Finite Group Databasebeta

← #98236

Subgroups — #98236

Order 512.

751 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 751 conjugacy classes of subgroups — too many to draw. Only its 160 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
1Q8×C4⋊Q8C211yesmaximal, characteristic
2#9691C212yesmaximal
3#8587C211yesmaximal, characteristic
4#8564C212yesmaximal
5#7571C211yesmaximal, characteristic
6C2×Q82C412yes—
7#2868—21no—
8#2868C2211yescharacteristic
9#2567—22no—
10#2567C412yes—
11#2551—22no—
12#2551—24no—
13Q8×C4⋊C4—22no—
14C4×C4⋊Q8—21no—
15C4×C4⋊Q8C2211yescharacteristic
16C42×Q8C2211yescharacteristic
17#1842C2212yes—
18C4×C4⋊C8C2212yes—
19C2×#333—21no—
20C2×#333—24no—
21C2×Q8⋊C8—22no—
22#1296—21no—
23#1279—22no—
24#1272—22no—
25Q82—44no—
26Q82—44no—
27C2×C4⋊Q8—24no—
28C2×C4⋊Q8—24no—
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8C2×C412yes—
32C2×C4⋊Q8D411yescharacteristic
33C2×C4⋊Q8D414yes—
34C2×C4×Q8—22no—
35C2×C4×Q8—22no—
36C2×C4×Q8C2×C411yescharacteristic
37C2×C4×Q8C2×C412yes—
38C2×C4×Q8D412yes—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—22no—
41C2×C4⋊C8—22no—
42C2×C4⋊C8—22no—
43C2×C4×C8—22no—
44#395—28no—
45#395—24no—
46#391—22no—
47#391D412yes—
48#389—24no—
49#389—28no—
50#389—24no—
51#389—22no—
52#384—24no—
53#384—22no—
54#384—21no—
55#384—21no—
56#384—22no—
57#384D411yescharacteristic
58C4×C4⋊C4—24no—
59C4×C4⋊C4—22no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4C2×C411yescharacteristic
63C4×C4⋊C4D412yes—
64C43—21no—
65C43C2311yescharacteristic
66#336—22no—
67#333—48no—
68#333—42no—
69Q8⋊C8—44no—
70C22×Q8—24no—
71C22×Q8C22⋊C411yescharacteristic
72C22×Q8C22⋊C412yes—
73C22×Q8C22⋊C414yes—
74C22×C8—42no—
75C22×C8—42no—
76C4⋊Q8—22no—
77C4⋊Q8—48no—
78C4⋊Q8—42no—
79C4⋊Q8—42no—
80C4⋊Q8—44no—
81C4⋊Q8—44no—
82C4⋊Q8—416no—
83C4⋊Q8—44no—
84C4⋊Q8—44no—
85C4⋊Q8—84no—
86C4⋊Q8—84no—
87C4⋊Q8SD1612yes—
88C4⋊Q8Q1612yes—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8—22no—
93C4×Q8—48no—
94C4×Q8—42no—
95C4×Q8—42no—
96C4×Q8—44no—
97C4×Q8—44no—
98C4×Q8SD1614yes—
99C4×Q8Q1614yes—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—22no—
102C2×C4⋊C4—22no—
103C2×C4⋊C4—24no—
104C2×C4⋊C4—24no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—22no—
107C2×C4⋊C4—22no—
108C2×C4⋊C4—22no—
109C2×C4⋊C4—24no—
110C2×C4⋊C4—28no—
111C2×C4⋊C4—24no—
112C2×C4⋊C4C22⋊C411yescharacteristic
113C2×C4⋊C4C22⋊C412yes—
114C2×C4⋊C4C22⋊C412yes—
115C2×C42—21no—
116C2×C42—22no—
117C2×C42—24no—
118C2×C42—22no—
119C2×C42—21no—
120C2×C42—22no—
121C2×C42C22×C411yescharacteristic
122C2×C42C2×D412yes—
123C2×C42C2×D411yescharacteristic
124C2×C42C2×D412yes—
125C2×C42C2×D411yescharacteristic
126C4⋊C8—24no—
127C4⋊C8—42no—
128C4⋊C8—44no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C4⋊C8—42no—
132C4⋊C8—42no—
133C4×C8—42no—
134C4×C8—42no—
135#95—28no—
136#95—21no—
137#95—22no—
138#95—24no—
139C2×Q8—21no—
140C2×Q8—22no—
141C2×Q8—22no—
142C2×Q8—48no—
143C2×Q8—42no—
144C2×Q8—42no—
145C2×Q8—44no—
146C2×Q8—48no—
147C2×Q8—88no—
148C2×Q8#10012yes—
149C2×Q8#10314yes—
150C2×Q8#10314yes—
151C22×C4—21no—
152C22×C4—22no—
153C22×C4—22no—
154C22×C4—21no—
155C22×C4—24no—
156C22×C4—22no—
157C22×C4C2×C22⋊C411yescharacteristic
158C22×C4C2×C22⋊C411yescharacteristic
159C22×C4C2×C22⋊C411yescharacteristic
160C22×C4#12012yes—
161C22×C4#12012yes—
162C2×C8—44no—
163C2×C8—82no—
164C2×C8—82no—
165C2×C8—82no—
166C2×C8—82no—
167C2×C8—82no—
168C4⋊C4—24no—
169C4⋊C4—21no—
170C4⋊C4—22no—
171C4⋊C4—24no—
172C4⋊C4—24no—
173C4⋊C4—24no—
174C4⋊C4—24no—
175C4⋊C4—42no—
176C4⋊C4—44no—
177C4⋊C4—48no—
178C4⋊C4—44no—
179C4⋊C4—48no—
180C4⋊C4—42no—
181C4⋊C4—42no—
182C4⋊C4—44no—
183C4⋊C4—42no—
184C4⋊C4—44no—
185C4⋊C4—42no—
186C4⋊C4—84no—
187C4⋊C4—88no—
188C4⋊C4#10012yes—
189C4⋊C4#10314yes—
190C42—24no—
191C42—21no—
192C42—21no—
193C42—24no—
194C42—24no—
195C42—22no—
196C42—22no—
197C42—41no—
198C42—42no—
199C42—48no—
200C42—42no—
201C42—42no—
202C42—42no—
203C42—42no—
204C42C2×SD1612yes—
205C42C2×SD1611yescharacteristic
206C42C2×Q1612yes—
207C42C2×Q1611yescharacteristic
208C42#13712yes—
209C42#13712yes—
210C42#13712yes—
211C23#37911yescharacteristic
212Q8—24no—
213Q8—22no—
214Q8—44no—
215Q8—44no—
216Q8—164no—
217Q8—168no—
218Q8#33314yescomplemented
219C2×C4—22no—
220C2×C4—21no—
221C2×C4—22no—
222C2×C4—21no—
223C2×C4—22no—
224C2×C4—24no—
225C2×C4—22no—
226C2×C4—21no—
227C2×C4—22no—
228C2×C4—24no—
229C2×C4—24no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—42no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—48no—
236C2×C4—44no—
237C2×C4—44no—
238C2×C4—42no—
239C2×C4C2×#10011yescharacteristic
240C2×C4C2×#10011yescharacteristic
241C2×C4C2×#10311yescharacteristic
242C2×C4C2×#10312yes—
243C2×C4#41911yescharacteristic
244C2×C4#41912yes—
245C2×C4#44812yes—
246C2×C4#44812yes—
247C2×C4#44812yes—
248C2×C4#45112yes—
249C2×C4#45112yes—
250C2×C4#45112yes—
251C2×C4#45312yes—
252C2×C4#45512yes—
253C8—84no—
254C8—162no—
255C22#178311yescharacteristic
256C22#178711yescharacteristic
257C22#178712yes—
258C22#179212yes—
259C22#179611yescharacteristic
260C4—24no—
261C4—21no—
262C4—21no—
263C4—22no—
264C4—22no—
265C4—41no—
266C4—42no—
267C4—44no—
268C4—44no—
269C4—84no—
270C4—82no—
271C4C2×#33311yescharacteristic
272C4#154812yes—
273C4#155811yescharacteristic
274C4#162312yes—
275C4#163312yes—
276C4#164512yes—
277C4#164512yes—
278C2#834211yescharacteristic
279C2#839111yescharacteristic
280C2#842512yes—
281C2#845112yes—
282C2#846911yescharacteristic
subgroupG/H
center, socle, prime radicalC23#379
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
218. Q8#333semidirect

Maximal subgroups

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

subgroupindex
Q8×C4⋊Q82
#96912
#85872
#85642
#75712