Finite Group Databasebeta

← #455288

Subgroups — #455288

Order 512.

1194 conjugacy classes of subgroups.

The subgroup lattice has 517 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#477C211yesmaximal, characteristic
2C2×#2074C212yesmaximal
3C2×#1865C211yesmaximal, characteristic
4#15230C211yesmaximal, characteristic
5#13931C211yesmaximal, characteristic
6#11384C211yesmaximal, characteristic
7#11325C212yesmaximal
8#11256C212yesmaximal
9#11255C211yesmaximal, characteristic
10#11227C211yesmaximal, characteristic
11#11190C212yesmaximal
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#477—28no—
15C2×#477—24no—
16C22×#107C2212yes—
17C22×#106C2211yescharacteristic
18C22×C4⋊C8C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2211yescharacteristic
21C22×#103C2212yes—
22C2×#391C2212yes—
23C2×#389C2211yescharacteristic
24C2×#382C2212yes—
25#2074—28no—
26#1865—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2211yescharacteristic
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1446—22no—
36#1445—22no—
37#1441—21no—
38#1441—21no—
39C4⋊#95C2212yes—
40C4⋊#95C2211yescharacteristic
41C4⋊#95C2212yes—
42#1434—21no—
43#1434—21no—
44C4×#95C2211yescharacteristic
45C23×Q8—21no—
46C23×Q8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2311yescharacteristic
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—42no—
54C2×C4×Q8—22no—
55C2×C4×Q8—24no—
56C2×C4×Q8—42no—
57C2×C4×Q8D414yes—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2311yescharacteristic
66C22×C42—21no—
67C22×C42C2312yes—
68C22×C42C2311yescharacteristic
69#477—232no—
70C2×#107—24no—
71C2×#107—24no—
72C2×#107—24no—
73C2×#106—22no—
74C2×#106—42no—
75C2×C4⋊C8—24no—
76C2×C4⋊C8—22no—
77C2×#103—24no—
78C2×#103—24no—
79C2×#103—22no—
80C2×#103—22no—
81C2×#103—22no—
82C2×#103—24no—
83C2×#103—22no—
84C2×#103D418yes—
85#391—24no—
86#391—42no—
87#389—22no—
88#389—22no—
89#382—24no—
90#382—42no—
91C2×#95—21no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—21no—
102C2×#95D411yescharacteristic
103C2×#95D411yescharacteristic
104C2×#95D412yes—
105#340—24no—
106#340—24no—
107#340—22no—
108#340—24no—
109#340—42no—
110#336—24no—
111#336—24no—
112#336—22no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—21no—
120C22×Q8—21no—
121C22×Q8—44no—
122C22×Q8—44no—
123C22×Q8—42no—
124C22×Q8—81no—
125C22×Q8—81no—
126C22×Q8C2×D414yes—
127C22×Q8C2×D412yes—
128C23×C4—21no—
129C23×C4—22no—
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—21no—
133C23×C4—22no—
134C23×C4C2×D412yes—
135C23×C4C2×D412yes—
136C23×C4C2×D411yescharacteristic
137C23×C4C2×D411yescharacteristic
138C23×C4C2411yescharacteristic
139C22×C8—21no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—21no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—21no—
150C22×C8—22no—
151C22×C8—22no—
152C22×C8—21no—
153C22×C8—21no—
154C22×C8C2×D414yes—
155C22×C8#4112yes—
156C4⋊Q8—28no—
157C4⋊Q8—48no—
158C4⋊Q8—44no—
159C4×Q8—28no—
160C4×Q8—44no—
161C4×Q8—48no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
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—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—42no—
190C2×C4⋊C4—42no—
191C2×C4⋊C4—42no—
192C2×C4⋊C4C2×D412yes—
193C2×C4⋊C4C2×D414yes—
194C2×C4⋊C4#4112yes—
195C2×C42—22no—
196C2×C42—22no—
197C2×C42—24no—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—44no—
205C2×C42C2×D412yes—
206C2×C42#4112yes—
207#107—216no—
208#106—44no—
209C4⋊C8—28no—
210#103—28no—
211#103—216no—
212#103—44no—
213#95—22no—
214#95—22no—
215#95—42no—
216#95—44no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—42no—
223#95—81no—
224#95—82no—
225#95—82no—
226#95—82no—
227C24C22×D411yescharacteristic
228C2×Q8—22no—
229C2×Q8—24no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—48no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—41no—
243C2×Q8—41no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—84no—
247C2×Q8—84no—
248C2×Q8—88no—
249C2×Q8—88no—
250C2×Q8#12014yes—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—24no—
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—42no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—41no—
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—41no—
302C22×C4—41no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4#12112yes—
308C22×C4#12112yes—
309C22×C4#13512yes—
310C22×C4#13712yes—
311C22×C4C22×D411yescharacteristic
312C22×C4C22×D412yes—
313C22×C4C2×#4111yescharacteristic
314C22×C4C2×#4111yescharacteristic
315C22×C4C2×#4111yescharacteristic
316C22×C4H(5,𝔽2)12yes—
317C22×C42−1+412yes—
318C22×C42−1+411yescharacteristic
319C22×C42−1+411yescharacteristic
320C22×C42−1+412yes—
321C2×C8—24no—
322C2×C8—22no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—24no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—41no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C2×C8—42no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8—41no—
336C2×C8—42no—
337C2×C8—42no—
338C4⋊C4—24no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—24no—
343C4⋊C4—22no—
344C4⋊C4—48no—
345C4⋊C4—48no—
346C4⋊C4—42no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—41no—
351C4⋊C4—41no—
352C4⋊C4—41no—
353C4⋊C4—42no—
354C4⋊C4—42no—
355C4⋊C4—42no—
356C4⋊C4—42no—
357C4⋊C4—42no—
358C4⋊C4—84no—
359C42—22no—
360C42—22no—
361C42—24no—
362C42—48no—
363C42—42no—
364C42—42no—
365C42—42no—
366C42—42no—
367C23C2×#12111yescharacteristic
368C23#53612yes—
369C23D4212yes—
370C23#54711yescharacteristic
371C23#54811yescharacteristic
372C23C2×#13511yescharacteristic
373C23C2×#13711yescharacteristic
374C23#57512yes—
375C23#57611yescharacteristic
376C23#57711yescharacteristic
377C23#57812yes—
378Q8—24no—
379Q8—24no—
380Q8—42no—
381Q8—42no—
382Q8—44no—
383Q8—44no—
384Q8—816no—
385C2×C4—22no—
386C2×C4—24no—
387C2×C4—24no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—24no—
391C2×C4—22no—
392C2×C4—22no—
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—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—44no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—81no—
423C2×C4—81no—
424C2×C4—82no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—81no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4—82no—
442C2×C4—81no—
443C2×C4—81no—
444C2×C4#46414yes—
445C2×C4C2×#12011yescharacteristic
446C2×C4#52511yescharacteristic
447C2×C4#53411yescharacteristic
448C2×C4#53611yescharacteristic
449C2×C4#53611yescharacteristic
450C2×C4#53611yescharacteristic
451C2×C4#53612yes—
452C2×C4#53612yes—
453C2×C4#53912yes—
454C2×C4#54112yes—
455C2×C4#54212yes—
456C2×C4#54612yes—
457C2×C4#54712yes—
458C2×C4#54712yes—
459C2×C4#55211yescharacteristic
460C2×C4#55611yescharacteristic
461C2×C4#56312yes—
462C2×C4#56312yes—
463C8—42no—
464C8—44no—
465C8—81no—
466C22#241011yescharacteristic
467C22#241511yescharacteristic
468C22#262111yescharacteristic
469C22#267412yes—
470C22#268012yes—
471C22C2×#46411yescharacteristic
472C22#304011yescharacteristic
473C22#304411yescharacteristic
474C22#304511yescharacteristic
475C22#323312yes—
476C22#323712yes—
477C22#324212yes—
478C22#324312yes—
479C22#326812yes—
480C22#327012yes—
481C22#327512yes—
482C22#328512yes—
483C22#334011yescharacteristic
484C22#334912yes—
485C22#335711yescharacteristic
486C22#336111yescharacteristic
487C22#337211yescharacteristic
488C22#337712yes—
489C4—24no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—84no—
494C4—82no—
495C4—82no—
496C4—82no—
497C4—161no—
498C4—162no—
499C4#240611yescharacteristic
500C4#261411yescharacteristic
501C4#263811yescharacteristic
502C4#268912yes—
503C4#269812yes—
504C4#281711yescharacteristic
505C2#1626211yescharacteristic
506C2#2436111yescharacteristic
507C2#2437111yescharacteristic
508C2#2467911yescharacteristic
509C2#2469911yescharacteristic
510C2#2986711yescharacteristic
511C2#2987511yescharacteristic
512C2#3082012yes—
513C2#3083612yes—
514C2#3108512yes—
515C2#3109912yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C4×#95 ◁ 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
C22×#4772
C2×#20742
C2×#18652
#152302
#139312
#113842
#113252
#112562
#112552
#112272
#111902