Finite Group Databasebeta

← #194799

Subgroups — #194799

Order 512.

1940 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 1940 conjugacy classes of subgroups — too many to draw. Only its 266 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2397C211yesmaximal, characteristic
2#34186C211yesmaximal, characteristic, complemented
3#20738C212yesmaximal, complemented
4C2×#1466C211yesmaximal, characteristic, complemented
5#9024C214yesmaximal, complemented
6#9020C214yesmaximal, complemented
7#8022C212yesmaximal, complemented
8C22×C4×D4—22no—
9C22×C4×D4C2211yescharacteristic
10C22×C4×D4C2211yescharacteristic
11C2×#432C2212yes—
12#2397—24no—
13#2397C418yescomplemented
14C2×#399—21no—
15C2×#399C2212yes—
16C2×#392—21no—
17C2×#392C2212yes—
18#2357—24no—
19#2354—24no—
20#2312C2214yes—
21#2309—21no—
22#2285C2212yes—
23#2280—22no—
24#2274—21no—
25C2×#376C2211yescharacteristic
26#1824C2214yes—
27#1802—24no—
28#1786C2214yes—
29#1762—24no—
30#1466—24no—
31C2×#323C2214yes—
32#1294C2218yes—
33C23×D4—22no—
34C23×D4D411yescharacteristic
35C23×D4C2311yescharacteristic
36C24×C4—21no—
37C24×C4D412yes—
38C2×C4×D4—216no—
39C2×C4×D4—22no—
40C2×C4×D4—22no—
41C2×C4×D4—22no—
42C2×C4×D4—22no—
43C2×C4×D4—22no—
44C2×C4×D4—24no—
45C2×C4×D4—24no—
46C2×C4×D4—416no—
47C2×C4×D4C2×C414yes—
48C2×C4×D4C2×C414yes—
49C2×C4×D4D414yes—
50C2×#117—22no—
51C2×#117—24no—
52C22×C4⋊C4—22no—
53C22×C4⋊C4D411yescharacteristic
54C22×C4⋊C4C2311yescharacteristic
55C22×C22⋊C4—22no—
56C22×C22⋊C4—21no—
57C22×C22⋊C4—22no—
58C22×C22⋊C4C2312yes—
59C22×C42—21no—
60C22×C42C2311yescharacteristic
61#432—24no—
62#432—24no—
63C2×C4⋊C8—22no—
64C2×#102—28no—
65C2×C22⋊C8C2314yes—
66C2×#97—22no—
67#399—28no—
68#399—24no—
69#399C2×C418yes—
70#392—28no—
71#392—24no—
72#392C2×C418yes—
73#390—28no—
74#388—216no—
75#384D414yescomplemented
76#382—22no—
77#381—24no—
78#380—24no—
79#380—22no—
80#380C2314yes—
81#379—24no—
82C4×C4⋊C4—28no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4D414yescomplemented
85C4×C22⋊C4—24no—
86C4×C22⋊C4—24no—
87#376—22no—
88#376C2×C414yes—
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—22no—
99C22×D4—21no—
100C22×D4—22no—
101C22×D4—44no—
102C22×D4—48no—
103C22×D4—44no—
104C22×D4—48no—
105C22×D4—48no—
106C22×D4C22⋊C414yes—
107C22×D4C22×C412yes—
108C23×C4—22no—
109C23×C4—22no—
110C23×C4—22no—
111C23×C4—24no—
112C23×C4—21no—
113C23×C4—24no—
114C23×C4—48no—
115C23×C4—48no—
116C23×C4—41no—
117C23×C4—41no—
118C23×C4—42no—
119C23×C4C22⋊C418yes—
120C23×C4C2×D411yescharacteristic
121C23×C4C2×D411yescharacteristic
122C23×C4C2×D412yes—
123C23×C4C2411yescharacteristic
124C22×C8—24no—
125C4×D4—28no—
126C4×D4—28no—
127C4×D4—28no—
128C4×D4—28no—
129C4×D4—28no—
130C4×D4—28no—
131C4×D4—464no—
132C4×D4—44no—
133C4×D4—44no—
134#117—48no—
135#117—88no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—28no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—48no—
145C2×C4⋊C4C22⋊C414yes—
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C2×D418yes—
148C2×C22⋊C4—24no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—24no—
152C2×C22⋊C4—28no—
153C2×C22⋊C4—24no—
154C2×C22⋊C4—28no—
155C2×C22⋊C4—28no—
156C2×C22⋊C4—48no—
157C2×C22⋊C4—48no—
158C2×C22⋊C4—42no—
159C2×C22⋊C4—42no—
160C2×C22⋊C4C22×C414yes—
161C2×C22⋊C4C2×D414yes—
162C2×C42—28no—
163C2×C42—24no—
164C2×C42—24no—
165C2×C42—28no—
166C2×C42C22×C412yes—
167C2×C42C2×D412yes—
168C2×C42C2×D412yes—
169C2×C42C2×D412yes—
170C2×C42#4114yes—
171C4⋊C8—44no—
172#102—416no—
173C22⋊C8—24no—
174C22⋊C8—24no—
175#97—44no—
176#95—28no—
177#95—24no—
178#95—24no—
179#95—22no—
180#95—24no—
181#95C22×C414yes—
182C24—24no—
183C24—22no—
184C24—22no—
185C24—24no—
186C24—22no—
187C24—24no—
188C24—44no—
189C24—44no—
190C24—44no—
191C24—42no—
192C24—41no—
193C24—41no—
194C24—44no—
195C24C2×C22⋊C414yes—
196C24C22×D411yescharacteristic
197C2×D4—28no—
198C2×D4—24no—
199C2×D4—28no—
200C2×D4—48no—
201C2×D4—416no—
202C2×D4—48no—
203C2×D4—44no—
204C2×D4—44no—
205C2×D4—44no—
206C2×D4—44no—
207C2×D4—48no—
208C2×D4—44no—
209C2×D4—416no—
210C2×D4—416no—
211C2×D4—464no—
212C2×D4—42no—
213C22×C4—24no—
214C22×C4—24no—
215C22×C4—28no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—28no—
220C22×C4—28no—
221C22×C4—22no—
222C22×C4—28no—
223C22×C4—42no—
224C22×C4—416no—
225C22×C4—416no—
226C22×C4—44no—
227C22×C4—44no—
228C22×C4—48no—
229C22×C4—44no—
230C22×C4—44no—
231C22×C4—41no—
232C22×C4—41no—
233C22×C4—44no—
234C22×C4—44no—
235C22×C4—88no—
236C22×C4—84no—
237C22×C4—84no—
238C22×C4—88no—
239C22×C4—84no—
240C22×C4C2×C22⋊C412yes—
241C22×C4C2×C22⋊C412yes—
242C22×C4C2×C22⋊C414yes—
243C22×C4#12011yescharacteristic
244C22×C4#12012yes—
245C22×C4#12011yescharacteristic
246C22×C4#12312yes—
247C22×C4#12312yes—
248C22×C4C23×C411yescharacteristic
249C22×C4C22×D412yes—
250C22×C4C22×D411yescharacteristic
251C22×C4C2×#4112yes—
252C2×C8—48no—
253C2×C8—44no—
254C4⋊C4—28no—
255C4⋊C4—24no—
256C4⋊C4—48no—
257C4⋊C4—416no—
258C4⋊C4—48no—
259C4⋊C4—42no—
260C4⋊C4—48no—
261C22⋊C4—28no—
262C22⋊C4—28no—
263C22⋊C4—416no—
264C22⋊C4—416no—
265C22⋊C4—48no—
266C22⋊C4—48no—
267C22⋊C4—88no—
268C22⋊C4—88no—
269C42—216no—
270C42—48no—
271C42—88no—
272C42C4×D414yes—
273C42C4×D414yes—
274C42#12114yes—
275C42#12114yes—
276C23—22no—
277C23—22no—
278C23—24no—
279C23—24no—
280C23—24no—
281C23—24no—
282C23—44no—
283C23—44no—
284C23—44no—
285C23—416no—
286C23—416no—
287C23—42no—
288C23—44no—
289C23—44no—
290C23—44no—
291C23—44no—
292C23—44no—
293C23—44no—
294C23—44no—
295C23—42no—
296C23—82no—
297C23—82no—
298C23—88no—
299C23—82no—
300C23—88no—
301C23—82no—
302C23—82no—
303C23—82no—
304C23C22×C22⋊C411yescharacteristic
305C23C2×#12011yescharacteristic
306C23C2×#12311yescharacteristic
307C23#57514yes—
308D4—28no—
309D4—416no—
310D4—48no—
311D4—44no—
312D4—464no—
313C2×C4—24no—
314C2×C4—24no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—22no—
318C2×C4—24no—
319C2×C4—24no—
320C2×C4—24no—
321C2×C4—28no—
322C2×C4—24no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—416no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—416no—
331C2×C4—44no—
332C2×C4—44no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—48no—
336C2×C4—42no—
337C2×C4—88no—
338C2×C4—88no—
339C2×C4—82no—
340C2×C4—82no—
341C2×C4—82no—
342C2×C4—88no—
343C2×C4#38614yes—
344C2×C4#38612yes—
345C2×C4#38612yes—
346C2×C4#45314yes—
347C2×C4C2×C4×D412yes—
348C2×C4C2×#12112yes—
349C8—84no—
350C22—28no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—48no—
356C22—48no—
357C22—44no—
358C22—44no—
359C22—44no—
360C22—44no—
361C22—84no—
362C22—84no—
363C22—88no—
364C22—88no—
365C22—88no—
366C22—88no—
367C22—82no—
368C22—82no—
369C22C2×#38611yescharacteristic
370C22#289612yes—
371C22#300512yes—
372C22C2×#45312yes—
373C22#302512yes—
374C22#309312yes—
375C4—24no—
376C4—22no—
377C4—22no—
378C4—22no—
379C4—48no—
380C4—44no—
381C4—88no—
382C4—82no—
383C4#188714yes—
384C2—22no—
385C2—22no—
386C2—84no—
387C2—84no—
388C2#1819014yes—
389C2#2091011yescharacteristic
390C2C2×#188711yescharacteristic
391C2#2095911yescharacteristic
subgroupG/H
center, socleC23C2×#120
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC2×#2397C2
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. #20738C2semidirect
4. C2×#1466C2semidirect
5. #9024C2semidirect
6. #9020C2semidirect
7. #8022C2semidirect
13. #2397C4semidirect
75. #384D4semidirect
84. C4×C4⋊C4D4semidirect

Maximal subgroups

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

subgroupindex
C2×#23972
#341862
#207382
C2×#14662
#90242
#90202
#80222