Finite Group Databasebeta

← #161144

Subgroups — #161144

Order 512.

1006 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34026C211yesmaximal, characteristic
2#20810C212yesmaximal, complemented
3#7615C214yesmaximal, complemented
4#7614C214yesmaximal, complemented
5#7598C212yesmaximal, complemented
6#7597C212yesmaximal
7C2×C4×#41C411yescharacteristic
8C2×#432—22no—
9#2298—22no—
10#2297C411yescharacteristic
11#2297C2211yescharacteristic
12#2296C411yescharacteristic
13#2296C2211yescharacteristic
14#2295C412yes—
15#2295C2212yes—
16#2293C412yes—
17#2293C2212yes—
18#2290—22no—
19#2290—22no—
20C4×#117—22no—
21C4×#117C411yescharacteristic
22C2×C43C2211yescharacteristic
23#1842—22no—
24#1831C2214yes—
25#1797—22no—
26#1797C2214yes—
27#1767—22no—
28#1767C2214yes—
29#1272C2214yes—
30#1271C2218yes—
31#1270C2214yes—
32C22×#41C2×C411yescharacteristic
33C4×#41—24no—
34C4×#41—24no—
35C2×C4×Q8C2×C412yes—
36C2×C4×D4—22no—
37C2×C4×D4C2×C412yes—
38C2×#117—21no—
39C2×#117—22no—
40C2×#117—22no—
41C2×#117C2×C411yescharacteristic
42C2×#117C2×C411yescharacteristic
43C2×#117C2×C411yescharacteristic
44C22×C42—21no—
45C22×C42C2×C411yescharacteristic
46C22×C42C2×C411yescharacteristic
47C22×C42D412yes—
48C22×C42D412yes—
49C22×C42C2311yescharacteristic
50#432—48no—
51C2×C4⋊C8—24no—
52C2×C4⋊C8—24no—
53C2×C4⋊C8—24no—
54C2×C22⋊C8—24no—
55C2×#97—24no—
56#391C2×C411yescharacteristic
57#391C2×C412yes—
58#391C2311yescharacteristic
59#390C2×C411yescharacteristic
60#390C2×C412yes—
61#390C2311yescharacteristic
62#389C2×C412yes—
63#389C2×C412yes—
64#389C2×C412yes—
65#389C2312yes—
66#388—22no—
67#388—22no—
68#388—22no—
69#388—22no—
70#387—22no—
71#387—22no—
72#387—22no—
73#387—22no—
74#386—21no—
75#386—21no—
76#386—22no—
77#385—21no—
78#385—21no—
79#385—22no—
80C4×C4⋊C4—22no—
81C4×C4⋊C4—22no—
82C4×C4⋊C4—22no—
83C4×C4⋊C4—22no—
84C4×C4⋊C4C2×C412yes—
85C4×C4⋊C4C2×C412yes—
86C4×C22⋊C4—22no—
87C4×C22⋊C4—22no—
88C4×C22⋊C4—22no—
89C4×C22⋊C4—22no—
90C4×C22⋊C4—22no—
91#376—22no—
92#376—21no—
93#376—22no—
94C43—21no—
95C43C2×C412yes—
96C43D412yes—
97C43D412yes—
98C43C2312yes—
99#336—24no—
100#336C2318yes—
101C2×#41—22no—
102C2×#41—42no—
103C2×#41—42no—
104C2×#41M1614yes—
105C22×Q8C22×C411yescharacteristic
106C22×D4—21no—
107C22×D4C22×C411yescharacteristic
108C23×C4—22no—
109C23×C4—21no—
110C23×C4C22⋊C412yes—
111C23×C4C22⋊C411yescharacteristic
112C23×C4C22⋊C411yescharacteristic
113C23×C4C22×C411yescharacteristic
114C23×C4C2×D412yes—
115C22×C8—28no—
116C4×Q8—24no—
117C4×Q8—24no—
118C4×D4—216no—
119C4×D4—24no—
120C4×D4—24no—
121#117—28no—
122#117—22no—
123#117—48no—
124#117—44no—
125#117—44no—
126#117—42no—
127#117—42no—
128#117M1614yes—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4C22×C412yes—
135C2×C4⋊C4C22×C411yescharacteristic
136C2×C4⋊C4C22×C411yescharacteristic
137C2×C4⋊C4C22×C412yes—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—21no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—22no—
146C2×C42—28no—
147C2×C42—21no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—22no—
151C2×C42—24no—
152C2×C42—24no—
153C2×C42—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42—21no—
157C2×C42—42no—
158C2×C42—44no—
159C2×C42—42no—
160C2×C42C22⋊C412yes—
161C2×C42C22⋊C411yescharacteristic
162C2×C42C22⋊C411yescharacteristic
163C2×C42C22⋊C412yes—
164C2×C42C22⋊C412yes—
165C2×C42C22⋊C412yes—
166C2×C42C22⋊C412yes—
167C2×C42C22×C412yes—
168C2×C42C22×C411yescharacteristic
169C2×C42C22×C412yes—
170C2×C42C2×D412yes—
171C2×C42C2×D412yes—
172C2×C42C2×D412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D412yes—
175C2×C42C2411yescharacteristic
176C4⋊C8—48no—
177C4⋊C8—48no—
178C4⋊C8—48no—
179C22⋊C8—48no—
180#97—48no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191C24—21no—
192C24C2×C22⋊C411yescharacteristic
193#41—48no—
194#41—48no—
195C2×Q8—21no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8C2×M1612yes—
199C2×D4—24no—
200C2×D4—21no—
201C2×D4—41no—
202C2×D4—41no—
203C2×D4—42no—
204C2×D4—42no—
205C2×D4C2×M1612yes—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—28no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—24no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—44no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—44no—
230C22×C4C2×C22⋊C412yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C2×C22⋊C412yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×M1612yes—
237C22×C4C2×M1612yes—
238C22×C4C23×C411yescharacteristic
239C22×C4C22×D412yes—
240C2×C8—48no—
241C2×C8—48no—
242C2×C8—48no—
243C4⋊C4—28no—
244C4⋊C4—22no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—41no—
248C4⋊C4—44no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C4⋊C4—44no—
253C4⋊C4C2×M1614yes—
254C22⋊C4—28no—
255C22⋊C4—24no—
256C22⋊C4—44no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—44no—
262C22⋊C4—44no—
263C42—28no—
264C42—22no—
265C42—22no—
266C42—22no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—22no—
271C42—24no—
272C42—24no—
273C42—24no—
274C42—24no—
275C42—24no—
276C42—24no—
277C42—42no—
278C42—41no—
279C42—42no—
280C42—42no—
281C42—41no—
282C42—44no—
283C42—42no—
284C42—42no—
285C42—44no—
286C42AGL(1,ℤ/8)14yes—
287C42#13714yes—
288C23—21no—
289C23—21no—
290C23—22no—
291C23—22no—
292C23—22no—
293C23—41no—
294C23—41no—
295C23—42no—
296C23—42no—
297C23#40712yes—
298C23C22×C22⋊C411yescharacteristic
299Q8—42no—
300Q8—42no—
301D4—42no—
302D4—42no—
303D4—48no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—21no—
310C2×C4—22no—
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—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—44no—
325C2×C4—44no—
326C2×C4—48no—
327C2×C4—416no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—42no—
334C2×C4—48no—
335C2×C4—42no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4#40712yes—
343C2×C4#41012yes—
344C2×C4#41312yes—
345C2×C4#41714yes—
346C2×C4#41714yes—
347C2×C4#42114yes—
348C2×C4C22×M1611yescharacteristic
349C2×C4C22×M1611yescharacteristic
350C2×C4C2×AGL(1,ℤ/8)12yes—
351C2×C4C2×#13712yes—
352C8—88no—
353C22—24no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—42no—
358C22—44no—
359C22—42no—
360C22—81no—
361C22—81no—
362C22—82no—
363C22C2×#40711yescharacteristic
364C22C2×#41011yescharacteristic
365C22C2×#41311yescharacteristic
366C22C2×#41712yes—
367C22C2×#42112yes—
368C4—24no—
369C4—24no—
370C4—24no—
371C4—22no—
372C4—22no—
373C4—21no—
374C4—21no—
375C4—42no—
376C4—44no—
377C4—42no—
378C4—42no—
379C4—81no—
380C4—81no—
381C4—82no—
382C4#146414yes—
383C2—42no—
384C2—82no—
385C2C2×#146411yescharacteristic
386C2#1745112yes—
387C2#1756312yes—
388C2#1767811yescharacteristic
389C2#1775211yescharacteristic
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C2×C43 ◁ G
LECSG ▷ C2×C42 ▷ C23 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #20810C2semidirect
3. #7615C2semidirect
4. #7614C2semidirect
5. #7598C2semidirect

Maximal subgroups

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

subgroupindex
#340262
#208102
#76152
#76142
#75982
#75972