Finite Group Databasebeta

← #163972

Subgroups — #163972

Order 512.

1008 conjugacy classes of subgroups.

The subgroup lattice has 383 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2#22643C212yesmaximal, complemented
3#7649C212yesmaximal, complemented
4#7635C212yesmaximal, complemented
5#7624C214yesmaximal, complemented
6#7597C211yesmaximal, characteristic
7#7575C212yesmaximal, complemented
8#7563C211yesmaximal, characteristic
9C2×#536C411yescharacteristic
10C2×#520C411yescharacteristic
11C2×#519C411yescharacteristic
12#2362—22no—
13#2360—22no—
14#2359—24no—
15#2327C412yes—
16#2326C412yes—
17#2317C411yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#124C2212yes—
20#2297C2211yescharacteristic
21C2×#391C2211yescharacteristic
22C4×#117C2211yescharacteristic
23#1939—22no—
24#1924—24no—
25#1839C2214yes—
26#1797C2212yes—
27#1770—22no—
28#1767C2214yes—
29C4×C22⋊C8C2212yes—
30#1272C2212yes—
31#1272C2214yes—
32#1272C2214yes—
33#1272C2212yes—
34#1270C2211yescharacteristic
35#1270C2211yescharacteristic
36#1270C2212yes—
37C22×#41C2×C411yescharacteristic
38C23×Q8C2×C411yescharacteristic
39#536—24no—
40#536—24no—
41C2×C4⋊Q8C2×C412yes—
42C2×#124C2×C412yes—
43C2×#123—22no—
44C2×C22⋊Q8—22no—
45#520—44no—
46#519—44no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×C4×Q8C2×C412yes—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C412yes—
52C2×C4×D4—22no—
53C2×C4×D4C2×C412yes—
54C2×#117—21no—
55C2×#117—22no—
56C2×#117C2×C411yescharacteristic
57C2×#117C2×C411yescharacteristic
58C2×#117C2×C411yescharacteristic
59C22×C4⋊C4—21no—
60C22×C4⋊C4C2×C411yescharacteristic
61C22×C42C2311yescharacteristic
62C2×C4⋊C8—22no—
63C2×C4⋊C8—24no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×#97—24no—
67C2×C4×C8—22no—
68#391—22no—
69#391C2×C412yes—
70#391C2×C411yescharacteristic
71#391C2311yescharacteristic
72#391C2312yes—
73#390C2×C411yescharacteristic
74#390C2311yescharacteristic
75#389—22no—
76#389—24no—
77#388—24no—
78#388—24no—
79#387—24no—
80#387—24no—
81#386—22no—
82#385—22no—
83#385—24no—
84#384C2×C412yes—
85#382C2×C412yes—
86C4×C4⋊C4—24no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—24no—
90C4×C22⋊C4—22no—
91C4×C22⋊C4D414yes—
92#376C2×C412yes—
93C2×#95—22no—
94C43C2312yes—
95#336—24no—
96#336—24no—
97#336C2314yes—
98#336C2312yes—
99#336C2312yes—
100C2×#41—22no—
101C2×#41—42no—
102C2×#41—42no—
103C2×#41M1614yes—
104C22×Q8—21no—
105C22×Q8—22no—
106C22×Q8—42no—
107C22×Q8—42no—
108C22×Q8M1614yes—
109C22×Q8C22×C412yes—
110C22×Q8C22×C411yescharacteristic
111C22×D4—21no—
112C22×D4C22×C411yescharacteristic
113C23×C4—21no—
114C23×C4—21no—
115C23×C4—21no—
116C23×C4—22no—
117C23×C4C22×C411yescharacteristic
118C23×C4C2×D412yes—
119C22×C8—22no—
120C22×C8—22no—
121C22×C8—24no—
122C4⋊Q8—24no—
123C4⋊Q8—24no—
124#124—24no—
125#124—24no—
126#123—216no—
127C22⋊Q8—216no—
128C4×Q8—44no—
129C4×Q8—44no—
130C4×Q8—44no—
131C4×Q8—44no—
132C4×Q8—44no—
133C4×D4—44no—
134C4×D4—48no—
135#117—22no—
136#117—22no—
137#117—24no—
138#117—48no—
139#117—44no—
140#117—42no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—44no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4C22⋊C411yescharacteristic
150C2×C4⋊C4C22⋊C412yes—
151C2×C4⋊C4C22⋊C411yescharacteristic
152C2×C4⋊C4C22×C412yes—
153C2×C4⋊C4C22×C411yescharacteristic
154C2×C4⋊C4C22×C411yescharacteristic
155C2×C22⋊C4—24no—
156C2×C22⋊C4—21no—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4C22⋊C412yes—
161C2×C22⋊C4C22⋊C412yes—
162C2×C42—22no—
163C2×C42—22no—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—42no—
167C2×C42C22×C412yes—
168C2×C42C22×C412yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C2×D414yes—
171C2×C42C2×D412yes—
172C2×C42C2×D414yes—
173C2×C42C2411yescharacteristic
174C4⋊C8—44no—
175C4⋊C8—48no—
176C22⋊C8—44no—
177C22⋊C8—82no—
178#97—84no—
179C4×C8—44no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—24no—
184#95—22no—
185#95C22⋊C414yes—
186#95C22⋊C414yes—
187C24—21no—
188C24C2×C22⋊C411yescharacteristic
189#41—48no—
190#41—48no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—24no—
194C2×Q8—48no—
195C2×Q8—48no—
196C2×Q8—42no—
197C2×Q8—41no—
198C2×Q8—42no—
199C2×Q8—41no—
200C2×Q8—42no—
201C2×Q8—42no—
202C2×Q8C2×M1614yes—
203C2×Q8C2×M1612yes—
204C2×D4—24no—
205C2×D4—21no—
206C2×D4—41no—
207C2×D4—41no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4C2×M1612yes—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—21no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4—84no—
231C22×C4—82no—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C412yes—
236C22×C4C2×C22⋊C411yescharacteristic
237C22×C4C2×M1612yes—
238C22×C4C2×M1612yes—
239C22×C4C23×C411yescharacteristic
240C22×C4C22×D412yes—
241C2×C8—24no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—44no—
247C2×C8—44no—
248C2×C8—44no—
249C2×C8—42no—
250C4⋊C4—216no—
251C4⋊C4—24no—
252C4⋊C4—22no—
253C4⋊C4—42no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—44no—
260C4⋊C4—84no—
261C4⋊C4#10014yes—
262C22⋊C4—216no—
263C22⋊C4—24no—
264C22⋊C4—24no—
265C22⋊C4—44no—
266C22⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—22no—
271C42—22no—
272C42—24no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—44no—
277C42—41no—
278C42—41no—
279C42—44no—
280C42—44no—
281C42—44no—
282C42—42no—
283C23—21no—
284C23—21no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—41no—
290C23—42no—
291C23—42no—
292C23#40712yes—
293C23C22×C22⋊C411yescharacteristic
294Q8—44no—
295Q8—42no—
296Q8—44no—
297Q8—42no—
298Q8—48no—
299D4—42no—
300D4—42no—
301D4—48no—
302C2×C4—22no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—24no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—24no—
315C2×C4—24no—
316C2×C4—22no—
317C2×C4—24no—
318C2×C4—22no—
319C2×C4—48no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—48no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—44no—
331C2×C4—82no—
332C2×C4—82no—
333C2×C4—164no—
334C2×C4—162no—
335C2×C4#40712yes—
336C2×C4#41012yes—
337C2×C4C2×#10014yes—
338C2×C4C2×#10012yes—
339C2×C4C22×M1611yescharacteristic
340C2×C4C22×M1611yescharacteristic
341C2×C4#57712yes—
342C2×C4#57812yes—
343C8—44no—
344C8—84no—
345C8—82no—
346C22—24no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—42no—
351C22—44no—
352C22—42no—
353C22—81no—
354C22—81no—
355C22—82no—
356C22C2×#40711yescharacteristic
357C22C2×#41011yescharacteristic
358C22C22×#10011yescharacteristic
359C22#289812yes—
360C22#290112yes—
361C4—24no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—44no—
367C4—42no—
368C4—42no—
369C4—42no—
370C4—44no—
371C4—81no—
372C4—82no—
373C4—81no—
374C4#146614yes—
375C2—42no—
376C2—82no—
377C2C2×#146611yescharacteristic
378C2#1745412yes—
379C2#1761212yes—
380C2#1768511yescharacteristic
381C2#1777311yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C4×#117 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #22643C2semidirect
3. #7649C2semidirect
4. #7635C2semidirect
5. #7624C2semidirect
7. #7575C2semidirect

Maximal subgroups

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

subgroupindex
#342762
#226432
#76492
#76352
#76242
#75972
#75752
#75632