Finite Group Databasebeta

← #430991

Subgroups — #430991

Order 512.

1718 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2542C211yesmaximal, characteristic
2C2×#2093C211yesmaximal, characteristic
3C2×#2057C212yesmaximal, complemented
4C2×#1862C212yesmaximal
5C2×#1845C211yesmaximal, characteristic, complemented
6#15151C211yesmaximal, characteristic, complemented
7#11373C211yesmaximal, characteristic, complemented
8#11324C212yesmaximal, complemented
9#11123C212yesmaximal
10#10925C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic, complemented
12C22×C4⋊Q8C2211yescharacteristic
13C22×C22⋊Q8—22no—
14C22×C4×D4C2211yescharacteristic
15C22×#107C2212yes—
16C22×#107C2212yes—
17C22×#107C2212yes—
18C22×#102C2212yes—
19C22×#102C2212yes—
20C22×C4×C8C2211yescharacteristic
21#2542—24no—
22#2542—24no—
23C2×#399—22no—
24C2×#398—22no—
25C2×#390C2211yescharacteristic
26C2×#390C2211yescharacteristic
27C2×#389C2212yes—
28C2×#384C2211yescharacteristic
29C2×#384C2211yescharacteristic
30C2×#384C2211yescharacteristic
31C2×C4×C4⋊C4C2211yescharacteristic
32#2093—24no—
33#2057—28no—
34#1862—28no—
35#1845—24no—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#340C2212yes—
39C2×#336C2212yes—
40C2×#336C2211yescharacteristic
41C2×#336C2212yes—
42#1445—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×D4C2311yescharacteristic
47C24×C4—21no—
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—42no—
54C2×C22⋊Q8—216no—
55C2×C22⋊Q8—48no—
56C2×C22⋊Q8—48no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4—24no—
60C2×C4×D4—42no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4—22no—
63C22×C4⋊C4—22no—
64C22×C4⋊C4—22no—
65C22×C4⋊C4—22no—
66C22×C4⋊C4D412yes—
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2312yes—
72C22×C22⋊C4—21no—
73C22×C22⋊C4—22no—
74C22×C22⋊C4—22no—
75C22×C42C2311yescharacteristic
76C22×C42C2311yescharacteristic
77C22×C42C2311yescharacteristic
78C2×#107—24no—
79C2×#107—24no—
80C2×#107—24no—
81C2×#107—24no—
82C2×#107—24no—
83C2×#107—24no—
84C2×#107—42no—
85C2×#107—42no—
86C2×#107—42no—
87C2×#102—24no—
88C2×#102—24no—
89C2×#102—24no—
90C2×#102—24no—
91C2×#102—42no—
92C2×#102Q818yescomplemented
93C2×C4×C8—24no—
94C2×C4×C8—22no—
95#399—216no—
96#398—48no—
97#390—22no—
98#390—22no—
99#390—22no—
100#390D414yes—
101#389—24no—
102#389—24no—
103#384—22no—
104#384—22no—
105#384—22no—
106#384—22no—
107#384—22no—
108#384D414yes—
109C4×C4⋊C4—22no—
110C4×C4⋊C4—22no—
111C2×#95—22no—
112C2×#95—24no—
113C2×#95—22no—
114C2×#95D412yes—
115#340—24no—
116#340—24no—
117#340—24no—
118#336—24no—
119#336—22no—
120#336D418yes—
121C25—21no—
122C22×Q8—24no—
123C22×Q8—44no—
124C22×Q8—42no—
125C22×Q8—42no—
126C22×Q8—44no—
127C22×D4—22no—
128C22×D4—21no—
129C22×D4—22no—
130C22×D4—22no—
131C22×D4—22no—
132C22×D4—22no—
133C22×D4C2×D412yes—
134C22×D4C2×Q814yes—
135C23×C4—24no—
136C23×C4—21no—
137C23×C4—21no—
138C23×C4—21no—
139C23×C4—22no—
140C23×C4—22no—
141C23×C4—42no—
142C23×C4—44no—
143C23×C4—42no—
144C23×C4—44no—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D412yes—
147C23×C4C2×D412yes—
148C23×C4C2×D411yescharacteristic
149C23×C4C2411yescharacteristic
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8C2×D414yes—
161C22×C8C2×D414yes—
162C22×C8C2×Q814yes—
163C22×C8#4114yes—
164C4⋊Q8—48no—
165C4⋊Q8—48no—
166C22⋊Q8—464no—
167C4×D4—48no—
168C4×D4—48no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—28no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—28no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—44no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—44no—
198C2×C4⋊C4—44no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4—44no—
203C2×C4⋊C4—48no—
204C2×C4⋊C4—44no—
205C2×C4⋊C4D818yes—
206C2×C4⋊C4C2×D412yes—
207C2×C4⋊C4C2×D412yes—
208C2×C4⋊C4C2×D412yes—
209C2×C4⋊C4C2×Q814yes—
210C2×C22⋊C4—24no—
211C2×C22⋊C4—28no—
212C2×C22⋊C4—28no—
213C2×C22⋊C4—42no—
214C2×C22⋊C4—44no—
215C2×C22⋊C4—42no—
216C2×C22⋊C4—44no—
217C2×C22⋊C4—44no—
218C2×C22⋊C4—44no—
219C2×C42—21no—
220C2×C42—21no—
221C2×C42—22no—
222C2×C42—22no—
223C2×C42—22no—
224C2×C42—21no—
225C2×C42—21no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42C2×D412yes—
229C2×C42C2×D412yes—
230C2×C42C2×D414yes—
231#107—48no—
232#107—48no—
233#107—48no—
234#102—216no—
235#102—48no—
236C4×C8—28no—
237#95—24no—
238#95—44no—
239#95—44no—
240#95—48no—
241C24—22no—
242C24—24no—
243C24—41no—
244C24—41no—
245C24—42no—
246C24—42no—
247C24—42no—
248C24—44no—
249C24C22×D411yescharacteristic
250C2×Q8—416no—
251C2×Q8—48no—
252C2×Q8—84no—
253C2×Q8—88no—
254C2×Q8—84no—
255C2×D4—24no—
256C2×D4—22no—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—42no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4C4⋊Q814yes—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—24no—
276C22×C4—21no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—24no—
280C22×C4—22no—
281C22×C4—24no—
282C22×C4—24no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—24no—
287C22×C4—48no—
288C22×C4—416no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—41no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—44no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—44no—
309C22×C4—44no—
310C22×C4—84no—
311C22×C4—88no—
312C22×C4—84no—
313C22×C4#12014yes—
314C22×C4#12712yes—
315C22×C4#12712yes—
316C22×C4C2×D814yes—
317C22×C4C2×D814yes—
318C22×C4C2×D814yes—
319C22×C4AGL(1,ℤ/8)14yes—
320C22×C4C22×D412yes—
321C22×C4C22×D411yescharacteristic
322C22×C4C22×D411yescharacteristic
323C22×C4C22×Q812yes—
324C22×C4C2×#4112yes—
325C22×C4H(5,𝔽2)12yes—
326C22×C42−1+412yes—
327C22×C42−1+412yes—
328C2×C8—24no—
329C2×C8—24no—
330C2×C8—24no—
331C2×C8—24no—
332C2×C8—24no—
333C2×C8—24no—
334C2×C8—24no—
335C2×C8—24no—
336C2×C8—24no—
337C2×C8—42no—
338C2×C8—42no—
339C2×C8—42no—
340C2×C8—42no—
341C2×C8C22⋊Q818yescomplemented
342C4⋊C4—216no—
343C4⋊C4—24no—
344C4⋊C4—24no—
345C4⋊C4—24no—
346C4⋊C4—24no—
347C4⋊C4—24no—
348C4⋊C4—416no—
349C4⋊C4—48no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—42no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—416no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—88no—
361C4⋊C4—88no—
362C22⋊C4—48no—
363C22⋊C4—416no—
364C22⋊C4—416no—
365C42—22no—
366C42—22no—
367C42—24no—
368C42—24no—
369C42—24no—
370C42—41no—
371C42—41no—
372C42—42no—
373C23—24no—
374C23—48no—
375C23—48no—
376C23—42no—
377C23—42no—
378C23—44no—
379C23—44no—
380C23—44no—
381C23—44no—
382C23—84no—
383C23—84no—
384C23—81no—
385C23—81no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23C2×#12011yescharacteristic
390C23C2×#12711yescharacteristic
391C23#53511yescharacteristic
392C23#54712yes—
393C23D4×Q812yes—
394C23C22×D812yes—
395C23C2×AGL(1,ℤ/8)12yes—
396C23#57814yes—
397Q8—816no—
398D4—28no—
399D4—44no—
400D4—44no—
401D4—44no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—22no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—28no—
411C2×C4—24no—
412C2×C4—22no—
413C2×C4—24no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—48no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—816no—
428C2×C4—84no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—81no—
435C2×C4—81no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—84no—
444C2×C4—84no—
445C2×C4#44718yes—
446C2×C4#49314yes—
447C2×C4C2×C22⋊Q812yes—
448C2×C4#52511yescharacteristic
449C2×C4C2×C4⋊Q811yescharacteristic
450C2×C4#53611yescharacteristic
451C2×C4#53612yes—
452C2×C4#53611yescharacteristic
453C2×C4#53912yes—
454C2×C4#54412yes—
455C2×C4D4212yes—
456C2×C4D4212yes—
457C2×C4#54712yes—
458C2×C4#54712yes—
459C2×C4D4×Q812yes—
460C2×C4D4×Q812yes—
461C2×C4#55412yes—
462C2×C4#56312yes—
463C8—28no—
464C8—44no—
465C22—44no—
466C22—48no—
467C22—84no—
468C22—88no—
469C22—84no—
470C22—82no—
471C22—84no—
472C22—82no—
473C22#240611yescharacteristic
474C22#243311yescharacteristic
475C22#243411yescharacteristic
476C22#254212yes—
477C22#272012yes—
478C22C2×#44712yes—
479C22#301112yes—
480C22C2×#49311yescharacteristic
481C22#315111yescharacteristic
482C22#315612yes—
483C22#320512yes—
484C22#322312yes—
485C22#333414yes—
486C22#334314yes—
487C22Q8×D814yes—
488C22#338614yes—
489C4—28no—
490C4—24no—
491C4—44no—
492C4—42no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—162no—
498C4#244012yes—
499C4#244912yes—
500C4#267712yes—
501C4#268012yes—
502C2—84no—
503C2#1543011yescharacteristic
504C2#2408812yes—
505C2#2572111yescharacteristic
506C2#2573511yescharacteristic
507C2#2583812yes—
508C2#2852712yes—
509C2#2854512yes—
510C2#3234214yes—
subgroupG/H
center, socleC24C22×D4
commutator, Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
LCSG ▷ C23×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
3. C2×#2057C2semidirect
5. C2×#1845C2semidirect
6. #15151C2semidirect
7. #11373C2semidirect
8. #11324C2semidirect
11. #10874C2semidirect
92. C2×#102Q8semidirect
341. C2×C8C22⋊Q8semidirect

Maximal subgroups

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

subgroupindex
C2×#25422
C2×#20932
C2×#20572
C2×#18622
C2×#18452
#151512
#113732
#113242
#111232
#109252
#108742