Finite Group Databasebeta

← #266715

Subgroups — #266715

Order 512.

1214 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 1214 conjugacy classes of subgroups — too many to draw. Only its 200 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
1#59817C211yesmaximal, characteristic, complemented
2#28279C212yesmaximal, complemented
3C4×#493C211yesmaximal, characteristic
4#10573C212yesmaximal, complemented
5#10348C212yesmaximal, complemented
6#9870C212yesmaximal, complemented
7#9061C212yesmaximal
8#8920C212yesmaximal, complemented
9#8585C211yesmaximal, characteristic, complemented
10C2×D4×Q8—22no—
11C2×#493C2211yescharacteristic
12C2×#474—22no—
13C2×C4×D8—22no—
14#2683—22no—
15#2551C2212yes—
16#2538—22no—
17#2452—24no—
18#2409—22no—
19#2366C2212yes—
20C4×C4⋊Q8C2212yes—
21C4×#127C2211yescharacteristic
22C4×#127C2211yescharacteristic
23#2067—24no—
24#1982C2214yes—
25#1970—22no—
26#1957C2211yescharacteristic
27#1957C2211yescharacteristic
28#1947C2214yes—
29#1845C2211yescharacteristic
30#1842C2212yes—
31#1802—22no—
32C42×C8C2211yescharacteristic
33C2×#331C2212yes—
34#1296C2212yes—
35#1279C2214yes—
36#1279C2214yes—
37C23×Q8—22no—
38C22×D8—22no—
39D4×Q8—48no—
40D4×Q8—48no—
41C2×C4⋊Q8—24no—
42C2×C4⋊Q8D412yes—
43C2×#127C2311yescharacteristic
44C2×#127C2311yescharacteristic
45C2×C22⋊Q8—24no—
46C2×C22⋊Q8—28no—
47C2×C4×Q8—22no—
48C2×C4×D4—24no—
49C2×C4×D4—22no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4D412yes—
53#493—22no—
54#493Q814yescomplemented
55#474—48no—
56C4×D8—48no—
57C2×#107—24no—
58C2×#107—22no—
59C2×C4⋊C8—22no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8C2314yes—
62C2×#102—22no—
63C2×#102—24no—
64C2×#102—22no—
65C2×#102—24no—
66C2×C4×C8—21no—
67C2×C4×C8—21no—
68C2×C4×C8C2311yescharacteristic
69C2×C4×C8C2311yescharacteristic
70#396—28no—
71#395—24no—
72#393—24no—
73#391—22no—
74#390—24no—
75#390—22no—
76#390—22no—
77#390—21no—
78#390D412yes—
79#389—24no—
80#389—22no—
81#389D414yes—
82#386—24no—
83#385—24no—
84#384D412yes—
85#384C2311yescharacteristic
86#384C2311yescharacteristic
87#380—22no—
88C4×C4⋊C4—22no—
89C4×C4⋊C4C2314yes—
90C43C2311yescharacteristic
91#340—24no—
92#331—24no—
93C22×Q8—24no—
94C22×Q8—22no—
95C22×Q8—416no—
96C22×Q8—84no—
97C22×D4—22no—
98C22×D4—21no—
99C22×D4—22no—
100C22×D4C2×D412yes—
101C23×C4—24no—
102C23×C4—22no—
103C23×C4—41no—
104C2×D8—28no—
105C2×D8—44no—
106C2×D8—44no—
107C22×C8—21no—
108C22×C8—21no—
109C22×C8—24no—
110C4⋊Q8—24no—
111C4⋊Q8—48no—
112C4⋊Q8—48no—
113C4⋊Q8D818yescomplemented
114#127—21no—
115#127—22no—
116#127—22no—
117#127C2×Q812yes—
118#127C2×Q812yescomplemented
119#127#4112yes—
120C22⋊Q8—432no—
121C22⋊Q8—416no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×D4—44no—
125C4×D4—44no—
126C4×D4—48no—
127C4×D4—48no—
128C4×D4—44no—
129C4×D4—44no—
130C4×D4—44no—
131C4×D4—44no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—28no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4C2×D414yes—
141C2×C4⋊C4C2×D414yes—
142C2×C4⋊C4C2×D412yes—
143C2×C22⋊C4—28no—
144C2×C22⋊C4—24no—
145C2×C22⋊C4—24no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C22⋊C4—42no—
149C2×C42—24no—
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D412yes—
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D412yes—
154C2×C42C2411yescharacteristic
155#107—44no—
156#107—48no—
157C4⋊C8—24no—
158C4⋊C8—24no—
159C4⋊C8—44no—
160C4⋊C8—82no—
161#102—44no—
162#102—44no—
163#102—48no—
164#102—84no—
165C4×C8—24no—
166C4×C8—21no—
167C4×C8—22no—
168C4×C8—42no—
169C4×C8C2×Q812yescomplemented
170#95—24no—
171#95—24no—
172#95—24no—
173C24—22no—
174C24—41no—
175C2×Q8—28no—
176C2×Q8—416no—
177C2×Q8—44no—
178C2×Q8—44no—
179C2×Q8—84no—
180C2×Q8—832no—
181C2×D4—22no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—24no—
185C2×D4—24no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—41no—
191C2×D4—42no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—42no—
195C2×D4—42no—
196C2×D4—42no—
197C2×D4—42no—
198C2×D4—42no—
199C2×D4—42no—
200C2×D4C22⋊Q814yes—
201C22×C4—24no—
202C22×C4—24no—
203C22×C4—22no—
204C22×C4—416no—
205C22×C4—48no—
206C22×C4—44no—
207C22×C4—41no—
208C22×C4—84no—
209C22×C4—84no—
210C22×C4—82no—
211C22×C4#12011yescharacteristic
212C22×C4#12012yes—
213C22×C4#12011yescharacteristic
214C22×C4C22×D411yescharacteristic
215C22×C4C22×D412yes—
216D8—416no—
217C2×C8—22no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—44no—
224C2×C8—44no—
225C2×C8—44no—
226C4⋊C4—24no—
227C4⋊C4—28no—
228C4⋊C4—24no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—416no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—48no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—82no—
240C4⋊C4C2×D818yes—
241C22⋊C4—416no—
242C22⋊C4—48no—
243C22⋊C4—82no—
244C22⋊C4—82no—
245C22⋊C4—84no—
246C22⋊C4—84no—
247C42—22no—
248C42—22no—
249C42—22no—
250C42—41no—
251C42—44no—
252C42—44no—
253C42C2×D814yes—
254C42AGL(1,ℤ/8)14yes—
255C42C22×Q811yescharacteristic
256C42C2×#4111yescharacteristic
257C422−1+412yes—
258C23—24no—
259C23—44no—
260C23—44no—
261C23—42no—
262C23—42no—
263C23—42no—
264C23—82no—
265C23—82no—
266C23—81no—
267C23—81no—
268C23C2×#12011yescharacteristic
269Q8—416no—
270Q8—816no—
271D4—44no—
272D4—44no—
273D4—44no—
274D4—44no—
275D4—44no—
276D4—44no—
277D4—44no—
278D4—44no—
279D4—82no—
280D4—82no—
281D4—82no—
282D4—82no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—28no—
288C2×C4—22no—
289C2×C4—21no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—44no—
295C2×C4—42no—
296C2×C4—44no—
297C2×C4—816no—
298C2×C4—88no—
299C2×C4—82no—
300C2×C4—81no—
301C2×C4—164no—
302C2×C4#44714yes—
303C2×C4#44714yes—
304C2×C4#45312yes—
305C2×C4C2×C22⋊Q811yescharacteristic
306C2×C4#53611yescharacteristic
307C2×C4#54812yes—
308C2×C4D4×Q812yes—
309C2×C4C22×D812yes—
310C2×C4C2×AGL(1,ℤ/8)12yes—
311C2×C4#57712yes—
312C2×C4#57812yes—
313C8—24no—
314C8—42no—
315C8—84no—
316C22—48no—
317C22—44no—
318C22—82no—
319C22—82no—
320C22—84no—
321C22—84no—
322C22—82no—
323C22—161no—
324C22—161no—
325C22—162no—
326C22#240911yescharacteristic
327C22C2×#44712yes—
328C22#301312yes—
329C22C2×#45311yescharacteristic
330C22#301811yescharacteristic
331C4—24no—
332C4—21no—
333C4—21no—
334C4—41no—
335C4—48no—
336C4—82no—
337C4—84no—
338C4—161no—
339C4#161914yes—
340C4#335812yes—
341C4#336712yes—
342C4Q8×D812yes—
343C4#338412yes—
344C2—84no—
345C2—162no—
346C2C2×#161911yescharacteristic
347C2#1867412yes—
348C2#1868811yescharacteristic
349C2#2431211yescharacteristic
350C2#2432612yes—
subgroupG/H
center, socleC23C2×#120
commutator, Frattini, cosocleC2×C42C24
prime radicalC2×#493C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59817C2semidirect
2. #28279C2semidirect
4. #10573C2semidirect
5. #10348C2semidirect
6. #9870C2semidirect
8. #8920C2semidirect
9. #8585C2semidirect
54. #493Q8semidirect
113. C4⋊Q8D8semidirect
118. #127C2×Q8semidirect
169. C4×C8C2×Q8semidirect

Maximal subgroups

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

subgroupindex
#598172
#282792
C4×#4932
#105732
#103482
#98702
#90612
#89202
#85852