Finite Group Databasebeta

← #365521

Subgroups — #365521

Order 512.

1282 conjugacy classes of subgroups.

The subgroup lattice has 506 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×#1847C211yesmaximal, characteristic
5#13925C211yesmaximal, characteristic
6#11325C212yesmaximal
7#11325C212yesmaximal
8#10908C212yesmaximal
9#10907C212yesmaximal
10#10849C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×C4×Q8C2211yescharacteristic
13C22×#107C2212yes—
14C22×#107C2211yescharacteristic
15C22×#106C2212yes—
16C22×#106C2211yescharacteristic
17C22×C4⋊C8C2211yescharacteristic
18C22×#97C2211yescharacteristic
19#2355—24no—
20#2355—24no—
21C2×#391—21no—
22C2×#387—22no—
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25C2×#382C2211yescharacteristic
26C2×C4×C4⋊C4—21no—
27C2×C4×C4⋊C4C2212yes—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#1985C418yescomplemented
31#1949—24no—
32#1847—24no—
33C2×#340C2212yes—
34C2×#340C2211yescharacteristic
35C2×#340C2211yescharacteristic
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×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C23×C8C2312yes—
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—24no—
72C2×#107—24no—
73C2×#107—22no—
74C2×#107—42no—
75C2×#107—42no—
76C2×#106—24no—
77C2×#106—24no—
78C2×#106—22no—
79C2×#106—42no—
80C2×#106—42no—
81C2×C4⋊C8—22no—
82C2×C4⋊C8—22no—
83C2×C4⋊C8C2×C414yes—
84C2×#97—22no—
85C2×#97—22no—
86C2×#97C2×C414yes—
87#391—28no—
88#387—216no—
89#384—22no—
90#384D414yes—
91#382—24no—
92#382—24no—
93#382—22no—
94#382—22no—
95C4×C4⋊C4—28no—
96C4×C4⋊C4—24no—
97C4×C4⋊C4—24no—
98C4×C4⋊C4—22no—
99C4×C4⋊C4—22no—
100C4×C4⋊C4—22no—
101C4×C4⋊C4D414yes—
102C2×#95—22no—
103C2×#95—21no—
104C2×#95—22no—
105C2×#95—22no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—21no—
109C2×#95—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95—22no—
113#340—24no—
114#340—22no—
115#340—22no—
116#340—42no—
117#340—42no—
118#340Q818yescomplemented
119#336C2×C418yes—
120#336C2×C418yes—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—42no—
124C22×Q8—42no—
125C22×Q8C4⋊C414yes—
126C23×C4—21no—
127C23×C4—22no—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4C2×D412yes—
133C23×C4C2×D412yes—
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C23×C4C2411yescharacteristic
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—21no—
142C22×C8—21no—
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×C8C22×C414yes—
152C22×C8C2×Q814yes—
153C22×C8#4114yes—
154C4×Q8—28no—
155C4×Q8—28no—
156C4×Q8—28no—
157C4×Q8—28no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—24no—
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—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—81no—
189C2×C4⋊C4—81no—
190C2×C4⋊C4C4⋊C414yes—
191C2×C4⋊C4C4⋊C418yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2×D412yes—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×Q814yes—
196C2×C4⋊C4C2×Q814yes—
197C2×C42—22no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—24no—
202C2×C42—28no—
203C2×C42—44no—
204C2×C42—42no—
205C2×C42—81no—
206C2×C42—81no—
207C2×C42—82no—
208C2×C42—82no—
209C2×C42C22×C414yes—
210C2×C42C22×C412yes—
211C2×C42C2×D412yes—
212C2×C42C2×D412yes—
213C2×C42C2×Q814yes—
214C2×C42C2×Q812yes—
215C2×C42#4112yes—
216C2×C42#4112yes—
217#107—48no—
218#107—44no—
219#106—48no—
220#106—44no—
221C4⋊C8—28no—
222#97—28no—
223#95—28no—
224#95—28no—
225#95—28no—
226#95—24no—
227#95—44no—
228#95—44no—
229#95—42no—
230#95—42no—
231#95—42no—
232#95—82no—
233#95—82no—
234C24C22×D411yescharacteristic
235C2×Q8—28no—
236C2×Q8—48no—
237C2×Q8—48no—
238C2×Q8—44no—
239C2×Q8—44no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—42no—
264C22×C4—41no—
265C22×C4—41no—
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—42no—
275C22×C4—42no—
276C22×C4—42no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—41no—
283C22×C4—41no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4C2×C4⋊C414yes—
293C22×C4C2×C4⋊C414yes—
294C22×C4C2×C4⋊C412yes—
295C22×C4C2×C4⋊C412yes—
296C22×C4#12012yes—
297C22×C4#12012yes—
298C22×C4#12312yes—
299C22×C4#12312yes—
300C22×C4C23×C411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×D411yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C22×Q811yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—24no—
314C2×C8—24no—
315C2×C8—24no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—41no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8C4×Q818yes—
327C4⋊C4—22no—
328C4⋊C4—24no—
329C4⋊C4—28no—
330C4⋊C4—216no—
331C4⋊C4—22no—
332C4⋊C4—24no—
333C4⋊C4—22no—
334C4⋊C4—24no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—48no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—41no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—84no—
350C4⋊C4C4⋊Q814yes—
351C42—24no—
352C42—24no—
353C42—24no—
354C42—24no—
355C42—48no—
356C42—84no—
357C42—88no—
358C42C4×D414yes—
359C42C4×D414yes—
360C42C22⋊Q814yes—
361C42C22⋊Q814yes—
362C23C22×C4⋊C411yescharacteristic
363C23C2×#12011yescharacteristic
364C23C2×#12311yescharacteristic
365C23D4×Q812yes—
366C23#55012yes—
367C23#57512yes—
368C23#57512yes—
369C23#57611yescharacteristic
370C23#57711yescharacteristic
371C23#57711yescharacteristic
372C23#57811yescharacteristic
373Q8—416no—
374C2×C4—24no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—24no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—42no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—44no—
396C2×C4—44no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—44no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—42no—
409C2×C4—44no—
410C2×C4—44no—
411C2×C4—42no—
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—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—81no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4#38514yes—
433C2×C4#38514yes—
434C2×C4C2×C4×D411yescharacteristic
435C2×C4C2×C4×D411yescharacteristic
436C2×C4C2×C4×Q812yes—
437C2×C4#52012yes—
438C2×C4C2×C22⋊Q811yescharacteristic
439C2×C4C2×C22⋊Q811yescharacteristic
440C2×C4C2×C4⋊Q811yescharacteristic
441C2×C4#53211yescharacteristic
442C2×C4#53312yes—
443C2×C4#53511yescharacteristic
444C2×C4#53611yescharacteristic
445C2×C4#54212yes—
446C2×C4#54612yes—
447C2×C4D4×Q812yes—
448C2×C4#55212yes—
449C2×C4#55412yes—
450C2×C4#55712yes—
451C2×C4Q8212yes—
452C8—28no—
453C8—42no—
454C8—42no—
455C22C2×#38511yescharacteristic
456C22Q8×C4⋊C412yes—
457C22#242312yes—
458C22#243111yescharacteristic
459C22#243411yescharacteristic
460C22#291112yes—
461C22#291211yescharacteristic
462C22#291311yescharacteristic
463C22#300511yescharacteristic
464C22#300511yescharacteristic
465C22#301211yescharacteristic
466C22#301311yescharacteristic
467C22#309312yes—
468C22#309611yescharacteristic
469C22#309711yescharacteristic
470C22#338812yes—
471C22#338812yes—
472C22#338912yes—
473C22#339012yes—
474C22#340112yes—
475C22#340112yes—
476C22#340212yes—
477C22#340312yes—
478C4—24no—
479C4—24no—
480C4—44no—
481C4—44no—
482C4—44no—
483C4—82no—
484C4—84no—
485C4—84no—
486C4—161no—
487C4—161no—
488C4C4×C4⋊Q811yescharacteristic
489C4#232611yescharacteristic
490C4#236012yes—
491C4#245112yes—
492C4#256511yescharacteristic
493C4#256711yescharacteristic
494C2#1397611yescharacteristic
495C2#2084511yescharacteristic
496C2#2084911yescharacteristic
497C2#2273012yes—
498C2#2273112yes—
499C2#2506912yes—
500C2#2507112yes—
501C2#2563811yescharacteristic
502C2#2563911yescharacteristic
503C2#2584511yescharacteristic
504C2#2584611yescharacteristic
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
30. #1985C4semidirect
118. #340Q8semidirect

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×#18472
#139252
#113252
#109082
#109072
#108492
#107032