Finite Group Databasebeta

← #461625

Subgroups — #461625

Order 512.

1126 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1878C211yesmaximal, characteristic
2C2×#1803C211yesmaximal, characteristic
3#15240C211yesmaximal, characteristic
4#13935C211yesmaximal, characteristic
5#11357C211yesmaximal, characteristic
6#11255C211yesmaximal, characteristic
7#11208C212yesmaximal
8#11077C211yesmaximal, characteristic
9#11067C212yesmaximal
10#11052C212yesmaximal
11#11024C212yesmaximal
12C23×Q16C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×#103C2212yes—
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#391C2212yes—
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#382C2212yes—
25#1878—24no—
26#1803—24no—
27C2×#340C2212yes—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2212yes—
34C2×#336C2212yes—
35#1445—21no—
36#1445—21no—
37#1445—22no—
38#1441—22no—
39C4⋊#95—21no—
40C4⋊#95C2211yescharacteristic
41#1434—21no—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×Q8C2311yescharacteristic
45C22×Q16—22no—
46C22×Q16—22no—
47C22×Q16—42no—
48C22×Q16—42no—
49C22×Q16Q814yes—
50C23×C8C2312yes—
51C23×C8C2311yescharacteristic
52C23×C8C2311yescharacteristic
53C2×C4×Q8—22no—
54C2×C4×Q8—24no—
55C2×C4×Q8—42no—
56C2×C4×Q8D414yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—22no—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2312yes—
64C22×C4⋊C4C2311yescharacteristic
65C22×C42—21no—
66C22×C42C2312yes—
67C22×C42C2311yescharacteristic
68C2×C4⋊C8—22no—
69C2×C4⋊C8—42no—
70C2×#103—24no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77C2×#103—42no—
78C2×#103—42no—
79C2×#103—42no—
80#391—24no—
81#391—22no—
82#391—22no—
83#391—42no—
84#391—42no—
85#391—42no—
86#389—24no—
87#389—42no—
88#384—22no—
89#384—22no—
90#382—42no—
91#382—42no—
92C2×#95—21no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—21no—
101C2×#95—21no—
102C2×#95—21no—
103C2×#95D411yescharacteristic
104C2×#95D411yescharacteristic
105C2×#95D412yes—
106#340—24no—
107#340—22no—
108#340—42no—
109#340—42no—
110#340—42no—
111#336—22no—
112#336—42no—
113#336—42no—
114C22×Q8—21no—
115C22×Q8—21no—
116C22×Q8—21no—
117C22×Q8—21no—
118C22×Q8—21no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8—21no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—22no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—21no—
130C22×Q8C2×D412yes—
131C22×Q8C2×Q812yes—
132C22×Q8C2×Q812yes—
133C22×Q8#4112yes—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—22no—
138C23×C4—22no—
139C23×C4—21no—
140C23×C4C2×D412yes—
141C23×C4C2×D412yes—
142C23×C4C2×D411yescharacteristic
143C23×C4C2×D411yescharacteristic
144C23×C4C2411yescharacteristic
145C2×Q16—28no—
146C2×Q16—48no—
147C2×Q16—44no—
148C2×Q16—44no—
149C2×Q16—48no—
150C22×C8—21no—
151C22×C8—21no—
152C22×C8—22no—
153C22×C8—22no—
154C22×C8—21no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—21no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8C2×Q812yes—
167C4×Q8—28no—
168C4×Q8—44no—
169C4×Q8—48no—
170C2×C4⋊C4—21no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
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—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—42no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—82no—
198C2×C4⋊C4C2×D412yes—
199C2×C4⋊C4#4114yes—
200C2×C42—22no—
201C2×C42—22no—
202C2×C42—24no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—22no—
208C2×C42—22no—
209C2×C42—44no—
210C2×C42C2×D412yes—
211C2×C42#4112yes—
212C4⋊C8—44no—
213#103—48no—
214#103—44no—
215#103—48no—
216#103—44no—
217#95—22no—
218#95—22no—
219#95—42no—
220#95—44no—
221#95—44no—
222#95—44no—
223#95—44no—
224#95—44no—
225#95—44no—
226#95—44no—
227#95—42no—
228#95—42no—
229#95—42no—
230#95—81no—
231C24C22×D411yescharacteristic
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—41no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—41no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—41no—
258C2×Q8—41no—
259C2×Q8—41no—
260C2×Q8—41no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—42no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8C22⋊Q814yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—24no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—21no—
278C22×C4—21no—
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—22no—
290C22×C4—22no—
291C22×C4—22no—
292C22×C4—22no—
293C22×C4—22no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—41no—
298C22×C4—41no—
299C22×C4—41no—
300C22×C4—42no—
301C22×C4—41no—
302C22×C4—42no—
303C22×C4—41no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—42no—
307C22×C4—42no—
308C22×C4—42no—
309C22×C4—42no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—42no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—42no—
321C22×C4—42no—
322C22×C4#12112yes—
323C22×C4#12112yes—
324C22×C4#13512yes—
325C22×C4#13712yes—
326C22×C4C22×D411yescharacteristic
327C22×C4C22×Q811yescharacteristic
328C22×C4C2×#4111yescharacteristic
329C22×C4C2×#4112yes—
330C22×C4C2×#4111yescharacteristic
331C22×C4H(5,𝔽2)12yes—
332C22×C4H(5,𝔽2)12yes—
333C22×C4H(5,𝔽2)12yes—
334C22×C4H(5,𝔽2)11yescharacteristic
335C22×C42−1+411yescharacteristic
336Q16—416no—
337C2×C8—22no—
338C2×C8—24no—
339C2×C8—41no—
340C2×C8—41no—
341C2×C8—42no—
342C2×C8—41no—
343C2×C8—42no—
344C2×C8—41no—
345C2×C8—42no—
346C2×C8—41no—
347C2×C8—42no—
348C2×C8—42no—
349C2×C8—42no—
350C2×C8—42no—
351C2×C8—42no—
352C2×C8—42no—
353C2×C8—42no—
354C4⋊C4—22no—
355C4⋊C4—24no—
356C4⋊C4—48no—
357C4⋊C4—41no—
358C4⋊C4—42no—
359C4⋊C4—42no—
360C4⋊C4—41no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—42no—
366C4⋊C4—42no—
367C4⋊C4—42no—
368C4⋊C4—84no—
369C4⋊C4—88no—
370C4⋊C4—84no—
371C4⋊C4#12314yes—
372C42—22no—
373C42—22no—
374C42—24no—
375C42—48no—
376C42—42no—
377C42—42no—
378C42—42no—
379C42—42no—
380C23C2×#12111yescharacteristic
381C23#53512yes—
382C23#54611yescharacteristic
383C23#54612yes—
384C23D4×Q811yescharacteristic
385C23C2×#13511yescharacteristic
386C23C2×#13711yescharacteristic
387C23#57512yes—
388C23#57711yescharacteristic
389C23#57811yescharacteristic
390C23#57812yes—
391Q8—24no—
392Q8—24no—
393Q8—42no—
394Q8—44no—
395Q8—44no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—44no—
400Q8—81no—
401Q8—82no—
402Q8—81no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—24no—
409C2×C4—24no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—44no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—44no—
420C2×C4—44no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—44no—
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—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—81no—
439C2×C4—81no—
440C2×C4—82no—
441C2×C4—82no—
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—82no—
453C2×C4—82no—
454C2×C4—81no—
455C2×C4—82no—
456C2×C4—81no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4#46414yes—
461C2×C4C2×C22⋊Q811yescharacteristic
462C2×C4C2×#12311yescharacteristic
463C2×C4#53411yescharacteristic
464C2×C4#53511yescharacteristic
465C2×C4#53511yescharacteristic
466C2×C4#53711yescharacteristic
467C2×C4#53812yes—
468C2×C4#54012yes—
469C2×C4#54012yes—
470C2×C4#54212yes—
471C2×C4#54312yes—
472C2×C4#55112yes—
473C2×C4#55211yescharacteristic
474C2×C4#55312yes—
475C2×C4#55412yes—
476C2×C4#55411yescharacteristic
477C2×C4#56012yes—
478C2×C4#56212yes—
479C8—42no—
480C8—81no—
481C8—82no—
482C22#241211yescharacteristic
483C22#241711yescharacteristic
484C22#261911yescharacteristic
485C22#269212yes—
486C22#270312yes—
487C22C2×#46411yescharacteristic
488C22#304011yescharacteristic
489C22#304411yescharacteristic
490C22#304611yescharacteristic
491C22#321212yes—
492C22#321612yes—
493C22#322312yes—
494C22#322512yes—
495C22#330012yes—
496C22#330411yescharacteristic
497C22#331212yes—
498C22#331312yes—
499C22#331512yes—
500C22#332312yes—
501C22#332811yescharacteristic
502C22#338311yescharacteristic
503C22#338711yescharacteristic
504C22#339012yes—
505C4—24no—
506C4—44no—
507C4—42no—
508C4—44no—
509C4—82no—
510C4—82no—
511C4—82no—
512C4—162no—
513C4—162no—
514C4—161no—
515C4#248711yescharacteristic
516C4#262411yescharacteristic
517C4#262611yescharacteristic
518C4#274112yes—
519C4#274812yes—
520C4#281311yescharacteristic
521C2#1639811yescharacteristic
522C2#2452911yescharacteristic
523C2#2454511yescharacteristic
524C2#2481811yescharacteristic
525C2#2483211yescharacteristic
526C2#2979711yescharacteristic
527C2#2980511yescharacteristic
528C2#3152812yes—
529C2#3154812yes—
530C2#3184412yes—
531C2#3185612yes—
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
C2×#18782
C2×#18032
#152402
#139352
#113572
#112552
#112082
#110772
#110672
#110522
#110242