Finite Group Databasebeta

← #134871

Subgroups — #134871

Order 512.

498 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 74 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#21442C211yesmaximal, characteristic
2#11820C212yesmaximal
3#8620C211yesmaximal, characteristic, complemented
4#8608C211yesmaximal, characteristic, complemented
5#7320C212yesmaximal, complemented
6C22×#131C411yescharacteristic, complemented
7C2×C8×D4—21no—
8C2×C8×D4C2211yescharacteristic
9#2388C2211yescharacteristic
10#1870—21no—
11#1870C2211yescharacteristic
12#1851—21no—
13#1851C411yescharacteristic, complemented
14#1807—21no—
15#1799C2212yes—
16#1780C411yescharacteristic, complemented
17C8×C22⋊C4—21no—
18C8×C22⋊C4C411yescharacteristic, complemented
19#1568—21no—
20#1466C2212yes—
21#1351—22no—
22#1343—22no—
23#1290—21no—
24#1289—21no—
25C22×#41C2×C411yescharacteristic
26C2×#131—44no—
27C2×#131—41no—
28C2×#131—41no—
29C22×M16—21no—
30C22×M16C2×C411yescharacteristic
31C23×C8—21no—
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×C4×D4—21no—
35C2×C4×D4C2311yescharacteristic
36C8×D4—41no—
37C8×D4—41no—
38C8×D4—44no—
39#432—22no—
40C2×#106—41no—
41C2×C4⋊C8—21no—
42C2×C4⋊C8D411yescharacteristic
43C2×#99—22no—
44C2×C22⋊C8—21no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8C2×C411yescharacteristic
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8D411yescharacteristic
50C2×C4×C8—21no—
51C2×C4×C8D411yescharacteristic
52#395Q811yescharacteristic
53#394—21no—
54#394Q811yescharacteristic
55#393D411yescharacteristic
56#392—21no—
57#392D411yescharacteristic
58#380—22no—
59C4×C22⋊C4—21no—
60C4×C22⋊C4C2×C411yescharacteristic
61#342—22no—
62#340—41no—
63#336—21no—
64#336—21no—
65#336—21no—
66#335—41no—
67#334—41no—
68#323—22no—
69#323—22no—
70C2×#41—42no—
71C2×#41—42no—
72C2×#41—41no—
73C2×#41—41no—
74C22×Q8C4⋊C411yescharacteristic
75C22×D4—21no—
76C22×D4—21no—
77C22×D4C4⋊C411yescharacteristic
78C22×D4#4111yescharacteristic
79C23×C4—21no—
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82#131—44no—
83#131—44no—
84#131—44no—
85#131—44no—
86C2×M16—42no—
87C2×M16—42no—
88C2×M16—42no—
89C2×M16—42no—
90C2×M16—41no—
91C22×C8—24no—
92C22×C8—21no—
93C22×C8—21no—
94C22×C8—21no—
95C22×C8—21no—
96C22×C8—21no—
97C22×C8—21no—
98C22×C8—42no—
99C22×C8—42no—
100C22×C8—42no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8C22⋊C411yescharacteristic
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C4×D4—41no—
108C4×D4—41no—
109C4×D4—42no—
110C4×D4—42no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—42no—
113C2×C4⋊C4C2×Q811yescharacteristic
114C2×C22⋊C4—21no—
115C2×C22⋊C4—21no—
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—42no—
120C2×C22⋊C4C2×D411yescharacteristic
121C2×C22⋊C4#4111yescharacteristic
122C2×C42—21no—
123C2×C42C2×D411yescharacteristic
124#106—82no—
125C4⋊C8—41no—
126C4⋊C8—42no—
127C4⋊C8—42no—
128#99—82no—
129C22⋊C8—41no—
130C22⋊C8—41no—
131C22⋊C8—42no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—41no—
135C22⋊C8—42no—
136#97—42no—
137C4×C8—21no—
138C4×C8—21no—
139C4×C8—42no—
140#95—21no—
141#95—21no—
142#95—42no—
143#95C4⋊C411yescharacteristic
144#95C4⋊C411yescharacteristic
145C24—21no—
146C24#11711yescharacteristic
147C24#12111yescharacteristic
148#41—42no—
149#41—44no—
150#41—44no—
151#41—42no—
152#41—42no—
153#41—42no—
154C2×Q8—42no—
155C2×Q8—41no—
156C2×D4—22no—
157C2×D4—41no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C2×D4—42no—
163C2×D4—42no—
164C2×D4—42no—
165C2×D4—42no—
166C2×D4—42no—
167C22×C4—22no—
168C22×C4—22no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
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—82no—
185C22×C4C2×C22⋊C411yescharacteristic
186C22×C4C2×C4⋊C411yescharacteristic
187C22×C4#12111yescharacteristic
188C22×C4C22⋊Q811yescharacteristic
189C22×C4C22⋊Q811yescharacteristic
190M16—42no—
191M16—48no—
192M16—42no—
193C2×C8—22no—
194C2×C8—22no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—44no—
199C2×C8—42no—
200C2×C8—41no—
201C2×C8—41no—
202C2×C8—42no—
203C2×C8—44no—
204C2×C8—42no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—41no—
208C2×C8—41no—
209C2×C8—41no—
210C2×C8—41no—
211C2×C8—81no—
212C2×C8—82no—
213C2×C8#9911yescharacteristic
214C2×C8#9911yescharacteristic
215C4⋊C4—41no—
216C4⋊C4—42no—
217C4⋊C4—82no—
218C4⋊C4—82no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—42no—
222C22⋊C4—42no—
223C22⋊C4—42no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—82no—
227C22⋊C4—82no—
228C42—21no—
229C42—21no—
230C42—41no—
231C42—41no—
232C23—21no—
233C23—21no—
234C23—22no—
235C23—21no—
236C23—21no—
237C23—21no—
238C23—41no—
239C23—41no—
240C23—41no—
241C23—41no—
242C23—41no—
243C23—41no—
244C23—41no—
245C23—41no—
246C23—41no—
247C23—41no—
248C23#38011yescharacteristic
249C23#47112yes—
250Q8—42no—
251Q8—42no—
252D4—44no—
253D4—42no—
254D4—48no—
255D4—44no—
256D4—44no—
257D4—42no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—81no—
288C2×C4—162no—
289C2×C4—162no—
290C2×C4—162no—
291C2×C4C2×#9911yescharacteristic
292C2×C4#41011yescharacteristic
293C8—21no—
294C8—42no—
295C8—41no—
296C8—41no—
297C8—41no—
298C8—81no—
299C8—162no—
300C8AΣL(1,𝔽16)12yes—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—42no—
305C22—42no—
306C22—42no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22—42no—
313C22—81no—
314C22—81no—
315C22—81no—
316C22—81no—
317C22—81no—
318C22—81no—
319C22—81no—
320C22#179911yescharacteristic
321C22#181411yescharacteristic
322C22#181511yescharacteristic
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—42no—
328C4—41no—
329C4—41no—
330C4—81no—
331C4—322no—
332C4C2×AΣL(1,𝔽16)11yescharacteristic
333C4#212011yescharacteristic
334C2—21no—
335C2—21no—
336C2—42no—
337C2—81no—
338C2—82no—
339C2#862611yescharacteristic
340C2#1279311yescharacteristic
341C2#1280711yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radical#2388C22
LCSG ▷ C23×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C8×D4 ◁ G
LECSG ▷ C2×C4×D4 ▷ C23 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #8620C2semidirect
4. #8608C2semidirect
5. #7320C2semidirect
6. C22×#131C4semidirect
13. #1851C4semidirect
16. #1780C4semidirect
18. C8×C22⋊C4C4semidirect

Maximal subgroups

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

subgroupindex
#214422
#118202
#86202
#86082
#73202