Finite Group Databasebeta

← #235782

Subgroups — #235782

Order 512.

1366 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3230C211yesmaximal, characteristic
2C22×#487C211yesmaximal, characteristic
3#35281C211yesmaximal, characteristic
4#21313C211yesmaximal, characteristic
5C2×#1549C211yesmaximal, characteristic
6C2×#1493C211yesmaximal, characteristic
7#10059C218yesmaximal, complemented
8#7890C211yesmaximal, characteristic
9C2×#533C2211yescharacteristic
10C22×C4⋊Q8C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3230—24no—
13#3230—24no—
14C2×#500—21no—
15C2×#487—24no—
16C2×#487—21no—
17C2×#487—24no—
18C2×#487—22no—
19C2×#487C2214yes—
20C2×#484—22no—
21C2×#477C2214yes—
22C2×#431C2211yescharacteristic
23C2×#423C2211yescharacteristic
24C22×#103—22no—
25C22×C4×C8C2211yescharacteristic
26#2868—21no—
27#2862—21no—
28#2660—22no—
29#2567—22no—
30#2546C2214yes—
31#2502—22no—
32#2434—22no—
33#2317—22no—
34C4×#117C2211yescharacteristic
35#1988—22no—
36#1987C2214yes—
37#1736—22no—
38#1549—24no—
39C2×#332C2214yes—
40#1493—24no—
41C2×Q8⋊C8C2214yes—
42#1279C2214yes—
43C23×Q8—21no—
44C23×Q8—21no—
45C22×M16—21no—
46C23×C8—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533—22no—
51C2×C4⋊Q8—24no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—21no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8—22no—
56C2×C4⋊Q8—22no—
57C2×C4⋊Q8—21no—
58C2×C4⋊Q8—22no—
59C2×C4⋊Q8—21no—
60C2×C4⋊Q8—21no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—22no—
63C2×C4⋊Q8—22no—
64C2×C4⋊Q8D411yescharacteristic
65C2×C4⋊Q8D414yes—
66C2×C4⋊Q8D411yescharacteristic
67C2×C4⋊Q8D412yes—
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2312yes—
70C2×#125—21no—
71C2×C22⋊Q8—21no—
72C2×C22⋊Q8D412yes—
73C2×C4×Q8—21no—
74C2×C4×Q8C2312yes—
75C2×#117—21no—
76C2×#117D412yes—
77C22×C4⋊C4—21no—
78C22×C4⋊C4—21no—
79C22×C4⋊C4—21no—
80C22×C4⋊C4D412yes—
81C22×C42C2311yescharacteristic
82#500—28no—
83#487—216no—
84#487—28no—
85#487—28no—
86#487—28no—
87#484—216no—
88#477—28no—
89#477—28no—
90#431—22no—
91#431D414yes—
92C2×#107—24no—
93#423—22no—
94#423—22no—
95C2×C4⋊C8—21no—
96C2×C4⋊C8C2312yes—
97C2×C4⋊C8C2312yes—
98C2×#103—216no—
99C2×#103—24no—
100C2×#103—28no—
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#332—28no—
124Q8⋊C8—28no—
125C22×Q8—21no—
126C22×Q8—22no—
127C22×Q8—22no—
128C22×Q8—24no—
129C22×Q8—22no—
130C22×Q8—48no—
131C22×Q8—41no—
132C22×Q8—42no—
133C22×Q8—42no—
134C22×Q8—42no—
135C22×Q8—84no—
136C22×Q8C2×D412yes—
137C23×C4—22no—
138C23×C4—41no—
139C23×C4C2×D411yescharacteristic
140C23×C4C2×D411yescharacteristic
141C23×C4C2×D411yescharacteristic
142C2×M16—42no—
143C2×M16—44no—
144C22×C8—24no—
145C22×C8—22no—
146C22×C8—22no—
147C22×C8—22no—
148C22×C8—44no—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—22no—
152C4⋊Q8—24no—
153C4⋊Q8—22no—
154C4⋊Q8—24no—
155C4⋊Q8—24no—
156C4⋊Q8—22no—
157C4⋊Q8—22no—
158C4⋊Q8—22no—
159C4⋊Q8—22no—
160C4⋊Q8—22no—
161C4⋊Q8—22no—
162C4⋊Q8—22no—
163C4⋊Q8—22no—
164C4⋊Q8—22no—
165C4⋊Q8—42no—
166C4⋊Q8—42no—
167C4⋊Q8—42no—
168C4⋊Q8—42no—
169C4⋊Q8—42no—
170C4⋊Q8—42no—
171C4⋊Q8—42no—
172C4⋊Q8—88no—
173C4⋊Q8C2×D414yes—
174#125—42no—
175#125—42no—
176C22⋊Q8—24no—
177C22⋊Q8—24no—
178C22⋊Q8—42no—
179C22⋊Q8—42no—
180C4×Q8—22no—
181C4×Q8—22no—
182C4×Q8—22no—
183C4×Q8—22no—
184C4×Q8—42no—
185C4×Q8—42no—
186#117—24no—
187#117—42no—
188#117—42no—
189#117Q1618yes—
190C2×C4⋊C4—24no—
191C2×C4⋊C4—24no—
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—216no—
205C2×C4⋊C4—41no—
206C2×C4⋊C4—41no—
207C2×C4⋊C4—42no—
208C2×C4⋊C4—42no—
209C2×C4⋊C4—42no—
210C2×C4⋊C4—42no—
211C2×C4⋊C4C2×D414yes—
212C2×C4⋊C4C2×D412yes—
213C2×C4⋊C4C2×D414yes—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C42—22no—
217C2×C42—22no—
218C2×C42—22no—
219C2×C42—22no—
220C2×C42C2×D411yescharacteristic
221C2×C42C2×D411yescharacteristic
222C2×C42C2×D412yes—
223C2×C42C2×D412yes—
224C2×C42C2×D412yes—
225C2×C42C2×D411yescharacteristic
226C2×C42#4112yes—
227C2×C42C2411yescharacteristic
228#107—216no—
229C4⋊C8—22no—
230C4⋊C8—22no—
231C4⋊C8—22no—
232C4⋊C8—42no—
233C4⋊C8C2×D414yes—
234#103—232no—
235#103—216no—
236C22⋊C8—24no—
237C4×C8—24no—
238C4×C8—24no—
239C4×C8—22no—
240C4×C8C2×D414yes—
241#95—216no—
242#95—22no—
243#95—22no—
244C24#12011yescharacteristic
245C2×Q8—22no—
246C2×Q8—24no—
247C2×Q8—22no—
248C2×Q8—28no—
249C2×Q8—48no—
250C2×Q8—416no—
251C2×Q8—42no—
252C2×Q8—42no—
253C2×Q8—42no—
254C2×Q8—48no—
255C2×Q8—42no—
256C2×Q8—44no—
257C2×Q8—44no—
258C2×Q8—44no—
259C2×Q8—84no—
260C2×Q8—88no—
261C2×Q8—88no—
262C2×Q8—88no—
263C2×Q8—84no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—22no—
268C22×C4—24no—
269C22×C4—22no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—84no—
274C22×C4—84no—
275C22×C4#12012yes—
276C22×C4#12012yes—
277C22×C4#12011yescharacteristic
278C22×C4#12011yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12112yes—
281C22×C4#12112yes—
282C22×C4#12112yes—
283C22×C4#12712yes—
284C22×C4C2×Q1614yes—
285C22×C4#13714yes—
286C22×C4C22×D411yescharacteristic
287C22×C4C22×D411yescharacteristic
288C22×C4C22×D411yescharacteristic
289M16—84no—
290C2×C8—24no—
291C2×C8—24no—
292C2×C8—44no—
293C2×C8—48no—
294C2×C8—44no—
295C2×C8—42no—
296C2×C8—44no—
297C4⋊C4—24no—
298C4⋊C4—24no—
299C4⋊C4—24no—
300C4⋊C4—22no—
301C4⋊C4—24no—
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—44no—
311C4⋊C4—44no—
312C4⋊C4—44no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—44no—
317C4⋊C4—42no—
318C4⋊C4—816no—
319C4⋊C4C2×Q1618yes—
320C22⋊C4—28no—
321C22⋊C4—44no—
322C42—28no—
323C42—42no—
324C42—42no—
325C42—44no—
326C42—44no—
327C42#12012yes—
328C42#12012yes—
329C42#12112yes—
330C42#12112yes—
331C42#12112yes—
332C42#12712yes—
333C42C22×D411yescharacteristic
334C42C22×D411yescharacteristic
335C42C22×D411yescharacteristic
336C42C2×#4111yescharacteristic
337C23—22no—
338C23—22no—
339C23#39212yes—
340C23#45114yes—
341C23C2×#12011yescharacteristic
342Q8—48no—
343Q8—44no—
344Q8—44no—
345Q8—84no—
346Q8—88no—
347Q8—84no—
348Q8—84no—
349Q8—88no—
350Q8—84no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—44no—
364C2×C4—44no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—88no—
368C2×C4—88no—
369C2×C4—84no—
370C2×C4#39212yes—
371C2×C4#39212yes—
372C2×C4#39212yes—
373C2×C4#45114yes—
374C2×C4#45414yes—
375C2×C4#46218yes—
376C2×C4#49414yes—
377C2×C4#49714yes—
378C2×C4C2×#12011yescharacteristic
379C2×C4C2×#12011yescharacteristic
380C2×C4C2×#12111yescharacteristic
381C2×C4C2×#12111yescharacteristic
382C2×C4C2×#12111yescharacteristic
383C2×C4C2×#12711yescharacteristic
384C2×C4C22×Q1612yes—
385C2×C4C2×#13712yes—
386C2×C4#57714yes—
387C8—44no—
388C8—44no—
389C8—82no—
390C22—22no—
391C22—22no—
392C22—28no—
393C22#201614yes—
394C22C2×#39211yescharacteristic
395C22C2×#45112yes—
396C22#300912yes—
397C22C2×#45412yes—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—48no—
402C4—84no—
403C4—82no—
404C4—82no—
405C4—84no—
406C4#200314yes—
407C4#201614yes—
408C4#301012yes—
409C4C2×#46212yes—
410C4#305712yes—
411C4C2×#49411yescharacteristic
412C4C2×#49711yescharacteristic
413C2—24no—
414C2#1864314yes—
415C2#2338811yescharacteristic
416C2C2×#200311yescharacteristic
417C2C2×#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. #10059C2semidirect

Maximal subgroups

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

subgroupindex
C2×#32302
C22×#4872
#352812
#213132
C2×#15492
C2×#14932
#100592
#78902