Finite Group Databasebeta

← #98969

Subgroups — #98969

Order 512.

422 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 110 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#21597C211yesmaximal, characteristic
2#10864C211yesmaximal, characteristic
3#8596C211yesmaximal, characteristic
4#7617C212yesmaximal
5#7045C212yesmaximal
6C2×#446C411yescharacteristic
7C22×#108—21no—
8C22×C4×C8C2211yescharacteristic
9C4×C22⋊Q8C2211yescharacteristic
10#1984C411yescharacteristic
11#1939C411yescharacteristic
12#1916—21no—
13#1850—21no—
14#1797C2212yes—
15#1797C2211yescharacteristic
16#1771—21no—
17#1770C411yescharacteristic
18#1752—21no—
19C2×#341—21no—
20#1728C2212yes—
21C2×#335—21no—
22#1277—21no—
23#1272—22no—
24#1271—22no—
25C22×M16—22no—
26C23×C8D411yescharacteristic
27C23×C8D411yescharacteristic
28C2×C22⋊Q8Q811yescharacteristic
29C2×C4×Q8C2×C411yescharacteristic
30C22×C42C2311yescharacteristic
31#446—42no—
32C2×#108—24no—
33C2×#108—42no—
34C2×#108—42no—
35C2×C4⋊C8—21no—
36C2×C4⋊C8—21no—
37C2×C4⋊C8—22no—
38C2×C4⋊C8C2×C411yescharacteristic
39C2×C4⋊C8C2×C411yescharacteristic
40C2×C22⋊C8—21no—
41C2×C22⋊C8—22no—
42C2×#97—21no—
43C2×#97—21no—
44C2×C4×C8—42no—
45C2×C4×C8C2×C411yescharacteristic
46C2×C4×C8C2×C411yescharacteristic
47C2×C4×C8D411yescharacteristic
48C2×C4×C8D411yescharacteristic
49#391D411yescharacteristic
50#389D411yescharacteristic
51#387—21no—
52#385—21no—
53#380Q811yescharacteristic
54C4×C4⋊C4—21no—
55C4×C4⋊C4C2×C411yescharacteristic
56C4×C22⋊C4—21no—
57#341—42no—
58#336—21no—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—22no—
64#336—22no—
65#336—22no—
66#335—42no—
67C22×Q8C4⋊C411yescharacteristic
68C23×C4C2×D411yescharacteristic
69C23×C4C2×Q811yescharacteristic
70C23×C4#4111yescharacteristic
71C2×M16—24no—
72C2×M16—42no—
73C2×M16—42no—
74C2×M16—42no—
75C2×M16—42no—
76C2×M16—42no—
77C22×C8—22no—
78C22×C8—22no—
79C22×C8—21no—
80C22×C8—21no—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—41no—
84C22×C8—41no—
85C22×C8—42no—
86C22×C8—42no—
87C22×C8C22⋊C411yescharacteristic
88C22×C8C22⋊C411yescharacteristic
89C22×C8C22⋊C411yescharacteristic
90C22×C8C22⋊C411yescharacteristic
91C22×C8SD1612yes—
92C22×C8Q1612yes—
93C22⋊Q8—42no—
94C4×Q8—41no—
95C4×Q8—41no—
96C2×C4⋊C4—21no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C4⋊C411yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C42—21no—
105C2×C42—21no—
106C2×C42—41no—
107C2×C42—41no—
108C2×C42C22×C411yescharacteristic
109C2×C42C2×D411yescharacteristic
110C2×C42C2×D411yescharacteristic
111C2×C42#4111yescharacteristic
112#108—48no—
113C4⋊C8—41no—
114C4⋊C8—41no—
115C4⋊C8—42no—
116C4⋊C8—42no—
117C4⋊C8—42no—
118C4⋊C8—42no—
119C22⋊C8—41no—
120C22⋊C8—41no—
121C22⋊C8—82no—
122#97—42no—
123#97—42no—
124C4×C8—22no—
125C4×C8—22no—
126C4×C8—41no—
127C4×C8—41no—
128#95—21no—
129#95—21no—
130#95—21no—
131#95—21no—
132#95—21no—
133C24C22⋊Q811yescharacteristic
134C2×Q8—22no—
135C2×Q8—41no—
136C2×Q8—41no—
137C22×C4—21no—
138C22×C4—21no—
139C22×C4—21no—
140C22×C4—21no—
141C22×C4—22no—
142C22×C4—21no—
143C22×C4—21no—
144C22×C4—21no—
145C22×C4—21no—
146C22×C4—21no—
147C22×C4—21no—
148C22×C4—41no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4—41no—
152C22×C4C2×C22⋊C411yescharacteristic
153C22×C4C2×C4⋊C411yescharacteristic
154C22×C4#11711yescharacteristic
155C22×C4#12111yescharacteristic
156C22×C4#12111yescharacteristic
157C22×C4C22⋊Q811yescharacteristic
158C22×C4C2×SD1611yescharacteristic
159C22×C4C2×Q1611yescharacteristic
160C22×C4AGL(1,ℤ/8)11yescharacteristic
161C22×C4#13711yescharacteristic
162M16—44no—
163M16—44no—
164M16—82no—
165M16—82no—
166C2×C8—21no—
167C2×C8—21no—
168C2×C8—24no—
169C2×C8—24no—
170C2×C8—21no—
171C2×C8—21no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—21no—
175C2×C8—21no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—44no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—82no—
186C2×C8—82no—
187C2×C8—82no—
188C2×C8—82no—
189C2×C8—82no—
190C2×C8#9912yes—
191C2×C8#10312yes—
192C2×C8#10312yes—
193C2×C8#10412yes—
194C4⋊C4—22no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—41no—
200C4⋊C4—41no—
201C22⋊C4—42no—
202C22⋊C4—42no—
203C42—21no—
204C42—21no—
205C42—21no—
206C42—21no—
207C42—41no—
208C42—41no—
209C42—41no—
210C42—41no—
211C42—41no—
212C42—41no—
213C23—21no—
214C23—21no—
215C23—21no—
216C23—21no—
217C23—21no—
218C23—21no—
219C23#38011yescharacteristic
220C23#47511yescharacteristic
221C23#47711yescharacteristic
222Q8—82no—
223C2×C4—21no—
224C2×C4—21no—
225C2×C4—21no—
226C2×C4—21no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—21no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—44no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—41no—
254C2×C4—41no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4C2×#9911yescharacteristic
259C2×C4#41011yescharacteristic
260C2×C4C2×#10311yescharacteristic
261C2×C4#41711yescharacteristic
262C2×C4C2×#10411yescharacteristic
263C2×C4#42111yescharacteristic
264C2×C4#43012yes—
265C2×C4#46011yescharacteristic
266C2×C4#46211yescharacteristic
267C2×C4#47211yescharacteristic
268C2×C4#47311yescharacteristic
269C8—22no—
270C8—41no—
271C8—41no—
272C8—84no—
273C8—81no—
274C8—81no—
275C8—162no—
276C8#32814yes—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—21no—
281C22—21no—
282C22—21no—
283C22—41no—
284C22—41no—
285C22—41no—
286C22—41no—
287C22#179911yescharacteristic
288C22#180311yescharacteristic
289C22#180611yescharacteristic
290C22#181111yescharacteristic
291C22#181311yescharacteristic
292C22#181411yescharacteristic
293C22#181611yescharacteristic
294C4—21no—
295C4—21no—
296C4—22no—
297C4—21no—
298C4—21no—
299C4—41no—
300C4—41no—
301C4—81no—
302C4—81no—
303C4—81no—
304C4—81no—
305C4C2×#32811yescharacteristic
306C4#150911yescharacteristic
307C4#151311yescharacteristic
308C4#151811yescharacteristic
309C2—41no—
310C2—41no—
311C2#857211yescharacteristic
312C2#861311yescharacteristic
313C2#862811yescharacteristic
314C2#866311yescharacteristic
315C2#867311yescharacteristic
316C2#868511yescharacteristic
317C2#869311yescharacteristic
subgroupG/H
center, socleC23#380
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
LCSG ▷ C2×C42 ▷ C23 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#215972
#108642
#85962
#76172
#70452