Finite Group Databasebeta

← #409434

Subgroups — #409434

Order 512.

1020 conjugacy classes of subgroups.

The subgroup lattice has 443 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2095C211yesmaximal, characteristic
2C2×#2084C211yesmaximal, characteristic
3C2×#1933C211yesmaximal, characteristic
4C2×#1834C211yesmaximal, characteristic
5#15197C211yesmaximal, characteristic
6#15185C211yesmaximal, characteristic
7#11379C212yesmaximal
8#11122C212yesmaximal
9#10928C212yesmaximal
10#10863C212yesmaximal
11#10696C211yesmaximal, characteristic
12C22×#107C2212yes—
13C22×#106C2212yes—
14C22×#103C2211yescharacteristic
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#97C2212yes—
18C22×C4×C8C2211yescharacteristic
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2211yescharacteristic
24#2095—24no—
25#2084—24no—
26#1933—24no—
27#1834—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2211yescharacteristic
35#1445—22no—
36#1442—22no—
37#1442—22no—
38#1441—22no—
39C4⋊#95C2212yes—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8C2311yescharacteristic
44C23×C8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2312yes—
47C22×C4⋊C4—21no—
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4C2312yes—
53C22×C4⋊C4C2311yescharacteristic
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2312yes—
58C22×C42C2311yescharacteristic
59C2×#107—24no—
60C2×#107—24no—
61C2×#107—24no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—42no—
65C2×#103—22no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—24no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#97—24no—
74C2×#97—24no—
75C2×#97—42no—
76C2×C4×C8—24no—
77C2×C4×C8D414yes—
78#391—24no—
79#391—22no—
80#391—22no—
81#391—42no—
82#389—22no—
83#389—22no—
84#384—22no—
85#384—22no—
86#384—22no—
87#384—22no—
88C2×#95—21no—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95—24no—
93C2×#95—24no—
94C2×#95—24no—
95C2×#95D412yes—
96C2×#95D412yes—
97#340—24no—
98#340—24no—
99#340—24no—
100#340—22no—
101#340—22no—
102#336—24no—
103#336D414yes—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—22no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8#4112yes—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4—22no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4C2×D414yes—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2411yescharacteristic
124C22×C8—22no—
125C22×C8—22no—
126C22×C8—22no—
127C22×C8—21no—
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
134C22×C8—21no—
135C22×C8C2×D412yes—
136C22×C8C2×D412yes—
137C22×C8C2×D414yes—
138C22×C8#4114yes—
139C22×C8#4114yes—
140C22×C8#4114yes—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
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—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—44no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—82no—
174C2×C4⋊C4—82no—
175C2×C4⋊C4—82no—
176C2×C4⋊C4#4112yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187#107—48no—
188#106—48no—
189#103—28no—
190#103—44no—
191#103—48no—
192#97—48no—
193C4×C8—28no—
194#95—24no—
195#95—24no—
196#95—48no—
197#95—42no—
198#95—42no—
199#95—42no—
200#95—84no—
201#95—84no—
202#95—84no—
203C24C22×D411yescharacteristic
204C2×Q8—22no—
205C2×Q8—24no—
206C2×Q8—22no—
207C2×Q8—24no—
208C2×Q8—41no—
209C2×Q8—42no—
210C2×Q8—41no—
211C2×Q8—42no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—41no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—42no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—24no—
227C22×C4—22no—
228C22×C4—24no—
229C22×C4—24no—
230C22×C4—22no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—41no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—42no—
240C22×C4—41no—
241C22×C4—42no—
242C22×C4—42no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—44no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—44no—
258C22×C4—44no—
259C22×C4—44no—
260C22×C4—42no—
261C22×C4#12712yes—
262C22×C4#12712yes—
263C22×C4#13714yes—
264C22×C4#13714yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C22×D411yescharacteristic
267C22×C4C2×#4112yes—
268C22×C4C2×#4111yescharacteristic
269C22×C4C2×#4111yescharacteristic
270C22×C4C2×#4112yes—
271C22×C4C2×#4112yes—
272C22×C4H(5,𝔽2)11yescharacteristic
273C22×C42−1+412yes—
274C22×C42−1+411yescharacteristic
275C2×C8—24no—
276C2×C8—22no—
277C2×C8—22no—
278C2×C8—24no—
279C2×C8—24no—
280C2×C8—24no—
281C2×C8—24no—
282C2×C8—24no—
283C2×C8—24no—
284C2×C8—24no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—42no—
289C2×C8—42no—
290C2×C8#12314yes—
291C2×C8#12418yes—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—41no—
295C4⋊C4—41no—
296C4⋊C4—41no—
297C4⋊C4—42no—
298C4⋊C4—42no—
299C4⋊C4—42no—
300C4⋊C4—42no—
301C4⋊C4—42no—
302C4⋊C4—41no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—84no—
311C4⋊C4—84no—
312C4⋊C4—84no—
313C4⋊C4—84no—
314C4⋊C4—84no—
315C42—22no—
316C42—22no—
317C42—24no—
318C42—42no—
319C42—42no—
320C42—42no—
321C42—42no—
322C23C2×#12711yescharacteristic
323C23#53212yes—
324C23#54611yescharacteristic
325C23#54712yes—
326C23#54811yescharacteristic
327C23C2×#13712yes—
328C23C2×#13712yes—
329C23#57612yes—
330C23#57712yes—
331Q8—42no—
332Q8—42no—
333Q8—44no—
334Q8—81no—
335Q8—82no—
336Q8—81no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—82no—
355C2×C4—81no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—81no—
359C2×C4—82no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—82no—
363C2×C4—81no—
364C2×C4—84no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—82no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4—84no—
376C2×C4—81no—
377C2×C4—82no—
378C2×C4#49714yes—
379C2×C4#49714yes—
380C2×C4C2×#12311yescharacteristic
381C2×C4#52511yescharacteristic
382C2×C4C2×#12412yes—
383C2×C4#52912yes—
384C2×C4#52912yes—
385C2×C4#53211yescharacteristic
386C2×C4#53211yescharacteristic
387C2×C4#53611yescharacteristic
388C2×C4#53711yescharacteristic
389C2×C4#53912yes—
390C2×C4#54112yes—
391C2×C4#54611yescharacteristic
392C2×C4#54712yes—
393C2×C4#54811yescharacteristic
394C2×C4#55112yes—
395C2×C4#55212yes—
396C2×C4#55212yes—
397C2×C4#55212yes—
398C8—28no—
399C8—24no—
400C8—42no—
401C22#250111yescharacteristic
402C22#251912yes—
403C22#254011yescharacteristic
404C22#255312yes—
405C22#285811yescharacteristic
406C22C2×#49711yescharacteristic
407C22C2×#49711yescharacteristic
408C22#315411yescharacteristic
409C22#315511yescharacteristic
410C22#317914yes—
411C22#318514yes—
412C22#332912yes—
413C22#333312yes—
414C22#334414yes—
415C22#334614yes—
416C22#337212yes—
417C22#337512yes—
418C4—24no—
419C4—24no—
420C4—82no—
421C4—82no—
422C4—84no—
423C4—162no—
424C4—161no—
425C4—162no—
426C4—161no—
427C4#239511yescharacteristic
428C4#248812yes—
429C4#248912yes—
430C4#249811yescharacteristic
431C4#261111yescharacteristic
432C4#266411yescharacteristic
433C2#1502011yescharacteristic
434C2#2793311yescharacteristic
435C2#2793811yescharacteristic
436C2#2804814yes—
437C2#2841611yescharacteristic
438C2#2841811yescharacteristic
439C2#2887114yes—
440C2#3324311yescharacteristic
441C2#3325511yescharacteristic
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×#20952
C2×#20842
C2×#19332
C2×#18342
#151972
#151852
#113792
#111222
#109282
#108632
#106962