Finite Group Databasebeta

← #461657

Subgroups — #461657

Order 512.

1104 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2072C212yesmaximal
2C2×#1865C211yesmaximal, characteristic
3C2×#1803C211yesmaximal, characteristic
4#15145C211yesmaximal, characteristic
5#13935C211yesmaximal, characteristic
6#11373C211yesmaximal, characteristic
7#11255C211yesmaximal, characteristic
8#11227C211yesmaximal, characteristic
9#11220C212yesmaximal
10#11207C212yesmaximal
11#11190C212yesmaximal
12C22×C4×Q8C2211yescharacteristic
13C22×#106C2211yescharacteristic
14C22×C4⋊C8C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×#103C2211yescharacteristic
18C2×#398—21no—
19C2×#398—21no—
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#389C2212yes—
23C2×#384C2211yescharacteristic
24C2×#384C2212yes—
25#2072—28no—
26#1865—24no—
27#1803—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2212yes—
32C2×#340C2212yes—
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36C2×#336C2212yes—
37#1444—22no—
38#1441—22no—
39C4⋊#95—21no—
40C4⋊#95C2212yes—
41C4⋊#95C2211yescharacteristic
42#1434—21no—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C2×C4×Q8—22no—
49C2×C4×Q8—24no—
50C2×C4×Q8—42no—
51C2×C4×Q8D414yes—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—22no—
56C22×C4⋊C4—22no—
57C22×C4⋊C4—22no—
58C22×C4⋊C4—21no—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C4⋊C4C2312yes—
62C22×C4⋊C4C2311yescharacteristic
63C22×C4⋊C4C2311yescharacteristic
64C22×C42—21no—
65C22×C42C2312yes—
66C22×C42C2311yescharacteristic
67C2×#106—22no—
68C2×#106—24no—
69C2×C4⋊C8—22no—
70C2×C4⋊C8—42no—
71C2×#103—22no—
72C2×#103—22no—
73C2×#103—24no—
74C2×#103—24no—
75C2×#103—22no—
76C2×#103—24no—
77C2×#103—42no—
78C2×#103D414yes—
79#398—44no—
80#398—44no—
81#391—24no—
82#391—42no—
83#389—24no—
84#389—22no—
85#389—22no—
86#389—42no—
87#384—24no—
88#384—22no—
89#384—22no—
90#384—42no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—21no—
98C2×#95—21no—
99C2×#95D411yescharacteristic
100C2×#95D412yes—
101C2×#95D411yescharacteristic
102#340—24no—
103#340—24no—
104#340—24no—
105#340—22no—
106#340—42no—
107#340—42no—
108#336—24no—
109#336—22no—
110#336—42no—
111C22×Q8—21no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8C2×D412yes—
119C22×Q8C2×D412yes—
120C22×Q8#4112yes—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4—22no—
125C23×C4—21no—
126C23×C4—22no—
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D412yes—
129C23×C4C2×D412yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—21no—
133C22×C8—21no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—21no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—21no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—21no—
148C22×C8C2×D412yes—
149C4×Q8—28no—
150C4×Q8—44no—
151C4×Q8—48no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
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—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—42no—
182C2×C4⋊C4—42no—
183C2×C4⋊C4—42no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—42no—
186C2×C4⋊C4—42no—
187C2×C4⋊C4—42no—
188C2×C4⋊C4—42no—
189C2×C4⋊C4—44no—
190C2×C4⋊C4—82no—
191C2×C4⋊C4—82no—
192C2×C4⋊C4—81no—
193C2×C4⋊C4—81no—
194C2×C4⋊C4—82no—
195C2×C4⋊C4C2×D412yes—
196C2×C4⋊C4C2×D412yes—
197C2×C4⋊C4#4114yes—
198C2×C42—22no—
199C2×C42—22no—
200C2×C42—24no—
201C2×C42—22no—
202C2×C42—22no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—22no—
207C2×C42—44no—
208C2×C42C2×D412yes—
209C2×C42#4112yes—
210#106—44no—
211C4⋊C8—44no—
212#103—28no—
213#103—44no—
214#103—48no—
215#95—22no—
216#95—24no—
217#95—44no—
218#95—44no—
219#95—44no—
220#95—44no—
221#95—42no—
222#95—44no—
223#95—42no—
224#95—81no—
225#95SD1614yes—
226C24C22×D411yescharacteristic
227C2×Q8—22no—
228C2×Q8—22no—
229C2×Q8—22no—
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—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8#12114yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—22no—
262C22×C4—21no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—42no—
272C22×C4—41no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—42no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—41no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—42no—
294C22×C4—42no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—42no—
299C22×C4#12112yes—
300C22×C4#12112yes—
301C22×C4C2×SD1614yes—
302C22×C4C2×SD1612yes—
303C22×C4AGL(1,ℤ/8)12yes—
304C22×C4#13714yes—
305C22×C4C22×D411yescharacteristic
306C22×C4C22×D411yescharacteristic
307C22×C4C2×#4111yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4111yescharacteristic
310C22×C4H(5,𝔽2)12yes—
311C22×C4H(5,𝔽2)11yescharacteristic
312C22×C42−1+412yes—
313C22×C42−1+412yes—
314C22×C42−1+411yescharacteristic
315C2×C8—22no—
316C2×C8—24no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—42no—
320C2×C8—42no—
321C2×C8—41no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—41no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—41no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C2×C8—42no—
332C4⋊C4—22no—
333C4⋊C4—24no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—48no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—42no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—42no—
348C4⋊C4—42no—
349C4⋊C4—42no—
350C4⋊C4—88no—
351C4⋊C4—88no—
352C4⋊C4—84no—
353C4⋊C4—88no—
354C4⋊C4—84no—
355C4⋊C4—84no—
356C4⋊C4#12314yes—
357C42—22no—
358C42—22no—
359C42—24no—
360C42—48no—
361C42—42no—
362C42—42no—
363C42—42no—
364C42—42no—
365C23C2×#12111yescharacteristic
366C23#53612yes—
367C23D4211yescharacteristic
368C23#54612yes—
369C23#54811yescharacteristic
370C23C22×SD1611yescharacteristic
371C23C2×AGL(1,ℤ/8)11yescharacteristic
372C23C2×#13712yes—
373C23#57611yescharacteristic
374C23#57711yescharacteristic
375C23#57812yes—
376Q8—24no—
377Q8—24no—
378Q8—42no—
379Q8—44no—
380Q8—44no—
381Q8—42no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—22no—
385C2×C4—24no—
386C2×C4—22no—
387C2×C4—24no—
388C2×C4—24no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—22no—
394C2×C4—24no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—44no—
405C2×C4—44no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—44no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—42no—
418C2×C4—81no—
419C2×C4—81no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—81no—
425C2×C4—82no—
426C2×C4—82no—
427C2×C4—82no—
428C2×C4—82no—
429C2×C4—82no—
430C2×C4—82no—
431C2×C4—82no—
432C2×C4—82no—
433C2×C4—82no—
434C2×C4—81no—
435C2×C4—82no—
436C2×C4—81no—
437C2×C4—82no—
438C2×C4—82no—
439C2×C4—82no—
440C2×C4—82no—
441C2×C4#46014yes—
442C2×C4C2×#12111yescharacteristic
443C2×C4C2×#12311yescharacteristic
444C2×C4#53411yescharacteristic
445C2×C4#53611yescharacteristic
446C2×C4#53611yescharacteristic
447C2×C4#53712yes—
448C2×C4#53711yescharacteristic
449C2×C4#53912yes—
450C2×C4#54112yes—
451C2×C4#54112yes—
452C2×C4#54212yes—
453C2×C4#54711yescharacteristic
454C2×C4#54712yes—
455C2×C4#55112yes—
456C2×C4#55212yes—
457C2×C4#55311yescharacteristic
458C2×C4#56212yes—
459C2×C4#56312yes—
460C8—42no—
461C8—82no—
462C8—81no—
463C22#241211yescharacteristic
464C22#241611yescharacteristic
465C22#262111yescharacteristic
466C22#267212yes—
467C22#269012yes—
468C22C2×#46011yescharacteristic
469C22#303911yescharacteristic
470C22#304411yescharacteristic
471C22#304511yescharacteristic
472C22#322912yes—
473C22#323912yes—
474C22#324212yes—
475C22#324312yes—
476C22#327012yes—
477C22#327411yescharacteristic
478C22D4×SD1611yescharacteristic
479C22#329812yes—
480C22#330812yes—
481C22#331412yes—
482C22#333112yes—
483C22#336211yescharacteristic
484C22#336811yescharacteristic
485C22#337412yes—
486C4—24no—
487C4—44no—
488C4—44no—
489C4—44no—
490C4—42no—
491C4—44no—
492C4—82no—
493C4—162no—
494C4—162no—
495C4—161no—
496C4#248611yescharacteristic
497C4#261811yescharacteristic
498C4#262511yescharacteristic
499C4#272212yes—
500C4#274712yes—
501C4#281411yescharacteristic
502C2#1639911yescharacteristic
503C2#2452811yescharacteristic
504C2#2454811yescharacteristic
505C2#2472811yescharacteristic
506C2#2474611yescharacteristic
507C2#2986611yescharacteristic
508C2#2987511yescharacteristic
509C2#3063312yes—
510C2#3065912yes—
511C2#3145612yes—
512C2#3147612yes—
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×#18652
C2×#18032
#151452
#139352
#113732
#112552
#112272
#112202
#112072
#111902