Finite Group Databasebeta

← #205777

Subgroups — #205777

Order 512.

1161 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#21938C211yesmaximal, characteristic
3C2×#1996C212yesmaximal
4#20526C211yesmaximal, characteristic
5C2×#1520C212yesmaximal
6#9784C218yesmaximal, complemented
7C22×C22⋊Q8C2212yes—
8#3189—24no—
9#3189—22no—
10#3189—22no—
11C2×#484C414yescomplemented
12C2×#467C2214yes—
13C2×#451C414yescomplemented
14C22×C22⋊C8C2211yescharacteristic
15C22×C22⋊C8C2211yescharacteristic
16C2×#380C2212yes—
17C2×C4×C22⋊C4C2211yescharacteristic
18#1996—24no—
19#1996C2218yes—
20#1853—24no—
21#1807C2214yes—
22#1752—22no—
23#1752C2214yes—
24#1749—22no—
25C2×#342—24no—
26#1520—24no—
27#1520C2218yes—
28C2×#328—24no—
29C23×Q8C2×C412yes—
30C24×C4C2311yescharacteristic
31C22×Q16C2×C414yescomplemented
32C23×C8—21no—
33C23×C8C2312yes—
34C2×C22⋊Q8—22no—
35C2×C22⋊Q8—22no—
36C2×C22⋊Q8—22no—
37C2×C22⋊Q8—22no—
38C2×C22⋊Q8—22no—
39C2×C22⋊Q8—22no—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8C2×C414yes—
45C2×C22⋊Q8C2314yes—
46C22×C4⋊C4—22no—
47C22×C4⋊C4—22no—
48C22×C4⋊C4C2×C412yes—
49C22×C22⋊C4—22no—
50C22×C22⋊C4—22no—
51C22×C22⋊C4D412yes—
52C22×C42—21no—
53#484—28no—
54#484—28no—
55#467—28no—
56#467—28no—
57#451—28no—
58#451—28no—
59C2×#107C2×C414yescomplemented
60C2×#106—24no—
61C2×#103C2×C418yescomplemented
62C2×C22⋊C8—21no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8C2×C414yes—
65C2×C22⋊C8D414yes—
66C2×C22⋊C8D414yes—
67C2×C22⋊C8D412yes—
68C2×C22⋊C8D412yes—
69C2×C22⋊C8C2312yes—
70C2×#97—24no—
71#380—22no—
72#380—22no—
73#380—22no—
74#380C2314yes—
75C4×C22⋊C4—21no—
76C4×C22⋊C4—21no—
77C4×C22⋊C4D412yes—
78C4×C22⋊C4C2312yes—
79C2×#95—21no—
80C2×#95—22no—
81#342—216no—
82#340—24no—
83#336—24no—
84#328—216no—
85C25C2×D411yescharacteristic
86C22×Q8—22no—
87C22×Q8—22no—
88C22×Q8—22no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—84no—
92C22×Q8C22×C414yes—
93C23×C4—22no—
94C23×C4—22no—
95C23×C4—22no—
96C23×C4—42no—
97C23×C4—82no—
98C23×C4C22×C412yes—
99C23×C4C2×D411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D412yes—
102C23×C4C2×D411yescharacteristic
103C23×C4C2×D412yes—
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C23×C4#4112yes—
107C23×C4C2411yescharacteristic
108C2×Q16—28no—
109C2×Q16—84no—
110C22×C8—22no—
111C22×C8—24no—
112C22×C8—82no—
113C22×C8—82no—
114C22×C8C22×C414yes—
115C22×C8C2×D414yes—
116C22×C8C2×D414yes—
117C22⋊Q8—24no—
118C22⋊Q8—24no—
119C22⋊Q8—24no—
120C22⋊Q8—24no—
121C22⋊Q8—24no—
122C22⋊Q8—24no—
123C22⋊Q8—24no—
124C22⋊Q8—24no—
125C22⋊Q8—42no—
126C22⋊Q8—42no—
127C22⋊Q8—42no—
128C22⋊Q8—42no—
129C22⋊Q8—42no—
130C22⋊Q8—42no—
131C22⋊Q8—42no—
132C22⋊Q8—42no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—24no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—42no—
142C2×C4⋊C4—42no—
143C2×C4⋊C4—42no—
144C2×C4⋊C4—42no—
145C2×C4⋊C4—42no—
146C2×C4⋊C4—84no—
147C2×C4⋊C4C22×C414yes—
148C2×C22⋊C4—28no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—42no—
154C2×C22⋊C4—42no—
155C2×C22⋊C4—42no—
156C2×C22⋊C4—42no—
157C2×C22⋊C4—48no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4C2×D414yes—
160C2×C42—22no—
161C2×C42—22no—
162C2×C42—82no—
163#107—28no—
164#106—84no—
165#103—216no—
166C22⋊C8—24no—
167C22⋊C8—22no—
168C22⋊C8C22⋊C418yes—
169C22⋊C8C22⋊C418yes—
170C22⋊C8C2×D414yes—
171#97—84no—
172#95—22no—
173#95—22no—
174#95—24no—
175#95—42no—
176C24—81no—
177C24—81no—
178C24C4×D412yes—
179C24#12111yescharacteristic
180C24#12111yescharacteristic
181C24#12712yes—
182C24#13718yes—
183C24C22×D411yescharacteristic
184C2×Q8—24no—
185C2×Q8—24no—
186C2×Q8—48no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—84no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C22×C4—24no—
195C22×C4—24no—
196C22×C4—24no—
197C22×C4—42no—
198C22×C4—44no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—44no—
207C22×C4—44no—
208C22×C4—81no—
209C22×C4—82no—
210C22×C4—81no—
211C22×C4—82no—
212C22×C4—81no—
213C22×C4—82no—
214C22×C4—82no—
215C22×C4—84no—
216C22×C4—84no—
217C22×C4—84no—
218C22×C4—84no—
219C22×C4—84no—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C4×D412yes—
223C22×C4C4×D412yes—
224C22×C4C4×D412yes—
225C22×C4#12112yes—
226C22×C4#12112yes—
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12412yes—
230C22×C4#12412yes—
231C22×C4#12712yes—
232C22×C4C23×C411yescharacteristic
233C22×C4C22×D411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4C22×D411yescharacteristic
236C22×C4C2×#4111yescharacteristic
237C22×C4C2×#4111yescharacteristic
238Q16—88no—
239C2×C8—24no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C2×C8—84no—
245C2×C8C2×C22⋊C418yes—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—48no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—84no—
256C22⋊C4—28no—
257C22⋊C4—44no—
258C22⋊C4—44no—
259C22⋊C4—44no—
260C22⋊C4—44no—
261C22⋊C4—44no—
262C22⋊C4—48no—
263C22⋊C4—84no—
264C42—44no—
265C42—82no—
266C23—42no—
267C23—41no—
268C23—42no—
269C23—41no—
270C23—41no—
271C23—42no—
272C23—41no—
273C23—42no—
274C23—41no—
275C23—41no—
276C23—88no—
277C23#39012yes—
278C23#39012yes—
279C23#44118yes—
280C23#47014yes—
281C23#47014yes—
282C23#49714yes—
283C23#49714yes—
284C23C2×C4×D411yescharacteristic
285C23C2×#12111yescharacteristic
286C23C2×#12711yescharacteristic
287C23C2×#13714yes—
288Q8—48no—
289Q8—84no—
290Q8—84no—
291Q8—84no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—84no—
319C2×C4—88no—
320C2×C4—84no—
321C2×C4—84no—
322C2×C4—84no—
323C2×C4—84no—
324C2×C4—84no—
325C2×C4—168no—
326C2×C4#39012yes—
327C2×C4#39012yes—
328C2×C4C22×C22⋊C411yescharacteristic
329C2×C4C2×C4×D411yescharacteristic
330C2×C4C2×#12111yescharacteristic
331C2×C4C2×#12411yescharacteristic
332C8—28no—
333C8—44no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—22no—
338C22—22no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—22no—
343C22—22no—
344C22—44no—
345C22—44no—
346C22—44no—
347C22—48no—
348C22—44no—
349C22—44no—
350C22#197114yes—
351C22#197114yes—
352C22C2×#39011yescharacteristic
353C22C2×#44112yes—
354C22C2×#47012yes—
355C22C2×#49712yes—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—22no—
360C4—22no—
361C4—84no—
362C4—84no—
363C4—84no—
364C4—164no—
365C4#288312yes—
366C2—22no—
367C2—22no—
368C2—22no—
369C2—24no—
370C2—24no—
371C2#2121311yescharacteristic
372C2C2×#197111yescharacteristic
373C2#2124411yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25C2×D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C24×C4 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
6. #9784C2semidirect
11. C2×#484C4semidirect
13. C2×#451C4semidirect
31. C22×Q16C2×C4semidirect
59. C2×#107C2×C4semidirect
61. C2×#103C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31892
#219382
C2×#19962
#205262
C2×#15202
#97842