Finite Group Databasebeta

← #194773

Subgroups — #194773

Order 512.

1658 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2536C211yesmaximal, characteristic
2#34182C211yesmaximal, characteristic, complemented
3#20732C212yesmaximal
4C2×#1461C211yesmaximal, characteristic, complemented
5#9027C214yesmaximal, complemented
6#9019C214yesmaximal
7#8026C212yesmaximal, complemented
8C22×C22⋊Q8—21no—
9C2×#519—21no—
10C22×C4×Q8C2211yescharacteristic
11C22×C4×D4C2211yescharacteristic
12C2×#431C2212yes—
13#2536—24no—
14#2536C418yescomplemented
15C2×#398C2212yes—
16C2×#394C2212yes—
17#2351—24no—
18Q8×C4⋊C4—24no—
19#2316—21no—
20#2316C2214yes—
21C2×#388—22no—
22#2285—22no—
23#2285C2212yes—
24C2×#382C2211yescharacteristic
25C2×#380—21no—
26#1823C2214yes—
27#1802—24no—
28#1786C2214yes—
29C4×#102—24no—
30#1461—24no—
31C2×#324C2214yes—
32#1294C2218yes—
33C23×Q8D411yescharacteristic
34C23×D4C2311yescharacteristic
35C24×C4—21no—
36C2×C22⋊Q8—28no—
37C2×C22⋊Q8—44no—
38C2×C22⋊Q8—44no—
39#519—44no—
40#519—44no—
41C2×C4×Q8—28no—
42C2×C4×Q8—22no—
43C2×C4×Q8—22no—
44C2×C4×Q8—24no—
45C2×C4×Q8C2×C414yes—
46C2×C4×Q8D414yes—
47C2×C4×D4—22no—
48C2×C4×D4—22no—
49C2×C4×D4—22no—
50C2×C4×D4—24no—
51C2×C4×D4C2×C414yes—
52C2×#117—22no—
53C2×#117—24no—
54C22×C4⋊C4—22no—
55C22×C4⋊C4—21no—
56C22×C4⋊C4D411yescharacteristic
57C22×C4⋊C4C2311yescharacteristic
58C22×C4⋊C4C2312yes—
59C22×C22⋊C4—22no—
60C22×C22⋊C4—22no—
61C22×C22⋊C4—21no—
62C22×C42D412yes—
63C22×C42C2311yescharacteristic
64#431—24no—
65#431—24no—
66C2×C4⋊C8—22no—
67C2×#102—28no—
68C2×C22⋊C8C2314yes—
69C2×C4×C8—22no—
70#398—24no—
71#398C2×C418yes—
72#394—24no—
73#394C2×C418yes—
74#389—28no—
75#389—216no—
76#388—216no—
77#384—24no—
78#384D414yes—
79#382—24no—
80#382—22no—
81#382—22no—
82#382C2×C414yes—
83#380—24no—
84#380—22no—
85#380—28no—
86#380C2314yes—
87C4×C4⋊C4—24no—
88C4×C4⋊C4—24no—
89C4×C4⋊C4—22no—
90C4×C4⋊C4D414yes—
91C2×#95—22no—
92C2×#95—21no—
93C2×#95C2312yes—
94#340—28no—
95#324—28no—
96C25—21no—
97C22×Q8—22no—
98C22×Q8—22no—
99C22×Q8—22no—
100C22×Q8—22no—
101C22×Q8C22⋊C414yes—
102C22×Q8D814yes—
103C22×Q8SD1614yes—
104C22×D4—22no—
105C22×D4—21no—
106C22×D4—22no—
107C22×D4—44no—
108C22×D4C22×C412yes—
109C23×C4—22no—
110C23×C4—21no—
111C23×C4—24no—
112C23×C4—42no—
113C23×C4—42no—
114C23×C4—42no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—42no—
118C23×C4C2×D412yes—
119C23×C4C2×D412yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2411yescharacteristic
123C22×C8—24no—
124C22⋊Q8—432no—
125C4×Q8—28no—
126C4×Q8—28no—
127C4×Q8—28no—
128C4×Q8—28no—
129C4×Q8—432no—
130C4×D4—28no—
131C4×D4—28no—
132C4×D4—44no—
133C4×D4—44no—
134#117—48no—
135#117—48no—
136#117—44no—
137#117—44no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—28no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—28no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—28no—
151C2×C4⋊C4—42no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—42no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C414yes—
158C2×C4⋊C4C2×D414yes—
159C2×C4⋊C4C2×D418yes—
160C2×C22⋊C4—24no—
161C2×C22⋊C4—24no—
162C2×C22⋊C4—28no—
163C2×C22⋊C4—28no—
164C2×C22⋊C4—48no—
165C2×C22⋊C4—41no—
166C2×C22⋊C4—41no—
167C2×C22⋊C4—42no—
168C2×C22⋊C4—44no—
169C2×C22⋊C4—44no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—24no—
173C2×C42—28no—
174C2×C42—24no—
175C2×C42C22⋊C418yes—
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42#4114yes—
181C4⋊C8—44no—
182#102—416no—
183C22⋊C8—24no—
184C22⋊C8—24no—
185C4×C8—44no—
186#95—28no—
187#95—28no—
188#95—24no—
189#95—24no—
190#95—22no—
191#95—24no—
192#95C22×C414yes—
193C24—24no—
194C24—22no—
195C24—42no—
196C24—42no—
197C24—42no—
198C24—41no—
199C24—41no—
200C24—41no—
201C24—41no—
202C24—41no—
203C24—41no—
204C24C22×D411yescharacteristic
205C2×Q8—28no—
206C2×Q8—28no—
207C2×Q8—28no—
208C2×Q8—28no—
209C2×Q8—28no—
210C2×Q8—28no—
211C2×Q8#102116yescomplemented
212C2×D4—28no—
213C2×D4—24no—
214C2×D4—44no—
215C2×D4—44no—
216C2×D4—48no—
217C2×D4—44no—
218C2×D4—42no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—22no—
227C22×C4—28no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4—44no—
232C22×C4—44no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—42no—
236C22×C4—41no—
237C22×C4—42no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—48no—
242C22×C4—48no—
243C22×C4—48no—
244C22×C4—84no—
245C22×C4—84no—
246C22×C4—84no—
247C22×C4—84no—
248C22×C4C2×C22⋊C414yes—
249C22×C4C2×C22⋊C414yes—
250C22×C4C2×C22⋊C412yes—
251C22×C4C2×C22⋊C412yes—
252C22×C4#12011yescharacteristic
253C22×C4#12011yescharacteristic
254C22×C4#12012yes—
255C22×C4#12312yes—
256C22×C4#12312yes—
257C22×C4C2×D814yes—
258C22×C4C2×D812yes—
259C22×C4C2×SD1614yes—
260C22×C4C2×SD1612yes—
261C22×C4AGL(1,ℤ/8)12yes—
262C22×C4#13712yes—
263C22×C4C23×C411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C22×D411yescharacteristic
266C22×C4C2×#4112yes—
267C2×C8—48no—
268C2×C8—44no—
269C4⋊C4—28no—
270C4⋊C4—28no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—416no—
276C4⋊C4—42no—
277C4⋊C4—48no—
278C4⋊C4—416no—
279C4⋊C4—48no—
280C4⋊C4—88no—
281C22⋊C4—416no—
282C22⋊C4—44no—
283C22⋊C4—44no—
284C22⋊C4—44no—
285C22⋊C4—44no—
286C22⋊C4—416no—
287C42—28no—
288C42—28no—
289C42—416no—
290C42—48no—
291C42C4×D414yes—
292C42C4×D414yes—
293C42#12114yes—
294C42#12114yes—
295C23—22no—
296C23—22no—
297C23—24no—
298C23—42no—
299C23—44no—
300C23—44no—
301C23—44no—
302C23—44no—
303C23—44no—
304C23—42no—
305C23—42no—
306C23—42no—
307C23—42no—
308C23—44no—
309C23—42no—
310C23—82no—
311C23—82no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—81no—
316C23—81no—
317C23—81no—
318C23—81no—
319C23—82no—
320C23C22×C22⋊C411yescharacteristic
321C23C2×#12011yescharacteristic
322C23C2×#12311yescharacteristic
323C23C22×D811yescharacteristic
324C23C22×SD1611yescharacteristic
325C23C2×AGL(1,ℤ/8)11yescharacteristic
326C23C2×#13711yescharacteristic
327Q8—232no—
328D4—28no—
329D4—48no—
330D4—44no—
331C2×C4—24no—
332C2×C4—24no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—24no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—48no—
354C2×C4—48no—
355C2×C4—48no—
356C2×C4—42no—
357C2×C4—82no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—81no—
361C2×C4—81no—
362C2×C4—81no—
363C2×C4—81no—
364C2×C4—816no—
365C2×C4#38612yes—
366C2×C4#38612yes—
367C2×C4#38614yes—
368C2×C4C2×#10218yes—
369C2×C4C2×#10214yes—
370C2×C4#41714yes—
371C2×C4#44714yes—
372C2×C4#44814yes—
373C2×C4#48014yes—
374C2×C4#48114yes—
375C2×C4C2×C4×D412yes—
376C2×C4C2×#12112yes—
377C8—84no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—22no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—44no—
387C22—44no—
388C22—44no—
389C22—82no—
390C22—82no—
391C22—84no—
392C22—84no—
393C22—84no—
394C22—84no—
395C22—82no—
396C22—82no—
397C22C2×#38611yescharacteristic
398C22C22×#10211yescharacteristic
399C22C2×#41711yescharacteristic
400C22C2×#44711yescharacteristic
401C22C2×#44811yescharacteristic
402C22#302012yes—
403C22#302512yes—
404C22C2×#48011yescharacteristic
405C22C2×#48111yescharacteristic
406C4—28no—
407C4—22no—
408C4—22no—
409C4—22no—
410C4—22no—
411C4—44no—
412C4—84no—
413C4—88no—
414C4—82no—
415C4#187418yes—
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2—84no—
421C2#1821812yes—
422C2#1822012yes—
423C2C2×#187411yescharacteristic
424C2#2095411yescharacteristic
425C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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
2. #34182C2semidirect
4. C2×#1461C2semidirect
5. #9027C2semidirect
7. #8026C2semidirect
14. #2536C4semidirect
211. C2×Q8#102semidirect

Maximal subgroups

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

subgroupindex
C2×#25362
#341822
#207322
C2×#14612
#90272
#90192
#80262