Finite Group Databasebeta

← #235774

Subgroups — #235774

Order 512.

1398 conjugacy classes of subgroups.

The subgroup lattice has 421 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3209C211yesmaximal, characteristic
2C22×#487C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic
7#10065C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3209—24no—
13#3209—24no—
14C2×#494—21no—
15C2×#487—24no—
16C2×#487—21no—
17C2×#487—22no—
18C2×#487—24no—
19C2×#487C2214yes—
20C2×#462C2214yes—
21C2×#451—22no—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#103—22no—
25C22×C4×C8C2211yescharacteristic
26#2868—21no—
27#2862—21no—
28#2660—22no—
29#2567C2214yes—
30#2546—22no—
31#2502—22no—
32#2434—22no—
33#2317—22no—
34C4×#117C2211yescharacteristic
35#1988C2214yes—
36#1987—22no—
37#1736—22no—
38#1548—24no—
39C2×#333C2214yes—
40#1493—24no—
41C2×Q8⋊C8C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×Q8D412yes—
45C22×Q16—24no—
46C22×M16—21no—
47C23×C8—21no—
48#533—22no—
49#533—22no—
50#533—22no—
51#533—22no—
52C2×C4⋊Q8—24no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—21no—
55C2×C4⋊Q8—21no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—22no—
60C2×C4⋊Q8—22no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8—22no—
65C2×C4⋊Q8D411yescharacteristic
66C2×C4⋊Q8D412yes—
67C2×C4⋊Q8D414yes—
68C2×C4⋊Q8D411yescharacteristic
69C2×C4⋊Q8C2312yes—
70C2×C4⋊Q8C2312yes—
71C2×#125—21no—
72C2×C22⋊Q8—21no—
73C2×C22⋊Q8D412yes—
74C2×C4×Q8—21no—
75C2×C4×Q8C2312yes—
76C2×#117—21no—
77C2×#117D412yes—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4—21no—
81C22×C4⋊C4—21no—
82C22×C42C2311yescharacteristic
83#494—28no—
84#487—216no—
85#487—28no—
86#487—28no—
87#487—28no—
88#462—28no—
89#462—28no—
90#451—216no—
91#431—22no—
92#431D414yes—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—216no—
99C2×#103—28no—
100C2×#103—24no—
101C2×C22⋊C8—21no—
102C2×C4×C8—21no—
103C2×C4×C8—22no—
104C2×C4×C8—21no—
105C2×C4×C8D414yes—
106C2×C4×C8C2312yes—
107#398—28no—
108#395—28no—
109#391—24no—
110#391—24no—
111#389—28no—
112#385—28no—
113#384—24no—
114#384—24no—
115#382—24no—
116C4×C4⋊C4D412yes—
117C4×C4⋊C4C2312yes—
118C4×C22⋊C4—21no—
119C4×C22⋊C4—21no—
120#376—21no—
121C43—21no—
122#340—28no—
123#333—28no—
124Q8⋊C8—28no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—24no—
128C22×Q8—24no—
129C22×Q8—24no—
130C22×Q8—22no—
131C22×Q8—48no—
132C22×Q8—41no—
133C22×Q8—42no—
134C22×Q8—84no—
135C22×Q8C2×D412yes—
136C22×Q8C2×D414yes—
137C23×C4—22no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×Q16—216no—
143C2×Q16—88no—
144C2×M16—42no—
145C2×M16—44no—
146C22×C8—24no—
147C22×C8—22no—
148C22×C8—22no—
149C22×C8—22no—
150C22×C8—44no—
151C4⋊Q8—24no—
152C4⋊Q8—22no—
153C4⋊Q8—24no—
154C4⋊Q8—24no—
155C4⋊Q8—22no—
156C4⋊Q8—24no—
157C4⋊Q8—22no—
158C4⋊Q8—24no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—22no—
166C4⋊Q8—22no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—42no—
173C4⋊Q8—42no—
174C4⋊Q8—88no—
175C4⋊Q8C2×D414yes—
176#125—42no—
177#125—42no—
178C22⋊Q8—24no—
179C22⋊Q8—24no—
180C22⋊Q8—42no—
181C22⋊Q8—42no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—22no—
185C4×Q8—22no—
186C4×Q8—42no—
187C4×Q8—42no—
188#117—24no—
189#117—42no—
190#117—42no—
191#117Q1618yescomplemented
192C2×C4⋊C4—24no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—22no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—22no—
205C2×C4⋊C4—216no—
206C2×C4⋊C4—42no—
207C2×C4⋊C4—41no—
208C2×C4⋊C4—41no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4—42no—
212C2×C4⋊C4—42no—
213C2×C4⋊C4—42no—
214C2×C4⋊C4C2×D412yes—
215C2×C4⋊C4C2×D414yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42—22no—
222C2×C42C2×D411yescharacteristic
223C2×C42C2×D411yescharacteristic
224C2×C42C2×D412yes—
225C2×C42C2×D412yes—
226C2×C42C2×D412yes—
227C2×C42C2×D411yescharacteristic
228C2×C42#4112yes—
229C2×C42C2411yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—22no—
233C4⋊C8—42no—
234C4⋊C8C2×D414yes—
235#103—232no—
236#103—216no—
237C22⋊C8—24no—
238C4×C8—24no—
239C4×C8—24no—
240C4×C8—22no—
241C4×C8C2×D414yes—
242#95—216no—
243#95—22no—
244#95—22no—
245C24#12011yescharacteristic
246C2×Q8—24no—
247C2×Q8—24no—
248C2×Q8—22no—
249C2×Q8—22no—
250C2×Q8—24no—
251C2×Q8—28no—
252C2×Q8—48no—
253C2×Q8—416no—
254C2×Q8—42no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—48no—
260C2×Q8—48no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—84no—
264C2×Q8—88no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—24no—
270C22×C4—22no—
271C22×C4—42no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—84no—
275C22×C4—84no—
276C22×C4#12012yes—
277C22×C4#12012yes—
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12112yes—
284C22×C4#12712yes—
285C22×C4C2×Q1614yes—
286C22×C4#13714yes—
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289C22×C4C22×D411yescharacteristic
290Q16—816no—
291M16—84no—
292C2×C8—24no—
293C2×C8—24no—
294C2×C8—44no—
295C2×C8—48no—
296C2×C8—44no—
297C2×C8—42no—
298C2×C8—44no—
299C4⋊C4—24no—
300C4⋊C4—24no—
301C4⋊C4—22no—
302C4⋊C4—24no—
303C4⋊C4—22no—
304C4⋊C4—42no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4—42no—
308C4⋊C4—42no—
309C4⋊C4—42no—
310C4⋊C4—42no—
311C4⋊C4—42no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—44no—
315C4⋊C4—44no—
316C4⋊C4—42no—
317C4⋊C4—42no—
318C4⋊C4—44no—
319C4⋊C4—42no—
320C4⋊C4—816no—
321C4⋊C4C2×Q1618yes—
322C22⋊C4—28no—
323C22⋊C4—44no—
324C42—28no—
325C42—42no—
326C42—42no—
327C42—44no—
328C42—44no—
329C42#12012yes—
330C42#12012yes—
331C42#12112yes—
332C42#12112yes—
333C42#12112yes—
334C42#12712yes—
335C42C22×D411yescharacteristic
336C42C22×D411yescharacteristic
337C42C22×D411yescharacteristic
338C42C2×#4111yescharacteristic
339C23—22no—
340C23—22no—
341C23#39212yes—
342C23#45114yes—
343C23C2×#12011yescharacteristic
344Q8—48no—
345Q8—44no—
346Q8—48no—
347Q8—44no—
348Q8—84no—
349Q8—88no—
350Q8—84no—
351Q8—88no—
352Q8—84no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—28no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—44no—
367C2×C4—44no—
368C2×C4—48no—
369C2×C4—88no—
370C2×C4—88no—
371C2×C4—84no—
372C2×C4#39212yes—
373C2×C4#39212yes—
374C2×C4#39212yes—
375C2×C4#45114yes—
376C2×C4#45514yes—
377C2×C4#46218yes—
378C2×C4#49414yes—
379C2×C4#49714yes—
380C2×C4C2×#12011yescharacteristic
381C2×C4C2×#12011yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12111yescharacteristic
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12711yescharacteristic
386C2×C4C22×Q1612yes—
387C2×C4C2×#13712yes—
388C2×C4#57714yes—
389C8—44no—
390C8—44no—
391C8—82no—
392C22—22no—
393C22—22no—
394C22—28no—
395C22#201614yes—
396C22C2×#39211yescharacteristic
397C22C2×#45112yes—
398C22#301012yes—
399C22C2×#45512yes—
400C4—22no—
401C4—21no—
402C4—21no—
403C4—48no—
404C4—82no—
405C4—84no—
406C4—84no—
407C4—82no—
408C4#200414yes—
409C4#201614yes—
410C4#300912yes—
411C4C2×#46212yes—
412C4#305712yes—
413C4C2×#49411yescharacteristic
414C4C2×#49711yescharacteristic
415C2—24no—
416C2#1863314yes—
417C2#2338811yescharacteristic
418C2C2×#200411yescharacteristic
419C2C2×#201611yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC24#120
LCSG ▷ C2×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
7. #10065C2semidirect
191. #117Q16semidirect

Maximal subgroups

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

subgroupindex
C2×#32092
C22×#4872
#352812
#213132
C2×#15482
C2×#14932
#100652
#78902