Finite Group Databasebeta

← #472092

Subgroups — #472092

Order 512.

1120 conjugacy classes of subgroups.

The subgroup lattice has 454 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#15185C211yesmaximal, characteristic
6#15143C211yesmaximal, characteristic
7#11378C212yesmaximal
8#11373C212yesmaximal
9#11122C212yesmaximal
10#11050C212yesmaximal
11#10797C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×#106C2212yes—
14C22×#103C2211yescharacteristic
15C22×#103C2212yes—
16C22×#103C2211yescharacteristic
17C22×C4×C8C2211yescharacteristic
18C2×#395—22no—
19C2×#391C2211yescharacteristic
20C2×#391C2212yes—
21C2×#389C2211yescharacteristic
22C2×#384C2211yescharacteristic
23C2×#384C2212yes—
24#2084—24no—
25#2024—24no—
26#1990—24no—
27#1933—24no—
28C2×#340C2212yes—
29C2×#340C2212yes—
30C2×#340C2212yes—
31C2×#340C2211yescharacteristic
32C2×#340C2211yescharacteristic
33C2×#340C2212yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2211yescharacteristic
37#1445—22no—
38#1444—22no—
39#1442—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—24no—
56C22×C4⋊C4—21no—
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2311yescharacteristic
60C22×C4⋊C4C2311yescharacteristic
61C22×C4⋊C4C2312yes—
62C22×C42C2312yes—
63C22×C42C2311yescharacteristic
64C2×#106—24no—
65C2×#106—24no—
66C2×#106—42no—
67C2×#103—22no—
68C2×#103—22no—
69C2×#103—22no—
70C2×#103—22no—
71C2×#103—24no—
72C2×#103—24no—
73C2×#103—42no—
74C2×#103—42no—
75C2×C4×C8—42no—
76C2×C4×C8D414yes—
77#395—48no—
78#391—24no—
79#391—22no—
80#391—22no—
81#391—42no—
82#389—22no—
83#389—22no—
84#384—24no—
85#384—22no—
86#384—22no—
87#384—42no—
88C2×#95—24no—
89C2×#95—21no—
90C2×#95—24no—
91C2×#95—21no—
92C2×#95—21no—
93C2×#95—24no—
94C2×#95D412yes—
95C2×#95D412yes—
96#340—24no—
97#340—22no—
98#340—22no—
99#340—24no—
100#340—24no—
101#340—24no—
102#336—42no—
103#336—42no—
104#336D414yes—
105C22×Q8—21no—
106C22×Q8—21no—
107C22×Q8—21no—
108C22×Q8—22no—
109C22×Q8—21no—
110C22×Q8—22no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—21no—
114C22×Q8—44no—
115C22×Q8—42no—
116C22×Q8—44no—
117C22×Q8—82no—
118C22×Q8#4112yes—
119C23×C4—22no—
120C23×C4—21no—
121C23×C4—21no—
122C23×C4—21no—
123C23×C4—22no—
124C23×C4—21no—
125C23×C4C2×D414yes—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2411yescharacteristic
129C22×C8—21no—
130C22×C8—21no—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—21no—
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—21no—
143C22×C8C2×D414yes—
144C22×C8C2×D412yes—
145C22×C8C2×D412yes—
146C4⋊Q8—28no—
147C4⋊Q8—48no—
148C4⋊Q8—44no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—21no—
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—22no—
173C2×C4⋊C4—44no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—48no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—42no—
178C2×C4⋊C4—42no—
179C2×C4⋊C4—48no—
180C2×C4⋊C4—44no—
181C2×C4⋊C4—44no—
182C2×C4⋊C4—84no—
183C2×C4⋊C4—82no—
184C2×C4⋊C4#4114yes—
185C2×C4⋊C4#4112yes—
186C2×C42—22no—
187C2×C42—22no—
188C2×C42—22no—
189C2×C42—22no—
190C2×C42—22no—
191C2×C42—22no—
192C2×C42—22no—
193C2×C42—22no—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196#106—48no—
197#103—28no—
198#103—48no—
199#103—44no—
200C4×C8—44no—
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—48no—
215C2×Q8—41no—
216C2×Q8—42no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—41no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—42no—
224C2×Q8—42no—
225C2×Q8—42no—
226C2×Q8—42no—
227C2×Q8—42no—
228C2×Q8—42no—
229C2×Q8—816no—
230C2×Q8—84no—
231C2×Q8—84no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—22no—
245C22×C4—41no—
246C22×C4—41no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—41no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—42no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—44no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4#12712yes—
271C22×C4#12712yes—
272C22×C4C2×SD1618yes—
273C22×C4C2×SD1614yes—
274C22×C4#13714yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C2×#4111yescharacteristic
278C22×C4C2×#4112yes—
279C22×C4C2×#4111yescharacteristic
280C22×C4H(5,𝔽2)12yes—
281C22×C4H(5,𝔽2)11yescharacteristic
282C22×C42−1+412yes—
283C22×C42−1+412yes—
284C22×C42−1+411yescharacteristic
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—24no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—42no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—22no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—48no—
307C4⋊C4—42no—
308C4⋊C4—41no—
309C4⋊C4—42no—
310C4⋊C4—41no—
311C4⋊C4—41no—
312C4⋊C4—42no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—42no—
319C4⋊C4—42no—
320C4⋊C4—42no—
321C4⋊C4—816no—
322C4⋊C4—84no—
323C4⋊C4—84no—
324C4⋊C4—84no—
325C42—22no—
326C42—22no—
327C42—24no—
328C42—42no—
329C42—42no—
330C42—42no—
331C42—42no—
332C23C2×#12711yescharacteristic
333C23#53612yes—
334C23#54611yescharacteristic
335C23#54612yes—
336C23#54811yescharacteristic
337C23C22×SD1612yes—
338C23C2×#13712yes—
339C23#57612yes—
340C23#57812yes—
341Q8—42no—
342Q8—42no—
343Q8—44no—
344Q8—816no—
345Q8—81no—
346Q8—82no—
347Q8—81no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—28no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—28no—
356C2×C4—28no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—81no—
368C2×C4—82no—
369C2×C4—82no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—82no—
373C2×C4—84no—
374C2×C4—82no—
375C2×C4—81no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—81no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—81no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—81no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—81no—
388C2×C4—82no—
389C2×C4#49214yes—
390C2×C4#49714yes—
391C2×C4#52511yescharacteristic
392C2×C4#52511yescharacteristic
393C2×C4#53411yescharacteristic
394C2×C4#53611yescharacteristic
395C2×C4#53611yescharacteristic
396C2×C4#53612yes—
397C2×C4#53711yescharacteristic
398C2×C4#53912yes—
399C2×C4#54112yes—
400C2×C4#54212yes—
401C2×C4#54212yes—
402C2×C4#54612yes—
403C2×C4#54611yescharacteristic
404C2×C4#54811yescharacteristic
405C2×C4#55212yes—
406C2×C4#55612yes—
407C2×C4#56212yes—
408C2×C4#56312yes—
409C8—42no—
410C8—44no—
411C8—42no—
412C22#253911yescharacteristic
413C22#254011yescharacteristic
414C22#265811yescharacteristic
415C22#268712yes—
416C22#271612yes—
417C22C2×#49211yescharacteristic
418C22C2×#49711yescharacteristic
419C22#315511yescharacteristic
420C22#315611yescharacteristic
421C22#322914yes—
422C22#324314yes—
423C22#329514yes—
424C22#329812yes—
425C22#331214yes—
426C22#333312yes—
427C22#335912yes—
428C22#337512yes—
429C4—24no—
430C4—24no—
431C4—48no—
432C4—82no—
433C4—84no—
434C4—82no—
435C4—162no—
436C4—161no—
437C4—161no—
438C4#259911yescharacteristic
439C4#263711yescharacteristic
440C4#263811yescharacteristic
441C4#269012yes—
442C4#270012yes—
443C4#281411yescharacteristic
444C2#1681211yescharacteristic
445C2#2833211yescharacteristic
446C2#2834511yescharacteristic
447C2#2840011yescharacteristic
448C2#2841811yescharacteristic
449C2#3028011yescharacteristic
450C2#3028911yescharacteristic
451C2#3135414yes—
452C2#3222814yes—
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
#151852
#151432
#113782
#113732
#111222
#110502
#107972