Finite Group Databasebeta

← #472091

Subgroups — #472091

Order 512.

1056 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×#2084C211yesmaximal, characteristic
2C2×#2024C211yesmaximal, characteristic
3C2×#1990C211yesmaximal, characteristic
4C2×#1933C211yesmaximal, characteristic
5#15230C211yesmaximal, characteristic
6#15185C211yesmaximal, characteristic
7#11378C212yesmaximal
8#11372C212yesmaximal
9#11122C212yesmaximal
10#11052C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#391C2211yescharacteristic
19C2×#391C2212yes—
20C2×#389C2211yescharacteristic
21C2×#384C2211yescharacteristic
22C2×#382C2212yes—
23#2084—24no—
24#2024—24no—
25#1990—24no—
26#1933—24no—
27C2×#340C2212yes—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2211yescharacteristic
31C2×#340C2211yescharacteristic
32C2×#340C2212yes—
33C2×#336C2212yes—
34C2×#336C2212yes—
35C2×#336C2211yescharacteristic
36#1446—22no—
37#1445—22no—
38#1442—22no—
39#1441—22no—
40C4⋊#95C2212yes—
41C4⋊#95C2212yes—
42C4×#95C2211yescharacteristic
43C23×Q8—21no—
44C23×Q8C2311yescharacteristic
45C23×C8C2311yescharacteristic
46C23×C8C2311yescharacteristic
47C23×C8C2312yes—
48C2×C4⋊Q8—24no—
49C2×C4⋊Q8—22no—
50C2×C4⋊Q8—22no—
51C2×C4⋊Q8—42no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4C2311yescharacteristic
57C22×C4⋊C4C2312yes—
58C22×C4⋊C4C2311yescharacteristic
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2312yes—
61C22×C42C2312yes—
62C22×C42C2311yescharacteristic
63C2×#106—24no—
64C2×#106—24no—
65C2×#106—42no—
66C2×#103—22no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—24no—
71C2×#103—24no—
72C2×#103—42no—
73C2×#103—42no—
74C2×C4×C8—42no—
75C2×C4×C8D414yes—
76#391—24no—
77#391—22no—
78#391—22no—
79#391—42no—
80#389—22no—
81#389—22no—
82#384—22no—
83#384—22no—
84#382—24no—
85#382—42no—
86C2×#95—24no—
87C2×#95—21no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—21no—
91C2×#95—24no—
92C2×#95—24no—
93C2×#95D412yes—
94C2×#95D412yes—
95#340—24no—
96#340—22no—
97#340—22no—
98#340—24no—
99#340—24no—
100#340—24no—
101#336—42no—
102#336—42no—
103#336D414yes—
104C22×Q8—21no—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—22no—
108C22×Q8—21no—
109C22×Q8—22no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—21no—
113C22×Q8—44no—
114C22×Q8—42no—
115C22×Q8—44no—
116C22×Q8—82no—
117C22×Q8#4112yes—
118C23×C4—22no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—21no—
123C23×C4—21no—
124C23×C4C2×D414yes—
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C23×C4C2411yescharacteristic
128C22×C8—21no—
129C22×C8—21no—
130C22×C8—22no—
131C22×C8—22no—
132C22×C8—21no—
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—21no—
142C22×C8C2×D414yes—
143C22×C8C2×D412yes—
144C22×C8C2×D412yes—
145C4⋊Q8—28no—
146C4⋊Q8—48no—
147C4⋊Q8—44no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—21no—
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—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—44no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4—44no—
179C2×C4⋊C4—82no—
180C2×C4⋊C4#4114yes—
181C2×C4⋊C4#4112yes—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42C2×D412yes—
191C2×C42C2×D412yes—
192#106—48no—
193#103—28no—
194#103—48no—
195#103—44no—
196C4×C8—44no—
197#95—24no—
198#95—24no—
199#95—48no—
200#95—48no—
201#95—42no—
202#95—42no—
203#95—42no—
204#95—84no—
205#95—84no—
206C24C22×D411yescharacteristic
207C2×Q8—22no—
208C2×Q8—24no—
209C2×Q8—22no—
210C2×Q8—24no—
211C2×Q8—48no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—41no—
215C2×Q8—42no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8—41no—
220C2×Q8—42no—
221C2×Q8—42no—
222C2×Q8—42no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—816no—
227C2×Q8—84no—
228C2×Q8—84no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—24no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—41no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—44no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4#12712yes—
269C22×C4#12712yes—
270C22×C4#13514yes—
271C22×C4#13714yes—
272C22×C4C22×D411yescharacteristic
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4111yescharacteristic
275C22×C4C2×#4112yes—
276C22×C4C2×#4111yescharacteristic
277C22×C4H(5,𝔽2)12yes—
278C22×C4H(5,𝔽2)11yescharacteristic
279C22×C42−1+412yes—
280C22×C42−1+412yes—
281C22×C42−1+411yescharacteristic
282C2×C8—22no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—24no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—24no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—22no—
303C4⋊C4—48no—
304C4⋊C4—42no—
305C4⋊C4—41no—
306C4⋊C4—42no—
307C4⋊C4—41no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—41no—
313C4⋊C4—42no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—84no—
319C4⋊C4—84no—
320C4⋊C4—84no—
321C42—22no—
322C42—22no—
323C42—24no—
324C42—42no—
325C42—42no—
326C42—42no—
327C42—42no—
328C23C2×#12711yescharacteristic
329C23#53612yes—
330C23#54611yescharacteristic
331C23#54612yes—
332C23#54811yescharacteristic
333C23C2×#13512yes—
334C23C2×#13712yes—
335C23#57712yes—
336C23#57712yes—
337Q8—42no—
338Q8—42no—
339Q8—44no—
340Q8—816no—
341Q8—81no—
342Q8—82no—
343Q8—81no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—28no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—48no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—82no—
364C2×C4—82no—
365C2×C4—81no—
366C2×C4—81no—
367C2×C4—82no—
368C2×C4—81no—
369C2×C4—82no—
370C2×C4—84no—
371C2×C4—81no—
372C2×C4—84no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4—81no—
383C2×C4—82no—
384C2×C4—84no—
385C2×C4#49514yes—
386C2×C4#49714yes—
387C2×C4#52511yescharacteristic
388C2×C4#52511yescharacteristic
389C2×C4#53411yescharacteristic
390C2×C4#53611yescharacteristic
391C2×C4#53611yescharacteristic
392C2×C4#53612yes—
393C2×C4#53711yescharacteristic
394C2×C4#53912yes—
395C2×C4#54112yes—
396C2×C4#54212yes—
397C2×C4#54212yes—
398C2×C4#54612yes—
399C2×C4#54611yescharacteristic
400C2×C4#54811yescharacteristic
401C2×C4#55212yes—
402C2×C4#55612yes—
403C2×C4#56212yes—
404C2×C4#56312yes—
405C8—42no—
406C8—44no—
407C8—42no—
408C22#253911yescharacteristic
409C22#254011yescharacteristic
410C22#265811yescharacteristic
411C22#268712yes—
412C22#271612yes—
413C22C2×#49511yescharacteristic
414C22C2×#49711yescharacteristic
415C22#315411yescharacteristic
416C22#315411yescharacteristic
417C22#323114yes—
418C22#324414yes—
419C22#329614yes—
420C22#330412yes—
421C22#331614yes—
422C22#332912yes—
423C22#336512yes—
424C22#337212yes—
425C4—24no—
426C4—24no—
427C4—82no—
428C4—84no—
429C4—84no—
430C4—82no—
431C4—161no—
432C4—161no—
433C4—162no—
434C4#259911yescharacteristic
435C4#263711yescharacteristic
436C4#263811yescharacteristic
437C4#269012yes—
438C4#270012yes—
439C4#281411yescharacteristic
440C2#1681211yescharacteristic
441C2#2833011yescharacteristic
442C2#2834711yescharacteristic
443C2#2840211yescharacteristic
444C2#2841611yescharacteristic
445C2#3028111yescharacteristic
446C2#3029011yescharacteristic
447C2#3135314yes—
448C2#3222914yes—
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×#20842
C2×#20242
C2×#19902
C2×#19332
#152302
#151852
#113782
#113722
#111222
#110522
#107972