Finite Group Databasebeta

← #205420

Subgroups — #205420

Order 512.

970 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2#21839C211yesmaximal, characteristic
3#21833C211yesmaximal, characteristic
4#20537C211yesmaximal, characteristic
5C2×#1572C211yesmaximal, characteristic
6C2×#1556C211yesmaximal, characteristic
7C2×#1554C211yesmaximal, characteristic
8#9745C212yesmaximal, complemented
9#9742C212yesmaximal, complemented
10#9737C212yesmaximal, complemented
11#9733C212yesmaximal, complemented
12C2×#533C2211yescharacteristic
13C22×C4⋊Q8C2211yescharacteristic
14#3209—24no—
15#3209C418yescomplemented
16C2×#494C2211yescharacteristic
17C2×#494C2211yescharacteristic
18C2×#487C2211yescharacteristic
19C2×#487C2211yescharacteristic
20C2×#462—22no—
21C2×#451—22no—
22C2×#432C2211yescharacteristic
23C2×#432C2211yescharacteristic
24C2×#431C2211yescharacteristic
25C2×#431C2211yescharacteristic
26#2285C2211yescharacteristic
27#1992—22no—
28#1990—22no—
29#1988C2212yes—
30#1988C2212yes—
31#1988C2212yes—
32#1987C2212yes—
33#1825—22no—
34#1823—22no—
35#1771—22no—
36#1770C2212yes—
37C8×C4⋊C4C2212yes—
38#1750—22no—
39#1572—24no—
40C2×#335C2212yes—
41C2×#334C2212yes—
42#1556—24no—
43#1554—24no—
44C2×#333C2212yes—
45C2×#332C2212yes—
46C2×#332C2212yes—
47C2×#332C2212yes—
48C23×Q8—21no—
49C22×Q16—22no—
50C22×Q16—22no—
51#533—22no—
52#533—22no—
53#533—22no—
54#533C2×C414yes—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—21no—
57C2×C4⋊Q8—22no—
58C2×C4⋊Q8—21no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8C2×C414yes—
62C2×C4⋊Q8C2311yescharacteristic
63C2×C4⋊Q8C2311yescharacteristic
64C2×C4⋊Q8C2311yescharacteristic
65C2×C4⋊Q8C2311yescharacteristic
66C2×#125—21no—
67C2×C22⋊Q8—21no—
68C2×C22⋊Q8—21no—
69C2×C4×Q8—21no—
70C2×C4×Q8—21no—
71C2×#117—21no—
72C2×#117—21no—
73C22×C4⋊C4—21no—
74C22×C4⋊C4—21no—
75C22×C42C2311yescharacteristic
76#494—22no—
77#494—22no—
78#494C2×C414yes—
79#494C2×C414yes—
80#487—22no—
81#487—22no—
82#487C2×C414yes—
83#487C2×C414yes—
84#462—216no—
85#451—216no—
86#432—22no—
87#432—22no—
88#432—22no—
89#432D414yes—
90#431—22no—
91#431—22no—
92#431C2×C414yes—
93#431D414yes—
94C2×C4⋊C8D412yes—
95C2×C4⋊C8C2312yes—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—22no—
99C2×#103—22no—
100C2×C22⋊C8—21no—
101C2×C22⋊C8—21no—
102C2×C22⋊C8—21no—
103C2×C22⋊C8D412yes—
104C2×#97—21no—
105C2×#97—21no—
106C2×#97—22no—
107C2×C4×C8—21no—
108C2×C4×C8—22no—
109C2×C4×C8C2311yescharacteristic
110C2×C4×C8C2311yescharacteristic
111#384D412yes—
112#382—21no—
113#380—21no—
114#380D412yes—
115C4×C4⋊C4—21no—
116C4×C4⋊C4C2312yes—
117#340—24no—
118#340—24no—
119#340—24no—
120#340—24no—
121#336—22no—
122#336—22no—
123#335—24no—
124#334—24no—
125#333—24no—
126#332—24no—
127#332—24no—
128#332—24no—
129C22×Q8—24no—
130C22×Q8—21no—
131C22×Q8—21no—
132C22×Q8—21no—
133C22×Q8—21no—
134C22×Q8—41no—
135C22×Q8—42no—
136C22×Q8—82no—
137C22×Q8—82no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4#4111yescharacteristic
142C2×Q16—28no—
143C2×Q16—28no—
144C2×Q16—82no—
145C2×Q16—82no—
146C2×Q16—82no—
147C2×Q16—82no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—22no—
151C22×C8C2×D412yes—
152C22×C8C2×D412yes—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—21no—
157C4⋊Q8—21no—
158C4⋊Q8—21no—
159C4⋊Q8—21no—
160C4⋊Q8—21no—
161C4⋊Q8—21no—
162C4⋊Q8—21no—
163C4⋊Q8—21no—
164C4⋊Q8—21no—
165C4⋊Q8—21no—
166C4⋊Q8—21no—
167C4⋊Q8—21no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8C22×C412yes—
172C4⋊Q8C22×C412yes—
173C4⋊Q8C22×C412yes—
174C4⋊Q8C22×C412yes—
175#125—24no—
176#125—42no—
177C22⋊Q8—24no—
178C22⋊Q8—24no—
179C22⋊Q8—42no—
180C22⋊Q8—42no—
181C4×Q8—24no—
182C4×Q8—24no—
183C4×Q8—42no—
184C4×Q8—42no—
185#117—24no—
186#117—42no—
187#117—82no—
188C2×C4⋊C4—24no—
189C2×C4⋊C4—24no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—22no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—41no—
201C2×C4⋊C4—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4—41no—
204C2×C4⋊C4—42no—
205C2×C4⋊C4—42no—
206C2×C4⋊C4C2×D414yes—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—41no—
209C2×C42—22no—
210C2×C42—41no—
211C2×C42C22×C412yes—
212C2×C42C2×D411yescharacteristic
213C2×C42C2×D412yes—
214C2×C42C2×D411yescharacteristic
215C2×C42C2×D412yes—
216C2×C42#4111yescharacteristic
217C2×C42#4112yes—
218C2×C42C2411yescharacteristic
219C4⋊C8—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—22no—
223C4⋊C8—22no—
224C4⋊C8C22⋊C418yes—
225C4⋊C8C2×D414yes—
226#103—28no—
227#103—28no—
228C22⋊C8—24no—
229C22⋊C8—42no—
230C22⋊C8—42no—
231C22⋊C8C22⋊C418yes—
232#97—24no—
233#97—42no—
234#97—82no—
235C4×C8—42no—
236C4×C8—82no—
237C4×C8C22×C412yes—
238C4×C8C22×C412yes—
239C4×C8C2×D412yes—
240C4×C8C2×D412yes—
241#95—22no—
242#95—22no—
243C24#12111yescharacteristic
244C2×Q8—22no—
245C2×Q8—22no—
246C2×Q8—22no—
247C2×Q8—22no—
248C2×Q8—41no—
249C2×Q8—41no—
250C2×Q8—41no—
251C2×Q8—41no—
252C2×Q8—44no—
253C2×Q8—42no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—42no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—88no—
260C2×Q8—82no—
261C2×Q8—82no—
262C2×Q8—88no—
263C2×Q8—82no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—42no—
269C22×C4—41no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—41no—
273C22×C4—44no—
274C22×C4—82no—
275C22×C4—82no—
276C22×C4C2×C22⋊C412yes—
277C22×C4C4×D412yes—
278C22×C4C4×D412yes—
279C22×C4#12112yes—
280C22×C4#12111yescharacteristic
281C22×C4#12111yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12412yes—
284C22×C4#12712yes—
285C22×C4C22×D411yescharacteristic
286C22×C4C22×D411yescharacteristic
287C22×C4C2×#4111yescharacteristic
288Q16—88no—
289Q16—88no—
290C2×C8—22no—
291C2×C8—22no—
292C2×C8—22no—
293C2×C8—22no—
294C2×C8—24no—
295C2×C8—42no—
296C2×C8—42no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—44no—
300C2×C8—44no—
301C2×C8C2×C22⋊C414yes—
302C2×C8C2×C22⋊C414yes—
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—28no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—41no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—41no—
319C4⋊C4—41no—
320C4⋊C4—44no—
321C4⋊C4—44no—
322C4⋊C4—44no—
323C4⋊C4—44no—
324C4⋊C4—82no—
325C4⋊C4—82no—
326C22⋊C4—44no—
327C22⋊C4—82no—
328C42—42no—
329C42—42no—
330C42—44no—
331C42C2×C22⋊C412yes—
332C42C4×D412yes—
333C42C4×D412yes—
334C42#12112yes—
335C42#12412yes—
336C42#12712yes—
337C42C23×C411yescharacteristic
338C42C22×D411yescharacteristic
339C42C22×D411yescharacteristic
340C42C2×#4111yescharacteristic
341C23—21no—
342C23—21no—
343C23—22no—
344C23—22no—
345C23#39012yes—
346C23C2×#12111yescharacteristic
347Q8—44no—
348Q8—44no—
349Q8—44no—
350Q8—82no—
351Q8—82no—
352Q8—88no—
353Q8—82no—
354Q8—82no—
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—22no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—88no—
375C2×C4—81no—
376C2×C4—81no—
377C2×C4—81no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—82no—
382C2×C4#39012yes—
383C2×C4#39012yes—
384C2×C4#39012yes—
385C2×C4C22×C22⋊C411yescharacteristic
386C2×C4C2×C4×D411yescharacteristic
387C2×C4C2×C4×D411yescharacteristic
388C2×C4C2×#12111yescharacteristic
389C2×C4C2×#12411yescharacteristic
390C2×C4C2×#12711yescharacteristic
391C2×C4#57612yes—
392C2×C4#57712yes—
393C2×C4#57814yes—
394C8—24no—
395C8—24no—
396C8—44no—
397C8—82no—
398C8—82no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22—42no—
403C22—42no—
404C22C2×#39011yescharacteristic
405C22#304111yescharacteristic
406C22#304311yescharacteristic
407C22#305912yes—
408C22#306912yes—
409C4—21no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—82no—
415C4—82no—
416C4—82no—
417C4—82no—
418C4—84no—
419C4#289712yes—
420C4#295012yes—
421C4#304512yes—
422C4#315612yes—
423C2—42no—
424C2#1950712yes—
425C2#1952312yes—
426C2#2123211yescharacteristic
427C2#2125011yescharacteristic
428C2#2126311yescharacteristic
subgroupG/H
center, socleC23C2×#121
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#121
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
8. #9745C2semidirect
9. #9742C2semidirect
10. #9737C2semidirect
11. #9733C2semidirect
15. #3209C4semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
#218392
#218332
#205372
C2×#15722
C2×#15562
C2×#15542
#97452
#97422
#97372
#97332