—
386C23D4×Q812yes—
387C23C2×#13512yes—
388C23C2×#13512yes—
389C23#57611yescharacteristic
390C23#57711yescharacteristic
391C23#57711yescharacteristic
392C23#57811yescharacteristic
393Q8—232no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—22no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—24no—
401C2×C4—24no—
402C2×C4—24no—
403C2×C4—24no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—28no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—44no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—44no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4#38614yes—
454C2×C4#38614yes—
455C2×C4#41618yes—
456C2×C4C2×C4×D411yescharacteristic
457C2×C4C2×C4×D411yescharacteristic
458C2×C4C2×C4×Q812yes—
459C2×C4#51912yes—
460C2×C4C2×#12011yescharacteristic
461C2×C4C2×#12311yescharacteristic
462C2×C4C2×C4⋊Q811yescharacteristic
463C2×C4#53211yescharacteristic
464C2×C4#53312yes—
465C2×C4#53611yescharacteristic
466C2×C4#53611yescharacteristic
467C2×C4#54112yes—
468C2×C4#54712yes—
469C2×C4#54812yes—
470C2×C4D4×Q812yes—
471C2×C4D4×Q812yes—
472C2×C4#55612yes—
473C2×C4#55712yes—
474C8—24no—
475C8—42no—
476C8—44no—
477C22C2×#38611yescharacteristic
478C22Q8×C22⋊C412yes—
479C22#242912yes—
480C22#254511yescharacteristic
481C22#254611yescharacteristic
482C22C2×#41612yes—
483C22#289711yescharacteristic
484C22#289811yescharacteristic
485C22#305011yescharacteristic
486C22#305011yescharacteristic
487C22#305012yes—
488C22#306011yescharacteristic
489C22#306111yescharacteristic
490C22#306211yescharacteristic
491C22#306311yescharacteristic
492C22#336012yes—
493C22#336112yes—
494C22#336412yes—
495C22#336512yes—
496C22#338012yes—
497C22#338112yes—
498C22#338112yes—
499C22#338312yes—
500C4—24no—
501C4—28no—
502C4—24no—
503C4—44no—
504C4—48no—
505C4—84no—
506C4—82no—
507C4—82no—
508C4—161no—
509C4—161no—
510C4C4×C4⋊Q811yescharacteristic
511C4#232711yescharacteristic
512C4#236512yes—
513C4#243211yescharacteristic
514C4#243411yescharacteristic
515C4#244612yes—
516C2#1402411yescharacteristic
517C2#2096711yescharacteristic
518C2#2097611yescharacteristic
519C2#2251912yes—
520C2#2252112yes—
521C2#2547312yes—
522C2#2547412yes—
523C2#2864111yescharacteristic
524C2#2864211yescharacteristic
525C2#2867811yescharacteristic
526C2#2867911yescharacteristic
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
22. #2584C4semidirect
125. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C22×C8×Q82
C2×#25842
C2×#19912
C2×#19892
#139252
#112282
#112272
#108802
#108502
C4×#3402