Finite Group Databasebeta

← #480314

Subgroups — #480314

Order 512.

725 conjugacy classes of subgroups.

The subgroup lattice has 351 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60123C211yesmaximal, characteristic
2C2×#2165C212yesmaximal, complemented
3#11522C214yesmaximal, complemented
4#11521C214yesmaximal, complemented
5#11456C212yesmaximal, complemented
6#11436C212yesmaximal
7C2×C4×#41C411yescharacteristic
8#2923—22no—
9#2923—22no—
10#2922—22no—
11#2922—22no—
12#2921C412yes—
13#2921C2212yes—
14#2920C412yes—
15#2920C2212yes—
16#2905C411yescharacteristic
17#2905C2211yescharacteristic
18#2879C411yescharacteristic
19#2879C2211yescharacteristic
20C2×C4×M16C411yescharacteristic
21C2×C4×M16C2211yescharacteristic
22#2165—22no—
23#2165—22no—
24#2165C2214yes—
25#2165C2214yes—
26C2×#364—22no—
27C2×#349C2214yes—
28C2×#347—22no—
29#1327C2218yes—
30#1320C2218yes—
31C22×#41C2×C411yescharacteristic
32C22×M16C2×C411yescharacteristic
33C22×M16C2×C411yescharacteristic
34C22×M16C2×C411yescharacteristic
35C22×M16C2311yescharacteristic
36C4×#41—22no—
37C4×#41—22no—
38C4×#41—21no—
39C4×#41—21no—
40C4×#41—21no—
41C4×#41—21no—
42C2×C4×Q8—21no—
43C2×C4×D4—22no—
44C2×C4×D4—21no—
45C2×#117C2×C412yes—
46C2×#117C2×C411yescharacteristic
47C22×C42C2×C412yes—
48C22×C42C2×C411yescharacteristic
49C2×M32—24no—
50#433—24no—
51#433—24no—
52#433—24no—
53#433—24no—
54#432—24no—
55#432—24no—
56#431—24no—
57#431—24no—
58#424—22no—
59#424—22no—
60#424—22no—
61#424—22no—
62#423C2×C412yes—
63#423C2×C412yes—
64#423C2×C412yes—
65#423C2312yes—
66#408—22no—
67#408—22no—
68#408—22no—
69#408—22no—
70#407C2×C412yes—
71#407C2×C412yes—
72#407C2×C412yes—
73#407C2312yes—
74C4×M16—21no—
75C4×M16—21no—
76C4×M16C2×C412yes—
77C4×M16C2×C412yes—
78C4×M16C2×C411yescharacteristic
79C4×M16C2×C411yescharacteristic
80C4×M16D412yes—
81C4×M16D412yes—
82C4×M16C2311yescharacteristic
83C4×M16C2311yescharacteristic
84C2×#97—21no—
85C2×#97D412yes—
86C2×C4×C8—21no—
87C2×C4×C8D412yes—
88#364—24no—
89#364—24no—
90#349—24no—
91#349C2318yes—
92#347—24no—
93#347—24no—
94C2×#41—21no—
95C2×#41—21no—
96C2×#41—21no—
97C2×#41—21no—
98C2×#41—42no—
99C2×#41—42no—
100C22×Q8M1611yescharacteristic
101C22×D4M1611yescharacteristic
102C22×D4M1612yes—
103C23×C4—22no—
104C23×C4—21no—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C412yes—
107C2×M16—21no—
108C2×M16—21no—
109C2×M16—42no—
110C2×M16—42no—
111C2×M16—42no—
112C2×M16—42no—
113C2×M16C22⋊C412yes—
114C2×M16C22×C412yes—
115C2×M16C22×C412yes—
116C2×M16C22×C411yescharacteristic
117C2×M16C22×C411yescharacteristic
118C2×M16C22×C411yescharacteristic
119C2×M16C2×D412yes—
120C2×M16C2411yescharacteristic
121C22×C8—21no—
122C22×C8—21no—
123C22×C8C22⋊C412yes—
124C22×C8C2×D412yes—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×D4—24no—
130C4×D4—24no—
131C4×D4—24no—
132C4×D4—24no—
133C4×D4—22no—
134C4×D4—22no—
135C4×D4—22no—
136C4×D4—22no—
137#117—21no—
138#117—21no—
139#117—22no—
140#117—22no—
141#117—24no—
142#117M1612yes—
143#117M1612yes—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—21no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—21no—
148C2×C42—22no—
149C2×C42—21no—
150C2×C42—21no—
151C2×C42—21no—
152C2×C42—22no—
153C2×C42—22no—
154C2×C42—24no—
155C2×C42C22⋊C411yescharacteristic
156C2×C42C22⋊C411yescharacteristic
157C2×C42C22⋊C411yescharacteristic
158C2×C42C22⋊C411yescharacteristic
159C2×C42C22×C411yescharacteristic
160C2×C42C22×C411yescharacteristic
161C2×C42C22×C412yes—
162M32—28no—
163M32—44no—
164C2×C16—44no—
165C4⋊C8—24no—
166C4⋊C8—24no—
167C4⋊C8—24no—
168C4⋊C8—24no—
169C22⋊C8—24no—
170C22⋊C8—24no—
171C22⋊C8—24no—
172C22⋊C8—24no—
173#97—22no—
174#97—22no—
175#97C22⋊C412yes—
176#97C22⋊C412yes—
177#97C2×D412yes—
178#97C2×D412yes—
179C4×C8—22no—
180C4×C8—22no—
181C4×C8C22⋊C412yes—
182C4×C8C22⋊C412yes—
183C4×C8C2×D412yes—
184C4×C8C2×D412yes—
185C24C2×M1611yescharacteristic
186C24C2×M1612yes—
187#41—44no—
188#41—42no—
189#41—42no—
190#41—44no—
191#41—42no—
192#41—42no—
193C2×Q8—21no—
194C2×Q8—21no—
195C2×Q8—42no—
196C2×D4—22no—
197C2×D4—22no—
198C2×D4—21no—
199C2×D4—21no—
200C2×D4—42no—
201C2×D4—44no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—24no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—24no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—82no—
218C22×C4—84no—
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×M1611yescharacteristic
222C22×C4C2×M1611yescharacteristic
223C22×C4C2×M1611yescharacteristic
224C22×C4C2×M1612yes—
225C22×C4C23×C411yescharacteristic
226M16—42no—
227M16—41no—
228M16—41no—
229M16—41no—
230M16—42no—
231M16—41no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—42no—
236C2×C8—22no—
237C2×C8—22no—
238C2×C8—24no—
239C2×C8—22no—
240C2×C8—22no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8C2×C22⋊C412yes—
244C2×C8C2×C22⋊C412yes—
245C2×C8C2×C22⋊C412yes—
246C2×C8C22×D412yes—
247C4⋊C4—24no—
248C4⋊C4—24no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C22⋊C4—24no—
252C22⋊C4—24no—
253C22⋊C4—22no—
254C22⋊C4—22no—
255C42—24no—
256C42—24no—
257C42—22no—
258C42—21no—
259C42—21no—
260C42—21no—
261C42—21no—
262C42—21no—
263C42—21no—
264C42C2×C22⋊C411yescharacteristic
265C42C2×C22⋊C411yescharacteristic
266C42C2×C22⋊C411yescharacteristic
267C42C2×C22⋊C411yescharacteristic
268C42C2×M1614yes—
269C16—48no—
270C23—22no—
271C23—22no—
272C23—42no—
273C23—81no—
274C23—82no—
275C23#40711yescharacteristic
276C23#40711yescharacteristic
277C23C22×M1611yescharacteristic
278Q8—42no—
279Q8—42no—
280D4—42no—
281D4—44no—
282D4—44no—
283D4—42no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—21no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—24no—
297C2×C4—44no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—48no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—81no—
306C2×C4—82no—
307C2×C4—81no—
308C2×C4—81no—
309C2×C4—84no—
310C2×C4#40711yescharacteristic
311C2×C4#40711yescharacteristic
312C2×C4#41012yes—
313C2×C4#41312yes—
314C2×C4C22×C22⋊C411yescharacteristic
315C2×C4C22×M1611yescharacteristic
316C8—22no—
317C8—22no—
318C8—22no—
319C8—22no—
320C8—42no—
321C8—42no—
322C22—21no—
323C22—21no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—44no—
328C22—42no—
329C22—82no—
330C22—84no—
331C22#146412yes—
332C22C2×#40711yescharacteristic
333C4—22no—
334C4—21no—
335C4—21no—
336C4—41no—
337C4—41no—
338C4—44no—
339C4—41no—
340C4—42no—
341C4#146412yes—
342C4C2×#41011yescharacteristic
343C4C2×#41311yescharacteristic
344C2—21no—
345C2—22no—
346C2—41no—
347C2—41no—
348C2—82no—
349C2C2×#146411yescharacteristic
subgroupG/H
centerC4C2×#410
commutatorC22×C4C23×C4
Frattini, cosocleC2×M16C24
socleC2C2×#1464
prime radicalC22×#41C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4 ◁ C2×C42 ◁ C2×C4×M16 ◁ G
LECSG ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2165C2semidirect
3. #11522C2semidirect
4. #11521C2semidirect
5. #11456C2semidirect

Maximal subgroups

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

subgroupindex
#601232
C2×#21652
#115222
#115212
#114562
#114362