Finite Group Databasebeta

← #443433

Subgroups — #443433

Order 512.

1516 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2568C211yesmaximal, characteristic
2C2×#2567C211yesmaximal, characteristic
3C2×#2043C212yesmaximal
4C2×#2024C212yesmaximal
5#11380C211yesmaximal, characteristic
6#11067C214yesmaximal
7#10932C212yesmaximal
8#10885C212yesmaximal
9C22×C4⋊Q8C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4×Q8C2211yescharacteristic
12C22×#103C2214yes—
13C22×#103C2214yes—
14#2568—28no—
15#2567—28no—
16C2×#400—21no—
17C2×#400—21no—
18C2×#395—21no—
19C2×#395—21no—
20C2×#391—21no—
21C2×#391—21no—
22C2×#391C2212yes—
23C2×#391C2212yes—
24C2×#389C2212yes—
25C2×#389C2212yes—
26C2×#384—21no—
27C2×#382—21no—
28C2×C4×C4⋊C4C2211yescharacteristic
29C2×C4×C4⋊C4C2211yescharacteristic
30#2043—28no—
31#2024—28no—
32C2×#340C2214yes—
33C2×#340C2214yes—
34C2×#336C2212yes—
35C2×#336C2212yes—
36C2×#336C2212yes—
37C23×Q8—21no—
38C23×Q8D411yescharacteristic
39C23×Q8C2312yes—
40C23×C8C2314yes—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8—24no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8—24no—
45C2×C4⋊Q8—42no—
46C2×C4⋊Q8—42no—
47C2×C4×Q8—24no—
48C2×C4×Q8—24no—
49C2×C4×Q8—42no—
50C22×C4⋊C4—21no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—22no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4D411yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C4⋊C4C2312yes—
60C22×C4⋊C4C2312yes—
61C22×C42—21no—
62C22×C42—21no—
63C22×C42C2311yescharacteristic
64C22×C42C2311yescharacteristic
65C22×C42C2311yescharacteristic
66C2×#103—28no—
67C2×#103—28no—
68C2×#103—28no—
69C2×#103—28no—
70C2×#103—28no—
71C2×#103—44no—
72#400—28no—
73#400—28no—
74#395—28no—
75#395—28no—
76#391—28no—
77#391—28no—
78#391—24no—
79#391—24no—
80#391—24no—
81#391D418yes—
82#389—24no—
83#389—24no—
84#389—24no—
85#389D418yes—
86#384—28no—
87#382—28no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—24no—
90C2×#95—21no—
91C2×#95—22no—
92C2×#95—22no—
93C2×#95—21no—
94C2×#95—22no—
95C2×#95—22no—
96C2×#95—22no—
97C2×#95—22no—
98#340—28no—
99#340—28no—
100#336—24no—
101#336—24no—
102#336—42no—
103C22×Q8—24no—
104C22×Q8—22no—
105C22×Q8—22no—
106C22×Q8—22no—
107C22×Q8—22no—
108C22×Q8—22no—
109C22×Q8—24no—
110C22×Q8—24no—
111C22×Q8—24no—
112C22×Q8—24no—
113C22×Q8—24no—
114C22×Q8—24no—
115C22×Q8—41no—
116C22×Q8—41no—
117C22×Q8—42no—
118C22×Q8—42no—
119C22×Q8—84no—
120C22×Q8Q1614yes—
121C22×Q8C2×D414yes—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4C2×D411yescharacteristic
127C23×C4C2×D411yescharacteristic
128C23×C4C2×D411yescharacteristic
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2411yescharacteristic
133C22×C8—24no—
134C22×C8—24no—
135C22×C8—24no—
136C22×C8—24no—
137C22×C8—24no—
138C22×C8—24no—
139C22×C8—24no—
140C4⋊Q8—416no—
141C4⋊Q8—416no—
142C4×Q8—416no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
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—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—41no—
163C2×C4⋊C4—41no—
164C2×C4⋊C4—41no—
165C2×C4⋊C4—42no—
166C2×C4⋊C4—42no—
167C2×C4⋊C4—42no—
168C2×C4⋊C4—42no—
169C2×C4⋊C4—42no—
170C2×C4⋊C4—42no—
171C2×C4⋊C4—84no—
172C2×C4⋊C4—84no—
173C2×C4⋊C4C2×D414yes—
174C2×C4⋊C4C2×D414yes—
175C2×C4⋊C4C2×D414yes—
176C2×C4⋊C4#4118yes—
177C2×C42—22no—
178C2×C42—22no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—42no—
187C2×C42—42no—
188C2×C42—42no—
189C2×C42—42no—
190C2×C42C2×D414yes—
191C2×C42C2×D414yes—
192#103—416no—
193#103—416no—
194#95—24no—
195#95—24no—
196#95—28no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—28no—
201#95—28no—
202C24C22×D411yescharacteristic
203C2×Q8—28no—
204C2×Q8—28no—
205C2×Q8—24no—
206C2×Q8—24no—
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—44no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—44no—
213C2×Q8—44no—
214C2×Q8—48no—
215C2×Q8—42no—
216C2×Q8—42no—
217C2×Q8—44no—
218C2×Q8—44no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—44no—
222C2×Q8—44no—
223C2×Q8—44no—
224C2×Q8—44no—
225C2×Q8—44no—
226C2×Q8—44no—
227C2×Q8—88no—
228C2×Q8—88no—
229C2×Q8—88no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—21no—
239C22×C4—21no—
240C22×C4—24no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—21no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—28no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—24no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—44no—
267C22×C4—44no—
268C22×C4—44no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4#12012yes—
275C22×C4#12012yes—
276C22×C4#12012yes—
277C22×C4#12012yes—
278C22×C4C2×Q1612yes—
279C22×C4C2×Q1612yes—
280C22×C4C2×Q1612yes—
281C22×C4#13712yes—
282C22×C4#13712yes—
283C22×C4C22×D412yes—
284C22×C4C22×D412yes—
285C22×C4C2×#4114yes—
286C22×C4H(5,𝔽2)12yes—
287C22×C42−1+414yes—
288C2×C8—44no—
289C2×C8—44no—
290C2×C8—44no—
291C2×C8—44no—
292C2×C8—44no—
293C2×C8—44no—
294C2×C8—44no—
295C4⋊C4—24no—
296C4⋊C4—28no—
297C4⋊C4—28no—
298C4⋊C4—28no—
299C4⋊C4—44no—
300C4⋊C4—44no—
301C4⋊C4—44no—
302C4⋊C4—48no—
303C4⋊C4—48no—
304C4⋊C4—48no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—44no—
308C4⋊C4—88no—
309C4⋊C4—88no—
310C4⋊C4#12418yes—
311C42—28no—
312C42—28no—
313C42—48no—
314C42—48no—
315C42—44no—
316C23C2×#12011yescharacteristic
317C23C2×#12011yescharacteristic
318C23#53411yescharacteristic
319C23#54814yes—
320C23C22×Q1611yescharacteristic
321C23C2×#13711yescharacteristic
322C23C2×#13711yescharacteristic
323C23#57511yescharacteristic
324C23#57714yes—
325Q8—416no—
326Q8—48no—
327Q8—48no—
328Q8—816no—
329Q8—84no—
330Q8—84no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—28no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—48no—
352C2×C4—48no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—44no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—41no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—44no—
371C2×C4—42no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—44no—
376C2×C4—84no—
377C2×C4—84no—
378C2×C4—84no—
379C2×C4—84no—
380C2×C4—84no—
381C2×C4—84no—
382C2×C4#45114yes—
383C2×C4#45114yes—
384C2×C4C2×#12412yes—
385C2×C4#53212yes—
386C2×C4#53612yes—
387C2×C4#53612yes—
388C2×C4#54214yes—
389C2×C4D4214yes—
390C2×C4#54714yes—
391C2×C4#54814yes—
392C2×C4#55512yes—
393C2×C4#56312yes—
394C8—84no—
395C22#240311yescharacteristic
396C22#243212yes—
397C22#243612yes—
398C22#273112yes—
399C22C2×#45111yescharacteristic
400C22C2×#45111yescharacteristic
401C22#300711yescharacteristic
402C22#300711yescharacteristic
403C22#300912yes—
404C22#301012yes—
405C22#318911yescharacteristic
406C22#319511yescharacteristic
407C22#320012yes—
408C22#336314yes—
409C22#336914yes—
410C22#337214yes—
411C22#337614yes—
412C4—24no—
413C4—24no—
414C4—48no—
415C4—42no—
416C4—42no—
417C4—42no—
418C4—84no—
419C4—81no—
420C4—84no—
421C4—84no—
422C4#249114yes—
423C4#267414yes—
424C2#1576111yescharacteristic
425C2#2392111yescharacteristic
426C2#2392611yescharacteristic
427C2#2565112yes—
428C2#2566212yes—
429C2#2593612yes—
430C2#2594512yes—
431C2#3263112yes—
432C2#3264512yes—
subgroupG/H
center, socle, prime radicalC24C22×D4
commutator, Frattini, cosocleC23×C4C24
LCSG ▷ C23×C4 ▷ C2 ▷ C1
UCSC1 ◁ C24 ◁ C22×C4×Q8 ◁ 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×#25682
C2×#25672
C2×#20432
C2×#20242
#113802
#110672
#109322
#108852