Finite Group Databasebeta

← #136162

Subgroups — #136162

Order 512.

351 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 351 conjugacy classes of subgroups — too many to draw. Only its 80 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#22288C211yesmaximal, characteristic
2#11715C211yesmaximal, characteristic
3#11288C211yesmaximal, characteristic
4#9033C211yesmaximal, characteristic
5#8861C211yesmaximal, characteristic
6#7252C211yesmaximal, characteristic
7#7188C211yesmaximal, characteristic
8#2905C2211yescharacteristic
9#2396C2211yescharacteristic
10#1984—21no—
11#1939—21no—
12#1916—21no—
13#1916—21no—
14#1851—21no—
15#1798C2211yescharacteristic
16C8×C22⋊C4—21no—
17C2×#344—21no—
18#1740C2211yescharacteristic
19#1729—21no—
20#1729C2211yescharacteristic
21#1571—21no—
22#1487—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1352C412yes—
26#1351C412yes—
27#1303—21no—
28#1301—21no—
29C22×M16—21no—
30C22×M16D411yescharacteristic
31C23×C8D411yescharacteristic
32C2×C4×Q8D411yescharacteristic
33C2×#117—21no—
34C2×#117C2311yescharacteristic
35#431—21no—
36#431—21no—
37#424—42no—
38#423—21no—
39#423—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×C4⋊C8D411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C4×C8—21no—
46C2×C4×C8—21no—
47#400Q811yescharacteristic
48#399C2×C411yescharacteristic
49#398—21no—
50#396—21no—
51#395Q811yescharacteristic
52#393C2×C411yescharacteristic
53C4×C4⋊C4—21no—
54C4×C4⋊C4D411yescharacteristic
55C4×C22⋊C4—21no—
56#344—42no—
57#342—21no—
58#342C2×C412yes—
59#336—21no—
60#336—21no—
61#336—21no—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—42no—
67#335—42no—
68Q8⋊C8—42no—
69#324—21no—
70#324—21no—
71#323—21no—
72#323C2×C412yes—
73C22×Q8C4⋊C411yescharacteristic
74C23×C4C2×D411yescharacteristic
75C2×M16—21no—
76C2×M16—21no—
77C2×M16—41no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—22no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93C22×C8Q1612yes—
94C4×Q8—41no—
95C4×Q8—41no—
96#117—41no—
97#117—41no—
98#117—41no—
99#117—41no—
100#117—41no—
101#117—41no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—41no—
107C2×C4⋊C4C4⋊C411yescharacteristic
108C2×C4⋊C4C2×Q811yescharacteristic
109C2×C4⋊C4#4111yescharacteristic
110C2×C22⋊C4—21no—
111C2×C22⋊C4—21no—
112C2×C22⋊C4—22no—
113C2×C22⋊C4C22×C411yescharacteristic
114C2×C22⋊C4#4111yescharacteristic
115C2×C42—21no—
116C2×C42—21no—
117C2×C42—41no—
118C2×C42C2×D411yescharacteristic
119C2×C42C2×D411yescharacteristic
120C4⋊C8—22no—
121C4⋊C8—22no—
122C4⋊C8—41no—
123C4⋊C8—41no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—81no—
129C4⋊C8—81no—
130C22⋊C8—22no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C4×C8—41no—
135C4×C8—41no—
136C4×C8—81no—
137C4×C8—81no—
138#95—21no—
139#95—21no—
140#95—21no—
141#95—21no—
142#95C4⋊C411yescharacteristic
143#95C4⋊C411yescharacteristic
144C24C4×D411yescharacteristic
145C2×Q8—42no—
146C2×Q8—41no—
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—42no—
156C22×C4C2×C4⋊C411yescharacteristic
157C22×C4C4×D411yescharacteristic
158C22×C4#12011yescharacteristic
159C22×C4C22⋊Q811yescharacteristic
160C22×C4C22⋊Q811yescharacteristic
161C22×C4#12311yescharacteristic
162C22×C4C2×Q1611yescharacteristic
163C22×C4#13511yescharacteristic
164C22×C4AGL(1,ℤ/8)11yescharacteristic
165C22×C4#13711yescharacteristic
166M16—42no—
167M16—81no—
168M16—81no—
169M16—81no—
170M16—81no—
171M16—81no—
172C2×C8—22no—
173C2×C8—41no—
174C2×C8—41no—
175C2×C8—42no—
176C2×C8—42no—
177C2×C8—42no—
178C2×C8—42no—
179C2×C8—41no—
180C2×C8—41no—
181C2×C8—41no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—42no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—81no—
190C2×C8—81no—
191C4⋊C4—41no—
192C4⋊C4—41no—
193C4⋊C4—41no—
194C4⋊C4—42no—
195C4⋊C4—42no—
196C4⋊C4—42no—
197C4⋊C4—81no—
198C4⋊C4—81no—
199C4⋊C4—81no—
200C22⋊C4—41no—
201C22⋊C4—41no—
202C22⋊C4—42no—
203C22⋊C4—42no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C42—21no—
207C42—21no—
208C42—21no—
209C42—21no—
210C42—41no—
211C42—41no—
212C42—41no—
213C42—41no—
214C42—81no—
215C42—81no—
216C23—21no—
217C23—21no—
218C23—41no—
219C23—41no—
220C23#38511yescharacteristic
221C23C4×Q1611yescharacteristic
222C23#44011yescharacteristic
223Q8—42no—
224Q8—81no—
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—22no—
236C2×C4—22no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—41no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
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—82no—
256C2×C4—82no—
257C2×C4—82no—
258C2×C4#44911yescharacteristic
259C2×C4#45111yescharacteristic
260C2×C4#45411yescharacteristic
261C2×C4#45511yescharacteristic
262C2×C4B(4,𝔽2)11yescharacteristic
263C2×C4#45811yescharacteristic
264C2×C4#47711yescharacteristic
265C2×C4#47911yescharacteristic
266C8—42no—
267C8—82no—
268C8—81no—
269C8—82no—
270C8—81no—
271C8—81no—
272C8—161no—
273C22—21no—
274C22—21no—
275C22—21no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—41no—
280C22—41no—
281C22—81no—
282C22#186011yescharacteristic
283C22#186311yescharacteristic
284C22#186711yescharacteristic
285C4—21no—
286C4—21no—
287C4—42no—
288C4—41no—
289C4—42no—
290C4—41no—
291C4—41no—
292C4—41no—
293C4—81no—
294C4—81no—
295C4—162no—
296C4#159711yescharacteristic
297C4#160211yescharacteristic
298C4#161211yescharacteristic
299C4#161711yescharacteristic
300C2—21no—
301C2—21no—
302C2—41no—
303C2—41no—
304C2#886711yescharacteristic
305C2#1296211yescharacteristic
306C2#1296811yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C22⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC24C4×D4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Maximal subgroups

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

subgroupindex
#222882
#117152
#112882
#90332
#88612
#72522
#71882