Finite Group Databasebeta

← #493024

Subgroups — #493024

Order 512.

1231 conjugacy classes of subgroups.

The subgroup lattice has 365 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34276C211yesmaximal, characteristic
2C2×#2135C212yesmaximal
3#20397C212yesmaximal
4#12631C214yesmaximal, complemented
5C4×#356C214yesmaximal, complemented
6#12585C212yesmaximal
7C2×#536C2211yescharacteristic
8C2×#520C2211yescharacteristic
9C2×#519C411yescharacteristic
10C2×#413C2214yes—
11#2873—22no—
12#2362—22no—
13#2360—22no—
14#2359—24no—
15#2327C2212yes—
16#2326C412yes—
17#2317C411yescharacteristic
18C4×C4⋊Q8C2212yes—
19C4×#124C412yes—
20#2297C2211yescharacteristic
21C2×#391C411yescharacteristic
22C4×#117C411yescharacteristic
23#2135C4116yescomplemented
24C2×#356C2218yes—
25C4×#101C2218yes—
26C4×#100—24no—
27C2×#343—22no—
28#1394C2214yes—
29#1394C2214yes—
30C22×#41C2311yescharacteristic
31C23×Q8C2×C411yescharacteristic
32C22×M16—24no—
33#536—24no—
34#536C2×C418yes—
35C2×C4⋊Q8C2312yes—
36C2×#124C2×C412yes—
37C2×#123—22no—
38C2×C22⋊Q8—22no—
39#520—44no—
40#519—44no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8C2×C412yes—
45C2×C4×Q8C2312yes—
46C2×C4×D4C2×C412yes—
47C2×C4×D4D414yes—
48C2×#117—22no—
49C2×#117C2×C411yescharacteristic
50C2×#117C2×C411yescharacteristic
51C2×#117C2×C411yescharacteristic
52C2×#117D412yes—
53C22×C4⋊C4C2×C411yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C42C2×C411yescharacteristic
56#413—28no—
57#413C2×C4116yes—
58C2×#101C2318yes—
59C2×#100—24no—
60#405—48no—
61C4×M16—28no—
62#391—22no—
63#391C2×C412yes—
64#391C2×C412yes—
65#391C2311yescharacteristic
66#391C2311yescharacteristic
67#390C2×C411yescharacteristic
68#390C2×C411yescharacteristic
69#389—22no—
70#389—24no—
71#388—24no—
72#388—24no—
73#387—24no—
74#387—24no—
75#386—22no—
76#385—22no—
77#385—24no—
78#384C2×C412yes—
79#382C2×C412yes—
80C4×C4⋊C4—21no—
81C4×C4⋊C4—21no—
82C4×C4⋊C4—24no—
83C4×C22⋊C4—24no—
84C4×C22⋊C4—22no—
85C4×C22⋊C4—22no—
86#376C2×C412yes—
87C2×#95—22no—
88C43C2×C412yes—
89#356C2×C4132yescomplemented
90#343—28no—
91C2×#41—42no—
92C2×#41—42no—
93C2×#41C22×C414yes—
94C2×#41#4114yes—
95C22×Q8—21no—
96C22×Q8—22no—
97C22×Q8—42no—
98C22×Q8—42no—
99C22×Q8C4214yes—
100C22×Q8C22×C412yes—
101C22×Q8C2411yescharacteristic
102C22×D4C22×C411yescharacteristic
103C22×D4C2×D412yes—
104C23×C4—21no—
105C23×C4—21no—
106C23×C4C22⋊C412yes—
107C23×C4C22⋊C414yes—
108C23×C4C22×C411yescharacteristic
109C23×C4C2×D412yes—
110C2×M16—216no—
111C2×M16—28no—
112C2×M16—48no—
113C2×M16—44no—
114C22×C8—44no—
115C4⋊Q8—24no—
116C4⋊Q8C22×C418yes—
117#124—24no—
118#124C4218yes—
119#123—216no—
120C22⋊Q8—216no—
121C4×Q8—44no—
122C4×Q8—44no—
123C4×Q8—44no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×D4—44no—
127C4×D4—48no—
128#117—22no—
129#117—24no—
130#117—28no—
131#117—48no—
132#117—42no—
133#117C4214yes—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—24no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—42no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C411yescharacteristic
143C2×C4⋊C4C22⋊C411yescharacteristic
144C2×C4⋊C4C22×C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C2×D414yes—
148C2×C22⋊C4—21no—
149C2×C22⋊C4—24no—
150C2×C22⋊C4—21no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—22no—
153C2×C22⋊C4C22⋊C412yes—
154C2×C22⋊C4C22⋊C414yes—
155C2×C42—22no—
156C2×C42—22no—
157C2×C42—21no—
158C2×C42—22no—
159C2×C42—21no—
160C2×C42—22no—
161C2×C42—42no—
162C2×C42C22×C412yes—
163C2×C42C22×C412yes—
164C2×C42C22×C411yescharacteristic
165C2×C42C22×C411yescharacteristic
166C2×C42C2×D414yes—
167#101—28no—
168#101C22×C4116yes—
169#100—216no—
170#97—48no—
171C4×C8—48no—
172#95—24no—
173#95—24no—
174#95—22no—
175#95—22no—
176#95—24no—
177#95—22no—
178#95—22no—
179C24C2×C22⋊C412yes—
180C24C2×C22⋊C411yescharacteristic
181#41—48no—
182#41—44no—
183#41—44no—
184C2×Q8—22no—
185C2×Q8—24no—
186C2×Q8—41no—
187C2×Q8—41no—
188C2×Q8—48no—
189C2×Q8—48no—
190C2×Q8—42no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8C2×C4214yes—
195C2×Q8C23×C412yes—
196C2×Q8C2×#4112yes—
197C2×D4—41no—
198C2×D4—42no—
199C2×D4—41no—
200C2×D4—42no—
201C2×D4C2×C4212yes—
202C2×D4#11712yes—
203C2×D4C4×D418yes—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—24no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—44no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—81no—
222C22×C4—84no—
223C22×C4—82no—
224C22×C4C2×C4212yes—
225C22×C4C2×C22⋊C414yes—
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C412yes—
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4#11712yes—
230C22×C4C4×D418yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233M16—216no—
234M16—48no—
235M16—416no—
236C2×C8—48no—
237C2×C8—416no—
238C4⋊C4—216no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—44no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—84no—
249C4⋊C4#9914yes—
250C22⋊C4—24no—
251C22⋊C4—216no—
252C22⋊C4—24no—
253C22⋊C4—28no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C22⋊C4—82no—
257C42—22no—
258C42—28no—
259C42—24no—
260C42—41no—
261C42—41no—
262C42—44no—
263C42—44no—
264C42—44no—
265C42—42no—
266C42—42no—
267C42—82no—
268C42C2×C4214yes—
269C23—21no—
270C23—22no—
271C23—21no—
272C23—22no—
273C23—41no—
274C23—41no—
275C23—42no—
276C23—42no—
277C23C4×C22⋊C414yes—
278C23C4×C22⋊C412yes—
279C23C22×C22⋊C411yescharacteristic
280Q8—42no—
281Q8—42no—
282Q8—44no—
283Q8—44no—
284Q8—48no—
285D4—42no—
286D4—42no—
287D4—48no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—28no—
293C2×C4—28no—
294C2×C4—24no—
295C2×C4—24no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—24no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—48no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—44no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—84no—
319C2×C4—164no—
320C2×C4—162no—
321C2×C4C4×C22⋊C412yes—
322C2×C4C2×#9914yes—
323C2×C4C2×#9912yes—
324C2×C4#41012yes—
325C2×C4C22×C4211yescharacteristic
326C2×C4C2×#11711yescharacteristic
327C2×C4C2×C4×D414yes—
328C8—416no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—24no—
333C22—42no—
334C22—44no—
335C22—42no—
336C22—41no—
337C22—42no—
338C22—41no—
339C22—42no—
340C22—82no—
341C22C2×C4×C22⋊C411yescharacteristic
342C22C22×#9911yescharacteristic
343C22C2×#41011yescharacteristic
344C4—24no—
345C4—24no—
346C4—21no—
347C4—21no—
348C4—44no—
349C4—42no—
350C4—42no—
351C4—44no—
352C4—44no—
353C4—81no—
354C4—81no—
355C4—81no—
356C4C4×#9914yes—
357C2—21no—
358C2—21no—
359C2—42no—
360C2—44no—
361C2C2×C4×#9911yescharacteristic
362C2#3337111yescharacteristic
363C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC2×C4C22×C42
Frattini, cosocleC22×Q8C24
prime radicalC22×#41C23
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#520 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #12631C2semidirect
5. C4×#356C2semidirect
23. #2135C4semidirect
89. #356C2×C4semidirect

Maximal subgroups

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

subgroupindex
#342762
C2×#21352
#203972
#126312
C4×#3562
#125852