Finite Group Databasebeta

← #415359

Subgroups — #415359

Order 512.

1202 conjugacy classes of subgroups.

The subgroup lattice has 513 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2552C211yesmaximal, characteristic
2C2×#1992C211yesmaximal, characteristic
3C2×#1989C211yesmaximal, characteristic
4#14160C211yesmaximal, characteristic
5#11385C212yesmaximal
6#11385C212yesmaximal
7#11293C212yesmaximal
8#11292C212yesmaximal
9#10849C211yesmaximal, characteristic
10#10835C211yesmaximal, characteristic
11#10703C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×#125C2212yes—
14C22×C4⋊C8C2211yescharacteristic
15C22×#97C2211yescharacteristic
16#2552—24no—
17#2552—24no—
18C2×#398—22no—
19C2×#389—21no—
20C2×#389C2212yes—
21C2×#384—21no—
22C2×#384C2211yescharacteristic
23C2×C4×C4⋊C4C2211yescharacteristic
24C2×C4×C4⋊C4C2211yescharacteristic
25C2×C4×C4⋊C4C2211yescharacteristic
26#1992—24no—
27#1989—24no—
28C2×#340C2211yescharacteristic
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2211yescharacteristic
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#340C2212yes—
36C2×#340C2212yes—
37C2×#340C2212yes—
38C2×#336C2212yes—
39C2×#336C2212yes—
40#1441—21no—
41#1441—21no—
42#1436—22no—
43C4⋊#95C2212yes—
44C4⋊#95C2212yes—
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×#125—24no—
49C2×#125—24no—
50C2×#125—24no—
51C2×#125—24no—
52C2×#125—22no—
53C2×#125—22no—
54C2×#125—22no—
55C2×#125—22no—
56C2×#125—22no—
57C2×#125—42no—
58C2×#125—42no—
59C2×#125Q814yes—
60C22×C4⋊C4—22no—
61C22×C4⋊C4—22no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—21no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4—22no—
67C22×C4⋊C4D411yescharacteristic
68C22×C4⋊C4D411yescharacteristic
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2311yescharacteristic
71C22×C4⋊C4C2311yescharacteristic
72C22×C4⋊C4C2312yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C42—21no—
76C22×C42—21no—
77C22×C42C2312yes—
78C22×C42C2311yescharacteristic
79C2×C4⋊C8—22no—
80C2×C4⋊C8—22no—
81C2×C4⋊C8—22no—
82C2×#97—22no—
83C2×#97—22no—
84C2×#97—22no—
85#398—216no—
86#389—28no—
87#389—24no—
88#389—24no—
89#384—28no—
90#384—22no—
91#384D414yes—
92C4×C4⋊C4—22no—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—22no—
95C4×C4⋊C4—22no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4D414yes—
98C2×#95—21no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95—22no—
102C2×#95—21no—
103C2×#95—22no—
104C2×#95—22no—
105C2×#95D412yes—
106#340—22no—
107#340—22no—
108#340—22no—
109#340—22no—
110#340—24no—
111#340—24no—
112#340—24no—
113#340—42no—
114#340—42no—
115#340Q818yes—
116#336—42no—
117#336—42no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4C2×D412yes—
125C23×C4C2×D412yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—21no—
136C22×C8—21no—
137C22×C8—21no—
138C22×C8—21no—
139C22×C8—21no—
140C22×C8—21no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8C2×Q814yes—
146#125—28no—
147#125—28no—
148#125—48no—
149#125—48no—
150#125—48no—
151#125—48no—
152#125—44no—
153#125—44no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—28no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
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—28no—
186C2×C4⋊C4—24no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—44no—
191C2×C4⋊C4—44no—
192C2×C4⋊C4—44no—
193C2×C4⋊C4—44no—
194C2×C4⋊C4—42no—
195C2×C4⋊C4—42no—
196C2×C4⋊C4—42no—
197C2×C4⋊C4—42no—
198C2×C4⋊C4—81no—
199C2×C4⋊C4—81no—
200C2×C4⋊C4—82no—
201C2×C4⋊C4—82no—
202C2×C4⋊C4C2×D414yes—
203C2×C4⋊C4C2×D412yes—
204C2×C4⋊C4C2×D412yes—
205C2×C4⋊C4C2×Q812yes—
206C2×C4⋊C4C2×Q812yes—
207C2×C4⋊C4C2×Q814yes—
208C2×C4⋊C4C2×Q814yes—
209C2×C4⋊C4#4114yes—
210C2×C42—24no—
211C2×C42—22no—
212C2×C42—22no—
213C2×C42—22no—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—44no—
218C2×C42—42no—
219C2×C42—81no—
220C2×C42—81no—
221C2×C42C2×D412yes—
222C2×C42C2×D412yes—
223C2×C42C2×Q812yes—
224C2×C42#4112yes—
225C2×C42#4112yes—
226C2×C42#4112yes—
227C4⋊C8—44no—
228#97—44no—
229#95—28no—
230#95—24no—
231#95—24no—
232#95—44no—
233#95—44no—
234#95—42no—
235#95—82no—
236#95—82no—
237C24C22×D411yescharacteristic
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—24no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—41no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—42no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—42no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4—42no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—41no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—41no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4#12012yes—
295C22×C4#12012yes—
296C22×C4C22⋊Q812yes—
297C22×C4C22⋊Q812yes—
298C22×C4#12412yes—
299C22×C4#12412yes—
300C22×C4C22×D411yescharacteristic
301C22×C4C22×D411yescharacteristic
302C22×C4C22×Q811yescharacteristic
303C22×C4C22×Q812yes—
304C22×C4C2×#4111yescharacteristic
305C22×C4C2×#4111yescharacteristic
306C22×C4C2×#4111yescharacteristic
307C22×C4C2×#4112yes—
308C22×C4H(5,𝔽2)12yes—
309C22×C42−1+412yes—
310C2×C8—24no—
311C2×C8—24no—
312C2×C8—24no—
313C2×C8—42no—
314C2×C8—42no—
315C2×C8—42no—
316C2×C8—42no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—41no—
322C2×C8—41no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C4⋊C4—22no—
328C4⋊C4—22no—
329C4⋊C4—24no—
330C4⋊C4—24no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—24no—
336C4⋊C4—24no—
337C4⋊C4—24no—
338C4⋊C4—24no—
339C4⋊C4—44no—
340C4⋊C4—44no—
341C4⋊C4—48no—
342C4⋊C4—416no—
343C4⋊C4—416no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—41no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—42no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C4⋊C4—84no—
355C4⋊C4—88no—
356C4⋊C4#12414yes—
357C4⋊C4C4⋊Q814yes—
358C42—48no—
359C42—42no—
360C42—42no—
361C42—42no—
362C42—42no—
363C42—84no—
364C42C22⋊Q814yes—
365C42C22⋊Q814yes—
366C42#12414yes—
367C42#12414yes—
368C23C2×#12011yescharacteristic
369C23C2×C22⋊Q811yescharacteristic
370C23C2×#12411yescharacteristic
371C23#54612yes—
372C23D4×Q812yes—
373C23#57512yes—
374C23#57512yes—
375C23#57611yescharacteristic
376C23#57711yescharacteristic
377C23#57711yescharacteristic
378C23#57811yescharacteristic
379C2×C4—24no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—24no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—44no—
392C2×C4—44no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—44no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—44no—
403C2×C4—44no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—44no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—42no—
424C2×C4—82no—
425C2×C4—81no—
426C2×C4—81no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—81no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—81no—
433C2×C4—82no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4#39314yes—
440C2×C4#39314yes—
441C2×C4C2×C22⋊Q811yescharacteristic
442C2×C4C2×C22⋊Q811yescharacteristic
443C2×C4C2×#12411yescharacteristic
444C2×C4C2×#12411yescharacteristic
445C2×C4C2×#12411yescharacteristic
446C2×C4C2×C4⋊Q811yescharacteristic
447C2×C4#53312yes—
448C2×C4#53312yes—
449C2×C4#53511yescharacteristic
450C2×C4#53611yescharacteristic
451C2×C4#54212yes—
452C2×C4#54212yes—
453C2×C4#54612yes—
454C2×C4D4×Q812yes—
455C2×C4#55412yes—
456C2×C4#55612yes—
457C2×C4#55712yes—
458C2×C4Q8212yes—
459C8—44no—
460C8—81no—
461C8—81no—
462C22C2×#39311yescharacteristic
463C22#243411yescharacteristic
464C22#243511yescharacteristic
465C22#245112yes—
466C22#249212yes—
467C22#300511yescharacteristic
468C22#300511yescharacteristic
469C22#301211yescharacteristic
470C22#301311yescharacteristic
471C22#308012yes—
472C22#308311yescharacteristic
473C22#308411yescharacteristic
474C22#313611yescharacteristic
475C22#313711yescharacteristic
476C22#314212yes—
477C22#330712yes—
478C22#330912yes—
479C22#331512yes—
480C22#331712yes—
481C22#338812yes—
482C22#338812yes—
483C22#338912yes—
484C22#339012yes—
485C4—24no—
486C4—24no—
487C4—44no—
488C4—44no—
489C4—82no—
490C4—82no—
491C4—84no—
492C4—82no—
493C4—82no—
494C4—84no—
495C4#245112yes—
496C4#249212yes—
497C4#256711yescharacteristic
498C4#256811yescharacteristic
499C4#264011yescharacteristic
500C4#286211yescharacteristic
501C2#1519111yescharacteristic
502C2#2345311yescharacteristic
503C2#2345811yescharacteristic
504C2#2584511yescharacteristic
505C2#2584611yescharacteristic
506C2#2589211yescharacteristic
507C2#2589311yescharacteristic
508C2#2645712yes—
509C2#2645912yes—
510C2#2782912yes—
511C2#2783112yes—
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×#25522
C2×#19922
C2×#19892
#141602
#113852
#112932
#112922
#108492
#108352
#107032