Finite Group Databasebeta

← #195660

Subgroups — #195660

Order 512.

1198 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#3139C211yesmaximal, characteristic
2#34216C211yesmaximal, characteristic
3#21894C211yesmaximal, characteristic
4#21389C211yesmaximal, characteristic
5C2×#1548C211yesmaximal, characteristic
6#9060C214yesmaximal, complemented
7#9060C214yesmaximal, complemented
8#7903C211yesmaximal, characteristic
9#7896C211yesmaximal, characteristic
10C2×#533C2211yescharacteristic
11C22×C4⋊Q8C2211yescharacteristic
12#3139—24no—
13#3139C418yescomplemented
14C2×#491—22no—
15C2×#487C2212yes—
16C2×#487C2212yes—
17C2×#431C2211yescharacteristic
18C2×#423C2211yescharacteristic
19C2×#419—22no—
20C2×C4×M16C2211yescharacteristic
21#2369C2214yes—
22#2365—22no—
23#2345—22no—
24Q8×C22⋊C4—22no—
25#2331—22no—
26#2316—21no—
27C4×C4⋊Q8—22no—
28#2285C2211yescharacteristic
29#2274—21no—
30C4×#117C2211yescharacteristic
31#1988C2212yes—
32#1988C2212yes—
33#1981C2212yes—
34#1981C2212yes—
35#1894—22no—
36#1873—21no—
37#1873—21no—
38#1803—22no—
39C4×#103—22no—
40#1548—24no—
41C2×#333C2214yes—
42#1279C2214yes—
43#1279C2214yes—
44C23×Q8D412yes—
45C22×M16—21no—
46C22×M16—21no—
47#533—22no—
48#533—22no—
49#533—22no—
50#533C2×C414yes—
51C2×C4⋊Q8—21no—
52C2×C4⋊Q8—21no—
53C2×C4⋊Q8—22no—
54C2×C4⋊Q8—22no—
55C2×C4⋊Q8C2×C414yes—
56C2×C4⋊Q8D412yes—
57C2×C4⋊Q8D414yes—
58C2×C4⋊Q8C2311yescharacteristic
59C2×C4⋊Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×#125—21no—
62C2×C22⋊Q8—21no—
63C2×C22⋊Q8—21no—
64C2×C4×Q8—24no—
65C2×C4×Q8—21no—
66C2×C4×Q8—21no—
67C2×#117—21no—
68C2×#117—21no—
69C2×#117D411yescharacteristic
70C2×#117D411yescharacteristic
71C22×C4⋊C4—21no—
72C22×C4⋊C4—21no—
73C22×C42C2311yescharacteristic
74#491—216no—
75#487—24no—
76#487—24no—
77#487C2×C418yes—
78#487C2×C418yes—
79#431—22no—
80#431—22no—
81#423—22no—
82#423—22no—
83C2×C4⋊C8—21no—
84C2×C4⋊C8C2312yes—
85C2×C4⋊C8C2312yes—
86#419—216no—
87C2×#103—24no—
88C2×#103—24no—
89C2×#103—22no—
90C2×#103—22no—
91C2×#103—24no—
92C2×C22⋊C8—21no—
93C4×M16—22no—
94C4×M16C2×C414yes—
95C2×#97—21no—
96C2×C4×C8—21no—
97C2×C4×C8C2312yes—
98#391—24no—
99#391—28no—
100#389—24no—
101#389—24no—
102#389—24no—
103#387—28no—
104#384D412yes—
105#382—21no—
106#380—24no—
107#380—21no—
108#380D412yes—
109C4×C4⋊C4—24no—
110C4×C4⋊C4—24no—
111C4×C4⋊C4—24no—
112C4×C4⋊C4—21no—
113C4×C4⋊C4D412yes—
114C4×C4⋊C4C2312yes—
115C4×C4⋊C4C2312yes—
116C4×C22⋊C4—24no—
117C4×C22⋊C4—21no—
118C4×C22⋊C4—21no—
119#376—21no—
120C43—21no—
121#340—24no—
122#340—22no—
123#340—22no—
124#333—28no—
125C22×Q8—22no—
126C22×Q8—24no—
127C22×Q8—21no—
128C22×Q8—21no—
129C22×Q8—48no—
130C22×Q8C22⋊C418yes—
131C22×Q8C2×D414yes—
132C23×C4—22no—
133C23×C4C2×D411yescharacteristic
134C23×C4C2×D411yescharacteristic
135C23×C4C2×D411yescharacteristic
136C2×M16—24no—
137C2×M16—44no—
138C2×M16—42no—
139C2×M16—82no—
140C22×C8—22no—
141C22×C8—22no—
142C22×C8—22no—
143C4⋊Q8—24no—
144C4⋊Q8—24no—
145C4⋊Q8—24no—
146C4⋊Q8—24no—
147C4⋊Q8—24no—
148C4⋊Q8—21no—
149C4⋊Q8—21no—
150C4⋊Q8—22no—
151C4⋊Q8—22no—
152C4⋊Q8—21no—
153C4⋊Q8—21no—
154C4⋊Q8—21no—
155C4⋊Q8—21no—
156C4⋊Q8C22⋊C418yes—
157C4⋊Q8C22×C412yes—
158C4⋊Q8C22×C412yes—
159C4⋊Q8C22×C414yes—
160C4⋊Q8C2×D414yes—
161#125—24no—
162#125—42no—
163C22⋊Q8—24no—
164C22⋊Q8—24no—
165C22⋊Q8—42no—
166C22⋊Q8—42no—
167C4×Q8—24no—
168C4×Q8—24no—
169C4×Q8—42no—
170C4×Q8—42no—
171C4×Q8—88no—
172#117—24no—
173#117—24no—
174#117—42no—
175#117—44no—
176#117—42no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—22no—
190C2×C4⋊C4—22no—
191C2×C4⋊C4—28no—
192C2×C4⋊C4—41no—
193C2×C4⋊C4—41no—
194C2×C4⋊C4C2×D412yes—
195C2×C4⋊C4C2×D414yes—
196C2×C4⋊C4C2×D412yes—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—22no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—41no—
201C2×C42—21no—
202C2×C42—21no—
203C2×C42—22no—
204C2×C42—22no—
205C2×C42—22no—
206C2×C42—28no—
207C2×C42—41no—
208C2×C42C22×C412yes—
209C2×C42C2×D411yescharacteristic
210C2×C42C2×D412yes—
211C2×C42C2×D411yescharacteristic
212C2×C42C2×D411yescharacteristic
213C2×C42#4112yes—
214C2×C42#4112yes—
215C2×C42C2411yescharacteristic
216C4⋊C8—22no—
217C4⋊C8—22no—
218C4⋊C8—22no—
219C4⋊C8—22no—
220C4⋊C8—42no—
221#103—216no—
222#103—216no—
223#103—44no—
224#103—44no—
225#103—84no—
226C22⋊C8—42no—
227#97—24no—
228C4×C8—22no—
229C4×C8—42no—
230C4×C8C22×C414yes—
231#95—28no—
232#95—28no—
233#95—22no—
234#95—22no—
235#95—22no—
236#95—22no—
237C24#12011yescharacteristic
238C2×Q8—24no—
239C2×Q8—28no—
240C2×Q8—22no—
241C2×Q8—22no—
242C2×Q8—48no—
243C2×Q8—416no—
244C2×Q8—41no—
245C2×Q8—41no—
246C2×Q8—48no—
247C2×Q8—42no—
248C2×Q8—42no—
249C2×Q8—42no—
250C2×Q8—42no—
251C2×Q8—88no—
252C2×Q8C2×C22⋊C418yes—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—24no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—22no—
262C22×C4—24no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—84no—
266C22×C4C2×C22⋊C412yes—
267C22×C4C4×D412yes—
268C22×C4C4×D412yes—
269C22×C4#12011yescharacteristic
270C22×C4#12011yescharacteristic
271C22×C4#12011yescharacteristic
272C22×C4#12112yes—
273C22×C4#12112yes—
274C22×C4#12312yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D411yescharacteristic
277C22×C4C22×D411yescharacteristic
278M16—48no—
279M16—84no—
280C2×C8—24no—
281C2×C8—44no—
282C2×C8—42no—
283C2×C8—42no—
284C2×C8—42no—
285C2×C8—42no—
286C2×C8—44no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—24no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—28no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4—24no—
301C4⋊C4—24no—
302C4⋊C4—44no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—44no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—88no—
311C4⋊C4#13714yes—
312C22⋊C4—24no—
313C22⋊C4—42no—
314C22⋊C4—44no—
315C22⋊C4—44no—
316C42—24no—
317C42—42no—
318C42—44no—
319C42—42no—
320C42—42no—
321C42—44no—
322C42—44no—
323C42—88no—
324C42C2×C22⋊C412yes—
325C42C4×D412yes—
326C42C4×D412yes—
327C42#12112yes—
328C42#12112yes—
329C42#12312yes—
330C42C23×C411yescharacteristic
331C42C22×D411yescharacteristic
332C42C2×#4111yescharacteristic
333C42C2×#4111yescharacteristic
334C23—21no—
335C23—21no—
336C23—22no—
337C23—22no—
338C23#38612yes—
339C23C2×#12011yescharacteristic
340Q8—48no—
341Q8—416no—
342Q8—44no—
343Q8—44no—
344Q8—84no—
345Q8—82no—
346Q8—81no—
347Q8—81no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—24no—
353C2×C4—24no—
354C2×C4—24no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—24no—
360C2×C4—28no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—48no—
364C2×C4—42no—
365C2×C4—44no—
366C2×C4—42no—
367C2×C4—44no—
368C2×C4—42no—
369C2×C4—44no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—88no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—82no—
376C2×C4#38612yes—
377C2×C4#38612yes—
378C2×C4#38612yes—
379C2×C4#45514yes—
380C2×C4C22×C22⋊C411yescharacteristic
381C2×C4C2×C4×D411yescharacteristic
382C2×C4C2×C4×D411yescharacteristic
383C2×C4C2×#12111yescharacteristic
384C2×C4C2×#12111yescharacteristic
385C2×C4C2×#12311yescharacteristic
386C2×C4C2×#13712yes—
387C2×C4#57512yes—
388C2×C4#57712yes—
389C2×C4#57812yes—
390C8—44no—
391C8—82no—
392C8—82no—
393C22—22no—
394C22—22no—
395C22—22no—
396C22—42no—
397C22—42no—
398C22C2×#38611yescharacteristic
399C22#300411yescharacteristic
400C22#300511yescharacteristic
401C22#301312yes—
402C22C2×#45512yes—
403C4—22no—
404C4—22no—
405C4—21no—
406C4—21no—
407C4—44no—
408C4—48no—
409C4—82no—
410C4—84no—
411C4—82no—
412C4—82no—
413C4#188514yes—
414C4C4×#13712yes—
415C4#294712yes—
416C4#303712yes—
417C4#304311yescharacteristic
418C4#304411yescharacteristic
419C2—42no—
420C2#1857512yes—
421C2#1857712yes—
422C2#2092611yescharacteristic
423C2#2093011yescharacteristic
424C2C2×#188511yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC42C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24#120
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
6. #9060C2semidirect
7. #9060C2semidirect
13. #3139C4semidirect

Maximal subgroups

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

subgroupindex
C2×#31392
#342162
#218942
#213892
C2×#15482
#90602
#79032
#78962