Finite Group Databasebeta

← #136174

Subgroups — #136174

Order 512.

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

normal subgroups only

This group has 370 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#22298C211yesmaximal, characteristic
2#11715C211yesmaximal, characteristic
3#11345C211yesmaximal, characteristic
4#9033C211yesmaximal, characteristic
5#8861C211yesmaximal, characteristic
6#7252C211yesmaximal, characteristic
7#7188C211yesmaximal, characteristic
8#2905C2211yescharacteristic
9C2×#424—21no—
10#2396C2211yescharacteristic
11#1985—21no—
12#1939—21no—
13#1917—21no—
14#1916—21no—
15#1850—21no—
16#1798C2211yescharacteristic
17#1753—21no—
18C2×#344—21no—
19#1740C2211yescharacteristic
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#424—42no—
39#424—42no—
40#423—21no—
41#423—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—42no—
45C2×C4⋊C8D411yescharacteristic
46C2×C4⋊C8D411yescharacteristic
47C2×C22⋊C8—21no—
48C2×#97—21no—
49C2×C4×C8—21no—
50#400Q811yescharacteristic
51#399C2×C411yescharacteristic
52#398—21no—
53#396—21no—
54#395Q811yescharacteristic
55#393C2×C411yescharacteristic
56C4×C4⋊C4—21no—
57C4×C4⋊C4D411yescharacteristic
58C4×C22⋊C4—21no—
59#344—42no—
60#342—21no—
61#342C2×C412yes—
62#336—21no—
63#336—21no—
64#336—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#335—42no—
70Q8⋊C8—42no—
71#324—21no—
72#324—21no—
73#323—21no—
74#323C2×C412yes—
75C22×Q8C4⋊C411yescharacteristic
76C23×C4C2×D411yescharacteristic
77C2×M16—21no—
78C2×M16—41no—
79C2×M16—41no—
80C2×M16—41no—
81C2×M16—41no—
82C2×M16—41no—
83C2×M16—41no—
84C2×M16—41no—
85C2×M16—41no—
86C2×M16Q1612yes—
87C22×C8—21no—
88C22×C8—22no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—21no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—41no—
95C22×C8—41no—
96C4×Q8—41no—
97C4×Q8—41no—
98#117—22no—
99#117—22no—
100#117—41no—
101#117—41no—
102#117—41no—
103#117—41no—
104C2×C4⋊C4—21no—
105C2×C4⋊C4—21no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—41no—
109C2×C4⋊C4C4⋊C411yescharacteristic
110C2×C4⋊C4C2×Q811yescharacteristic
111C2×C4⋊C4#4111yescharacteristic
112C2×C22⋊C4—22no—
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4C22×C411yescharacteristic
116C2×C22⋊C4#4111yescharacteristic
117C2×C42—21no—
118C2×C42—21no—
119C2×C42—41no—
120C2×C42C2×D411yescharacteristic
121C2×C42C2×D411yescharacteristic
122C4⋊C8—22no—
123C4⋊C8—22no—
124C4⋊C8—41no—
125C4⋊C8—41no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—48no—
131C4⋊C8—81no—
132C4⋊C8—81no—
133C22⋊C8—22no—
134C22⋊C8—41no—
135C22⋊C8—41no—
136C22⋊C8—41no—
137#97—81no—
138C4×C8—41no—
139C4×C8—41no—
140C4×C8—81no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—21no—
145#95C4⋊C411yescharacteristic
146#95C4⋊C411yescharacteristic
147C24C4×D411yescharacteristic
148C2×Q8—41no—
149C2×Q8—81no—
150C22×C4—21no—
151C22×C4—21no—
152C22×C4—21no—
153C22×C4—21no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—42no—
159C22×C4C2×C4⋊C411yescharacteristic
160C22×C4C4×D411yescharacteristic
161C22×C4#12011yescharacteristic
162C22×C4C22⋊Q811yescharacteristic
163C22×C4C22⋊Q811yescharacteristic
164C22×C4#12311yescharacteristic
165C22×C4C2×Q1611yescharacteristic
166C22×C4#13511yescharacteristic
167C22×C4AGL(1,ℤ/8)11yescharacteristic
168C22×C4#13711yescharacteristic
169M16—42no—
170M16—81no—
171M16—81no—
172M16—81no—
173M16—81no—
174M16—81no—
175C2×C8—22no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—42no—
179C2×C8—42no—
180C2×C8—42no—
181C2×C8—42no—
182C2×C8—41no—
183C2×C8—41no—
184C2×C8—41no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—42no—
189C2×C8—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—81no—
193C2×C8—81no—
194C4⋊C4—41no—
195C4⋊C4—41no—
196C4⋊C4—41no—
197C4⋊C4—42no—
198C4⋊C4—42no—
199C4⋊C4—42no—
200C4⋊C4—42no—
201C4⋊C4—81no—
202C4⋊C4—81no—
203C22⋊C4—24no—
204C22⋊C4—41no—
205C22⋊C4—41no—
206C22⋊C4—42no—
207C22⋊C4—42no—
208C22⋊C4—81no—
209C42—21no—
210C42—21no—
211C42—21no—
212C42—21no—
213C42—42no—
214C42—42no—
215C42—41no—
216C42—41no—
217C42—41no—
218C42—41no—
219C23—21no—
220C23—21no—
221C23—41no—
222C23—41no—
223C23#38511yescharacteristic
224C23C4×Q1611yescharacteristic
225C23#44011yescharacteristic
226Q8—42no—
227Q8—81no—
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—22no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—41no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—41no—
249C2×C4—44no—
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×C4—41no—
259C2×C4—82no—
260C2×C4—82no—
261C2×C4#44911yescharacteristic
262C2×C4#45111yescharacteristic
263C2×C4#45311yescharacteristic
264C2×C4#45611yescharacteristic
265C2×C4B(4,𝔽2)11yescharacteristic
266C2×C4#45811yescharacteristic
267C2×C4#47711yescharacteristic
268C2×C4#47911yescharacteristic
269C8—42no—
270C8—82no—
271C8—81no—
272C8—81no—
273C8—82no—
274C8—81no—
275C8—161no—
276C22—21no—
277C22—21no—
278C22—21no—
279C22—21no—
280C22—41no—
281C22—41no—
282C22—41no—
283C22—41no—
284C22—81no—
285C22#185911yescharacteristic
286C22#186311yescharacteristic
287C22#186711yescharacteristic
288C4—21no—
289C4—21no—
290C4—42no—
291C4—41no—
292C4—41no—
293C4—41no—
294C4—41no—
295C4—81no—
296C4—84no—
297C4—81no—
298C4—81no—
299C4#160011yescharacteristic
300C4#160511yescharacteristic
301C4#160911yescharacteristic
302C4#161411yescharacteristic
303C2—21no—
304C2—21no—
305C2—41no—
306C2—41no—
307C2#886811yescharacteristic
308C2#1296311yescharacteristic
309C2#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
#222982
#117152
#113452
#90332
#88612
#72522
#71882