Finite Group Databasebeta

← #134807

Subgroups — #134807

Order 512.

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

normal subgroups only

This group has 554 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
1C8×#120C211yesmaximal, characteristic
2#11525C212yesmaximal
3#8622C211yesmaximal, characteristic, complemented
4#8612C211yesmaximal, characteristic, complemented
5#7247C212yesmaximal, complemented
6C24×C8C411yescharacteristic
7C2×C8×D4—21no—
8C2×C8×D4C2211yescharacteristic
9C4×#120C2211yescharacteristic
10#1869—21no—
11#1869C2211yescharacteristic
12#1850—21no—
13#1850C411yescharacteristic
14#1810—21no—
15#1779C411yescharacteristic
16C8×C22⋊C4—21no—
17C8×C22⋊C4C411yescharacteristic
18#1568—21no—
19#1466C2212yes—
20#1465C2212yes—
21#1326—22no—
22#1316—22no—
23#1293—21no—
24#1290—21no—
25C24×C4C2×C411yescharacteristic
26C23×C8—21no—
27C23×C8—21no—
28C23×C8—44no—
29C23×C8—41no—
30C23×C8—41no—
31C23×C8C2×C411yescharacteristic
32C23×C8C2×C411yescharacteristic
33C23×C8D411yescharacteristic
34C2×#120Q811yescharacteristic
35C2×C4×D4—21no—
36C2×C4×D4C2311yescharacteristic
37C8×D4—41no—
38C8×D4—41no—
39C8×D4—44no—
40#432—22no—
41C2×#108—41no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8D411yescharacteristic
44#407—22no—
45C2×C22⋊C8—21no—
46C2×C22⋊C8—21no—
47C2×C22⋊C8—21no—
48C2×C22⋊C8C2×C411yescharacteristic
49C2×C22⋊C8C2×C411yescharacteristic
50C2×C22⋊C8D411yescharacteristic
51C2×C4×C8—21no—
52C2×C4×C8D411yescharacteristic
53#386—21no—
54#386Q811yescharacteristic
55#385—21no—
56#385D411yescharacteristic
57#379D411yescharacteristic
58C4×C22⋊C4—21no—
59C4×C22⋊C4C2×C411yescharacteristic
60#341—41no—
61#336—21no—
62#336—21no—
63#336—21no—
64#335—41no—
65#334—41no—
66#324—22no—
67#323—22no—
68#323—22no—
69#323—22no—
70C25C4⋊C411yescharacteristic
71C22×D4—21no—
72C22×D4C2×Q811yescharacteristic
73C23×C4—21no—
74C23×C4—21no—
75C23×C4—42no—
76C23×C4—41no—
77C23×C4—42no—
78C23×C4—41no—
79C23×C4C4⋊C411yescharacteristic
80C23×C4C22×C411yescharacteristic
81C23×C4C2×D411yescharacteristic
82C2×M16—42no—
83C22×C8—24no—
84C22×C8—21no—
85C22×C8—21no—
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—21no—
90C22×C8—42no—
91C22×C8—42no—
92C22×C8—42no—
93C22×C8—42no—
94C22×C8—42no—
95C22×C8—42no—
96C22×C8—42no—
97C22×C8—44no—
98C22×C8—44no—
99C22×C8—44no—
100C22×C8—41no—
101C22×C8—41no—
102C22×C8—41no—
103C22×C8—44no—
104C22×C8C22⋊C411yescharacteristic
105C22×C8C22⋊C411yescharacteristic
106C22×C8C22⋊C411yescharacteristic
107C22×C8C22⋊C411yescharacteristic
108#120—22no—
109#120—22no—
110C4×D4—41no—
111C4×D4—41no—
112C4×D4—42no—
113C4×D4—42no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4#4111yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4C4⋊C411yescharacteristic
121C2×C22⋊C4C4⋊C411yescharacteristic
122C2×C22⋊C4C2×D411yescharacteristic
123C2×C22⋊C4#4111yescharacteristic
124C2×C42—21no—
125C2×C42C2×D411yescharacteristic
126#108—82no—
127C4⋊C8—41no—
128C4⋊C8—42no—
129C4⋊C8—42no—
130C22⋊C8—41no—
131C22⋊C8—41no—
132C22⋊C8—42no—
133C22⋊C8—42no—
134C22⋊C8—42no—
135C22⋊C8—41no—
136C22⋊C8—42no—
137C22⋊C8—42no—
138C22⋊C8—42no—
139#97—42no—
140C4×C8—21no—
141C4×C8—21no—
142C4×C8—42no—
143#95—21no—
144#95—21no—
145#95—21no—
146C24—21no—
147C24—21no—
148C24—42no—
149C24—41no—
150C24—42no—
151C24—41no—
152C24#10812yescomplemented
153C24C2×C4⋊C411yescharacteristic
154C24C22⋊Q811yescharacteristic
155C2×D4—24no—
156C2×D4—21no—
157C2×D4—21no—
158C2×D4—41no—
159C2×D4—41no—
160C2×D4—42no—
161C2×D4—42no—
162C22×C4—22no—
163C22×C4—22no—
164C22×C4—21no—
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—41no—
172C22×C4—41no—
173C22×C4—42no—
174C22×C4—42no—
175C22×C4—41no—
176C22×C4—42no—
177C22×C4—41no—
178C22×C4—41no—
179C22×C4—41no—
180C22×C4—42no—
181C22×C4—42no—
182C22×C4—44no—
183C22×C4—44no—
184C22×C4—42no—
185C22×C4—41no—
186C22×C4—41no—
187C22×C4—41no—
188C22×C4—42no—
189C22×C4—42no—
190C22×C4C2×C22⋊C411yescharacteristic
191C22×C4#11711yescharacteristic
192C22×C4#12111yescharacteristic
193C22×C4#12111yescharacteristic
194C22×C4C22⋊Q811yescharacteristic
195M16—82no—
196M16—82no—
197C2×C8—22no—
198C2×C8—22no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—44no—
203C2×C8—42no—
204C2×C8—41no—
205C2×C8—48no—
206C2×C8—42no—
207C2×C8—41no—
208C2×C8—42no—
209C2×C8—44no—
210C2×C8—42no—
211C2×C8—41no—
212C2×C8—41no—
213C2×C8—41no—
214C2×C8—42no—
215C2×C8—41no—
216C2×C8—41no—
217C2×C8—41no—
218C2×C8—81no—
219C2×C8—82no—
220C2×C8—82no—
221C2×C8#9911yescharacteristic
222C2×C8#9911yescharacteristic
223C4⋊C4—41no—
224C4⋊C4—42no—
225C22⋊C4—24no—
226C22⋊C4—21no—
227C22⋊C4—41no—
228C22⋊C4—41no—
229C22⋊C4—42no—
230C22⋊C4—42no—
231C22⋊C4—42no—
232C22⋊C4—41no—
233C22⋊C4#10812yescomplemented
234C42—21no—
235C42—21no—
236C42—41no—
237C42—41no—
238C23—22no—
239C23—22no—
240C23—21no—
241C23—22no—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—41no—
247C23—44no—
248C23—42no—
249C23—41no—
250C23—41no—
251C23—41no—
252C23—44no—
253C23—42no—
254C23—41no—
255C23—41no—
256C23—42no—
257C23—42no—
258C23—42no—
259C23—48no—
260C23—42no—
261C23—41no—
262C23—41no—
263C23#38011yescharacteristic
264C23C2×#10811yescharacteristic
265C23C2×#10811yescharacteristic
266D4—48no—
267D4—42no—
268D4—42no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—22no—
278C2×C4—21no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—44no—
286C2×C4—44no—
287C2×C4—42no—
288C2×C4—41no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—42no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—81no—
303C2×C4C2×#9911yescharacteristic
304C2×C4#41011yescharacteristic
305C8—21no—
306C8—42no—
307C8—41no—
308C8—41no—
309C8—41no—
310C8—81no—
311C8—162no—
312C8—162no—
313C8AΣL(1,𝔽16)12yes—
314C22—21no—
315C22—21no—
316C22—24no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—42no—
322C22—42no—
323C22—42no—
324C22—42no—
325C22—42no—
326C22—44no—
327C22—44no—
328C22—44no—
329C22—44no—
330C22—42no—
331C22—41no—
332C22—42no—
333C22—42no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—81no—
339C22#179911yescharacteristic
340C22#181011yescharacteristic
341C22#181311yescharacteristic
342C4—21no—
343C4—42no—
344C4—41no—
345C4—41no—
346C4—41no—
347C4—42no—
348C4—41no—
349C4—81no—
350C4C2×AΣL(1,𝔽16)11yescharacteristic
351C4#212011yescharacteristic
352C2—21no—
353C2—21no—
354C2—44no—
355C2—41no—
356C2—41no—
357C2—81no—
358C2#861211yescharacteristic
359C2#1279311yescharacteristic
360C2#1280511yescharacteristic
subgroupG/H
center, socleC22#1799
commutatorC23×C4C22×C4
Frattini, cosocleC2×C4×D4C23
prime radicalC2×#120Q8
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. #8622C2semidirect
4. #8612C2semidirect
5. #7247C2semidirect
152. C24#108semidirect
233. C22⋊C4#108semidirect

Maximal subgroups

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

subgroupindex
C8×#1202
#115252
#86222
#86122
#72472