Finite Group Databasebeta

← #97666

Subgroups — #97666

Order 512.

796 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 796 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C4×Q82C211yesmaximal, characteristic
2#9637C211yesmaximal, characteristic
3#8564C212yesmaximal
4C4×Q8⋊C8C212yesmaximal
5#7571C211yesmaximal, characteristic
6C2×Q82C411yescharacteristic
7#2369—22no—
8#2369—22no—
9#2369C411yescharacteristic
10Q8×C4⋊C4—22no—
11Q8×C4⋊C4C412yes—
12C4×C4⋊Q8—22no—
13C4×C4⋊Q8—22no—
14C4×C4⋊Q8C2211yescharacteristic
15C42×Q8—22no—
16C42×Q8C2212yes—
17#1842C2211yescharacteristic
18C4×C4⋊C8C2212yes—
19C42×C8C2211yescharacteristic
20C2×Q8⋊C8—22no—
21C2×Q8⋊C8—22no—
22C2×Q8⋊C8—22no—
23#1279—22no—
24#1279—22no—
25#1272—22no—
26Q82—22no—
27Q82—24no—
28Q82C814yescomplemented
29C2×C4⋊Q8—22no—
30C2×C4⋊Q8—22no—
31C2×C4⋊Q8C2×C411yescharacteristic
32C2×C4×Q8—22no—
33C2×C4×Q8—22no—
34C2×C4×Q8—22no—
35C2×C4×Q8C2×C412yes—
36C2×C4×Q8D412yes—
37C2×C4×Q8D412yes—
38C2×C4×Q8D412yes—
39C2×C4⋊C8—22no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—22no—
42C2×C4×C8—21no—
43C2×C4×C8—22no—
44C2×C4×C8—22no—
45#391—24no—
46#391—22no—
47#391—22no—
48#391D412yes—
49#389—24no—
50#389—24no—
51#389—22no—
52#389—22no—
53#389—24no—
54#389—22no—
55#384—22no—
56#384—22no—
57#384C2×C411yescharacteristic
58C4×C4⋊C4—24no—
59C4×C4⋊C4—24no—
60C4×C4⋊C4—24no—
61C4×C4⋊C4—22no—
62C4×C4⋊C4—22no—
63C4×C4⋊C4—22no—
64C4×C4⋊C4—22no—
65C4×C4⋊C4—22no—
66C4×C4⋊C4—22no—
67C4×C4⋊C4C2×C412yes—
68C4×C4⋊C4D412yes—
69C4×C4⋊C4D412yes—
70C43—22no—
71C43—22no—
72C43C2311yescharacteristic
73#336—22no—
74Q8⋊C8—28no—
75Q8⋊C8—44no—
76Q8⋊C8—44no—
77C22×Q8—22no—
78C22×Q8C22⋊C412yes—
79C22×Q8C22⋊C412yes—
80C22×C8—41no—
81C22×C8—42no—
82C22×C8—41no—
83C4⋊Q8—28no—
84C4⋊Q8—28no—
85C4⋊Q8—28no—
86C4⋊Q8—28no—
87C4⋊Q8—22no—
88C4⋊Q8C2×C812yes—
89C4⋊Q8M1612yes—
90C4×Q8—28no—
91C4×Q8—28no—
92C4×Q8—22no—
93C4×Q8—22no—
94C4×Q8—22no—
95C4×Q8—22no—
96C4×Q8—24no—
97C4×Q8—42no—
98C4×Q8—42no—
99C4×Q8—42no—
100C4×Q8—44no—
101C4×Q8—44no—
102C4×Q8—82no—
103C4×Q8—82no—
104C4×Q8C2×C814yes—
105C4×Q8M1614yes—
106C4×Q8SD1614yes—
107C4×Q8Q1614yes—
108C2×C4⋊C4—24no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4—22no—
111C2×C4⋊C4—22no—
112C2×C4⋊C4—24no—
113C2×C4⋊C4—22no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—24no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4C22⋊C412yes—
119C2×C4⋊C4C22⋊C412yes—
120C2×C4⋊C4C22⋊C412yes—
121C2×C4⋊C4C22⋊C412yes—
122C2×C42—22no—
123C2×C42—22no—
124C2×C42—22no—
125C2×C42—24no—
126C2×C42—22no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42C22×C411yescharacteristic
133C2×C42C2×D412yes—
134C2×C42C2×D411yescharacteristic
135C2×C42C2×D411yescharacteristic
136C2×C42C2×D412yes—
137C4⋊C8—24no—
138C4⋊C8—42no—
139C4⋊C8—42no—
140C4⋊C8—42no—
141C4⋊C8—42no—
142C4×C8—24no—
143C4×C8—22no—
144C4×C8—42no—
145C4×C8—41no—
146C4×C8—42no—
147C4×C8—42no—
148#95—24no—
149#95—24no—
150#95—22no—
151#95—24no—
152#95—22no—
153#95—22no—
154C2×Q8—28no—
155C2×Q8—22no—
156C2×Q8—22no—
157C2×Q8—22no—
158C2×Q8—22no—
159C2×Q8—42no—
160C2×Q8—42no—
161C2×Q8—82no—
162C2×Q8—82no—
163C2×Q8C22⋊C814yes—
164C2×Q8C22⋊C814yes—
165C2×Q8#10314yes—
166C2×Q8#10414yes—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—22no—
170C22×C4—22no—
171C22×C4—22no—
172C22×C4—22no—
173C22×C4C2×C22⋊C411yescharacteristic
174C22×C4C2×C22⋊C412yes—
175C22×C4#12011yescharacteristic
176C22×C4#12012yes—
177C22×C4#12011yescharacteristic
178C2×C8—42no—
179C2×C8—44no—
180C2×C8—44no—
181C2×C8—82no—
182C2×C8—82no—
183C2×C8—82no—
184C2×C8—81no—
185C4⋊C4—216no—
186C4⋊C4—24no—
187C4⋊C4—24no—
188C4⋊C4—24no—
189C4⋊C4—24no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—24no—
193C4⋊C4—24no—
194C4⋊C4—22no—
195C4⋊C4—22no—
196C4⋊C4—44no—
197C4⋊C4—42no—
198C4⋊C4—48no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—82no—
202C4⋊C4—82no—
203C4⋊C4—84no—
204C4⋊C4C22⋊C814yes—
205C4⋊C4#10314yes—
206C42—28no—
207C42—24no—
208C42—24no—
209C42—22no—
210C42—22no—
211C42—24no—
212C42—24no—
213C42—21no—
214C42—21no—
215C42—41no—
216C42—44no—
217C42—48no—
218C42—42no—
219C42—42no—
220C42—42no—
221C42—42no—
222C42—42no—
223C42—42no—
224C42—82no—
225C42—82no—
226C42C22×C811yescharacteristic
227C42C2×M1611yescharacteristic
228C42C2×M1612yes—
229C42C2×SD1612yes—
230C42C2×Q1612yes—
231C42#13712yes—
232C42#13712yes—
233C23#37911yescharacteristic
234Q8—24no—
235Q8—24no—
236Q8—24no—
237Q8—44no—
238Q8—82no—
239Q8—88no—
240Q8Q8⋊C818yescomplemented
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—24no—
245C2×C4—24no—
246C2×C4—22no—
247C2×C4—24no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—24no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—42no—
258C2×C4—42no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—42no—
262C2×C4—44no—
263C2×C4—42no—
264C2×C4C2×C22⋊C811yescharacteristic
265C2×C4C2×C22⋊C812yes—
266C2×C4#40711yescharacteristic
267C2×C4#40712yes—
268C2×C4C2×#10312yes—
269C2×C4#41912yes—
270C2×C4C2×#10412yes—
271C2×C4#42112yes—
272C2×C4#44812yes—
273C2×C4#44812yes—
274C2×C4#45112yes—
275C2×C4#45112yes—
276C2×C4#45311yescharacteristic
277C2×C4#45511yescharacteristic
278C8—44no—
279C8—84no—
280C8—161no—
281C22#177911yescharacteristic
282C22#178712yes—
283C22#178912yes—
284C22#179211yescharacteristic
285C22#179411yescharacteristic
286C4—28no—
287C4—21no—
288C4—21no—
289C4—22no—
290C4—22no—
291C4—44no—
292C4—44no—
293C4—44no—
294C4—41no—
295C4—82no—
296C4—82no—
297C4C2×Q8⋊C812yes—
298C4#147912yes—
299C4#148712yes—
300C4#149112yes—
301C4#162311yescharacteristic
302C4#163311yescharacteristic
303C4#164512yes—
304C2#831112yes—
305C2#831811yescharacteristic
306C2#843912yes—
307C2#844111yescharacteristic
308C2#845111yescharacteristic
subgroupG/H
centerC22×C4#120
commutatorC42C22×C8
Frattini, cosocleC43C23
socle, prime radicalC23#379
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C4 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
28. Q82C8semidirect
240. Q8Q8⋊C8semidirect

Maximal subgroups

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

subgroupindex
C4×Q822
#96372
#85642
C4×Q8⋊C82
#75712