Finite Group Databasebeta

← #194875

Subgroups — #194875

Order 512.

1532 conjugacy classes of subgroups.

The subgroup lattice has 429 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2547C211yesmaximal, characteristic
2#34057C211yesmaximal, characteristic
3#21840C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic
5#9028C214yesmaximal, complemented
6#9025C214yesmaximal, complemented
7#8021C212yesmaximal, complemented
8C22×C4⋊Q8C2211yescharacteristic
9C22×#124—21no—
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2547—24no—
13#2547C418yescomplemented
14C2×#397C2212yes—
15C2×#397C2212yes—
16#2318—24no—
17#2315C2212yes—
18#2313—21no—
19#2312C2212yes—
20#2308—21no—
21C2×#391—21no—
22#2290—24no—
23#2283C2212yes—
24#2280—21no—
25C2×#380—21no—
26C2×#380—21no—
27C2×C4×C4⋊C4C2211yescharacteristic
28C2×C4×C22⋊C4—21no—
29#1993—24no—
30#1987—24no—
31#1823C2218yes—
32#1466—24no—
33C2×#323C2212yes—
34C2×#323C2212yes—
35#1294C2218yes—
36C23×Q8—21no—
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×#124—28no—
46C2×#124—48no—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—24no—
50C2×C4×D4C2×C414yes—
51C2×C4×D4D414yes—
52C2×#117—22no—
53C22×C4⋊C4—21no—
54C22×C4⋊C4—21no—
55C22×C4⋊C4D411yescharacteristic
56C22×C4⋊C4C2312yes—
57C22×C22⋊C4—22no—
58C22×C22⋊C4—22no—
59C22×C22⋊C4—22no—
60C22×C22⋊C4C2311yescharacteristic
61C22×C22⋊C4C2311yescharacteristic
62C22×C42—21no—
63C22×C42—22no—
64C22×C42C2311yescharacteristic
65#432—24no—
66#432—24no—
67C2×C4⋊C8—22no—
68C2×C22⋊C8C2314yes—
69C2×#97—22no—
70#397—24no—
71#397—24no—
72#397C2×C418yes—
73#397C2×C418yescomplemented
74#391—28no—
75#388—28no—
76#387—28no—
77#386—28no—
78#385—28no—
79#383—22no—
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#340—216no—
103#323—24no—
104#323—24no—
105C25C2×D411yescharacteristic
106C25C2×D411yescharacteristic
107C22×Q8—24no—
108C22×Q8—44no—
109C22×Q8—44no—
110C22×Q8—42no—
111C22×Q8—42no—
112C22×D4—24no—
113C22×D4—22no—
114C22×D4—22no—
115C22×D4—42no—
116C22×D4C22⋊C414yes—
117C23×C4—21no—
118C23×C4—21no—
119C23×C4—21no—
120C23×C4—21no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—24no—
126C23×C4—44no—
127C23×C4—44no—
128C23×C4C22⋊C414yes—
129C23×C4C22⋊C414yes—
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C23×C4C2×D411yescharacteristic
133C23×C4C2×D411yescharacteristic
134C23×C4C2411yescharacteristic
135C22×C8—24no—
136C4⋊Q8—28no—
137C4⋊Q8—28no—
138C4⋊Q8—44no—
139C4⋊Q8—44no—
140#124—432no—
141C4×D4—28no—
142C4×D4—28no—
143C4×D4—28no—
144C4×D4—28no—
145#117—84no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—24no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4—41no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22×C414yes—
160C2×C4⋊C4C2×D418yes—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—22no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—28no—
167C2×C22⋊C4—28no—
168C2×C22⋊C4—24no—
169C2×C22⋊C4—24no—
170C2×C22⋊C4—24no—
171C2×C22⋊C4—44no—
172C2×C22⋊C4—48no—
173C2×C22⋊C4—44no—
174C2×C22⋊C4—48no—
175C2×C22⋊C4C22×C412yes—
176C2×C22⋊C4C22×C412yes—
177C2×C22⋊C4C2×D412yes—
178C2×C22⋊C4C2×D412yes—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—44no—
183C2×C42—84no—
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C2×C42C2×D412yes—
187C2×C42C2×D412yes—
188C2×C42#4114yes—
189C4⋊C8—44no—
190C22⋊C8—24no—
191C22⋊C8—24no—
192#97—44no—
193#95—24no—
194#95—24no—
195#95—28no—
196#95—28no—
197#95—24no—
198#95—24no—
199#95—24no—
200#95—22no—
201#95—24no—
202#95C22×C414yes—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—21no—
207C24—22no—
208C24—21no—
209C24—22no—
210C24—42no—
211C24—42no—
212C24—42no—
213C24—42no—
214C24C2×C22⋊C412yes—
215C24C2×C22⋊C412yes—
216C24AGL(1,ℤ/8)14yes—
217C24C22×D411yescharacteristic
218C2×Q8—48no—
219C2×Q8—48no—
220C2×Q8—48no—
221C2×Q8—88no—
222C2×Q8—88no—
223C2×D4—28no—
224C2×D4—28no—
225C2×D4—44no—
226C2×D4—48no—
227C2×D4—48no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—24no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—24no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—42no—
243C22×C4—48no—
244C22×C4—48no—
245C22×C4—44no—
246C22×C4—44no—
247C22×C4—42no—
248C22×C4—42no—
249C22×C4—48no—
250C22×C4—44no—
251C22×C4—44no—
252C22×C4—42no—
253C22×C4—42no—
254C22×C4—44no—
255C22×C4—44no—
256C22×C4—44no—
257C22×C4—84no—
258C22×C4—84no—
259C22×C4—82no—
260C22×C4—82no—
261C22×C4—84no—
262C22×C4—84no—
263C22×C4C2×C22⋊C412yes—
264C22×C4C2×C22⋊C412yes—
265C22×C4C2×C22⋊C412yes—
266C22×C4C2×C22⋊C412yes—
267C22×C4#12012yes—
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12312yes—
271C22×C4#12312yes—
272C22×C4C23×C411yescharacteristic
273C22×C4C22×D412yes—
274C22×C4C22×D411yescharacteristic
275C22×C4C2×#4112yes—
276C2×C8—48no—
277C2×C8—44no—
278C4⋊C4—28no—
279C4⋊C4—28no—
280C4⋊C4—48no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—44no—
285C4⋊C4—84no—
286C22⋊C4—24no—
287C22⋊C4—24no—
288C22⋊C4—24no—
289C22⋊C4—24no—
290C22⋊C4—48no—
291C22⋊C4—416no—
292C22⋊C4—44no—
293C22⋊C4—44no—
294C22⋊C4—416no—
295C22⋊C4—84no—
296C22⋊C4—84no—
297C42—48no—
298C42—48no—
299C42—84no—
300C42C4×D414yes—
301C42C4×D414yes—
302C42#12114yes—
303C42#12114yes—
304C23—22no—
305C23—22no—
306C23—24no—
307C23—22no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—48no—
318C23—48no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—81no—
326C23—81no—
327C23—84no—
328C23—81no—
329C23—81no—
330C23—84no—
331C23#41814yes—
332C23C22×C22⋊C411yescharacteristic
333C23C2×#12011yescharacteristic
334C23C2×#12311yescharacteristic
335C23C2×AGL(1,ℤ/8)12yes—
336C23#57512yes—
337Q8—816no—
338D4—416no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—22no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—24no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—44no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—44no—
367C2×C4—84no—
368C2×C4—84no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4—88no—
372C2×C4—816no—
373C2×C4—84no—
374C2×C4—84no—
375C2×C4#38614yes—
376C2×C4#38612yes—
377C2×C4#38612yes—
378C2×C4#45514yes—
379C2×C4C2×C4×D412yes—
380C2×C4C2×#12112yes—
381C8—84no—
382C22—24no—
383C22—24no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—24no—
389C22—44no—
390C22—44no—
391C22—48no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—84no—
398C22—82no—
399C22—82no—
400C22—84no—
401C22—84no—
402C22C2×#38611yescharacteristic
403C22C2×#41811yescharacteristic
404C22#289611yescharacteristic
405C22#300612yes—
406C22C2×#45512yes—
407C22#302812yes—
408C22#309412yes—
409C4—24no—
410C4—22no—
411C4—22no—
412C4—22no—
413C4—42no—
414C4—42no—
415C4—84no—
416C4—88no—
417C4—84no—
418C4#188514yes—
419C2—21no—
420C2—22no—
421C2—21no—
422C2—44no—
423C2—82no—
424C2#1834214yes—
425C2#2091211yescharacteristic
426C2C2×#188511yescharacteristic
427C2#2096211yescharacteristic
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. #21840C2semidirect
5. #9028C2semidirect
6. #9025C2semidirect
7. #8021C2semidirect
13. #2547C4semidirect
44. C2×C4⋊Q8C2×C4semidirect
73. #397C2×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×#25472
#340572
#218402
C2×#14662
#90282
#90252
#80212