Finite Group Databasebeta

← #409052

Subgroups — #409052

Order 512.

1420 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C211yesmaximal, characteristic
2C2×#2043C211yesmaximal, characteristic
3C2×#1956C212yesmaximal
4C2×#1956C211yesmaximal, characteristic
5C2×#1933C211yesmaximal, characteristic
6#15190C211yesmaximal, characteristic
7#15145C211yesmaximal, characteristic
8#11050C212yesmaximal
9#11010C212yesmaximal
10#10926C212yesmaximal
11C8×#95C211yesmaximal, characteristic
12C22×C4⋊Q8C2212yes—
13C22×C4⋊Q8C2211yescharacteristic
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C22×#103C2212yes—
18C22×#103C2211yescharacteristic
19C22×#103C2211yescharacteristic
20C22×#103C2212yes—
21C22×C4×C8C2212yes—
22C22×C4×C8C2211yescharacteristic
23C2×#398—22no—
24C2×#391C2212yes—
25C2×#391C2211yescharacteristic
26C2×#389C2211yescharacteristic
27C2×#384C2211yescharacteristic
28C2×#384C2212yes—
29#2072—24no—
30#2043—24no—
31#1956—28no—
32#1956—24no—
33#1933—24no—
34C2×#340C2211yescharacteristic
35C2×#340C2212yes—
36C2×#340C2211yescharacteristic
37C2×#336C2212yes—
38C2×#336C2211yescharacteristic
39#1444—22no—
40#1444—22no—
41#1444—22no—
42C4⋊#95C2212yes—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×Q8C2312yes—
46C23×C8C2312yes—
47C23×C8C2311yescharacteristic
48C23×C8C2311yescharacteristic
49C2×C4⋊Q8—24no—
50C2×C4⋊Q8—24no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—22no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—24no—
55C2×C4⋊Q8—24no—
56C2×C4⋊Q8—42no—
57C2×C4⋊Q8—42no—
58C2×C4⋊Q8—42no—
59C22×C4⋊C4—21no—
60C22×C4⋊C4—21no—
61C22×C4⋊C4—24no—
62C22×C4⋊C4—21no—
63C22×C4⋊C4—21no—
64C22×C4⋊C4—24no—
65C22×C4⋊C4—21no—
66C22×C4⋊C4C2311yescharacteristic
67C22×C4⋊C4C2312yes—
68C22×C4⋊C4C2311yescharacteristic
69C22×C4⋊C4C2311yescharacteristic
70C22×C42C2311yescharacteristic
71C22×C42C2312yes—
72C2×#103—22no—
73C2×#103—22no—
74C2×#103—24no—
75C2×#103—24no—
76C2×#103—24no—
77C2×#103—22no—
78C2×#103—24no—
79C2×#103—24no—
80C2×#103—24no—
81C2×#103—24no—
82C2×#103—24no—
83C2×#103—24no—
84C2×#103—24no—
85C2×#103—42no—
86C2×#103—42no—
87C2×#103—42no—
88C2×#103—42no—
89C2×#103—42no—
90C2×#103—42no—
91C2×#103D414yes—
92C2×C4×C8—24no—
93C2×C4×C8—24no—
94C2×C4×C8—42no—
95C2×C4×C8—42no—
96C2×C4×C8D414yes—
97#398—48no—
98#391—22no—
99#391—22no—
100#391—24no—
101#391—24no—
102#389—22no—
103#389—22no—
104#384—24no—
105#384—22no—
106#384—22no—
107#384—42no—
108C2×#95—24no—
109C2×#95—21no—
110C2×#95—24no—
111C2×#95—21no—
112C2×#95—21no—
113C2×#95D412yes—
114C2×#95D412yes—
115#340—24no—
116#340—22no—
117#340—22no—
118#336—24no—
119#336D414yes—
120C22×Q8—21no—
121C22×Q8—22no—
122C22×Q8—22no—
123C22×Q8—22no—
124C22×Q8—22no—
125C22×Q8—21no—
126C22×Q8—21no—
127C22×Q8—22no—
128C22×Q8—22no—
129C22×Q8—22no—
130C22×Q8—22no—
131C22×Q8—22no—
132C22×Q8—22no—
133C22×Q8—22no—
134C22×Q8—22no—
135C22×Q8—21no—
136C22×Q8—22no—
137C22×Q8—22no—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8—21no—
141C22×Q8—22no—
142C22×Q8C2×D412yes—
143C22×Q8#4114yes—
144C23×C4—21no—
145C23×C4—21no—
146C23×C4—22no—
147C23×C4—22no—
148C23×C4—21no—
149C23×C4—21no—
150C23×C4C2×D414yes—
151C23×C4C2×D411yescharacteristic
152C23×C4C2×D411yescharacteristic
153C23×C4C2411yescharacteristic
154C22×C8—22no—
155C22×C8—22no—
156C22×C8—22no—
157C22×C8—22no—
158C22×C8—22no—
159C22×C8—22no—
160C22×C8—22no—
161C22×C8—22no—
162C22×C8—22no—
163C22×C8—21no—
164C22×C8—21no—
165C22×C8—22no—
166C22×C8C2×D414yes—
167C22×C8C2×D412yes—
168C22×C8C2×D412yes—
169C22×C8C2×D412yes—
170C22×C8#4112yes—
171C4⋊Q8—28no—
172C4⋊Q8—216no—
173C4⋊Q8—48no—
174C4⋊Q8—48no—
175C4⋊Q8—48no—
176C4⋊Q8—44no—
177C4⋊Q8—48no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—21no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—21no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—48no—
198C2×C4⋊C4—48no—
199C2×C4⋊C4—44no—
200C2×C4⋊C4—44no—
201C2×C4⋊C4—48no—
202C2×C4⋊C4—42no—
203C2×C4⋊C4—42no—
204C2×C4⋊C4—48no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—48no—
209C2×C4⋊C4—44no—
210C2×C4⋊C4—44no—
211C2×C4⋊C4—84no—
212C2×C4⋊C4—82no—
213C2×C4⋊C4C2×D412yes—
214C2×C42—22no—
215C2×C42—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D412yes—
221C2×C42C2×D412yes—
222C2×C42#4114yes—
223C2×C42#4114yes—
224#103—28no—
225#103—48no—
226#103—48no—
227#103—48no—
228#103—44no—
229#103—48no—
230#103—48no—
231C4×C8—44no—
232C4×C8—48no—
233#95—48no—
234#95—42no—
235#95—42no—
236#95—42no—
237#95—48no—
238#95SD1618yes—
239#95Q1618yes—
240C24C22×D411yescharacteristic
241C2×Q8—22no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—22no—
247C2×Q8—24no—
248C2×Q8—24no—
249C2×Q8—24no—
250C2×Q8—24no—
251C2×Q8—42no—
252C2×Q8—41no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—42no—
262C2×Q8—42no—
263C2×Q8—41no—
264C2×Q8—42no—
265C2×Q8—42no—
266C2×Q8—42no—
267C2×Q8—41no—
268C2×Q8—42no—
269C2×Q8—41no—
270C2×Q8—42no—
271C2×Q8—42no—
272C2×Q8—42no—
273C2×Q8—41no—
274C2×Q8—42no—
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—
287C22×C4—24no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—24no—
292C22×C4—22no—
293C22×C4—24no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—24no—
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—41no—
308C22×C4—42no—
309C22×C4—41no—
310C22×C4—42no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—41no—
314C22×C4—41no—
315C22×C4—42no—
316C22×C4—42no—
317C22×C4—42no—
318C22×C4—42no—
319C22×C4—42no—
320C22×C4—44no—
321C22×C4—44no—
322C22×C4—42no—
323C22×C4—42no—
324C22×C4—42no—
325C22×C4#12418yes—
326C22×C4#12712yes—
327C22×C4#12712yes—
328C22×C4C2×SD1618yes—
329C22×C4C2×SD1614yes—
330C22×C4C2×Q1618yes—
331C22×C4C2×Q1614yes—
332C22×C4C22×D411yescharacteristic
333C22×C4C22×D411yescharacteristic
334C22×C4C22×D411yescharacteristic
335C22×C4C2×#4111yescharacteristic
336C22×C4C2×#4112yes—
337C22×C4C2×#4112yes—
338C22×C4C2×#4112yes—
339C22×C4H(5,𝔽2)12yes—
340C22×C4H(5,𝔽2)11yescharacteristic
341C22×C4H(5,𝔽2)11yescharacteristic
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—24no—
345C2×C8—24no—
346C2×C8—22no—
347C2×C8—24no—
348C2×C8—22no—
349C2×C8—22no—
350C2×C8—24no—
351C2×C8—24no—
352C2×C8—42no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C2×C8#12114yes—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—432no—
362C4⋊C4—48no—
363C4⋊C4—48no—
364C4⋊C4—41no—
365C4⋊C4—41no—
366C4⋊C4—42no—
367C4⋊C4—41no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—42no—
371C4⋊C4—41no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—816no—
377C4⋊C4—84no—
378C4⋊C4—84no—
379C4⋊C4—84no—
380C42—22no—
381C42—24no—
382C42—22no—
383C42—24no—
384C42—24no—
385C42—24no—
386C42—24no—
387C23C2×#12412yes—
388C23C2×#12711yescharacteristic
389C23D4211yescharacteristic
390C23#54612yes—
391C23#55011yescharacteristic
392C23C22×SD1612yes—
393C23C22×Q1612yes—
394C23#57712yes—
395C23#57812yes—
396Q8—44no—
397Q8—42no—
398Q8—44no—
399Q8—44no—
400Q8—44no—
401Q8—42no—
402Q8—44no—
403Q8—81no—
404Q8—82no—
405Q8—81no—
406Q8—82no—
407Q8—82no—
408Q8—82no—
409Q8—82no—
410C2×C4—28no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—28no—
414C2×C4—28no—
415C2×C4—28no—
416C2×C4—28no—
417C2×C4—28no—
418C2×C4—28no—
419C2×C4—28no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—44no—
423C2×C4—44no—
424C2×C4—44no—
425C2×C4—44no—
426C2×C4—44no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—82no—
430C2×C4—81no—
431C2×C4—82no—
432C2×C4—81no—
433C2×C4—81no—
434C2×C4—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—81no—
438C2×C4—82no—
439C2×C4—81no—
440C2×C4—84no—
441C2×C4—84no—
442C2×C4—82no—
443C2×C4—81no—
444C2×C4—82no—
445C2×C4—81no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—84no—
449C2×C4—84no—
450C2×C4—81no—
451C2×C4—82no—
452C2×C4#487116yes—
453C2×C4#49214yes—
454C2×C4#49414yes—
455C2×C4C2×#12111yescharacteristic
456C2×C4#52511yescharacteristic
457C2×C4#52512yes—
458C2×C4#52912yes—
459C2×C4#53211yescharacteristic
460C2×C4#53211yescharacteristic
461C2×C4#53212yes—
462C2×C4#53411yescharacteristic
463C2×C4#53511yescharacteristic
464C2×C4#53812yes—
465C2×C4#54012yes—
466C2×C4D4211yescharacteristic
467C2×C4#54612yes—
468C2×C4#54612yes—
469C2×C4#55011yescharacteristic
470C2×C4#55112yes—
471C2×C4#55312yes—
472C2×C4#55312yes—
473C8—24no—
474C8—42no—
475C8—44no—
476C22#247212yes—
477C22#249012yes—
478C22#253911yescharacteristic
479C22#254111yescharacteristic
480C22#285811yescharacteristic
481C22C2×#48714yes—
482C22#313714yes—
483C22C2×#49211yescharacteristic
484C22C2×#49411yescharacteristic
485C22#315411yescharacteristic
486C22#315611yescharacteristic
487C22D4×SD1612yes—
488C22D4×Q1612yes—
489C22#329814yes—
490C22#329914yes—
491C22#339212yes—
492C22#339512yes—
493C4—216no—
494C4—24no—
495C4—24no—
496C4—82no—
497C4—82no—
498C4—84no—
499C4—162no—
500C4—161no—
501C4—161no—
502C4#239211yescharacteristic
503C4#246412yes—
504C4#254311yescharacteristic
505C4#255412yes—
506C4#260311yescharacteristic
507C4#266111yescharacteristic
508C2#1500711yescharacteristic
509C2#2725814yes—
510C2#2774914yes—
511C2#2833311yescharacteristic
512C2#2833611yescharacteristic
513C2#2846111yescharacteristic
514C2#2846211yescharacteristic
515C2#3322511yescharacteristic
516C2#3322911yescharacteristic
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×#20722
C2×#20432
C2×#19562
C2×#19332
#151902
#151452
#110502
#110102
#109262
C8×#952