Finite Group Databasebeta

← #475600

Subgroups — #475600

Order 512.

1064 conjugacy classes of subgroups.

The subgroup lattice has 465 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2097C212yesmaximal
2C2×#2085C212yesmaximal
3C2×#2074C212yesmaximal
4#14703C211yesmaximal, characteristic
5#14687C211yesmaximal, characteristic
6#11372C212yesmaximal
7#11324C211yesmaximal, characteristic
8#11323C211yesmaximal, characteristic
9#11050C211yesmaximal, characteristic
10#11050C211yesmaximal, characteristic
11#10795C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#103C2212yes—
15C22×#103C2212yes—
16C2×#391C2211yescharacteristic
17C2×#391C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#389C2211yescharacteristic
20C2×#384C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23C2×#382C2212yes—
24C2×C43C2211yescharacteristic
25#2097—28no—
26#2085—28no—
27#2074—28no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2212yes—
32C2×#336C2211yescharacteristic
33C2×#336C2211yescharacteristic
34C2×#336C2212yes—
35C2×#336C2212yes—
36#1436—22no—
37C4⋊#95—22no—
38C4⋊#95—22no—
39C4⋊#95C2211yescharacteristic
40C4⋊#95C2211yescharacteristic
41#1434—22no—
42C23×Q8C2311yescharacteristic
43C23×C8C2312yes—
44C23×C8C2312yes—
45C22×C4⋊C4—22no—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—21no—
48C22×C4⋊C4C2312yes—
49C22×C4⋊C4C2312yes—
50C22×C4⋊C4C2311yescharacteristic
51C22×C4⋊C4C2311yescharacteristic
52C22×C4⋊C4C2311yescharacteristic
53C22×C42D412yes—
54C22×C42D412yes—
55C22×C42C2311yescharacteristic
56C22×C42C2311yescharacteristic
57C22×C42C2311yescharacteristic
58C2×#107—24no—
59C2×#107—24no—
60C2×#107—42no—
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—42no—
64C2×#103—24no—
65C2×#103—24no—
66C2×#103—24no—
67C2×#103—24no—
68C2×#103—42no—
69C2×#103—42no—
70#391—22no—
71#391—24no—
72#391—22no—
73#391—22no—
74#391D414yes—
75#389—24no—
76#384—42no—
77#384—42no—
78#382—24no—
79#382—24no—
80#382—24no—
81#382—24no—
82C2×#95—22no—
83C2×#95—21no—
84C2×#95—21no—
85C2×#95—22no—
86C2×#95—22no—
87C2×#95—22no—
88C2×#95—24no—
89C2×#95—22no—
90C2×#95—22no—
91C2×#95—21no—
92C43—22no—
93C43—22no—
94#340—24no—
95#340—24no—
96#340—24no—
97#340—24no—
98#336—24no—
99#336—22no—
100#336—22no—
101#336D418yes—
102C22×Q8—21no—
103C22×Q8—22no—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8C2×D412yes—
112C23×C4—21no—
113C23×C4—21no—
114C23×C4—22no—
115C23×C4—22no—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4C2×D412yes—
119C23×C4C2×D411yescharacteristic
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D412yes—
122C23×C4C2411yescharacteristic
123C22×C8—22no—
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—22no—
128C22×C8—22no—
129C22×C8—22no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8C2×D414yes—
136C22×C8C2×D414yes—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—44no—
161C2×C4⋊C4—44no—
162C2×C4⋊C4—44no—
163C2×C4⋊C4—44no—
164C2×C4⋊C4—44no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—44no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4C2×D412yes—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—21no—
172C2×C42—21no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—42no—
183C2×C42—42no—
184C2×C42C2×D412yes—
185C2×C42C2×D414yes—
186C2×C42#4112yes—
187C2×C42#4112yes—
188C2×C42#4112yes—
189#107—48no—
190#106—48no—
191#103—48no—
192#103—48no—
193#95—28no—
194#95—44no—
195#95—48no—
196#95—44no—
197#95—44no—
198#95—44no—
199#95—42no—
200#95—42no—
201#95—42no—
202#95—44no—
203C24C22×D411yescharacteristic
204C2×Q8—24no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—42no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—42no—
218C2×Q8—42no—
219C2×Q8—42no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—41no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—42no—
263C22×C4—44no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4#12114yes—
271C22×C4#12314yes—
272C22×C4#12712yes—
273C22×C4#12712yes—
274C22×C4#13514yes—
275C22×C4#13714yes—
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D412yes—
278C22×C4C2×#4111yescharacteristic
279C22×C4C2×#4111yescharacteristic
280C22×C4C2×#4111yescharacteristic
281C22×C4H(5,𝔽2)12yes—
282C22×C42−1+412yes—
283C22×C42−1+411yescharacteristic
284C22×C42−1+411yescharacteristic
285C22×C42−1+412yes—
286C2×C8—24no—
287C2×C8—24no—
288C2×C8—24no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C4⋊C4—24no—
301C4⋊C4—42no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—88no—
315C4⋊C4—88no—
316C4⋊C4—84no—
317C42—24no—
318C42—24no—
319C42—22no—
320C42—22no—
321C42—24no—
322C42—24no—
323C42—22no—
324C42—22no—
325C42—48no—
326C42—48no—
327C23C2×#12111yescharacteristic
328C23C2×#12311yescharacteristic
329C23C2×#12711yescharacteristic
330C23#53511yescharacteristic
331C23#53611yescharacteristic
332C23#53611yescharacteristic
333C23#53611yescharacteristic
334C23C2×#13512yes—
335C23C2×#13712yes—
336C23#57512yes—
337C23#57512yes—
338Q8—44no—
339Q8—44no—
340Q8—82no—
341Q8—82no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4—82no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—81no—
385C2×C4—82no—
386C2×C4—81no—
387C2×C4—84no—
388C2×C4—84no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—81no—
393C2×C4—84no—
394C2×C4—84no—
395C2×C4—82no—
396C2×C4—82no—
397C2×C4#48518yes—
398C2×C4#53612yes—
399C2×C4#53612yes—
400C2×C4#53612yes—
401C2×C4#53912yes—
402C2×C4#53912yes—
403C2×C4#54112yes—
404C2×C4D4212yes—
405C2×C4#54711yescharacteristic
406C2×C4#54711yescharacteristic
407C2×C4#54712yes—
408C2×C4#54712yes—
409C2×C4#55012yes—
410C2×C4#55211yescharacteristic
411C2×C4#55211yescharacteristic
412C2×C4#56312yes—
413C2×C4#56312yes—
414C8—44no—
415C8—44no—
416C22#248611yescharacteristic
417C22#250111yescharacteristic
418C22#254111yescharacteristic
419C22#262111yescharacteristic
420C22#262311yescharacteristic
421C22#265811yescharacteristic
422C22#282111yescharacteristic
423C22#303512yes—
424C22#303712yes—
425C22C2×#48512yes—
426C22#309312yes—
427C22#314711yescharacteristic
428C22#315011yescharacteristic
429C22#315211yescharacteristic
430C22#315311yescharacteristic
431C22#321112yes—
432C22#321512yes—
433C22#323112yes—
434C22#323212yes—
435C22#323312yes—
436C22#323612yes—
437C22#323712yes—
438C22#324012yes—
439C4—28no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—82no—
444C4—82no—
445C4—161no—
446C4—162no—
447C4—162no—
448C4—161no—
449C4#268012yes—
450C4#268012yes—
451C4#272112yes—
452C4#272112yes—
453C2#1706211yescharacteristic
454C2#2764912yes—
455C2#2790811yescharacteristic
456C2#2791511yescharacteristic
457C2#2844211yescharacteristic
458C2#2844711yescharacteristic
459C2#2987112yes—
460C2#2991312yes—
461C2#3028211yescharacteristic
462C2#3028411yescharacteristic
463C2#3284212yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C2×C43 ◁ 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×#20972
C2×#20852
C2×#20742
#147032
#146872
#113722
#113242
#113232
#110502
#107952