Finite Group Databasebeta

← #194775

Subgroups — #194775

Order 512.

1766 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2535C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#20732C212yesmaximal, complemented
4C2×#1460C211yesmaximal, characteristic, complemented
5#9028C214yesmaximal, complemented
6#9020C214yesmaximal, complemented
7#8026C212yesmaximal, complemented
8C22×#121—21no—
9C22×C4×D4C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#431C2212yes—
12#2535—24no—
13#2535C418yescomplemented
14C2×#397C2212yes—
15C2×#392C2212yes—
16#2357—24no—
17#2354—24no—
18#2312C2214yes—
19#2309—21no—
20C2×#386—22no—
21#2285C2212yes—
22C2×#382C2211yescharacteristic
23#2280—22no—
24C2×#380—21no—
25#2274—21no—
26#1823C2214yes—
27#1802—24no—
28#1786C2214yes—
29C4×#102—24no—
30#1460—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×D4—22no—
34C23×D4D411yescharacteristic
35C23×D4C2311yescharacteristic
36C24×C4—21no—
37C24×C4D412yes—
38C2×#121—28no—
39C2×#121—48no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—22no—
45C2×C4×D4—24no—
46C2×C4×D4—24no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4C2×C414yes—
49C2×C4×D4D414yes—
50C2×#117—22no—
51C2×#117—24no—
52C22×C4⋊C4—21no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4C2312yes—
59C22×C42C2311yescharacteristic
60#431—24no—
61#431—24no—
62C2×C4⋊C8—22no—
63C2×#102—28no—
64C2×C22⋊C8C2314yes—
65C2×C4×C8—22no—
66#397—24no—
67#397C2×C418yes—
68#392—24no—
69#392C2×C418yes—
70#390—28no—
71#388—216no—
72#386—216no—
73#384D414yescomplemented
74#382—22no—
75#382—22no—
76#382C2×C414yes—
77#381—24no—
78#380—24no—
79#380—22no—
80#380—28no—
81#380C2314yes—
82#379—24no—
83C4×C4⋊C4—28no—
84C4×C4⋊C4—22no—
85C4×C4⋊C4D414yescomplemented
86C4×C22⋊C4—24no—
87C4×C22⋊C4—24no—
88C2×#95—22no—
89C2×#95—21no—
90C2×#95C2312yes—
91#340—28no—
92#323—28no—
93C25—21no—
94C25C2×D412yes—
95C22×D4—28no—
96C22×D4—22no—
97C22×D4—22no—
98C22×D4—24no—
99C22×D4—24no—
100C22×D4—22no—
101C22×D4—21no—
102C22×D4—22no—
103C22×D4—44no—
104C22×D4—48no—
105C22×D4—48no—
106C22×D4—84no—
107C22×D4C22⋊C414yes—
108C22×D4C22×C412yes—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—22no—
114C23×C4—24no—
115C23×C4—42no—
116C23×C4—42no—
117C23×C4—44no—
118C23×C4—44no—
119C23×C4—84no—
120C23×C4C22⋊C418yes—
121C23×C4C2×D412yes—
122C23×C4C2×D411yescharacteristic
123C23×C4C2×D411yescharacteristic
124C23×C4C2411yescharacteristic
125C22×C8—24no—
126#121—432no—
127C4×D4—28no—
128C4×D4—28no—
129C4×D4—28no—
130C4×D4—28no—
131C4×D4—28no—
132C4×D4—28no—
133C4×D4—44no—
134C4×D4—44no—
135#117—44no—
136#117—44no—
137#117—88no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—24no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—44no—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C22×C412yes—
149C2×C4⋊C4C2×D418yes—
150C2×C22⋊C4—24no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—24no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—28no—
157C2×C22⋊C4—28no—
158C2×C22⋊C4—48no—
159C2×C22⋊C4—41no—
160C2×C22⋊C4—41no—
161C2×C22⋊C4—42no—
162C2×C22⋊C4—84no—
163C2×C22⋊C4C22×C414yes—
164C2×C22⋊C4C2×D414yes—
165C2×C42—24no—
166C2×C42—28no—
167C2×C42C22×C412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42C2×D412yes—
171C2×C42#4114yes—
172C4⋊C8—44no—
173#102—416no—
174C22⋊C8—24no—
175C22⋊C8—24no—
176C4×C8—44no—
177#95—28no—
178#95—28no—
179#95—24no—
180#95—24no—
181#95—22no—
182#95—24no—
183#95C22×C414yes—
184C24—24no—
185C24—22no—
186C24—22no—
187C24—24no—
188C24—22no—
189C24—24no—
190C24—44no—
191C24—44no—
192C24—42no—
193C24—42no—
194C24—42no—
195C24—41no—
196C24—41no—
197C24—42no—
198C24—42no—
199C24C2×C22⋊C414yes—
200C24C22×D411yescharacteristic
201C2×D4—28no—
202C2×D4—24no—
203C2×D4—28no—
204C2×D4—28no—
205C2×D4—28no—
206C2×D4—44no—
207C2×D4—44no—
208C2×D4—48no—
209C2×D4—44no—
210C2×D4—416no—
211C2×D4—416no—
212C2×D4—42no—
213C2×D4—44no—
214C2×D4—44no—
215C2×D4—48no—
216C2×D4—88no—
217C2×D4—832no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—28no—
221C22×C4—28no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—28no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—48no—
228C22×C4—44no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—42no—
234C22×C4—42no—
235C22×C4—44no—
236C22×C4—48no—
237C22×C4—48no—
238C22×C4—48no—
239C22×C4—84no—
240C22×C4—88no—
241C22×C4—88no—
242C22×C4—84no—
243C22×C4—84no—
244C22×C4—88no—
245C22×C4C2×C22⋊C414yes—
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C412yes—
248C22×C4#12011yescharacteristic
249C22×C4#12011yescharacteristic
250C22×C4#12012yes—
251C22×C4#12312yes—
252C22×C4#12312yes—
253C22×C4#13514yes—
254C22×C4AGL(1,ℤ/8)14yes—
255C22×C4C23×C411yescharacteristic
256C22×C4C22×D412yes—
257C22×C4C22×D411yescharacteristic
258C22×C4C2×#4112yes—
259C2×C8—48no—
260C2×C8—44no—
261C4⋊C4—28no—
262C4⋊C4—24no—
263C4⋊C4—48no—
264C4⋊C4—48no—
265C4⋊C4—42no—
266C4⋊C4—48no—
267C4⋊C4—48no—
268C22⋊C4—28no—
269C22⋊C4—28no—
270C22⋊C4—416no—
271C22⋊C4—44no—
272C22⋊C4—44no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—88no—
276C22⋊C4—88no—
277C22⋊C4—88no—
278C42—48no—
279C42—88no—
280C42C4×D414yes—
281C42C4×D414yes—
282C42#12114yes—
283C42#12114yes—
284C23—22no—
285C23—22no—
286C23—24no—
287C23—24no—
288C23—24no—
289C23—24no—
290C23—44no—
291C23—44no—
292C23—42no—
293C23—42no—
294C23—44no—
295C23—48no—
296C23—42no—
297C23—416no—
298C23—48no—
299C23—44no—
300C23—42no—
301C23—42no—
302C23—44no—
303C23—44no—
304C23—44no—
305C23—44no—
306C23—42no—
307C23—82no—
308C23—82no—
309C23—88no—
310C23—84no—
311C23—84no—
312C23—82no—
313C23—82no—
314C23—82no—
315C23—82no—
316C23C22×C22⋊C411yescharacteristic
317C23C2×#12011yescharacteristic
318C23C2×#12311yescharacteristic
319C23C2×#13512yes—
320C23C2×AGL(1,ℤ/8)12yes—
321D4—28no—
322D4—48no—
323D4—44no—
324D4—416no—
325D4—832no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—42no—
335C2×C4—416no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—48no—
345C2×C4—48no—
346C2×C4—42no—
347C2×C4—88no—
348C2×C4—82no—
349C2×C4—88no—
350C2×C4—82no—
351C2×C4—82no—
352C2×C4—816no—
353C2×C4#38612yes—
354C2×C4#38612yes—
355C2×C4#38614yes—
356C2×C4#41614yes—
357C2×C4#41814yes—
358C2×C4#44914yes—
359C2×C4#44914yes—
360C2×C4#48218yes—
361C2×C4C2×C4×D412yes—
362C2×C4C2×#12112yes—
363C8—84no—
364C22—24no—
365C22—24no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—21no—
370C22—44no—
371C22—48no—
372C22—44no—
373C22—44no—
374C22—44no—
375C22—44no—
376C22—44no—
377C22—84no—
378C22—82no—
379C22—82no—
380C22—88no—
381C22—88no—
382C22—88no—
383C22—88no—
384C22—82no—
385C22—82no—
386C22C2×#38611yescharacteristic
387C22C2×#41611yescharacteristic
388C22C2×#41811yescharacteristic
389C22C2×#44912yes—
390C22#301912yes—
391C22#302512yes—
392C22C2×#48212yes—
393C4—22no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—44no—
398C4—84no—
399C4—88no—
400C4—82no—
401C4#187518yes—
402C2—21no—
403C2—21no—
404C2—22no—
405C2—84no—
406C2—84no—
407C2#1822214yes—
408C2C2×#187511yescharacteristic
409C2#2095311yescharacteristic
410C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2535C2
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. #34186C2semidirect
3. #20732C2semidirect
4. C2×#1460C2semidirect
5. #9028C2semidirect
6. #9020C2semidirect
7. #8026C2semidirect
13. #2535C4semidirect
73. #384D4semidirect
85. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#25352
#341862
#207322
C2×#14602
#90282
#90202
#80262