Finite Group Databasebeta

← #194868

Subgroups — #194868

Order 512.

1488 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2563C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#21811C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9016C214yesmaximal, complemented
6#9013C214yesmaximal, complemented
7#8021C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×C4×D4C2211yescharacteristic
10C2×#432C2212yes—
11#2563—24no—
12#2563C418yescomplemented
13C2×#393C2212yes—
14C2×#393C2212yes—
15#2318—24no—
16#2315C2212yes—
17#2313—21no—
18#2312C2212yes—
19#2308—21no—
20C2×#390—21no—
21#2290—24no—
22#2283C2212yes—
23C2×#382—21no—
24#2280—21no—
25C2×#380—21no—
26C2×#380—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1981—24no—
30#1960—24no—
31#1787C2218yes—
32#1466—24no—
33C2×#323C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8C2312yes—
37C23×D4D411yescharacteristic
38C24×C4D411yescharacteristic
39C24×C4D411yescharacteristic
40C2×C4⋊Q8—22no—
41C2×C4⋊Q8—22no—
42C2×C4⋊Q8—22no—
43C2×C4⋊Q8—24no—
44C2×C4⋊Q8C2×C414yescomplemented
45C2×C4×D4—22no—
46C2×C4×D4—22no—
47C2×C4×D4—24no—
48C2×C4×D4C2×C414yes—
49C2×C4×D4D414yes—
50C2×#117—22no—
51C22×C4⋊C4—21no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4D411yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—22no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4C2311yescharacteristic
59C22×C22⋊C4C2311yescharacteristic
60C22×C42—21no—
61C22×C42—22no—
62C22×C42C2311yescharacteristic
63#432—24no—
64#432—24no—
65C2×C4⋊C8—22no—
66C2×#103—216no—
67C2×C22⋊C8C2314yes—
68C2×#97—22no—
69#393—24no—
70#393—24no—
71#393C2×C418yes—
72#393C2×C418yescomplemented
73#390—28no—
74#388—28no—
75#387—28no—
76#386—28no—
77#385—28no—
78#383—22no—
79#382—28no—
80#382D414yes—
81#381—22no—
82#381—22no—
83#380—22no—
84#380—28no—
85#380—28no—
86#380—22no—
87#380C2314yes—
88#379—22no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4C2×C414yescomplemented
91C4×C4⋊C4D414yes—
92C4×C22⋊C4—22no—
93C4×C22⋊C4—22no—
94C4×C22⋊C4—22no—
95C4×C22⋊C4—22no—
96C4×C22⋊C4—22no—
97#376—22no—
98C2×#95—22no—
99C2×#95—22no—
100C2×#95—22no—
101C2×#95C2312yes—
102#323—24no—
103#323—24no—
104C25C2×D411yescharacteristic
105C25C2×D411yescharacteristic
106C22×Q8—22no—
107C22×Q8—24no—
108C22×Q8—24no—
109C22×Q8—84no—
110C22×Q8C22×C414yes—
111C22×D4—24no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4—42no—
115C22×D4C22⋊C414yes—
116C23×C4—21no—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—24no—
125C23×C4—44no—
126C23×C4—44no—
127C23×C4C22⋊C414yes—
128C23×C4C22⋊C414yes—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2411yescharacteristic
134C22×C8—24no—
135C4⋊Q8—28no—
136C4⋊Q8—28no—
137C4⋊Q8—44no—
138C4⋊Q8—44no—
139C4×D4—28no—
140C4×D4—28no—
141C4×D4—28no—
142C4×D4—28no—
143#117—84no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—44no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—42no—
155C2×C4⋊C4—42no—
156C2×C4⋊C4C22⋊C414yes—
157C2×C4⋊C4C2×D418yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—84no—
171C2×C22⋊C4—84no—
172C2×C22⋊C4C22×C412yes—
173C2×C22⋊C4C22×C412yes—
174C2×C22⋊C4C2×D412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—24no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—44no—
180C2×C42—84no—
181C2×C42C22×C412yes—
182C2×C42C2×D412yes—
183C2×C42C2×D412yes—
184C2×C42C2×D412yes—
185C2×C42#4114yes—
186C4⋊C8—44no—
187#103—816no—
188C22⋊C8—24no—
189C22⋊C8—24no—
190#97—44no—
191#95—24no—
192#95—24no—
193#95—28no—
194#95—28no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—24no—
200#95C22×C414yes—
201C24—21no—
202C24—21no—
203C24—22no—
204C24—21no—
205C24—22no—
206C24—21no—
207C24—22no—
208C24—42no—
209C24—42no—
210C24—42no—
211C24—42no—
212C24C2×C22⋊C412yes—
213C24C2×C22⋊C412yes—
214C24#13714yes—
215C24C22×D411yescharacteristic
216C2×Q8—28no—
217C2×Q8—416no—
218C2×Q8—44no—
219C2×Q8—84no—
220C2×Q8—84no—
221C2×Q8—88no—
222C2×D4—28no—
223C2×D4—28no—
224C2×D4—44no—
225C2×D4—48no—
226C2×D4—48no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—28no—
241C22×C4—42no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—44no—
245C22×C4—44no—
246C22×C4—42no—
247C22×C4—42no—
248C22×C4—48no—
249C22×C4—44no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—44no—
253C22×C4—42no—
254C22×C4—42no—
255C22×C4—44no—
256C22×C4—84no—
257C22×C4—84no—
258C22×C4—82no—
259C22×C4—82no—
260C22×C4—84no—
261C22×C4—84no—
262C22×C4C2×C22⋊C412yes—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4#12012yes—
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12312yes—
270C22×C4#12312yes—
271C22×C4C23×C411yescharacteristic
272C22×C4C22×D412yes—
273C22×C4C22×D411yescharacteristic
274C22×C4C2×#4112yes—
275C2×C8—48no—
276C2×C8—44no—
277C4⋊C4—28no—
278C4⋊C4—48no—
279C4⋊C4—48no—
280C4⋊C4—48no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—84no—
284C22⋊C4—24no—
285C22⋊C4—24no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—48no—
289C22⋊C4—44no—
290C22⋊C4—44no—
291C22⋊C4—84no—
292C22⋊C4—88no—
293C22⋊C4—88no—
294C22⋊C4—84no—
295C42—48no—
296C42—48no—
297C42—84no—
298C42C4×D414yes—
299C42C4×D414yes—
300C42#12114yes—
301C42#12114yes—
302C23—22no—
303C23—22no—
304C23—24no—
305C23—22no—
306C23—24no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—22no—
311C23—42no—
312C23—42no—
313C23—42no—
314C23—42no—
315C23—48no—
316C23—48no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—81no—
324C23—81no—
325C23—84no—
326C23—81no—
327C23—81no—
328C23—84no—
329C23#41914yes—
330C23C22×C22⋊C411yescharacteristic
331C23C2×#12011yescharacteristic
332C23C2×#12311yescharacteristic
333C23C2×#13712yes—
334C23#57512yes—
335Q8—416no—
336Q8—88no—
337D4—416no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—44no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—44no—
366C2×C4—84no—
367C2×C4—84no—
368C2×C4—81no—
369C2×C4—81no—
370C2×C4—88no—
371C2×C4—84no—
372C2×C4—816no—
373C2×C4—84no—
374C2×C4#38614yes—
375C2×C4#38612yes—
376C2×C4#38612yes—
377C2×C4#45514yes—
378C2×C4C2×C4×D412yes—
379C2×C4C2×#12112yes—
380C8—84no—
381C22—24no—
382C22—24no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—24no—
388C22—44no—
389C22—44no—
390C22—48no—
391C22—42no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—82no—
397C22—84no—
398C22—82no—
399C22—84no—
400C22—84no—
401C22C2×#38611yescharacteristic
402C22C2×#41911yescharacteristic
403C22#289611yescharacteristic
404C22#300712yes—
405C22C2×#45512yes—
406C22#302712yes—
407C22#309512yes—
408C4—24no—
409C4—22no—
410C4—22no—
411C4—22no—
412C4—42no—
413C4—42no—
414C4—84no—
415C4—88no—
416C4—84no—
417C4#188514yes—
418C2—21no—
419C2—21no—
420C2—22no—
421C2—44no—
422C2—82no—
423C2#1833614yes—
424C2#2091111yescharacteristic
425C2C2×#188511yescharacteristic
426C2#2096111yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C23×C4 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #21811C2semidirect
5. #9016C2semidirect
6. #9013C2semidirect
7. #8021C2semidirect
12. #2563C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
72. #393C2×C4semidirect
90. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#25632
#340572
#218112
C2×#14662
#90162
#90132
#80212