Finite Group Databasebeta

← #134526

Subgroups — #134526

Order 512.

444 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 94 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#21492C211yesmaximal, characteristic
2#11691C212yesmaximal
3#10866C211yesmaximal, characteristic, complemented
4#8602C211yesmaximal, characteristic, complemented
5#7252C212yesmaximal, complemented
6C2×#435—21no—
7C2×C8×D4C411yescharacteristic, complemented
8C2×#427—21no—
9C2×#405C2211yescharacteristic
10#2392C2211yescharacteristic
11#1955C411yescharacteristic, complemented
12#1940C411yescharacteristic, complemented
13#1923—21no—
14#1869—21no—
15#1798C2211yescharacteristic
16#1770C411yescharacteristic, complemented
17#1752—21no—
18#1740C2212yes—
19C2×H(3,ℤ/4)—21no—
20#1564—21no—
21#1458C2212yes—
22#1344—22no—
23#1343—22no—
24#1275—21no—
25C22×M16D411yescharacteristic
26C23×C8—21no—
27C23×C8D411yescharacteristic
28C2×C4×D4—21no—
29C2×C4×D4C2×C411yescharacteristic
30C2×#117—22no—
31C2×#117C2311yescharacteristic
32#435—42no—
33#435—42no—
34C8×D4—41no—
35C8×D4—41no—
36#431—22no—
37#427—24no—
38#427—42no—
39C2×#107—41no—
40C2×#106—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8C2×C411yescharacteristic
43C2×C4⋊C8C2×C411yescharacteristic
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48#405—41no—
49#405—41no—
50C2×#97—21no—
51C2×#97C2×C411yescharacteristic
52C2×#97D411yescharacteristic
53C2×C4×C8C2×C411yescharacteristic
54C2×C4×C8D411yescharacteristic
55#398Q811yescharacteristic
56#397D411yescharacteristic
57#396—21no—
58#394—21no—
59#394Q811yescharacteristic
60#392D411yescharacteristic
61C4×C4⋊C4C2×C411yescharacteristic
62C4×C22⋊C4—21no—
63#342—22no—
64#342—22no—
65H(3,ℤ/4)—42no—
66#336—21no—
67#336—21no—
68#336—21no—
69#335—41no—
70#334—41no—
71#324—22no—
72#323—22no—
73C22×D4—21no—
74C22×D4C4⋊C411yescharacteristic
75C23×C4—21no—
76C23×C4C2×D411yescharacteristic
77C2×M16—41no—
78C2×M16—41no—
79C2×M16D812yes—
80C2×M16SD1612yes—
81C22×C8—21no—
82C22×C8—21no—
83C22×C8—21no—
84C22×C8—21no—
85C22×C8—22no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—42no—
89C22×C8—42no—
90C22×C8—42no—
91C22×C8C22⋊C411yescharacteristic
92C22×C8C22⋊C411yescharacteristic
93C22×C8C22⋊C411yescharacteristic
94C22×C8C22⋊C411yescharacteristic
95C4×D4—41no—
96C4×D4—41no—
97C4×D4—41no—
98C4×D4—41no—
99C4×D4—41no—
100C4×D4—41no—
101#117—24no—
102#117—41no—
103#117—41no—
104#117—42no—
105#117—42no—
106#117—42no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—42no—
110C2×C4⋊C4C4⋊C411yescharacteristic
111C2×C4⋊C4C2×Q811yescharacteristic
112C2×C4⋊C4#4111yescharacteristic
113C2×C22⋊C4—21no—
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—42no—
119C2×C22⋊C4C2×D411yescharacteristic
120C2×C22⋊C4#4111yescharacteristic
121C2×C42—21no—
122C2×C42—42no—
123C2×C42C22×C411yescharacteristic
124C2×C42C2×D411yescharacteristic
125#107—44no—
126#106—44no—
127C4⋊C8—41no—
128C4⋊C8—41no—
129C4⋊C8—42no—
130C4⋊C8—42no—
131C22⋊C8—41no—
132C22⋊C8—41no—
133C22⋊C8—41no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138#97—41no—
139#97—41no—
140#97—42no—
141C4×C8—21no—
142C4×C8—21no—
143C4×C8—21no—
144C4×C8—21no—
145C4×C8—42no—
146#95—21no—
147#95—21no—
148#95C4⋊C411yescharacteristic
149#95C4⋊C411yescharacteristic
150C24—21no—
151C24#12111yescharacteristic
152C2×D4—41no—
153C2×D4—42no—
154C2×D4—42no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—81no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—21no—
161C22×C4—21no—
162C22×C4—21no—
163C22×C4—21no—
164C22×C4—21no—
165C22×C4—41no—
166C22×C4—41no—
167C22×C4—41no—
168C22×C4—41no—
169C22×C4—41no—
170C22×C4—41no—
171C22×C4—82no—
172C22×C4C2×C22⋊C411yescharacteristic
173C22×C4C2×C4⋊C411yescharacteristic
174C22×C4#11711yescharacteristic
175C22×C4#12111yescharacteristic
176C22×C4C22⋊Q811yescharacteristic
177C22×C4C22⋊Q811yescharacteristic
178C22×C4C2×D811yescharacteristic
179C22×C4C2×SD1611yescharacteristic
180C22×C4AGL(1,ℤ/8)11yescharacteristic
181C22×C4#13711yescharacteristic
182M16—42no—
183M16—42no—
184C2×C8—22no—
185C2×C8—22no—
186C2×C8—22no—
187C2×C8—22no—
188C2×C8—21no—
189C2×C8—21no—
190C2×C8—21no—
191C2×C8—21no—
192C2×C8—21no—
193C2×C8—21no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—41no—
197C2×C8—41no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—84no—
203C2×C8—82no—
204C2×C8#9911yescharacteristic
205C2×C8#9911yescharacteristic
206C2×C8#10212yes—
207C2×C8#10212yes—
208C2×C8#10412yes—
209C4⋊C4—41no—
210C4⋊C4—41no—
211C4⋊C4—41no—
212C4⋊C4—44no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—82no—
216C4⋊C4—81no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—41no—
224C22⋊C4—41no—
225C22⋊C4—44no—
226C22⋊C4—81no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—21no—
231C42—21no—
232C42—44no—
233C42—41no—
234C42—41no—
235C42—82no—
236C23—21no—
237C23—21no—
238C23—22no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23#38011yescharacteristic
248C23#45911yescharacteristic
249C23#46111yescharacteristic
250D4—42no—
251D4—42no—
252D4—82no—
253D4—82no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—84no—
286C2×C4—162no—
287C2×C4—162no—
288C2×C4C2×#9911yescharacteristic
289C2×C4#41011yescharacteristic
290C2×C4C2×#10211yescharacteristic
291C2×C4#41711yescharacteristic
292C2×C4C2×#10411yescharacteristic
293C2×C4#42111yescharacteristic
294C2×C4#47411yescharacteristic
295C2×C4#47611yescharacteristic
296C8—21no—
297C8—21no—
298C8—22no—
299C8—22no—
300C8—22no—
301C8—81no—
302C8—81no—
303C8—162no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—21no—
308C22—41no—
309C22—41no—
310C22—42no—
311C22—42no—
312C22—41no—
313C22—41no—
314C22—42no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#179911yescharacteristic
321C22#180211yescharacteristic
322C22#180611yescharacteristic
323C4—21no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C4—164no—
333C4#150211yescharacteristic
334C4#150411yescharacteristic
335C4#151211yescharacteristic
336C4#151911yescharacteristic
337C2—21no—
338C2—21no—
339C2—41no—
340C2—41no—
341C2—82no—
342C2#857711yescharacteristic
343C2#1280511yescharacteristic
344C2#1280811yescharacteristic
subgroupG/H
centerC2×C4#410
commutatorC2×C42C22×C4
Frattini, cosocleC2×#117C23
socleC22#1799
prime radical#392D4
LCSG ▷ C2×C42 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#405 ◁ G
LECSG ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #10866C2semidirect
4. #8602C2semidirect
5. #7252C2semidirect
7. C2×C8×D4C4semidirect
11. #1955C4semidirect
12. #1940C4semidirect
16. #1770C4semidirect

Maximal subgroups

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

subgroupindex
#214922
#116912
#108662
#86022
#72522