Finite Group Databasebeta

← #206041

Subgroups — #206041

Order 512.

1093 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3189C211yesmaximal, characteristic
2#21948C211yesmaximal, characteristic
3#20525C211yesmaximal, characteristic
4C2×#1508C212yesmaximal
5#17397C212yesmaximal
6#9817C214yesmaximal, complemented
7#9814C214yesmaximal, complemented
8C22×C22⋊Q8C2212yes—
9#3189—24no—
10#3189—21no—
11#3189—21no—
12#3189—22no—
13C2×#484C414yescomplemented
14C2×#467C2214yes—
15C2×#451C414yescomplemented
16C2×#407C2212yes—
17C2×#407C2211yescharacteristic
18C22×C22⋊C8C2211yescharacteristic
19C2×C4×C22⋊C4C2211yescharacteristic
20#1855—24no—
21#1810C2214yes—
22#1753—22no—
23#1752C2212yes—
24C8×C22⋊C4C2212yes—
25#1749—21no—
26C4×C22⋊C8—21no—
27#1508—24no—
28#1508C2214yes—
29#1508C2214yes—
30C2×#328—24no—
31#1472—24no—
32#1472C2218yes—
33#1463—24no—
34C23×Q8C2×C412yes—
35C24×C4C2311yescharacteristic
36C22×Q16C2×C414yescomplemented
37C22×M16—21no—
38C22×M16—22no—
39C23×C8C2312yes—
40C2×C22⋊Q8—22no—
41C2×C22⋊Q8—22no—
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—22no—
46C2×C22⋊Q8—22no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C22⋊Q8—22no—
50C2×C22⋊Q8C2×C414yes—
51C2×C22⋊Q8C2314yes—
52C22×C4⋊C4—22no—
53C22×C4⋊C4C2×C412yes—
54C22×C22⋊C4—22no—
55C22×C22⋊C4—21no—
56C22×C42—21no—
57#484—24no—
58#484—24no—
59#484—28no—
60#467—24no—
61#467—24no—
62#467—24no—
63#467—24no—
64#451—24no—
65#451—24no—
66#451—28no—
67C2×#108—24no—
68C2×#107C2×C414yescomplemented
69C2×#103C2×C418yescomplemented
70#407—22no—
71#407—22no—
72#407—22no—
73#407—22no—
74#407D414yes—
75#407C2314yes—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—22no—
78C2×C22⋊C8—22no—
79C2×C22⋊C8C2×C414yes—
80C2×C22⋊C8D414yes—
81C2×C22⋊C8D411yescharacteristic
82C2×C22⋊C8D411yescharacteristic
83C2×C22⋊C8D412yes—
84C2×C22⋊C8C2312yes—
85C2×#97—22no—
86C2×C4×C8—22no—
87C4×C22⋊C4—21no—
88C4×C22⋊C4—21no—
89C4×C22⋊C4D412yes—
90C4×C22⋊C4C2311yescharacteristic
91C4×C22⋊C4C2311yescharacteristic
92C2×#95D412yes—
93#341—24no—
94#336—22no—
95#336—22no—
96#328—28no—
97#328—28no—
98#324—216no—
99C25#4111yescharacteristic
100C22×Q8—22no—
101C22×Q8—22no—
102C22×Q8—22no—
103C22×Q8—22no—
104C22×Q8—22no—
105C22×Q8—84no—
106C22×Q8C22×C414yes—
107C23×C4—22no—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—82no—
111C23×C4C22×C412yes—
112C23×C4C2×D411yescharacteristic
113C23×C4C2×D411yescharacteristic
114C23×C4C2×D412yes—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D412yes—
118C23×C4#4111yescharacteristic
119C23×C4#4112yes—
120C23×C4C2411yescharacteristic
121C2×Q16—24no—
122C2×Q16—24no—
123C2×Q16—84no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—42no—
127C2×M16—82no—
128C2×M16—84no—
129C22×C8—22no—
130C22×C8—42no—
131C22×C8—82no—
132C22×C8C22×C414yes—
133C22×C8C2×D412yes—
134C22×C8C2×D412yes—
135C22×C8C2×D414yes—
136C22⋊Q8—24no—
137C22⋊Q8—24no—
138C22⋊Q8—24no—
139C22⋊Q8—24no—
140C22⋊Q8—24no—
141C22⋊Q8—24no—
142C22⋊Q8—24no—
143C22⋊Q8—24no—
144C22⋊Q8—42no—
145C22⋊Q8—42no—
146C22⋊Q8—42no—
147C22⋊Q8—42no—
148C22⋊Q8—42no—
149C22⋊Q8—42no—
150C22⋊Q8—42no—
151C22⋊Q8—42no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—24no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—24no—
159C2×C4⋊C4—42no—
160C2×C4⋊C4—42no—
161C2×C4⋊C4—42no—
162C2×C4⋊C4—42no—
163C2×C4⋊C4C22×C414yes—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—42no—
171C2×C22⋊C4—42no—
172C2×C22⋊C4—42no—
173C2×C22⋊C4—42no—
174C2×C42—21no—
175C2×C42—21no—
176C2×C42—22no—
177C2×C42—82no—
178#108—84no—
179#107—24no—
180#107—24no—
181#103—28no—
182#103—28no—
183C22⋊C8—24no—
184C22⋊C8—22no—
185C22⋊C8—24no—
186C22⋊C8—24no—
187C22⋊C8—42no—
188C22⋊C8—42no—
189C22⋊C8C22⋊C414yes—
190C22⋊C8C22⋊C414yes—
191C22⋊C8C22⋊C418yes—
192C22⋊C8C2×D414yes—
193#97—82no—
194C4×C8—82no—
195#95—22no—
196#95C2×D412yes—
197#95C2×D412yes—
198C24—24no—
199C24—42no—
200C24—81no—
201C24C4×D412yes—
202C24#12111yescharacteristic
203C24#12111yescharacteristic
204C24#12412yes—
205C24C2×#4111yescharacteristic
206C2×Q8—24no—
207C2×Q8—24no—
208C2×Q8—48no—
209C2×Q8—42no—
210C2×Q8—42no—
211C2×Q8—84no—
212C2×Q8—84no—
213C2×Q8—84no—
214C2×Q8—84no—
215C2×Q8—84no—
216C22×C4—22no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—44no—
225C22×C4—44no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—81no—
232C22×C4—81no—
233C22×C4—82no—
234C22×C4—81no—
235C22×C4—82no—
236C22×C4—84no—
237C22×C4—82no—
238C22×C4—84no—
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C4×D412yes—
242C22×C4C4×D412yes—
243C22×C4C4×D412yes—
244C22×C4#12112yes—
245C22×C4#12112yes—
246C22×C4#12111yescharacteristic
247C22×C4#12111yescharacteristic
248C22×C4#12412yes—
249C22×C4#12712yes—
250C22×C4#12712yes—
251C22×C4C23×C411yescharacteristic
252C22×C4C22×D411yescharacteristic
253C22×C4C22×D411yescharacteristic
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C2×#4111yescharacteristic
257Q16—84no—
258Q16—84no—
259M16—48no—
260M16—84no—
261M16—84no—
262C2×C8—24no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—44no—
266C2×C8—44no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—44no—
270C2×C8—84no—
271C2×C8C2×C22⋊C414yes—
272C2×C8C2×C22⋊C414yes—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C22⋊C4—24no—
282C22⋊C4—44no—
283C22⋊C4—42no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—44no—
287C22⋊C4—44no—
288C42—42no—
289C42—42no—
290C42—82no—
291C23—22no—
292C23—24no—
293C23—24no—
294C23—22no—
295C23—24no—
296C23—22no—
297C23—41no—
298C23—42no—
299C23—41no—
300C23—41no—
301C23—41no—
302C23—41no—
303C23—42no—
304C23—42no—
305C23—88no—
306C23#39012yes—
307C23#39012yes—
308C23#47314yes—
309C23C2×C4×D411yescharacteristic
310C23C2×#12111yescharacteristic
311C23C2×#12411yescharacteristic
312Q8—48no—
313Q8—84no—
314Q8—84no—
315Q8—84no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—24no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—48no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—88no—
344C2×C4—84no—
345C2×C4—84no—
346C2×C4—84no—
347C2×C4—84no—
348C2×C4—84no—
349C2×C4#39012yes—
350C2×C4#39012yes—
351C2×C4C22×C22⋊C411yescharacteristic
352C2×C4C2×C4×D411yescharacteristic
353C2×C4C2×#12111yescharacteristic
354C2×C4C2×#12711yescharacteristic
355C8—24no—
356C8—24no—
357C8—44no—
358C8—84no—
359C22—21no—
360C22—21no—
361C22—22no—
362C22—21no—
363C22—22no—
364C22—21no—
365C22—22no—
366C22—21no—
367C22—22no—
368C22—21no—
369C22—21no—
370C22—22no—
371C22—44no—
372C22—44no—
373C22—48no—
374C22—82no—
375C22—84no—
376C22#197914yes—
377C22C2×#39011yescharacteristic
378C22#295612yes—
379C22C2×#47312yes—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—22no—
384C4—22no—
385C4—44no—
386C4—82no—
387C4—84no—
388C4—84no—
389C4#288312yes—
390C2—22no—
391C2—22no—
392C2—24no—
393C2—42no—
394C2—41no—
395C2#2121311yescharacteristic
396C2#2124111yescharacteristic
397C2C2×#197911yescharacteristic
subgroupG/H
center, socleC22C2×#390
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC25#41
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. #9817C2semidirect
7. #9814C2semidirect
13. C2×#484C4semidirect
15. C2×#451C4semidirect
36. C22×Q16C2×C4semidirect
68. C2×#107C2×C4semidirect
69. 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
#219482
#205252
C2×#15082
#173972
#98172
#98142