Finite Group Databasebeta

← #131965

Subgroups — #131965

Order 512.

564 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 564 conjugacy classes of subgroups — too many to draw. Only its 116 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×#3383C211yesmaximal, characteristic
2#13290C211yesmaximal, characteristic
3#12819C211yesmaximal, characteristic
4#11941C211yesmaximal, characteristic
5#11607C211yesmaximal, characteristic
6#11600C211yesmaximal, characteristic
7#8659C211yesmaximal, characteristic
8C2×#557—21no—
9#3383—22no—
10#3383—22no—
11#3383—22no—
12#3383C414yes—
13C2×#500C2211yescharacteristic
14C2×#499—21no—
15C2×#479—22no—
16C2×#475—21no—
17C2×C8×Q8C2211yescharacteristic
18C2×C4×Q16—21no—
19C2×C4×Q16C2211yescharacteristic
20#2150C2211yescharacteristic
21C2×#362—21no—
22C4×M32C2211yescharacteristic
23#1848C2211yescharacteristic
24#1564C2211yescharacteristic
25#1487—21no—
26#1387—21no—
27#1387—22no—
28#1365—22no—
29Q8⋊C16—22no—
30#1334—22no—
31#1301—21no—
32C22×Q16D411yescharacteristic
33#557—24no—
34#557—24no—
35#557—42no—
36#557—42no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—41no—
39C2×#125—22no—
40C2×#125—42no—
41C2×C4×Q8—22no—
42C2×C4×Q8—21no—
43C2×C4×Q8—41no—
44C2×C4×Q8D411yescharacteristic
45C2×M32—21no—
46C2×M32—21no—
47#500—21no—
48#500—21no—
49#500C2×C412yes—
50#500D412yes—
51#499—24no—
52#499—24no—
53#479—28no—
54#479—28no—
55#475—24no—
56#475—24no—
57C8×Q8—21no—
58C8×Q8—21no—
59C8×Q8C2×C412yes—
60C8×Q8D412yes—
61C4×Q16—24no—
62C4×Q16—24no—
63C4×Q16—21no—
64C4×Q16—21no—
65C4×Q16C2×C412yes—
66C4×Q16D412yes—
67#431—21no—
68C2×#108—21no—
69C2×#108—21no—
70C2×#107—21no—
71C2×#107D412yes—
72C2×#107D411yescharacteristic
73C2×#106—22no—
74#423—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8D411yescharacteristic
77C2×#103—22no—
78C2×#103—21no—
79C2×C4×C8—21no—
80C2×C4×C8C2311yescharacteristic
81C4⋊C16—21no—
82C4⋊C16—21no—
83#362—42no—
84#346—21no—
85C4×C16—21no—
86#335—21no—
87#334—21no—
88Q8⋊C8—42no—
89C22×Q8—21no—
90C22×Q8—41no—
91C22×Q8SD1611yescharacteristic
92C2×Q16—21no—
93C2×Q16—21no—
94C2×Q16—21no—
95C2×Q16C22⋊C412yes—
96C2×Q16SD1612yes—
97C2×Q16Q1612yes—
98C2×M16—41no—
99C22×C8—21no—
100C22×C8C2×D411yescharacteristic
101C22×C8C2×D411yescharacteristic
102C4⋊Q8—22no—
103C4⋊Q8—22no—
104C4⋊Q8—44no—
105C4⋊Q8—44no—
106C4⋊Q8—41no—
107C4⋊Q8—41no—
108#125—28no—
109#125—44no—
110#125—48no—
111#125—48no—
112C4×Q8—21no—
113C4×Q8—21no—
114C4×Q8—28no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—44no—
118C4×Q8—44no—
119C4×Q8—44no—
120C4×Q8—41no—
121C4×Q8—41no—
122C4×Q8C22⋊C412yes—
123C4×Q8D812yes—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—42no—
128C2×C4⋊C4—42no—
129C2×C4⋊C4—42no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4SD1611yescharacteristic
133C2×C42—21no—
134C2×C42—42no—
135C2×C42—41no—
136C2×C42C2×D411yescharacteristic
137M32—22no—
138M32—41no—
139M32—41no—
140M32—41no—
141C2×C16—41no—
142C2×C16—41no—
143#108—42no—
144#108—41no—
145#108—41no—
146#107—22no—
147#107—22no—
148#107—21no—
149#107—22no—
150#107C22⋊C412yes—
151#107C22⋊C414yes—
152#106—28no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—21no—
156C4⋊C8—41no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8C22⋊C412yes—
160#103—28no—
161#103—22no—
162#103—22no—
163C22⋊C8—41no—
164C4×C8—22no—
165C4×C8—22no—
166C4×C8—41no—
167C4×C8C22×C411yescharacteristic
168C4×C8C2×D411yescharacteristic
169C4×C8C2×D411yescharacteristic
170C4×C8C2×D411yescharacteristic
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—22no—
175C2×Q8—42no—
176C2×Q8—42no—
177C2×Q8—41no—
178C2×Q8—42no—
179C2×Q8—41no—
180C2×Q8—41no—
181C2×Q8—84no—
182C2×Q8#10212yes—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—42no—
188C22×C4#12011yescharacteristic
189C22×C4C2×SD1611yescharacteristic
190C22×C4AGL(1,ℤ/8)11yescharacteristic
191Q16—22no—
192Q16—22no—
193Q16—22no—
194Q16#10314yes—
195M16—81no—
196M16—81no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—22no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8C2×C22⋊C411yescharacteristic
205C2×C8C2×C22⋊C411yescharacteristic
206C2×C8#12011yescharacteristic
207C2×C8#12011yescharacteristic
208C2×C8C2×SD1611yescharacteristic
209C2×C8C2×Q1611yescharacteristic
210C2×C8#13711yescharacteristic
211C2×C8#13711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—22no—
214C4⋊C4—21no—
215C4⋊C4—22no—
216C4⋊C4—24no—
217C4⋊C4—48no—
218C4⋊C4—48no—
219C4⋊C4—48no—
220C4⋊C4—42no—
221C4⋊C4—42no—
222C4⋊C4—41no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4#10212yes—
227C42—22no—
228C42—22no—
229C42—48no—
230C42—42no—
231C42—42no—
232C42C2×C22⋊C411yescharacteristic
233C42#12011yescharacteristic
234C42C2×D811yescharacteristic
235C42AGL(1,ℤ/8)11yescharacteristic
236C16—42no—
237C16—81no—
238C23#45011yescharacteristic
239Q8—24no—
240Q8—24no—
241Q8—42no—
242Q8—42no—
243Q8—42no—
244Q8—42no—
245Q8—84no—
246Q8—84no—
247C2×C4—22no—
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—41no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—42no—
257C2×C4—44no—
258C2×C4—82no—
259C2×C4—84no—
260C2×C4#37911yescharacteristic
261C2×C4C2×#10211yescharacteristic
262C2×C4#41811yescharacteristic
263C2×C4#44711yescharacteristic
264C2×C4#44711yescharacteristic
265C2×C4#44811yescharacteristic
266C2×C4#45011yescharacteristic
267C2×C4#45111yescharacteristic
268C2×C4#45411yescharacteristic
269C2×C4#45611yescharacteristic
270C8—21no—
271C8—21no—
272C8—42no—
273C8—161no—
274C8—161no—
275C8C2×#10311yescharacteristic
276C8#41911yescharacteristic
277C8#44811yescharacteristic
278C8#45111yescharacteristic
279C22—21no—
280C22—21no—
281C22#163711yescharacteristic
282C22#165111yescharacteristic
283C22#178611yescharacteristic
284C4—21no—
285C4—21no—
286C4—21no—
287C4—42no—
288C4—42no—
289C4—44no—
290C4—84no—
291C4—82no—
292C4#162711yescharacteristic
293C4#164911yescharacteristic
294C4#178711yescharacteristic
295C4#179311yescharacteristic
296C4#214812yes—
297C2—21no—
298C2—21no—
299C2#844711yescharacteristic
300C2#1273811yescharacteristic
301C2#1275111yescharacteristic
subgroupG/H
center, socleC22#1786
commutatorC4×C8C22×C4
Frattini, cosocleC2×C4×C8C23
prime radicalC23#450
LCSG ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G
LECSG ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
C2×#33832
#132902
#128192
#119412
#116072
#116002
#86592