Finite Group Databasebeta

← #464764

Subgroups — #464764

Order 512.

1368 conjugacy classes of subgroups.

The subgroup lattice has 523 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2862C211yesmaximal, characteristic
2C2×#2097C212yesmaximal
3C2×#2058C211yesmaximal, characteristic
4C2×#2057C211yesmaximal, characteristic
5C2×#2043C211yesmaximal, characteristic
6C2×#2043C211yesmaximal, characteristic
7#14707C211yesmaximal, characteristic
8#11124C212yesmaximal
9#11122C212yesmaximal
10#11052C212yesmaximal
11#10795C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×#107C2212yes—
15C22×#106C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20#2862—28no—
21C2×#391—22no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2211yescharacteristic
25C2×#389C2211yescharacteristic
26C2×#382—22no—
27C2×#382C2211yescharacteristic
28C2×#382C2211yescharacteristic
29C2×C43C2211yescharacteristic
30#2097—28no—
31#2058—24no—
32#2057—24no—
33#2043—24no—
34#2043—24no—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#336C2212yes—
38C2×#336C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2211yescharacteristic
41#1436—22no—
42#1436—22no—
43C4⋊#95C2211yescharacteristic
44C4⋊#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2312yes—
47C23×C8C2312yes—
48C23×C8C2312yes—
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—42no—
56C2×C4⋊Q8—42no—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—21no—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2311yescharacteristic
66C22×C4⋊C4C2311yescharacteristic
67C22×C42D412yes—
68C22×C42D412yes—
69C22×C42C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2311yescharacteristic
72C2×#107—24no—
73C2×#107—24no—
74C2×#107—42no—
75C2×#106—24no—
76C2×#106—24no—
77C2×#106—42no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90#391—216no—
91#391—24no—
92#391—24no—
93#391—24no—
94#391D418yes—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#382—216no—
100#382—22no—
101#382—22no—
102#382—24no—
103C2×#95—22no—
104C2×#95—24no—
105C2×#95—24no—
106C2×#95—22no—
107C2×#95—22no—
108C2×#95—22no—
109C2×#95—21no—
110C2×#95—21no—
111C43—24no—
112#340—24no—
113#340—24no—
114#336—24no—
115#336—42no—
116#336D414yes—
117#336D414yes—
118C22×Q8—24no—
119C22×Q8—22no—
120C22×Q8—22no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—42no—
133C22×Q8—44no—
134C22×Q8—42no—
135C22×Q8—44no—
136C22×Q8C2×D414yes—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4—21no—
141C23×C4—21no—
142C23×C4—21no—
143C23×C4C2×D412yes—
144C23×C4C2×D412yes—
145C23×C4C2×D411yescharacteristic
146C23×C4C2×D411yescharacteristic
147C23×C4C2411yescharacteristic
148C22×C8—22no—
149C22×C8—22no—
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—
162C4⋊Q8—48no—
163C4⋊Q8—48no—
164C4⋊Q8—48no—
165C4⋊Q8—48no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—21no—
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—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—42no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—42no—
199C2×C4⋊C4—42no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—44no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4#4112yes—
204C2×C4⋊C4#4112yes—
205C2×C42—24no—
206C2×C42—24no—
207C2×C42—24no—
208C2×C42—22no—
209C2×C42—21no—
210C2×C42—21no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—24no—
214C2×C42—24no—
215C2×C42—22no—
216C2×C42—21no—
217C2×C42—21no—
218C2×C42—22no—
219C2×C42—42no—
220C2×C42C2×D414yes—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223#107—48no—
224#106—48no—
225#103—48no—
226#103—48no—
227#103—48no—
228#103—48no—
229#95—28no—
230#95—216no—
231#95—216no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—42no—
236#95—44no—
237C24C22×D411yescharacteristic
238C2×Q8—24no—
239C2×Q8—24no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—48no—
245C2×Q8—416no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—42no—
268C2×Q8—84no—
269C2×Q8—88no—
270C2×Q8—84no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
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—22no—
284C22×C4—22no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—24no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—42no—
294C22×C4—41no—
295C22×C4—41no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—41no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—44no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—44no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—42no—
321C22×C4#12014yes—
322C22×C4#12712yes—
323C22×C4#12712yes—
324C22×C4#13514yes—
325C22×C4#13714yes—
326C22×C4#13714yes—
327C22×C4C22×D411yescharacteristic
328C22×C4C22×D411yescharacteristic
329C22×C4C22×D412yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4C2×#4111yescharacteristic
332C22×C4H(5,𝔽2)11yescharacteristic
333C22×C4H(5,𝔽2)12yes—
334C22×C42−1+412yes—
335C22×C42−1+412yes—
336C22×C42−1+411yescharacteristic
337C2×C8—24no—
338C2×C8—24no—
339C2×C8—24no—
340C2×C8—24no—
341C2×C8—24no—
342C2×C8—24no—
343C2×C8—42no—
344C2×C8—42no—
345C2×C8—42no—
346C2×C8—42no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C4⋊C4—24no—
352C4⋊C4—24no—
353C4⋊C4—22no—
354C4⋊C4—22no—
355C4⋊C4—22no—
356C4⋊C4—22no—
357C4⋊C4—48no—
358C4⋊C4—48no—
359C4⋊C4—48no—
360C4⋊C4—42no—
361C4⋊C4—42no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—42no—
370C4⋊C4—84no—
371C4⋊C4—84no—
372C42—216no—
373C42—24no—
374C42—24no—
375C42—24no—
376C42—24no—
377C42—48no—
378C42—42no—
379C42—42no—
380C23C2×#12011yescharacteristic
381C23#52511yescharacteristic
382C23C2×#12711yescharacteristic
383C23#53511yescharacteristic
384C23#53611yescharacteristic
385C23#53611yescharacteristic
386C23#53711yescharacteristic
387C23C2×#13512yes—
388C23C2×#13712yes—
389C23C2×#13712yes—
390C23#57512yes—
391Q8—44no—
392Q8—44no—
393Q8—44no—
394Q8—44no—
395Q8—816no—
396Q8—82no—
397Q8—82no—
398Q8—82no—
399Q8—82no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—24no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—24no—
411C2×C4—24no—
412C2×C4—24no—
413C2×C4—24no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—44no—
428C2×C4—48no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—81no—
436C2×C4—81no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—84no—
441C2×C4—82no—
442C2×C4—82no—
443C2×C4—82no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4#49514yes—
455C2×C4#49714yes—
456C2×C4#53612yes—
457C2×C4#53612yes—
458C2×C4#53612yes—
459C2×C4#53712yes—
460C2×C4#54212yes—
461C2×C4#54212yes—
462C2×C4D4212yes—
463C2×C4D4212yes—
464C2×C4#54611yescharacteristic
465C2×C4#54611yescharacteristic
466C2×C4#54712yes—
467C2×C4#54712yes—
468C2×C4#54811yescharacteristic
469C2×C4#54811yescharacteristic
470C2×C4#56212yes—
471C2×C4#56312yes—
472C8—44no—
473C8—44no—
474C22#240611yescharacteristic
475C22#243311yescharacteristic
476C22#254011yescharacteristic
477C22#261611yescharacteristic
478C22#263211yescharacteristic
479C22#265811yescharacteristic
480C22#282011yescharacteristic
481C22#300212yes—
482C22#300412yes—
483C22#310712yes—
484C22#311612yes—
485C22C2×#49511yescharacteristic
486C22C2×#49711yescharacteristic
487C22#315211yescharacteristic
488C22#315311yescharacteristic
489C22#321112yes—
490C22#321812yes—
491C22#323112yes—
492C22#323112yes—
493C22#323612yes—
494C22#323712yes—
495C22#324812yes—
496C22#325212yes—
497C4—24no—
498C4—24no—
499C4—44no—
500C4—44no—
501C4—84no—
502C4—84no—
503C4—82no—
504C4—82no—
505C4—161no—
506C4—161no—
507C4#267212yes—
508C4#267212yes—
509C4#267412yes—
510C4#267412yes—
511C2#1651011yescharacteristic
512C2#2406512yes—
513C2#2572411yescharacteristic
514C2#2572611yescharacteristic
515C2#2837211yescharacteristic
516C2#2838411yescharacteristic
517C2#2969512yes—
518C2#3006712yes—
519C2#3028111yescharacteristic
520C2#3028411yescharacteristic
521C2#3282412yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#28622
C2×#20972
C2×#20582
C2×#20572
C2×#20432
#147072
#111242
#111222
#110522
#107952