Finite Group Databasebeta

← #224725

Subgroups — #224725

Order 512.

977 conjugacy classes of subgroups.

The subgroup lattice has 385 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36595C211yesmaximal, characteristic
2#22825C212yesmaximal, complemented
3#9707C212yesmaximal, complemented
4#9706C212yesmaximal
5#9699C212yesmaximal, complemented
6#8594C211yesmaximal, characteristic, complemented
7#8588C212yesmaximal, complemented
8#8234C212yesmaximal, complemented
9#8232C211yesmaximal, characteristic, complemented
10C2×#559C412yes—
11#2867C2211yescharacteristic
12#2858C2212yes—
13#2548—22no—
14#2548C412yes—
15#2544C414yes—
16#2445—22no—
17#2445—22no—
18#2366—22no—
19C4×C4⋊Q8C2211yescharacteristic
20C4×#127C2212yes—
21C42×Q8C2211yescharacteristic
22#1984—22no—
23#1917—22no—
24#1917—22no—
25#1842C2212yes—
26#1831C2212yes—
27#1831C2212yes—
28#1770—22no—
29#1767C2212yes—
30#1767C2212yes—
31C4×C4⋊C8C2212yes—
32C2×#333C2212yes—
33C2×#333C2211yescharacteristic
34C2×#333C2211yescharacteristic
35C2×#331C2212yes—
36C2×#331C2212yes—
37C2×#331C2212yes—
38C2×#331C2212yes—
39#1296C2211yescharacteristic
40#1296C2211yescharacteristic
41#1296C2212yes—
42#559—48no—
43C2×C4⋊Q8C2311yescharacteristic
44C2×#127—22no—
45C2×#127—22no—
46C2×#127C2×C412yes—
47C2×#127C2×C412yes—
48C2×#127C2312yes—
49C2×#124—22no—
50C2×#124C2×C414yes—
51C2×#124C2×C414yes—
52C2×#123—24no—
53C2×C4×Q8C2×C412yes—
54C2×C4×Q8C2×C411yescharacteristic
55C2×C4×Q8D412yes—
56C2×C4×D4—22no—
57C2×C4×D4—22no—
58C2×C4×D4—22no—
59C2×C4×D4C2×C414yes—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8C2312yes—
63C2×C4⋊C8C2312yes—
64C2×C4⋊C8C2312yes—
65C2×C4⋊C8C2312yes—
66C2×#97—22no—
67C2×#97—22no—
68C2×C4×C8—22no—
69#394—24no—
70#394—24no—
71#394—22no—
72#394—22no—
73#391C2×C412yes—
74#390—24no—
75#390—22no—
76#390—22no—
77#390—22no—
78#390—22no—
79#390—22no—
80#390—22no—
81#390C2×C414yes—
82#389—22no—
83#386—24no—
84#386—24no—
85#384C2311yescharacteristic
86C4×C4⋊C4—22no—
87C4×C4⋊C4C2×C412yes—
88C4×C4⋊C4C2×C411yescharacteristic
89C4×C4⋊C4D412yes—
90C4×C4⋊C4D412yes—
91C4×C4⋊C4D412yes—
92C43C2312yes—
93C43C2311yescharacteristic
94#336—22no—
95#336—22no—
96#336—22no—
97#336—22no—
98#336—22no—
99#336—22no—
100#336—22no—
101#333—24no—
102#333—41no—
103#333—41no—
104#331—24no—
105#331—24no—
106#331—42no—
107#331—42no—
108C22×Q8C22⋊C411yescharacteristic
109C22×Q8C22×C412yes—
110C22×D4—22no—
111C22×D4—22no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4C22×C414yes—
116C23×C4—24no—
117C22×C8—22no—
118C22×C8—22no—
119C22×C8—22no—
120C22×C8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—21no—
123C4⋊Q8—21no—
124#127—22no—
125#127—22no—
126#127—22no—
127#127—22no—
128#127—44no—
129#127—44no—
130#127—44no—
131#127—82no—
132#127—82no—
133#127—82no—
134#124—48no—
135#124—48no—
136#124—84no—
137#123—416no—
138C4×Q8—24no—
139C4×Q8—42no—
140C4×Q8—42no—
141C4×Q8—44no—
142C4×D4—48no—
143C4×D4—48no—
144C4×D4—82no—
145C4×D4—82no—
146C4×D4—82no—
147C4×D4—82no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22⋊C412yes—
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22⋊C412yes—
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C22⋊C4—24no—
158C2×C22⋊C4—24no—
159C2×C22⋊C4—24no—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C42—22no—
165C2×C42C22×C412yes—
166C2×C42C22×C411yescharacteristic
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—44no—
184C4⋊C8—82no—
185C4⋊C8#4114yes—
186#97—82no—
187#97—82no—
188C4×C8—44no—
189#95—24no—
190#95C22⋊C412yes—
191#95C22⋊C412yes—
192#95C22⋊C411yescharacteristic
193#95C22⋊C411yescharacteristic
194C24—24no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8#10014yes—
201C2×D4—24no—
202C2×D4—24no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—48no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—42no—
217C2×D4—42no—
218C2×D4—42no—
219C2×D4—42no—
220C2×D4—44no—
221C2×D4—84no—
222C2×D4—84no—
223C2×D4—84no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—48no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C411yescharacteristic
234C22×C4C2×C22⋊C411yescharacteristic
235C22×C4C2×C22⋊C412yes—
236C22×C4C23×C411yescharacteristic
237C22×C4C22×D412yes—
238C2×C8—24no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C4⋊C4—22no—
251C4⋊C4—24no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—48no—
262C22⋊C4—48no—
263C22⋊C4—48no—
264C22⋊C4—84no—
265C22⋊C4—84no—
266C22⋊C4—84no—
267C22⋊C4—84no—
268C42—22no—
269C42—22no—
270C42—22no—
271C42—22no—
272C42—22no—
273C42—22no—
274C42—22no—
275C42—22no—
276C42—22no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—24no—
281C42—22no—
282C42—42no—
283C42—41no—
284C42—44no—
285C42C2×#4112yes—
286C42H(5,𝔽2)11yescharacteristic
287C422−1+411yescharacteristic
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23C22×C22⋊C411yescharacteristic
296Q8—44no—
297Q8—82no—
298Q8—82no—
299D4—84no—
300D4—84no—
301D4—88no—
302D4—82no—
303D4—82no—
304D4—82no—
305D4—82no—
306D4—82no—
307D4—82no—
308D4—82no—
309D4—82no—
310D4—168no—
311D4—164no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—42no—
335C2×C4—44no—
336C2×C4—168no—
337C2×C4C2×#10012yes—
338C2×C4C2×#10014yes—
339C2×C4C4×#4111yescharacteristic
340C2×C4#52011yescharacteristic
341C2×C4#54812yes—
342C2×C4#55012yes—
343C2×C4#57512yes—
344C2×C4#57512yes—
345C2×C4#57612yes—
346C2×C4#57712yes—
347C8—44no—
348C8—82no—
349C8—82no—
350C8—82no—
351C22—84no—
352C22—84no—
353C22—84no—
354C22—84no—
355C22—84no—
356C22—84no—
357C22—84no—
358C22#234111yescharacteristic
359C22C22×#10011yescharacteristic
360C22#288811yescharacteristic
361C22#289612yes—
362C22#289712yes—
363C4—21no—
364C4—21no—
365C4—22no—
366C4—24no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—82no—
373C4—82no—
374C4#337012yes—
375C4#337112yes—
376C4#340112yes—
377C4#340312yes—
378C2—164no—
379C2#1775411yescharacteristic
380C2#1776612yes—
381C2#1778211yescharacteristic
382C2#2253611yescharacteristic
383C2#2254912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radical#2867C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22825C2semidirect
3. #9707C2semidirect
5. #9699C2semidirect
6. #8594C2semidirect
7. #8588C2semidirect
8. #8234C2semidirect
9. #8232C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#365952
#228252
#97072
#97062
#96992
#85942
#85882
#82342
#82322