Finite Group Databasebeta

← #425722

Subgroups — #425722

Order 512.

1534 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×#2546C211yesmaximal, characteristic
2C2×#2096C211yesmaximal, characteristic
3C2×#2043C212yesmaximal
4C2×#1956C211yesmaximal, characteristic
5C2×#1865C212yesmaximal
6#15151C211yesmaximal, characteristic
7#11122C212yesmaximal
8#11010C211yesmaximal, characteristic
9#11010C212yesmaximal
10#10926C211yesmaximal, characteristic
11#10874C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14C22×C4⋊Q8C2211yescharacteristic
15C22×C4×Q8C2211yescharacteristic
16C22×#106C2212yes—
17C22×#103C2212yes—
18C22×#103C2212yes—
19C22×#103C2212yes—
20C22×#103C2212yes—
21C22×#103C2212yes—
22C22×#103C2212yes—
23C22×C4×C8C2211yescharacteristic
24#2546—24no—
25#2546—24no—
26C2×#398—22no—
27C2×#398—22no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30C2×#389—22no—
31C2×#389C2212yes—
32C2×#384C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2096—24no—
35#2043—28no—
36#1956—24no—
37#1865—28no—
38C2×#340C2212yes—
39C2×#336C2211yescharacteristic
40C2×#336C2212yes—
41C2×#336C2212yes—
42#1445—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2211yescharacteristic
45C23×Q8C2311yescharacteristic
46C23×Q8C2311yescharacteristic
47C23×Q8C2311yescharacteristic
48C23×C8C2312yes—
49C23×C8C2312yes—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—22no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—24no—
58C2×C4⋊Q8—42no—
59C2×C4⋊Q8—42no—
60C2×C4⋊Q8—42no—
61C2×C4⋊Q8D414yes—
62C2×C4×Q8—22no—
63C2×C4×Q8—22no—
64C2×C4×Q8—24no—
65C2×C4×Q8—42no—
66C22×C4⋊C4—21no—
67C22×C4⋊C4—21no—
68C22×C4⋊C4—22no—
69C22×C4⋊C4—22no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—22no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2312yes—
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2312yes—
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81C2×#106—24no—
82C2×#106—24no—
83C2×#106—42no—
84C2×#103—24no—
85C2×#103—24no—
86C2×#103—24no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—24no—
90C2×#103—24no—
91C2×#103—24no—
92C2×#103—24no—
93C2×#103—24no—
94C2×#103—24no—
95C2×#103—24no—
96C2×#103—24no—
97C2×#103—42no—
98C2×#103—42no—
99C2×#103—42no—
100C2×#103—42no—
101C2×#103—42no—
102C2×C4×C8—22no—
103C2×C4×C8—42no—
104#398—216no—
105#398—48no—
106#391—22no—
107#391—22no—
108#391—22no—
109#391D414yes—
110#389—216no—
111#389—24no—
112#389—24no—
113#384—22no—
114#384—22no—
115C4×C4⋊C4—22no—
116C4×C4⋊C4—22no—
117C2×#95—22no—
118C2×#95—22no—
119C2×#95—22no—
120C2×#95—24no—
121C2×#95—22no—
122C2×#95D412yes—
123#340—24no—
124#336—22no—
125#336—42no—
126#336D418yes—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—21no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—22no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—22no—
141C22×Q8—22no—
142C22×Q8—22no—
143C22×Q8—22no—
144C22×Q8—22no—
145C22×Q8—22no—
146C22×Q8—22no—
147C22×Q8C2×D412yes—
148C22×Q8C2×D412yes—
149C22×Q8C2×D412yes—
150C22×Q8#4114yes—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—22no—
155C23×C4—21no—
156C23×C4—22no—
157C23×C4C2×D411yescharacteristic
158C23×C4C2×D412yes—
159C23×C4C2×D412yes—
160C23×C4C2×D411yescharacteristic
161C23×C4C2411yescharacteristic
162C22×C8—22no—
163C22×C8—22no—
164C22×C8—22no—
165C22×C8—22no—
166C22×C8—22no—
167C22×C8—22no—
168C22×C8—22no—
169C22×C8—22no—
170C22×C8—22no—
171C22×C8—22no—
172C22×C8—22no—
173C22×C8—22no—
174C22×C8C2×D414yes—
175C22×C8C2×D414yes—
176C4⋊Q8—28no—
177C4⋊Q8—44no—
178C4⋊Q8—48no—
179C4⋊Q8—48no—
180C4⋊Q8—48no—
181C4⋊Q8—48no—
182C4⋊Q8—44no—
183C4⋊Q8—44no—
184C4×Q8—48no—
185C4×Q8—48no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—24no—
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—24no—
195C2×C4⋊C4—24no—
196C2×C4⋊C4—28no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—28no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—48no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4—42no—
215C2×C4⋊C4—44no—
216C2×C4⋊C4—44no—
217C2×C4⋊C4—44no—
218C2×C4⋊C4—48no—
219C2×C4⋊C4—44no—
220C2×C4⋊C4SD1618yes—
221C2×C4⋊C4C2×D412yes—
222C2×C4⋊C4#4114yes—
223C2×C42—21no—
224C2×C42—21no—
225C2×C42—22no—
226C2×C42—22no—
227C2×C42—22no—
228C2×C42—24no—
229C2×C42—21no—
230C2×C42—21no—
231C2×C42—22no—
232C2×C42—22no—
233C2×C42—42no—
234C2×C42—42no—
235C2×C42C2×D412yes—
236C2×C42C2×D412yes—
237C2×C42C2×D414yes—
238#106—48no—
239#103—216no—
240#103—48no—
241#103—48no—
242#103—48no—
243#103—48no—
244#103—48no—
245C4×C8—44no—
246#95—28no—
247#95—28no—
248#95—24no—
249#95—44no—
250#95—44no—
251#95—48no—
252C24C22×D411yescharacteristic
253C2×Q8—22no—
254C2×Q8—24no—
255C2×Q8—22no—
256C2×Q8—24no—
257C2×Q8—22no—
258C2×Q8—22no—
259C2×Q8—22no—
260C2×Q8—24no—
261C2×Q8—24no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8—24no—
265C2×Q8—24no—
266C2×Q8—24no—
267C2×Q8—24no—
268C2×Q8—24no—
269C2×Q8—41no—
270C2×Q8—41no—
271C2×Q8—41no—
272C2×Q8—41no—
273C2×Q8—41no—
274C2×Q8—41no—
275C2×Q8—42no—
276C2×Q8—42no—
277C2×Q8—42no—
278C2×Q8—42no—
279C2×Q8—42no—
280C2×Q8—42no—
281C2×Q8—42no—
282C2×Q8—42no—
283C2×Q8—42no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—42no—
292C2×Q8—42no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—42no—
296C2×Q8—42no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—42no—
300C2×Q8#12314yes—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—24no—
304C22×C4—22no—
305C22×C4—24no—
306C22×C4—22no—
307C22×C4—21no—
308C22×C4—22no—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—24no—
312C22×C4—22no—
313C22×C4—24no—
314C22×C4—24no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—22no—
318C22×C4—24no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—42no—
322C22×C4—42no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—41no—
328C22×C4—41no—
329C22×C4—42no—
330C22×C4—42no—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—42no—
334C22×C4—44no—
335C22×C4—44no—
336C22×C4—44no—
337C22×C4—42no—
338C22×C4—42no—
339C22×C4—42no—
340C22×C4—44no—
341C22×C4—44no—
342C22×C4#12014yes—
343C22×C4#12712yes—
344C22×C4#12712yes—
345C22×C4C2×SD1614yes—
346C22×C4C2×SD1614yes—
347C22×C4C2×SD1614yes—
348C22×C4#13714yes—
349C22×C4C22×D411yescharacteristic
350C22×C4C22×D412yes—
351C22×C4C22×D411yescharacteristic
352C22×C4C2×#4112yes—
353C22×C4C2×#4112yes—
354C22×C4H(5,𝔽2)12yes—
355C22×C4H(5,𝔽2)12yes—
356C22×C42−1+412yes—
357C2×C8—24no—
358C2×C8—24no—
359C2×C8—24no—
360C2×C8—24no—
361C2×C8—24no—
362C2×C8—24no—
363C2×C8—42no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—42no—
367C2×C8—42no—
368C2×C8—42no—
369C2×C8—42no—
370C2×C8—42no—
371C4⋊C4—216no—
372C4⋊C4—24no—
373C4⋊C4—24no—
374C4⋊C4—24no—
375C4⋊C4—416no—
376C4⋊C4—48no—
377C4⋊C4—48no—
378C4⋊C4—416no—
379C4⋊C4—48no—
380C4⋊C4—42no—
381C4⋊C4—42no—
382C4⋊C4—42no—
383C4⋊C4—42no—
384C4⋊C4—42no—
385C4⋊C4—42no—
386C4⋊C4—42no—
387C4⋊C4—88no—
388C4⋊C4—88no—
389C42—22no—
390C42—22no—
391C42—24no—
392C42—24no—
393C42—24no—
394C42—48no—
395C42—41no—
396C42—41no—
397C42—42no—
398C23C2×#12011yescharacteristic
399C23C2×#12711yescharacteristic
400C23#53511yescharacteristic
401C23#54612yes—
402C23#54812yes—
403C23C22×SD1612yes—
404C23C2×#13712yes—
405C23#57814yes—
406Q8—28no—
407Q8—44no—
408Q8—44no—
409Q8—44no—
410Q8—44no—
411Q8—44no—
412Q8—44no—
413Q8—44no—
414Q8—82no—
415Q8—82no—
416Q8—82no—
417Q8—82no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—24no—
422C2×C4—24no—
423C2×C4—24no—
424C2×C4—28no—
425C2×C4—28no—
426C2×C4—24no—
427C2×C4—24no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—28no—
431C2×C4—42no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—44no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—44no—
440C2×C4—48no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—44no—
449C2×C4—44no—
450C2×C4—82no—
451C2×C4—82no—
452C2×C4—82no—
453C2×C4—82no—
454C2×C4—82no—
455C2×C4—82no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—82no—
460C2×C4—82no—
461C2×C4—82no—
462C2×C4—84no—
463C2×C4—84no—
464C2×C4#44818yes—
465C2×C4#49214yes—
466C2×C4C2×#12311yescharacteristic
467C2×C4#52512yes—
468C2×C4#53211yescharacteristic
469C2×C4#53411yescharacteristic
470C2×C4#53611yescharacteristic
471C2×C4#53712yes—
472C2×C4#53912yes—
473C2×C4#54012yes—
474C2×C4D4212yes—
475C2×C4D4212yes—
476C2×C4#54612yes—
477C2×C4#54612yes—
478C2×C4#54712yes—
479C2×C4#54812yes—
480C2×C4#55312yes—
481C2×C4#56212yes—
482C8—44no—
483C8—44no—
484C22#240511yescharacteristic
485C22#243211yescharacteristic
486C22#243311yescharacteristic
487C22#254012yes—
488C22#271112yes—
489C22C2×#44812yes—
490C22#301112yes—
491C22C2×#49211yescharacteristic
492C22#315211yescharacteristic
493C22#315612yes—
494C22#320612yes—
495C22#322312yes—
496C22#329814yes—
497C22#331214yes—
498C22#335914yes—
499C22#337414yes—
500C4—24no—
501C4—28no—
502C4—42no—
503C4—44no—
504C4—44no—
505C4—82no—
506C4—82no—
507C4—82no—
508C4—84no—
509C4—162no—
510C4#244312yes—
511C4#246412yes—
512C4#267112yes—
513C4#268112yes—
514C2#1534011yescharacteristic
515C2#2404112yes—
516C2#2564712yes—
517C2#2571811yescharacteristic
518C2#2573511yescharacteristic
519C2#2840012yes—
520C2#2842312yes—
521C2#3208214yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C4×C4⋊C4 ◁ 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×#25462
C2×#20962
C2×#20432
C2×#19562
C2×#18652
#151512
#111222
#110102
#109262
#108742