Finite Group Databasebeta

← #368916

Subgroups — #368916

Order 512.

1418 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#501C211yesmaximal, characteristic
2C2×#2369C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4C2×#1988C211yesmaximal, characteristic
5#14160C211yesmaximal, characteristic
6#11220C212yesmaximal
7#11219C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#125C2211yescharacteristic
14C2×#501—24no—
15C2×#501—24no—
16C2×#501C418yescomplemented
17C22×#107C2212yes—
18C22×#106C2212yes—
19C22×C4⋊C8C2211yescharacteristic
20C22×#97C2211yescharacteristic
21#2369—24no—
22#2369—24no—
23C2×#391—22no—
24C2×#389—21no—
25C2×#389C2212yes—
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C4⋊C4C2212yes—
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×C4×C4⋊C4C2211yescharacteristic
31#1989—24no—
32#1988—24no—
33C2×#340C2212yes—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2211yescharacteristic
38C2×#340C2211yescharacteristic
39C2×#340C2212yes—
40C2×#340C2212yes—
41C2×#336C2212yes—
42C2×#336C2212yes—
43#1441—21no—
44#1441—21no—
45#1436—22no—
46C4⋊#95C2212yes—
47C4⋊#95C2212yes—
48C23×Q8—21no—
49C23×C8C2311yescharacteristic
50C23×C8C2311yescharacteristic
51C23×C8C2312yes—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D414yes—
57C2×C4⋊Q8D414yes—
58C2×#125—22no—
59C2×#125—22no—
60C2×#125—22no—
61C2×#125—22no—
62C2×#125—22no—
63C2×#125C2×C414yes—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—22no—
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4D411yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2311yescharacteristic
73C22×C4⋊C4C2312yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42—22no—
78C22×C42—21no—
79C22×C42C2312yes—
80C22×C42C2311yescharacteristic
81#501—232no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107C2×C418yes—
85C2×#106—24no—
86C2×#106—24no—
87C2×#106C2×C418yes—
88C2×C4⋊C8—22no—
89C2×C4⋊C8—22no—
90C2×C4⋊C8—22no—
91C2×#97—22no—
92C2×#97—22no—
93C2×#97C2×C414yes—
94#391—216no—
95#389—28no—
96#389—24no—
97#389—24no—
98C4×C4⋊C4—28no—
99C4×C4⋊C4—24no—
100C4×C4⋊C4—24no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—22no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4D414yes—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—21no—
113C2×#95—22no—
114C2×#95—22no—
115C2×#95D412yes—
116#340—24no—
117#340—24no—
118#340—24no—
119#340—22no—
120#340—22no—
121#340—22no—
122#340—22no—
123#340Q818yescomplemented
124#336—24no—
125#336—24no—
126C22×Q8—24no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—81no—
130C22×Q8—81no—
131C22×Q8—81no—
132C22×Q8—81no—
133C23×C4—22no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—22no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—21no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8C22×C414yes—
159C22×C8C2×Q814yes—
160C22×C8#4114yes—
161C4⋊Q8—28no—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8C22⋊C4116yes—
165#125—28no—
166#125—28no—
167#125—44no—
168#125—44no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—24no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
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—42no—
206C2×C4⋊C4—82no—
207C2×C4⋊C4—82no—
208C2×C4⋊C4C22×C412yes—
209C2×C4⋊C4C22×C412yes—
210C2×C4⋊C4C22×C414yes—
211C2×C4⋊C4C2×D412yes—
212C2×C4⋊C4C2×D414yes—
213C2×C4⋊C4C2×D414yes—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×Q814yes—
216C2×C4⋊C4C2×Q814yes—
217C2×C42—24no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—28no—
225C2×C42—44no—
226C2×C42—42no—
227C2×C42—81no—
228C2×C42—81no—
229C2×C42—82no—
230C2×C42—82no—
231C2×C42C22×C412yes—
232C2×C42C2×D412yes—
233C2×C42C2×D412yes—
234C2×C42C2×D412yes—
235C2×C42#4112yes—
236C2×C42#4112yes—
237#107—216no—
238#106—216no—
239C4⋊C8—44no—
240#97—28no—
241#95—28no—
242#95—28no—
243#95—24no—
244#95—24no—
245#95—44no—
246#95—42no—
247#95—44no—
248#95—82no—
249#95—82no—
250C24C22×D411yescharacteristic
251C2×Q8—216no—
252C2×Q8—84no—
253C2×Q8—84no—
254C2×Q8—84no—
255C2×Q8—84no—
256C2×Q8—84no—
257C2×Q8—84no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—24no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—22no—
279C22×C4—22no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—42no—
296C22×C4—41no—
297C22×C4—41no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4C4×D412yes—
314C22×C4C4×D412yes—
315C22×C4#12012yes—
316C22×C4#12012yes—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4#13714yes—
320C22×C4C23×C411yescharacteristic
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q812yes—
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4112yes—
327C22×C4C2×#4111yescharacteristic
328C22×C42−1+412yes—
329C22×C42−1+412yes—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—42no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8C4×Q818yes—
347C4⋊C4—28no—
348C4⋊C4—24no—
349C4⋊C4—24no—
350C4⋊C4—24no—
351C4⋊C4—22no—
352C4⋊C4—24no—
353C4⋊C4—216no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—22no—
358C4⋊C4—22no—
359C4⋊C4—24no—
360C4⋊C4—22no—
361C4⋊C4—24no—
362C4⋊C4—44no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—48no—
371C4⋊C4—48no—
372C4⋊C4—88no—
373C4⋊C4C2×C22⋊C418yes—
374C4⋊C4C4⋊Q814yes—
375C42—48no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C42—84no—
381C42—88no—
382C42C2×C22⋊C414yes—
383C42C4×D414yes—
384C42#12114yes—
385C42#12314yes—
386C23C2×C4×D411yescharacteristic
387C23C2×#12011yescharacteristic
388C23C2×#12111yescharacteristic
389C23#54712yes—
390C23D4×Q812yes—
391C23C2×#13712yes—
392C23#57512yes—
393C23#57611yescharacteristic
394C23#57611yescharacteristic
395C23#57711yescharacteristic
396C23#57711yescharacteristic
397Q8—816no—
398C2×C4—24no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—28no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—44no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—44no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—44no—
432C2×C4—44no—
433C2×C4—44no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—44no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4#38614yes—
459C2×C4#38614yes—
460C2×C4C22×C22⋊C411yescharacteristic
461C2×C4C2×C4×D411yescharacteristic
462C2×C4C2×C4×Q812yes—
463C2×C4#51912yes—
464C2×C4C2×#12111yescharacteristic
465C2×C4C2×#12311yescharacteristic
466C2×C4C2×C4⋊Q811yescharacteristic
467C2×C4#53211yescharacteristic
468C2×C4#53312yes—
469C2×C4#53611yescharacteristic
470C2×C4#53611yescharacteristic
471C2×C4#54112yes—
472C2×C4#54812yes—
473C2×C4#54812yes—
474C2×C4D4×Q812yes—
475C2×C4D4×Q812yes—
476C2×C4#55612yes—
477C2×C4#55712yes—
478C8—28no—
479C8—81no—
480C8—81no—
481C22C2×#38611yescharacteristic
482C22#236512yes—
483C22#242912yes—
484C22#243211yescharacteristic
485C22#243411yescharacteristic
486C22C4×#13712yes—
487C22#294611yescharacteristic
488C22#294611yescharacteristic
489C22#300411yescharacteristic
490C22#300511yescharacteristic
491C22#301211yescharacteristic
492C22#301211yescharacteristic
493C22#303712yes—
494C22#304111yescharacteristic
495C22#304211yescharacteristic
496C22#334412yes—
497C22#334612yes—
498C22#334712yes—
499C22#334812yes—
500C22#338512yes—
501C22#338712yes—
502C22#338812yes—
503C22#338912yes—
504C4—24no—
505C4—24no—
506C4—44no—
507C4—48no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—84no—
512C4—82no—
513C4—82no—
514C4C4×C4⋊Q811yescharacteristic
515C4#232711yescharacteristic
516C4Q8×C22⋊C412yes—
517C4#244612yes—
518C4#254511yescharacteristic
519C4#254611yescharacteristic
520C2#1402411yescharacteristic
521C2#2092611yescharacteristic
522C2#2092911yescharacteristic
523C2#2316712yes—
524C2#2316912yes—
525C2#2546212yes—
526C2#2546412yes—
527C2#2571011yescharacteristic
528C2#2571211yescharacteristic
529C2#2584511yescharacteristic
530C2#2584711yescharacteristic
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
16. C2×#501C4semidirect
123. #340Q8semidirect

Maximal subgroups

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

subgroupindex
C22×#5012
C2×#23692
C2×#19892
C2×#19882
#141602
#112202
#112192
#109082
#109072
#108352
#107032