Finite Group Databasebeta

← #382643

Subgroups — #382643

Order 512.

1240 conjugacy classes of subgroups.

The subgroup lattice has 531 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2320C211yesmaximal, characteristic
2C2×#1843C211yesmaximal, characteristic
3C2×#1805C211yesmaximal, characteristic
4C2×#1803C211yesmaximal, characteristic
5#14348C211yesmaximal, characteristic
6#11228C212yesmaximal
7#11227C212yesmaximal
8#11077C212yesmaximal
9#11077C212yesmaximal
10#10850C211yesmaximal, characteristic
11#10849C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×C4⋊C8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×C4⋊C8C2212yes—
16C22×C4⋊C8C2212yes—
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20#2320—24no—
21#2320C418yescomplemented
22C2×#391C2212yes—
23C2×#389C2212yes—
24C2×#387—22no—
25C2×#384C2211yescharacteristic
26C2×#383—21no—
27C2×#382—21no—
28C2×#382C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30C2×#376C2211yescharacteristic
31#1843—24no—
32#1805—24no—
33#1803—24no—
34C2×#340C2212yes—
35C2×#340C2211yescharacteristic
36C2×#340C2211yescharacteristic
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#340C2212yes—
40C2×#340C2212yes—
41#1442—21no—
42#1442—21no—
43#1442—22no—
44C4⋊#95C2212yes—
45C4⋊#95C2212yes—
46C23×Q8C2311yescharacteristic
47C23×C8C2312yes—
48C23×C8C2311yescharacteristic
49C23×C8C2311yescharacteristic
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⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69C2×C4⋊C8—22no—
70C2×C4⋊C8—22no—
71C2×C4⋊C8—24no—
72C2×C4⋊C8—24no—
73C2×C4⋊C8—22no—
74C2×C4⋊C8—22no—
75C2×C4⋊C8—22no—
76C2×C4⋊C8—22no—
77C2×C4⋊C8—42no—
78C2×C4⋊C8—42no—
79C2×C4⋊C8—42no—
80C2×C4⋊C8—42no—
81C2×#103—24no—
82C2×#103—22no—
83C2×#103—22no—
84C2×#103—42no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88#391—24no—
89#391C2×C418yes—
90#389—24no—
91#389C2×C418yes—
92#387—216no—
93#384—22no—
94#384—22no—
95#383—28no—
96#382—28no—
97#382—22no—
98#382—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4C2×C414yes—
101#376C2×C414yes—
102#376D414yes—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95—22no—
110C2×#95—22no—
111C2×#95—22no—
112C2×#95—22no—
113C2×#95D411yescharacteristic
114C2×#95D411yescharacteristic
115C2×#95D412yes—
116#340—24no—
117#340—22no—
118#340—22no—
119#340—42no—
120#340—42no—
121#340—42no—
122#340—42no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8C22×C412yes—
131C22×Q8C2×D412yes—
132C22×Q8C2×Q812yes—
133C23×C4—21no—
134C23×C4—21no—
135C23×C4—22no—
136C23×C4—22no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4C2×D412yes—
140C23×C4C2×D412yes—
141C23×C4C2×D411yescharacteristic
142C23×C4C2×D411yescharacteristic
143C23×C4C2411yescharacteristic
144C22×C8—22no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—22no—
148C22×C8—21no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—21no—
152C22×C8—21no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C4×Q8—28no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8C4⋊C4116yes—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—28no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—82no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4—81no—
196C2×C4⋊C4—82no—
197C2×C4⋊C4—81no—
198C2×C4⋊C4C22×C414yes—
199C2×C4⋊C4C22×C412yes—
200C2×C4⋊C4C2×D412yes—
201C2×C4⋊C4C2×Q812yes—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—24no—
210C2×C42—81no—
211C2×C42—81no—
212C2×C42C22×C414yes—
213C2×C42C22×C412yes—
214C2×C42C2×D414yes—
215C2×C42C2×D412yes—
216C2×C42C2×D412yes—
217C2×C42C2×Q812yes—
218C2×C42#4112yes—
219C2×C42#4112yes—
220C4⋊C8—48no—
221C4⋊C8—48no—
222C4⋊C8—44no—
223C4⋊C8—44no—
224#103—48no—
225#103—44no—
226#103—44no—
227#95—24no—
228#95—28no—
229#95—28no—
230#95—28no—
231#95—42no—
232#95—42no—
233#95—82no—
234#95—82no—
235#95—82no—
236#95—82no—
237#95C22⋊C414yes—
238#95C22⋊C414yes—
239#95C22⋊C418yes—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—22no—
243C2×Q8—22no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—24no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8C2×C4⋊C414yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
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—22no—
282C22×C4—22no—
283C22×C4—24no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
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—41no—
304C22×C4—42no—
305C22×C4—41no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—42no—
313C22×C4C2×C22⋊C414yes—
314C22×C4C2×C22⋊C414yes—
315C22×C4C2×C22⋊C412yes—
316C22×C4C2×C22⋊C412yes—
317C22×C4#12112yes—
318C22×C4#12112yes—
319C22×C4C22⋊Q812yes—
320C22×C4C22⋊Q812yes—
321C22×C4C23×C411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×D411yescharacteristic
324C22×C4C22×Q811yescharacteristic
325C22×C4C2×#4111yescharacteristic
326C22×C4C2×#4111yescharacteristic
327C22×C4H(5,𝔽2)12yes—
328C22×C4H(5,𝔽2)12yes—
329C22×C42−1+412yes—
330C22×C42−1+412yes—
331C2×C8—42no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—42no—
335C2×C8—41no—
336C2×C8—41no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8—41no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C4⋊C4—22no—
349C4⋊C4—22no—
350C4⋊C4—22no—
351C4⋊C4—216no—
352C4⋊C4—24no—
353C4⋊C4—24no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—41no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—88no—
367C4⋊C4—84no—
368C4⋊C4—84no—
369C4⋊C4C2×C4⋊C414yes—
370C42—24no—
371C42—24no—
372C42—24no—
373C42—24no—
374C42—216no—
375C42—84no—
376C42C2×C4⋊C414yes—
377C42#11714yes—
378C42#12114yes—
379C42C22⋊Q814yes—
380C23C22×C22⋊C411yescharacteristic
381C23C2×#12111yescharacteristic
382C23C2×C22⋊Q811yescharacteristic
383C23#53712yes—
384C23#53712yes—
385C23#57611yescharacteristic
386C23#57712yes—
387C23#57711yescharacteristic
388C23#57711yescharacteristic
389C23#57812yes—
390C23#57811yescharacteristic
391Q8—24no—
392Q8—24no—
393Q8—44no—
394Q8—42no—
395Q8—42no—
396Q8—44no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—24no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—42no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—44no—
434C2×C4—44no—
435C2×C4—44no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—81no—
442C2×C4—81no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4—82no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4#38014yes—
458C2×C4#38014yes—
459C2×C4C22×C4⋊C411yescharacteristic
460C2×C4C2×#11711yescharacteristic
461C2×C4#52012yes—
462C2×C4#52012yes—
463C2×C4C2×#12111yescharacteristic
464C2×C4C2×C22⋊Q811yescharacteristic
465C2×C4#53511yescharacteristic
466C2×C4#53511yescharacteristic
467C2×C4#53512yes—
468C2×C4#53611yescharacteristic
469C2×C4#53611yescharacteristic
470C2×C4#53612yes—
471C2×C4#54012yes—
472C2×C4#54112yes—
473C2×C4#54212yes—
474C2×C4#54212yes—
475C2×C4#54412yes—
476C2×C4#54412yes—
477C8—82no—
478C8—81no—
479C8—81no—
480C22C2×#38011yescharacteristic
481C22#231112yes—
482C22#262011yescharacteristic
483C22#262011yescharacteristic
484C22#262912yes—
485C22#289711yescharacteristic
486C22#289812yes—
487C22#289811yescharacteristic
488C22#304311yescharacteristic
489C22#304411yescharacteristic
490C22#304511yescharacteristic
491C22#304611yescharacteristic
492C22#308212yes—
493C22#308311yescharacteristic
494C22#308411yescharacteristic
495C22#325712yes—
496C22#325812yes—
497C22#325812yes—
498C22#325812yes—
499C22#326012yes—
500C22#326112yes—
501C22#326212yes—
502C22#326312yes—
503C4—24no—
504C4—24no—
505C4—44no—
506C4—44no—
507C4—44no—
508C4—44no—
509C4—84no—
510C4—162no—
511C4—161no—
512C4—161no—
513C4#231412yes—
514C4#231711yescharacteristic
515C4#231711yescharacteristic
516C4#261911yescharacteristic
517C4#262111yescharacteristic
518C4#262812yes—
519C2#1434011yescharacteristic
520C2#2065111yescharacteristic
521C2#2065211yescharacteristic
522C2#2189612yes—
523C2#2189812yes—
524C2#2983511yescharacteristic
525C2#2983711yescharacteristic
526C2#2984011yescharacteristic
527C2#2984111yescharacteristic
528C2#3000412yes—
529C2#3000512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×#376 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #2320C4semidirect

Maximal subgroups

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

subgroupindex
C2×#23202
C2×#18432
C2×#18052
C2×#18032
#143482
#112282
#112272
#110772
#108502
#108492