Finite Group Databasebeta

← #106568

Subgroups — #106568

Order 512.

1005 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2973C211yesmaximal, characteristic
2#21740C211yesmaximal, characteristic
3#20722C211yesmaximal, characteristic
4#8112C212yesmaximal, complemented
5#7927C212yesmaximal, complemented
6C22×C4×D4C2211yescharacteristic
7#2973—24no—
8#2973—22no—
9#2973C414yescomplemented
10C2×#435—24no—
11C2×#432—21no—
12C2×#407C2211yescharacteristic
13C2×#407C2212yes—
14C22×C22⋊C8—21no—
15C22×C22⋊C8C2211yescharacteristic
16#2280C2212yes—
17#1870—22no—
18#1869C414yescomplemented
19#1819C412yes—
20#1817—21no—
21#1799—22no—
22#1781C412yes—
23#1779—21no—
24C4×#99—22no—
25#1306—24no—
26#1282—24no—
27C23×D4C2×C411yescharacteristic
28C24×C4D411yescharacteristic
29C24×C4C2311yescharacteristic
30C22×M16—22no—
31C22×M16—21no—
32C23×C8—22no—
33C23×C8C2×C412yes—
34C2×C4×D4—22no—
35C2×C4×D4—22no—
36C2×C4×D4—22no—
37C2×C4×D4—22no—
38C2×C4×D4C2×C412yes—
39C2×C4×D4C2×C412yes—
40C2×C4×D4D412yes—
41C2×C4×D4D412yes—
42C2×#117—22no—
43C22×C4⋊C4Q811yescharacteristic
44C22×C22⋊C4C2×C411yescharacteristic
45C22×C22⋊C4Q811yescharacteristic
46C22×C42D411yescharacteristic
47#435—216no—
48#435—216no—
49#432—24no—
50#432—24no—
51C2×C4⋊C8—22no—
52C2×#99—24no—
53#407—22no—
54#407—22no—
55#407—22no—
56#407—22no—
57#407C2×C414yes—
58#407D414yes—
59C2×C22⋊C8—22no—
60C2×C22⋊C8—24no—
61C2×C22⋊C8—24no—
62C2×C22⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8—22no—
68C2×C22⋊C8—24no—
69C2×C22⋊C8C2×C412yes—
70C2×C22⋊C8D414yes—
71C2×#97—22no—
72#381—22no—
73#380—22no—
74#380—22no—
75#379C2×C414yes—
76C4×C22⋊C4—22no—
77C4×C22⋊C4—22no—
78#342—24no—
79#336—24no—
80C25C4⋊C411yescharacteristic
81C25C22×C411yescharacteristic
82C22×D4—22no—
83C22×D4—22no—
84C22×D4—44no—
85C22×D4C4212yes—
86C22×D4C22⋊C412yes—
87C23×C4—21no—
88C23×C4—21no—
89C23×C4—21no—
90C23×C4—21no—
91C23×C4—82no—
92C23×C4—82no—
93C23×C4C22⋊C412yes—
94C23×C4C22⋊C412yes—
95C23×C4C4⋊C411yescharacteristic
96C23×C4C4⋊C411yescharacteristic
97C23×C4C4⋊C411yescharacteristic
98C23×C4C22×C411yescharacteristic
99C23×C4C22×C411yescharacteristic
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D412yes—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×D412yes—
105C23×C4C2×Q811yescharacteristic
106C23×C4#4112yes—
107C23×C4#4112yes—
108C2×M16—24no—
109C2×M16—24no—
110C2×M16—42no—
111C2×M16—84no—
112C2×M16—82no—
113C22×C8—24no—
114C22×C8—28no—
115C22×C8—22no—
116C22×C8—22no—
117C22×C8—24no—
118C22×C8—84no—
119C22×C8—82no—
120C4×D4—24no—
121C4×D4—24no—
122C4×D4—24no—
123C4×D4—24no—
124C4×D4—24no—
125C4×D4—24no—
126C4×D4—24no—
127C4×D4—24no—
128#117—84no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—42no—
132C2×C4⋊C4C4⋊C412yes—
133C2×C4⋊C4C4⋊C412yes—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—24no—
141C2×C22⋊C4C4212yes—
142C2×C22⋊C4C22⋊C412yes—
143C2×C22⋊C4C4⋊C412yes—
144C2×C22⋊C4C4⋊C412yes—
145C2×C42—21no—
146C2×C42—21no—
147C2×C42—21no—
148C2×C42—21no—
149C2×C42—42no—
150C2×C42C22⋊C412yes—
151C2×C42C22⋊C412yes—
152C4⋊C8—28no—
153#99—48no—
154C22⋊C8—28no—
155C22⋊C8—24no—
156C22⋊C8—24no—
157C22⋊C8—24no—
158C22⋊C8—24no—
159C22⋊C8—41no—
160C22⋊C8—41no—
161C22⋊C8—42no—
162C22⋊C8—41no—
163C22⋊C8—41no—
164C22⋊C8—84no—
165C22⋊C8C22⋊C418yes—
166#97—28no—
167#95—42no—
168#95—42no—
169C24—21no—
170C24—24no—
171C24—21no—
172C24—42no—
173C24—42no—
174C24—42no—
175C24—81no—
176C24—81no—
177C24#9512yes—
178C24C2×C4211yescharacteristic
179C24C2×C22⋊C411yescharacteristic
180C24C2×C4⋊C411yescharacteristic
181C24C4×D412yes—
182C24C4×D412yes—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—24no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—48no—
190C2×D4—48no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—44no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—44no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—44no—
208C22×C4—44no—
209C22×C4—42no—
210C22×C4—44no—
211C22×C4—82no—
212C22×C4—81no—
213C22×C4—81no—
214C22×C4—82no—
215C22×C4—81no—
216C22×C4—82no—
217C22×C4—82no—
218C22×C4—81no—
219C22×C4—82no—
220C22×C4—82no—
221C22×C4#9512yes—
222C22×C4#9512yes—
223C22×C4#9512yes—
224C22×C4C2×C22⋊C411yescharacteristic
225C22×C4C2×C22⋊C412yes—
226C22×C4C2×C22⋊C411yescharacteristic
227C22×C4C2×C4⋊C411yescharacteristic
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4#11712yes—
230C22×C4C4×D412yes—
231C22×C4C4×D412yes—
232C22×C4#12011yescharacteristic
233C22×C4#12012yes—
234C22×C4#12011yescharacteristic
235C22×C4#12112yes—
236C22×C4#12112yes—
237C22×C4C22⋊Q812yes—
238C22×C4C22⋊Q812yes—
239C22×C4#12311yescharacteristic
240C22×C4#12312yes—
241C22×C4#12311yescharacteristic
242M16—84no—
243M16—84no—
244M16—84no—
245C2×C8—28no—
246C2×C8—48no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—88no—
252C2×C8—84no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—84no—
256C22⋊C4—24no—
257C22⋊C4—24no—
258C22⋊C4—24no—
259C22⋊C4—24no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—84no—
263C42—22no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—84no—
268C23—24no—
269C23—24no—
270C23—22no—
271C23—22no—
272C23—22no—
273C23—22no—
274C23—24no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23—41no—
280C23—42no—
281C23—41no—
282C23—42no—
283C23—42no—
284C23—42no—
285C23—41no—
286C23—42no—
287C23—42no—
288C23—42no—
289C23—42no—
290C23—42no—
291C23—81no—
292C23—81no—
293C23—88no—
294C23—88no—
295C23#34414yes—
296C23C2×#9511yescharacteristic
297C23C4×C22⋊C412yes—
298C23#38511yescharacteristic
299C23#38511yescharacteristic
300C23#38612yes—
301D4—48no—
302D4—48no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—44no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—44no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—81no—
337C2×C4—81no—
338C2×C4—84no—
339C2×C4—84no—
340C2×C4—84no—
341C2×C4#37911yescharacteristic
342C2×C4#38012yes—
343C2×C4#38111yescharacteristic
344C2×C4#38512yes—
345C2×C4#38611yescharacteristic
346C2×C4#38611yescharacteristic
347C8—48no—
348C8—82no—
349C8—82no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—22no—
357C22—21no—
358C22—22no—
359C22—24no—
360C22—22no—
361C22—22no—
362C22—21no—
363C22—22no—
364C22—44no—
365C22—44no—
366C22—48no—
367C22—44no—
368C22—42no—
369C22—42no—
370C22—84no—
371C22—82no—
372C22—82no—
373C22—84no—
374C22—84no—
375C22C22⋊#9511yescharacteristic
376C22C2×#34412yes—
377C22#186812yes—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—22no—
382C4—22no—
383C4—42no—
384C4—42no—
385C4—42no—
386C4—164no—
387C4#191312yes—
388C2—22no—
389C2—22no—
390C2—42no—
391C2—42no—
392C2—41no—
393C2—44no—
394C2#1073711yescharacteristic
395C2#1074811yescharacteristic
396C2#1075411yescharacteristic
subgroupG/H
centerC2×C4#386
commutatorC24C2×C42
Frattini, cosocleC24×C4C23
socleC22C22⋊#95
prime radicalC23×D4C2×C4
LCSG ▷ C24 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C24×C4 ◁ G
LECSG ▷ C24×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8112C2semidirect
5. #7927C2semidirect
9. #2973C4semidirect
18. #1869C4semidirect

Maximal subgroups

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

subgroupindex
C2×#29732
#217402
#207222
#81122
#79272