Finite Group Databasebeta

← #136080

Subgroups — #136080

Order 512.

429 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 429 conjugacy classes of subgroups — too many to draw. Only its 78 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#21908C211yesmaximal, characteristic
2#11425C211yesmaximal, characteristic
3#11071C211yesmaximal, characteristic
4#8862C211yesmaximal, characteristic, complemented
5#8861C211yesmaximal, characteristic, complemented
6#7253C211yesmaximal, characteristic, complemented
7#7253C211yesmaximal, characteristic, complemented
8#2905C2211yescharacteristic
9C2×#407—21no—
10#2280C2211yescharacteristic
11#1924—21no—
12#1924—21no—
13#1923—21no—
14#1923—21no—
15#1798C2211yescharacteristic
16#1798C2211yescharacteristic
17#1779—21no—
18#1750—21no—
19C2×#344—21no—
20#1742C2211yescharacteristic
21#1571—21no—
22#1570—21no—
23#1458C2211yescharacteristic
24#1458C2211yescharacteristic
25#1317C412yes—
26#1316C412yes—
27#1301—21no—
28#1301—21no—
29C24×C4D411yescharacteristic
30C22×M16—21no—
31C22×M16—21no—
32C2×#117D411yescharacteristic
33C2×#117C2311yescharacteristic
34#431—21no—
35#431—21no—
36#424—42no—
37#423—21no—
38#423—21no—
39C2×C4⋊C8D411yescharacteristic
40C2×C4⋊C8D411yescharacteristic
41C2×C4⋊C8D411yescharacteristic
42C2×C4⋊C8D411yescharacteristic
43#407—42no—
44#407—42no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8—21no—
49C2×C22⋊C8—21no—
50C2×C22⋊C8—21no—
51C2×C22⋊C8—42no—
52C2×#97—21no—
53C2×C4×C8—21no—
54#381C2×C411yescharacteristic
55#381Q811yescharacteristic
56#380—21no—
57#380—21no—
58#379C2×C411yescharacteristic
59#379Q811yescharacteristic
60C4×C22⋊C4—21no—
61C4×C22⋊C4—21no—
62#344—42no—
63#343—21no—
64#343—21no—
65#336—21no—
66#336—21no—
67#336—21no—
68#336—21no—
69#335—42no—
70#334—42no—
71#324—21no—
72#324—21no—
73#323C2×C412yes—
74#323C2×C412yes—
75C25C4⋊C411yescharacteristic
76C23×C4—21no—
77C23×C4—21no—
78C23×C4—21no—
79C23×C4—21no—
80C23×C4—21no—
81C23×C4—21no—
82C23×C4—41no—
83C23×C4—41no—
84C23×C4—41no—
85C23×C4—41no—
86C23×C4C4⋊C411yescharacteristic
87C23×C4C2×D411yescharacteristic
88C2×M16—22no—
89C2×M16—41no—
90C2×M16—41no—
91C2×M16—41no—
92C2×M16—41no—
93C2×M16—41no—
94C2×M16—41no—
95C2×M16—41no—
96C2×M16—41no—
97C2×M16—41no—
98C2×M16—41no—
99C2×M16—41no—
100C22×C8—22no—
101C22×C8—21no—
102C22×C8—21no—
103C22×C8—21no—
104C22×C8—21no—
105C22×C8—41no—
106#117—41no—
107#117—41no—
108#117—41no—
109#117—41no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4#4111yescharacteristic
112C2×C4⋊C4#4111yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4C4⋊C411yescharacteristic
118C2×C22⋊C4C4⋊C411yescharacteristic
119C2×C22⋊C4C22×C411yescharacteristic
120C2×C22⋊C4C2×Q811yescharacteristic
121C2×C42—21no—
122C2×C42C2×D411yescharacteristic
123C2×C42C2×D411yescharacteristic
124C4⋊C8—22no—
125C4⋊C8—22no—
126C4⋊C8—41no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—41no—
130C4⋊C8—41no—
131C4⋊C8—41no—
132C4⋊C8—41no—
133C4⋊C8—41no—
134C22⋊C8—22no—
135C22⋊C8—22no—
136C22⋊C8—48no—
137C22⋊C8—41no—
138C22⋊C8—41no—
139C22⋊C8—41no—
140C22⋊C8—41no—
141C22⋊C8—81no—
142C22⋊C8—81no—
143C22⋊C8—81no—
144C22⋊C8—81no—
145#97—81no—
146C4×C8—41no—
147C4×C8—41no—
148C4×C8—81no—
149#95—21no—
150#95—21no—
151#95—21no—
152#95—21no—
153C24—21no—
154C24—21no—
155C24—21no—
156C24—41no—
157C24—41no—
158C24—42no—
159C24—81no—
160C24C2×C4⋊C411yescharacteristic
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—42no—
171C22×C4—42no—
172C22×C4—42no—
173C22×C4—42no—
174C22×C4—41no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—41no—
181C22×C4—41no—
182C22×C4—41no—
183C22×C4—41no—
184C22×C4—81no—
185C22×C4—81no—
186C22×C4—81no—
187C22×C4—81no—
188C22×C4—81no—
189C22×C4—81no—
190C22×C4C4×D411yescharacteristic
191C22×C4C4×D411yescharacteristic
192C22×C4#12011yescharacteristic
193C22×C4C22⋊Q811yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195C22×C4#12311yescharacteristic
196C22×C4#13511yescharacteristic
197C22×C4#13511yescharacteristic
198C22×C4AGL(1,ℤ/8)11yescharacteristic
199C22×C4#13711yescharacteristic
200M16—42no—
201M16—42no—
202M16—81no—
203M16—81no—
204M16—81no—
205M16—81no—
206M16—81no—
207M16—81no—
208C2×C8—42no—
209C2×C8—42no—
210C2×C8—42no—
211C2×C8—42no—
212C2×C8—42no—
213C2×C8—41no—
214C2×C8—41no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—41no—
220C2×C8—41no—
221C2×C8—41no—
222C2×C8—41no—
223C2×C8—42no—
224C2×C8—41no—
225C2×C8—41no—
226C2×C8—81no—
227C2×C8—81no—
228C4⋊C4—41no—
229C4⋊C4—41no—
230C4⋊C4—81no—
231C22⋊C4—41no—
232C22⋊C4—41no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—41no—
236C22⋊C4—41no—
237C22⋊C4—81no—
238C22⋊C4—81no—
239C42—21no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—41no—
244C42—41no—
245C23—21no—
246C23—21no—
247C23—22no—
248C23—41no—
249C23—41no—
250C23—41no—
251C23—42no—
252C23—41no—
253C23—41no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23—81no—
260C23—82no—
261C23—81no—
262C23—82no—
263C23—84no—
264C23—81no—
265C23#38511yescharacteristic
266C23#42711yescharacteristic
267C23#42811yescharacteristic
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—81no—
302C2×C4—81no—
303C2×C4#44911yescharacteristic
304C2×C4#45211yescharacteristic
305C2×C4#45311yescharacteristic
306C2×C4#45511yescharacteristic
307C2×C4B(4,𝔽2)11yescharacteristic
308C2×C4#45811yescharacteristic
309C2×C4#48211yescharacteristic
310C2×C4#48511yescharacteristic
311C8—82no—
312C8—82no—
313C8—82no—
314C8—81no—
315C8—82no—
316C8—81no—
317C8—81no—
318C8—81no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—41no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22—41no—
329C22—44no—
330C22—42no—
331C22—81no—
332C22—81no—
333C22—81no—
334C22—82no—
335C22—82no—
336C22—82no—
337C22—81no—
338C22—81no—
339C22#185411yescharacteristic
340C22#185611yescharacteristic
341C22#186711yescharacteristic
342C4—21no—
343C4—21no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—81no—
349C4—81no—
350C4—81no—
351C4#160011yescharacteristic
352C4#160611yescharacteristic
353C4#161211yescharacteristic
354C4#161811yescharacteristic
355C2—21no—
356C2—21no—
357C2—41no—
358C2—41no—
359C2—82no—
360C2#885811yescharacteristic
361C2#1296211yescharacteristic
362C2#1296311yescharacteristic
subgroupG/H
centerC2×C4B(4,𝔽2)
commutatorC2×C22⋊C4C22×C4
Frattini, cosocleC2×#117C23
socleC22#1867
prime radicalC25C4⋊C4
LCSG ▷ C2×C22⋊C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#117 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #8862C2semidirect
5. #8861C2semidirect
6. #7253C2semidirect
7. #7253C2semidirect

Maximal subgroups

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

subgroupindex
#219082
#114252
#110712
#88622
#88612
#72532