Finite Group Databasebeta

← #475620

Subgroups — #475620

Order 512.

1092 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2096C212yesmaximal
2C2×#2084C211yesmaximal, characteristic
3C2×#2072C211yesmaximal, characteristic
4C2×#1933C211yesmaximal, characteristic
5C2×#1844C211yesmaximal, characteristic
6#15185C211yesmaximal, characteristic
7#15145C211yesmaximal, characteristic
8#11378C212yesmaximal
9#11323C212yesmaximal
10#11122C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×#106C2212yes—
13C22×#106C2212yes—
14C22×#106C2212yes—
15C22×#103C2211yescharacteristic
16C22×#103C2212yes—
17C22×#103C2211yescharacteristic
18C22×C4×C8C2211yescharacteristic
19C2×#398—22no—
20C2×#391C2211yescharacteristic
21C2×#391C2212yes—
22C2×#389C2211yescharacteristic
23C2×#384C2211yescharacteristic
24C2×#384C2211yescharacteristic
25C2×#384C2212yes—
26#2096—28no—
27#2084—24no—
28#2072—24no—
29#1933—24no—
30#1844—24no—
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#340C2211yescharacteristic
34C2×#340C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C2×#336C2211yescharacteristic
38#1445—22no—
39#1444—22no—
40#1442—22no—
41C4⋊#95C2212yes—
42C4⋊#95C2212yes—
43C4×#95C2211yescharacteristic
44C23×Q8C2311yescharacteristic
45C23×C8C2312yes—
46C23×C8C2311yescharacteristic
47C23×C8C2311yescharacteristic
48C22×C4⋊C4—21no—
49C22×C4⋊C4—21no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—24no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4C2312yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2311yescharacteristic
59C22×C42C2312yes—
60C22×C42C2311yescharacteristic
61C2×#106—24no—
62C2×#106—24no—
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—24no—
67C2×#106—42no—
68C2×#106—42no—
69C2×#106—42no—
70C2×#103—22no—
71C2×#103—22no—
72C2×#103—24no—
73C2×#103—24no—
74C2×#103—22no—
75C2×#103—24no—
76C2×#103—42no—
77C2×#103D414yes—
78C2×C4×C8—24no—
79C2×C4×C8D414yes—
80#398—48no—
81#391—24no—
82#391—22no—
83#391—22no—
84#391—42no—
85#389—22no—
86#389—22no—
87#384—24no—
88#384—22no—
89#384—22no—
90#384—22no—
91#384—22no—
92#384—42no—
93C2×#95—24no—
94C2×#95—21no—
95C2×#95—24no—
96C2×#95—21no—
97C2×#95—21no—
98C2×#95—24no—
99C2×#95D412yes—
100C2×#95D412yes—
101#340—24no—
102#340—24no—
103#340—22no—
104#340—22no—
105#336—24no—
106#336—24no—
107#336D414yes—
108C22×Q8—21no—
109C22×Q8—21no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—21no—
114C22×Q8—22no—
115C22×Q8—22no—
116C22×Q8—22no—
117C22×Q8C2×D412yes—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D414yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—22no—
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C22×C8—22no—
136C22×C8—22no—
137C22×C8—22no—
138C22×C8—22no—
139C22×C8C2×D414yes—
140C22×C8C2×D412yes—
141C22×C8C2×D412yes—
142C22×C8C2×D412yes—
143C22×C8#4112yes—
144C22×C8#4114yes—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
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—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—48no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—44no—
180C2×C4⋊C4—84no—
181C2×C4⋊C4—82no—
182C2×C4⋊C4—82no—
183C2×C4⋊C4C2×D412yes—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42C2×D412yes—
193C2×C42C2×D412yes—
194#106—48no—
195#106—48no—
196#106—48no—
197#103—28no—
198#103—48no—
199#103—44no—
200C4×C8—28no—
201#95—24no—
202#95—48no—
203#95—42no—
204#95—42no—
205#95—42no—
206#95—48no—
207#95—84no—
208#95SD1618yes—
209C24C22×D411yescharacteristic
210C2×Q8—22no—
211C2×Q8—24no—
212C2×Q8—22no—
213C2×Q8—24no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—42no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—42no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—42no—
243C22×C4—42no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—44no—
260C22×C4—44no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—44no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4#12712yes—
267C22×C4#12712yes—
268C22×C4C2×SD1618yes—
269C22×C4C2×SD1614yes—
270C22×C4#13714yes—
271C22×C4C22×D411yescharacteristic
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4H(5,𝔽2)11yescharacteristic
277C22×C4H(5,𝔽2)11yescharacteristic
278C22×C42−1+412yes—
279C22×C42−1+412yes—
280C22×C42−1+412yes—
281C2×C8—22no—
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—24no—
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#12114yes—
297C2×C8#12314yes—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—41no—
301C4⋊C4—41no—
302C4⋊C4—42no—
303C4⋊C4—42no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—42no—
312C4⋊C4—42no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—816no—
317C4⋊C4—84no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C4⋊C4—84no—
322C42—22no—
323C42—22no—
324C42—24no—
325C42—42no—
326C42—42no—
327C42—42no—
328C42—42no—
329C23C2×#12711yescharacteristic
330C23#53612yes—
331C23D4211yescharacteristic
332C23#54712yes—
333C23#55011yescharacteristic
334C23C22×SD1612yes—
335C23C2×#13712yes—
336C23#57712yes—
337C23#57712yes—
338Q8—42no—
339Q8—44no—
340Q8—42no—
341Q8—81no—
342Q8—81no—
343Q8—82no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—28no—
352C2×C4—28no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—82no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—81no—
365C2×C4—82no—
366C2×C4—81no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—81no—
370C2×C4—82no—
371C2×C4—84no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—84no—
380C2×C4—81no—
381C2×C4—84no—
382C2×C4—84no—
383C2×C4—82no—
384C2×C4—82no—
385C2×C4#49214yes—
386C2×C4#49714yes—
387C2×C4C2×#12111yescharacteristic
388C2×C4C2×#12311yescharacteristic
389C2×C4#53511yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4#53611yescharacteristic
392C2×C4#53611yescharacteristic
393C2×C4#53612yes—
394C2×C4#53612yes—
395C2×C4#53912yes—
396C2×C4#53912yes—
397C2×C4#54112yes—
398C2×C4D4211yescharacteristic
399C2×C4#54712yes—
400C2×C4#54712yes—
401C2×C4#55011yescharacteristic
402C2×C4#55212yes—
403C2×C4#56312yes—
404C2×C4#56312yes—
405C8—24no—
406C8—24no—
407C8—44no—
408C22#250111yescharacteristic
409C22#254111yescharacteristic
410C22#265811yescharacteristic
411C22#268012yes—
412C22#272112yes—
413C22C2×#49211yescharacteristic
414C22C2×#49711yescharacteristic
415C22#315411yescharacteristic
416C22#315411yescharacteristic
417C22#322914yes—
418C22#324414yes—
419C22#327812yes—
420C22D4×SD1612yes—
421C22#333514yes—
422C22#334414yes—
423C22#339212yes—
424C22#340012yes—
425C4—24no—
426C4—24no—
427C4—48no—
428C4—82no—
429C4—82no—
430C4—161no—
431C4—162no—
432C4—162no—
433C4—161no—
434C4#248611yescharacteristic
435C4#262111yescharacteristic
436C4#262311yescharacteristic
437C4#268012yes—
438C4#272112yes—
439C4#282111yescharacteristic
440C2#1706211yescharacteristic
441C2#2792411yescharacteristic
442C2#2793311yescharacteristic
443C2#2846111yescharacteristic
444C2#2847311yescharacteristic
445C2#3028011yescharacteristic
446C2#3029011yescharacteristic
447C2#3108914yes—
448C2#3237614yes—
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×#20962
C2×#20842
C2×#20722
C2×#19332
C2×#18442
#151852
#151452
#113782
#113232
#111222
#107972