Finite Group Databasebeta

← #470432

Subgroups — #470432

Order 512.

1086 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#1927C212yesmaximal
2#15180C211yesmaximal, characteristic
3#13911C211yesmaximal, characteristic
4#11257C212yesmaximal
5#11202C211yesmaximal, characteristic
6#11122C212yesmaximal
7#11077C212yesmaximal
8#11050C212yesmaximal
9#11010C212yesmaximal
10C22×C4⋊Q8C2211yescharacteristic
11C22×#106C2212yes—
12C22×C4⋊C8C2211yescharacteristic
13C22×#103C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C22×#103C2212yes—
17C2×#391C2212yes—
18C2×#391C2212yes—
19C2×#384C2211yescharacteristic
20C2×#382—21no—
21C2×C4×C4⋊C4C2211yescharacteristic
22C2×C4×C4⋊C4C2211yescharacteristic
23#1927—28no—
24C2×#340C2212yes—
25C2×#340C2212yes—
26C2×#340C2212yes—
27C2×#336C2212yes—
28C2×#336C2212yes—
29C2×#336C2211yescharacteristic
30#1442—21no—
31#1442—21no—
32#1442—21no—
33#1442—21no—
34#1441—21no—
35#1441—21no—
36C4⋊#95—21no—
37C4⋊#95C2212yes—
38C4⋊#95C2212yes—
39C4⋊#95C2211yescharacteristic
40C23×Q8C2312yes—
41C23×C8C2312yes—
42C23×C8C2312yes—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—42no—
45C2×C4⋊Q8—41no—
46C2×C4⋊Q8—41no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4D411yescharacteristic
52C22×C4⋊C4D411yescharacteristic
53C22×C4⋊C4C2312yes—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2312yes—
56C22×C42—21no—
57C22×C42C2311yescharacteristic
58C22×C42C2311yescharacteristic
59C22×C42C2311yescharacteristic
60C2×#106—24no—
61C2×#106—42no—
62C2×#106—42no—
63C2×C4⋊C8—41no—
64C2×C4⋊C8—42no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—24no—
69C2×#103—24no—
70C2×#103—42no—
71C2×#103—42no—
72C2×#103—42no—
73C2×#103—42no—
74C2×#103—42no—
75C2×#103—42no—
76C2×#103—42no—
77#391—24no—
78#391—24no—
79#391—24no—
80#391—42no—
81#384—41no—
82#384—41no—
83#382—44no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4—41no—
86C4×C4⋊C4—41no—
87C4×C4⋊C4D414yes—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—22no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98C2×#95—22no—
99C2×#95D411yescharacteristic
100C2×#95D411yescharacteristic
101#340—24no—
102#340—24no—
103#340—42no—
104#336—24no—
105#336—42no—
106#336—41no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—22no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8—22no—
118C22×Q8—22no—
119C22×Q8—22no—
120C22×Q8#4114yes—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2411yescharacteristic
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8—22no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C22×C8—22no—
144C22×C8—22no—
145C22×C8#4114yes—
146C4⋊Q8—44no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4—41no—
170C2×C4⋊C4—41no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4SD1614yes—
181C2×C4⋊C4C2×D414yes—
182C2×C42—21no—
183C2×C42—21no—
184C2×C42—21no—
185C2×C42—21no—
186C2×C42—24no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—21no—
191C2×C42—21no—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42—22no—
195C2×C42—42no—
196C2×C42—42no—
197C2×C42C2×D412yes—
198C2×C42#4112yes—
199#106—48no—
200C4⋊C8—44no—
201#103—216no—
202#103—48no—
203#103—48no—
204#103—48no—
205#95—24no—
206#95—41no—
207#95—41no—
208#95—44no—
209#95—44no—
210#95—42no—
211#95—44no—
212#95—44no—
213#95—44no—
214#95—44no—
215#95—42no—
216#95—82no—
217#95—82no—
218C24C22×D411yescharacteristic
219C2×Q8—24no—
220C2×Q8—24no—
221C2×Q8—24no—
222C2×Q8—24no—
223C2×Q8—24no—
224C2×Q8—24no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—42no—
230C2×Q8—42no—
231C2×Q8—42no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—42no—
235C2×Q8—42no—
236C2×Q8—42no—
237C2×Q8—42no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—42no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8—42no—
245C2×Q8—42no—
246C2×Q8—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—21no—
251C22×C4—21no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4—21no—
256C22×C4—22no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—22no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—42no—
275C22×C4—42no—
276C22×C4—42no—
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—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—42no—
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—42no—
300C22×C4#12112yes—
301C22×C4#12112yes—
302C22×C4C2×SD1612yes—
303C22×C4C2×SD1612yes—
304C22×C4C2×SD1612yes—
305C22×C4#13712yes—
306C22×C4#13712yes—
307C22×C4C22×D411yescharacteristic
308C22×C4C2×#4112yes—
309C22×C4C2×#4112yes—
310C22×C4C2×#4111yescharacteristic
311C22×C4H(5,𝔽2)12yes—
312C22×C4H(5,𝔽2)12yes—
313C22×C4H(5,𝔽2)11yescharacteristic
314C22×C4H(5,𝔽2)11yescharacteristic
315C22×C42−1+411yescharacteristic
316C22×C42−1+411yescharacteristic
317C2×C8—24no—
318C2×C8—24no—
319C2×C8—24no—
320C2×C8—42no—
321C2×C8—42no—
322C2×C8—42no—
323C2×C8—42no—
324C2×C8—42no—
325C2×C8—42no—
326C2×C8—42no—
327C2×C8—42no—
328C2×C8—42no—
329C2×C8—42no—
330C2×C8—42no—
331C4⋊C4—24no—
332C4⋊C4—24no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—48no—
336C4⋊C4—48no—
337C4⋊C4—42no—
338C4⋊C4—42no—
339C4⋊C4—42no—
340C4⋊C4—42no—
341C4⋊C4—42no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—42no—
346C4⋊C4—42no—
347C4⋊C4—84no—
348C4⋊C4—84no—
349C42—22no—
350C42—22no—
351C42—22no—
352C42—22no—
353C42—42no—
354C42—42no—
355C42—41no—
356C42—41no—
357C42—41no—
358C42—41no—
359C42—48no—
360C23C2×#12111yescharacteristic
361C23#53411yescharacteristic
362C23#53611yescharacteristic
363C23#54612yes—
364C23#55012yes—
365C23C22×SD1611yescharacteristic
366C23C2×#13711yescharacteristic
367C23C2×#13711yescharacteristic
368C23#57511yescharacteristic
369C23#57712yes—
370C23#57812yes—
371Q8—44no—
372Q8—44no—
373Q8—44no—
374Q8—44no—
375Q8—82no—
376Q8—82no—
377Q8—82no—
378Q8—82no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—24no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—24no—
393C2×C4—42no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—44no—
407C2×C4—44no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—44no—
411C2×C4—41no—
412C2×C4—44no—
413C2×C4—44no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—82no—
417C2×C4—82no—
418C2×C4—82no—
419C2×C4—82no—
420C2×C4—82no—
421C2×C4—82no—
422C2×C4—82no—
423C2×C4—82no—
424C2×C4—82no—
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—82no—
435C2×C4—82no—
436C2×C4—82no—
437C2×C4—82no—
438C2×C4#46114yes—
439C2×C4#52512yes—
440C2×C4#53512yes—
441C2×C4#53711yescharacteristic
442C2×C4#53711yescharacteristic
443C2×C4#53812yes—
444C2×C4#53812yes—
445C2×C4#53912yes—
446C2×C4#53912yes—
447C2×C4#53911yescharacteristic
448C2×C4#53911yescharacteristic
449C2×C4#55111yescharacteristic
450C2×C4#55212yes—
451C2×C4#55312yes—
452C2×C4#55511yescharacteristic
453C2×C4#55512yes—
454C2×C4H(3,𝔽4)11yescharacteristic
455C2×C4H(3,𝔽4)11yescharacteristic
456C2×C4#56212yes—
457C8—44no—
458C8—82no—
459C22#241412yes—
460C22#261811yescharacteristic
461C22#268411yescharacteristic
462C22#269412yes—
463C22#273311yescharacteristic
464C22C2×#46111yescharacteristic
465C22#303711yescharacteristic
466C22#304612yes—
467C22#318711yescharacteristic
468C22#319511yescharacteristic
469C22#320011yescharacteristic
470C22#320211yescharacteristic
471C22#322911yescharacteristic
472C22#323611yescharacteristic
473C22#324212yes—
474C22#329812yes—
475C22#331312yes—
476C22#331612yes—
477C22#332912yes—
478C22#339212yes—
479C22#339812yes—
480C22#339812yes—
481C22#340112yes—
482C4—24no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—81no—
488C4—84no—
489C4—84no—
490C4—162no—
491C4—162no—
492C4#260911yescharacteristic
493C4#263612yes—
494C4#278112yes—
495C4#278211yescharacteristic
496C4#278511yescharacteristic
497C4#281911yescharacteristic
498C2#1668411yescharacteristic
499C2#2463012yes—
500C2#2464112yes—
501C2#2975611yescharacteristic
502C2#2976611yescharacteristic
503C2#3126211yescharacteristic
504C2#3127211yescharacteristic
505C2#3157212yes—
506C2#3159512yes—
507C2#3266911yescharacteristic
508C2#3267911yescharacteristic
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×#19272
#151802
#139112
#112572
#112022
#111222
#110772
#110502
#110102