Finite Group Databasebeta

← #200631

Subgroups — #200631

Order 512.

1246 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34955C211yesmaximal, characteristic
2#34059C211yesmaximal, characteristic, complemented
3#17544C212yesmaximal
4#17365C211yesmaximal, characteristic, complemented
5#9414C218yesmaximal, complemented
6#8048C212yesmaximal, complemented
7C2×#532C2211yescharacteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2212yes—
10#2592—22no—
11#2591C2214yes—
12#2568C412yescomplemented
13#2548C412yescomplemented
14#2504—21no—
15#2436—21no—
16#2398C414yescomplemented
17#2320—24no—
18#2316—21no—
19#2313C2214yes—
20#2310—21no—
21C4×#124—22no—
22#2297—22no—
23#2295—24no—
24#2283—22no—
25C4×#117C2211yescharacteristic
26C4×#117C2212yes—
27#1507C2218yes—
28#1505—24no—
29#1502—24no—
30#1467—22no—
31#1463C2214yes—
32#1461—22no—
33#1297C2218yes—
34C22×#41—21no—
35C23×Q8D411yescharacteristic
36#532—22no—
37#532—22no—
38#532—22no—
39#532—22no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×#127C2×C411yescharacteristic
42C2×#124—22no—
43C2×#124—21no—
44C2×#121—21no—
45C2×#121C2312yes—
46C2×C4×Q8—22no—
47C2×C4×Q8—22no—
48C2×C4×Q8—22no—
49C2×C4×Q8—22no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×Q8D412yes—
53C2×C4×Q8D412yes—
54C2×C4×D4—21no—
55C2×C4×D4C2×C412yes—
56C2×#117—22no—
57C2×#117—21no—
58C22×C4⋊C4D411yescharacteristic
59C22×C4⋊C4C2312yes—
60C22×C42D412yes—
61C22×C42C2311yescharacteristic
62#432—22no—
63#432—22no—
64#431—22no—
65#431—22no—
66#423—22no—
67#407C2314yes—
68C4×M16—22no—
69#400C2×C418yes—
70#397—24no—
71#394C2×C418yes—
72#393—24no—
73#391—28no—
74#391—22no—
75#391—24no—
76#390—22no—
77#390—24no—
78#389—28no—
79#388—28no—
80#387—216no—
81#386—24no—
82#386—24no—
83#383—24no—
84#382—24no—
85#382—22no—
86#381—24no—
87#380—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4—21no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4D414yes—
93C4×C22⋊C4—21no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—28no—
96C4×C22⋊C4C2312yes—
97C4×C22⋊C4C2314yes—
98#376—24no—
99#376C2×C412yes—
100#376D414yes—
101C43—22no—
102C43—21no—
103#327—216no—
104#324—28no—
105C2×#41—48no—
106C2×#41—41no—
107C2×#41—41no—
108C2×#41—41no—
109C2×#41—41no—
110C22×Q8—21no—
111C22×Q8—22no—
112C22×Q8—22no—
113C22×Q8—44no—
114C22×Q8C22⋊C412yes—
115C22×Q8C22⋊C412yes—
116C22×D4—22no—
117C22×D4—21no—
118C22×D4—21no—
119C22×D4C22×C412yes—
120C23×C4—22no—
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D412yes—
124C23×C4C2×D412yes—
125C23×C4C2411yescharacteristic
126C2×M16—24no—
127C4⋊Q8—22no—
128C4⋊Q8—22no—
129#127—22no—
130#127—22no—
131#124—48no—
132#124—42no—
133#124—42no—
134#121—24no—
135#121—24no—
136#121—42no—
137#121—42no—
138C4×Q8—24no—
139C4×Q8—24no—
140C4×Q8—24no—
141C4×Q8—24no—
142C4×Q8—24no—
143C4×Q8—24no—
144C4×Q8—24no—
145C4×Q8—24no—
146C4×D4—24no—
147C4×D4—24no—
148C4×D4—42no—
149C4×D4—42no—
150#117—48no—
151#117—44no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—28no—
160C2×C4⋊C4C22⋊C412yes—
161C2×C4⋊C4C22⋊C412yes—
162C2×C4⋊C4C22×C412yes—
163C2×C4⋊C4C22×C414yes—
164C2×C4⋊C4C2×D414yes—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—22no—
168C2×C22⋊C4—28no—
169C2×C22⋊C4—22no—
170C2×C22⋊C4—28no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—22no—
174C2×C42—22no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—24no—
180C2×C42—22no—
181C2×C42C22⋊C414yes—
182C2×C42C22⋊C414yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42#4112yes—
190C2×C42#4112yes—
191C4⋊C8—42no—
192C4⋊C8—42no—
193C22⋊C8—24no—
194C22⋊C8—24no—
195#97—42no—
196C4×C8—42no—
197#95—28no—
198#95—28no—
199#95—28no—
200#95—22no—
201#95—28no—
202#95—28no—
203#95—24no—
204#95C22×C414yes—
205#95C2×D418yes—
206C24—22no—
207C24C22×D411yescharacteristic
208#41—84no—
209#41—84no—
210#41—84no—
211#41—84no—
212C2×Q8—24no—
213C2×Q8—24no—
214C2×Q8—24no—
215C2×Q8—24no—
216C2×Q8—44no—
217C2×Q8—48no—
218C2×Q8—48no—
219C2×Q8—44no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×D4—24no—
223C2×D4—44no—
224C2×D4—48no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—41no—
229C2×D4—41no—
230C2×D4—44no—
231C2×D4—82no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—44no—
242C22×C4—48no—
243C22×C4—42no—
244C22×C4—42no—
245C22×C4—81no—
246C22×C4—82no—
247C22×C4C2×C22⋊C414yes—
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C414yes—
252C22×C4C2×C22⋊C411yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12012yes—
255C22×C4#12011yescharacteristic
256C22×C4#12311yescharacteristic
257C22×C4#12312yes—
258C22×C4#12311yescharacteristic
259C22×C4C23×C411yescharacteristic
260C22×C4C22×D412yes—
261C22×C4C22×D411yescharacteristic
262C22×C4C2×#4112yes—
263M16—84no—
264C2×C8—44no—
265C4⋊C4—28no—
266C4⋊C4—28no—
267C4⋊C4—24no—
268C4⋊C4—24no—
269C4⋊C4—24no—
270C4⋊C4—44no—
271C4⋊C4—48no—
272C4⋊C4—42no—
273C4⋊C4—88no—
274C22⋊C4—416no—
275C22⋊C4—48no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—44no—
279C22⋊C4—88no—
280C42—24no—
281C42—24no—
282C42—24no—
283C42—24no—
284C42—44no—
285C42—48no—
286C42—44no—
287C42—48no—
288C42—82no—
289C42—84no—
290C42C4×D412yes—
291C42C4×D412yes—
292C42C4×D412yes—
293C42C4×D412yes—
294C42#12112yes—
295C42#12112yes—
296C42#12112yes—
297C42#12112yes—
298C23—22no—
299C23—22no—
300C23—41no—
301C23—44no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23C22×C22⋊C411yescharacteristic
306C23C2×#12011yescharacteristic
307C23C2×#12311yescharacteristic
308Q8—48no—
309Q8—48no—
310Q8—82no—
311Q8—82no—
312D4—84no—
313D4—82no—
314D4—84no—
315D4—84no—
316D4—82no—
317D4—84no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—44no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—42no—
346C2×C4—84no—
347C2×C4—82no—
348C2×C4—84no—
349C2×C4#38611yescharacteristic
350C2×C4#38611yescharacteristic
351C2×C4#38612yes—
352C2×C4#38611yescharacteristic
353C2×C4#38612yes—
354C2×C4#38611yescharacteristic
355C2×C4#42114yes—
356C2×C4#45414yes—
357C2×C4C2×C4×D412yes—
358C2×C4C2×#12112yes—
359C8—84no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—88no—
369C22—82no—
370C22—82no—
371C22—82no—
372C22C2×#38611yescharacteristic
373C22C2×#42112yes—
374C22C2×#45412yes—
375C4—22no—
376C4—22no—
377C4—21no—
378C4—21no—
379C4—22no—
380C4—44no—
381C4—44no—
382C4—88no—
383C4—84no—
384C4—82no—
385C4#188812yes—
386C4#188812yes—
387C4#189714yes—
388C2—22no—
389C2—22no—
390C2—41no—
391C2—162no—
392C2C2×#188811yescharacteristic
393C2C2×#189711yescharacteristic
394C2#2101411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
socleC22C2×#386
prime radicalC2×#532C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #34059C2semidirect
4. #17365C2semidirect
5. #9414C2semidirect
6. #8048C2semidirect
12. #2568C4semidirect
13. #2548C4semidirect
16. #2398C4semidirect

Maximal subgroups

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

subgroupindex
#349552
#340592
#175442
#173652
#94142
#80482