Finite Group Databasebeta

← #136304

Subgroups — #136304

Order 512.

341 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 341 conjugacy classes of subgroups — too many to draw. Only its 66 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#23315C211yesmaximal, characteristic
2#11727C211yesmaximal, characteristic
3#9011C211yesmaximal, characteristic
4#8864C211yesmaximal, characteristic
5#8852C211yesmaximal, characteristic
6#7283C211yesmaximal, characteristic
7#7204C211yesmaximal, characteristic
8#2973C2211yescharacteristic
9#2395C2211yescharacteristic
10#1955C2211yescharacteristic
11#1939—21no—
12#1923—21no—
13#1923—21no—
14#1917—21no—
15#1869C2211yescharacteristic
16#1850—21no—
17#1850—21no—
18#1826—21no—
19#1798—21no—
20C4×#99C2211yescharacteristic
21C4×#99C2211yescharacteristic
22#1573—21no—
23#1460C2211yescharacteristic
24#1352C411yescharacteristic, complemented
25#1352C411yescharacteristic, complemented
26#1344C411yescharacteristic, complemented
27#1344C411yescharacteristic, complemented
28#1302—21no—
29#1297—21no—
30#1282—21no—
31C23×C8D411yescharacteristic
32C2×C4×D4C2311yescharacteristic
33C2×#117D411yescharacteristic
34#435—41no—
35#435—41no—
36#432—21no—
37#431—21no—
38C2×C4⋊C8—21no—
39C2×C4⋊C8—21no—
40C2×C4⋊C8—21no—
41C2×C4⋊C8—41no—
42C2×C4⋊C8D411yescharacteristic
43C2×#99—21no—
44C2×#99—21no—
45#407—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8D411yescharacteristic
51C2×#97D411yescharacteristic
52C2×C4×C8—21no—
53#400Q811yescharacteristic
54#399C2×C411yescharacteristic
55#398Q811yescharacteristic
56#397—21no—
57#397C2×C411yescharacteristic
58#394—21no—
59C4×C4⋊C4—21no—
60C4×C22⋊C4—21no—
61C4×C22⋊C4—21no—
62C4×C22⋊C4—21no—
63C4×C22⋊C4D411yescharacteristic
64#342—21no—
65#342C2×C411yescharacteristic
66#342C2×C411yescharacteristic
67#339—41no—
68#339—41no—
69#336—21no—
70#336—21no—
71#336—21no—
72#336—21no—
73#336—21no—
74#336—41no—
75#335—41no—
76#334—41no—
77#323—21no—
78#323C2×C411yescharacteristic
79#323C2×C411yescharacteristic
80C22×D4#4111yescharacteristic
81C23×C4C2×D411yescharacteristic
82C23×C4C2×D411yescharacteristic
83C2×M16—21no—
84C2×M16—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—81no—
93C4×D4—41no—
94C4×D4—41no—
95#117—41no—
96#117—41no—
97C2×C4⋊C4—21no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4C4⋊C411yescharacteristic
100C2×C4⋊C4C4⋊C411yescharacteristic
101C2×C4⋊C4C2×Q811yescharacteristic
102C2×C22⋊C4—21no—
103C2×C22⋊C4—21no—
104C2×C22⋊C4—21no—
105C2×C22⋊C4—21no—
106C2×C22⋊C4—21no—
107C2×C22⋊C4—21no—
108C2×C22⋊C4—41no—
109C2×C22⋊C4C22×C411yescharacteristic
110C2×C22⋊C4#4111yescharacteristic
111C2×C42—21no—
112C2×C42—21no—
113C2×C42—41no—
114C2×C42—41no—
115C2×C42C2×D411yescharacteristic
116C4⋊C8—21no—
117C4⋊C8—21no—
118C4⋊C8—41no—
119C4⋊C8—41no—
120C4⋊C8—81no—
121C4⋊C8—81no—
122C4⋊C8—81no—
123C4⋊C8—82no—
124#99—22no—
125#99—42no—
126#99—41no—
127C22⋊C8—21no—
128C22⋊C8—21no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—81no—
134C22⋊C8—81no—
135C22⋊C8—81no—
136#97—41no—
137#97—41no—
138C4×C8—41no—
139C4×C8—81no—
140#95—21no—
141#95—21no—
142#95—21no—
143#95—21no—
144#95—41no—
145#95—41no—
146#95C4⋊C411yescharacteristic
147#95C4⋊C411yescharacteristic
148C24C4×D411yescharacteristic
149C24#12311yescharacteristic
150C2×D4—21no—
151C2×D4—21no—
152C2×D4—41no—
153C2×D4—41no—
154C22×C4—21no—
155C22×C4—21no—
156C22×C4—21no—
157C22×C4—21no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—41no—
163C22×C4—41no—
164C22×C4—41no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—81no—
169C22×C4C2×C4⋊C411yescharacteristic
170C22×C4C4×D411yescharacteristic
171C22×C4#12011yescharacteristic
172C22×C4C22⋊Q811yescharacteristic
173C22×C4C22⋊Q811yescharacteristic
174M16—81no—
175M16—81no—
176C2×C8—41no—
177C2×C8—41no—
178C2×C8—41no—
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—41no—
190C2×C8—41no—
191C2×C8—41no—
192C2×C8—41no—
193C2×C8—81no—
194C2×C8—81no—
195C2×C8—81no—
196C4⋊C4—41no—
197C4⋊C4—41no—
198C4⋊C4—41no—
199C4⋊C4—42no—
200C4⋊C4—81no—
201C22⋊C4—22no—
202C22⋊C4—41no—
203C22⋊C4—41no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—41no—
207C22⋊C4—41no—
208C22⋊C4—41no—
209C22⋊C4—42no—
210C22⋊C4—41no—
211C22⋊C4—41no—
212C22⋊C4—81no—
213C22⋊C4—81no—
214C42—21no—
215C42—21no—
216C42—41no—
217C42—41no—
218C42—41no—
219C42—41no—
220C42—81no—
221C42—81no—
222C42—81no—
223C42—81no—
224C23—21no—
225C23—21no—
226C23—21no—
227C23—21no—
228C23—21no—
229C23—41no—
230C23—41no—
231C23—41no—
232C23—41no—
233C23#38511yescharacteristic
234C23#44311yescharacteristic
235C23#44411yescharacteristic
236D4—81no—
237D4—81no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—22no—
241C2×C4—21no—
242C2×C4—21no—
243C2×C4—21no—
244C2×C4—21no—
245C2×C4—21no—
246C2×C4—21no—
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—42no—
258C2×C4—41no—
259C2×C4—41no—
260C2×C4—41no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—81no—
267C2×C4—81no—
268C2×C4—81no—
269C2×C4—81no—
270C2×C4—81no—
271C2×C4—81no—
272C2×C4—82no—
273C2×C4—81no—
274C2×C4—161no—
275C2×C4—161no—
276C2×C4B(4,𝔽2)11yescharacteristic
277C2×C4#45811yescharacteristic
278C8—81no—
279C8—81no—
280C8—81no—
281C8—81no—
282C8—81no—
283C8—81no—
284C8—81no—
285C8—81no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—81no—
298C22—81no—
299C22—81no—
300C22#186111yescharacteristic
301C22#186611yescharacteristic
302C22#186711yescharacteristic
303C4—21no—
304C4—42no—
305C4—41no—
306C4—41no—
307C4—41no—
308C4—41no—
309C4—81no—
310C4—82no—
311C4—81no—
312C4—81no—
313C4—161no—
314C4—162no—
315C4#220011yescharacteristic
316C4#220111yescharacteristic
317C2—21no—
318C2—21no—
319C2—41no—
320C2—41no—
321C2—81no—
322C2#887511yescharacteristic
323C2#1296111yescharacteristic
324C2#1296811yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C22⋊C4C22×C4
Frattini, cosocleC2×C4×D4C23
socleC22#1867
prime radicalC22×D4#41
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
24. #1352C4semidirect
25. #1352C4semidirect
26. #1344C4semidirect
27. #1344C4semidirect

Maximal subgroups

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

subgroupindex
#233152
#117272
#90112
#88642
#88522
#72832
#72042