Finite Group Databasebeta

← #391845

Subgroups — #391845

Order 512.

961 conjugacy classes of subgroups.

The subgroup lattice has 363 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35668C211yesmaximal, characteristic
2#20691C211yesmaximal, characteristic
3#20639C211yesmaximal, characteristic
4#11388C212yesmaximal
5#11347C212yesmaximal
6#11225C212yesmaximal
7#11212C212yesmaximal
8#11162C212yesmaximal
9#11156C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#519C2211yescharacteristic
12#2905C2212yes—
13C22×#101C2212yes—
14#2568C412yescomplemented
15#2567C412yescomplemented
16#2565C412yescomplemented
17#2565C412yescomplemented
18#2453—22no—
19#2451—22no—
20#2438—24no—
21#2317C2211yescharacteristic
22#2310C2211yescharacteristic
23#2310C2212yes—
24#2296C2211yescharacteristic
25C2×#391C2211yescharacteristic
26#2285C2211yescharacteristic
27#1820—24no—
28#1801—22no—
29#1801—22no—
30#1742C2212yes—
31#1740C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C23×Q8C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—22no—
45#519—44no—
46C2×C4×Q8C2×C412yes—
47C2×C4×Q8C2×C412yes—
48C2×C4×Q8D412yes—
49C2×C4×Q8D412yes—
50C2×#117D414yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#424—44no—
59#423—24no—
60C2×#101—24no—
61C2×#101—44no—
62#400—24no—
63#398—24no—
64#398—24no—
65#395—24no—
66#391—22no—
67#391—22no—
68#391—24no—
69#391—44no—
70#391C2×C412yes—
71#391C2×C412yes—
72#391C2×C412yes—
73#391C2×C412yes—
74#389—21no—
75#389—21no—
76#389—24no—
77#389—22no—
78#387—24no—
79#387—24no—
80#385—22no—
81#385—22no—
82#384C2×C411yescharacteristic
83#384C2×C412yes—
84#384C2×C411yescharacteristic
85#382C2×C411yescharacteristic
86#382C2×C412yes—
87#382C2×C411yescharacteristic
88#381—22no—
89#381—24no—
90#380—22no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4C2×C412yes—
93C4×C4⋊C4C2×C412yes—
94C2×#95—22no—
95#343—22no—
96#343—22no—
97#342—24no—
98#342—24no—
99#340—44no—
100#339—44no—
101#339—44no—
102H(3,ℤ/4)—44no—
103C22×Q8—22no—
104C22×Q8—24no—
105C22×Q8—22no—
106C22×Q8—24no—
107C22×Q8—22no—
108C22×Q8C22×C412yes—
109C22×Q8C2×D412yes—
110C23×C4—21no—
111C23×C4—21no—
112C23×C4C2×D412yes—
113C23×C4C2411yescharacteristic
114C2×M16—24no—
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C22×C8—24no—
121C4⋊Q8—44no—
122C4⋊Q8—44no—
123#125—44no—
124#125—44no—
125C22⋊Q8—48no—
126C4×Q8—44no—
127C4×Q8—44no—
128C4×Q8—44no—
129C4×Q8—44no—
130#117—24no—
131#117—24no—
132#117—22no—
133#117—22no—
134#117—24no—
135#117—24no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—48no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—22no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—24no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4C22⋊C414yes—
153C2×C4⋊C4C22⋊C414yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C411yescharacteristic
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D411yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—24no—
164C2×C22⋊C4C2×D412yes—
165C2×C22⋊C4C2×D412yes—
166C2×C42—22no—
167C2×C42—22no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—24no—
171C2×C42C22⋊C414yes—
172C2×C42C22⋊C414yes—
173C2×C42C22×C412yes—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C412yes—
177C2×C42C2×D412yes—
178C4⋊C8—48no—
179#101—48no—
180#95—24no—
181#95—24no—
182#95—22no—
183#95—22no—
184#95—22no—
185#95—22no—
186#95—24no—
187#95—24no—
188#95—24no—
189#95—48no—
190C24C22×D411yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—24no—
194C2×Q8—24no—
195C2×Q8—22no—
196C2×Q8—24no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×Q8—44no—
200C2×Q8—44no—
201C2×Q8—44no—
202C2×Q8—44no—
203C2×Q8—84no—
204C22×C4—21no—
205C22×C4—24no—
206C22×C4—24no—
207C22×C4—24no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—21no—
211C22×C4—21no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—81no—
218C22×C4—82no—
219C22×C4C2×C22⋊C414yes—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C412yes—
224C22×C4C23×C411yescharacteristic
225C22×C4C22×D411yescharacteristic
226C22×C4H(5,𝔽2)14yes—
227C22×C4H(5,𝔽2)12yes—
228C22×C42−1+414yes—
229C22×C42−1+412yes—
230M16—44no—
231M16—44no—
232M16—44no—
233M16—44no—
234C2×C8—44no—
235C2×C8—44no—
236C2×C8—44no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—24no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—44no—
245C4⋊C4—44no—
246C4⋊C4—44no—
247C4⋊C4—44no—
248C4⋊C4—44no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C4⋊C4—82no—
259C22⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—44no—
264C22⋊C4—44no—
265C42—24no—
266C42—24no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—44no—
271C42—44no—
272C42—42no—
273C42—41no—
274C42—41no—
275C42—44no—
276C42—44no—
277C23—41no—
278C23—41no—
279C23C22×C22⋊C411yescharacteristic
280C23#53412yes—
281C23#53612yes—
282C23#53712yes—
283Q8—44no—
284Q8—44no—
285Q8—48no—
286Q8—164no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—24no—
293C2×C4—24no—
294C2×C4—24no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—48no—
298C2×C4—42no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4—44no—
307C2×C4—42no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—84no—
315C2×C4—84no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—162no—
319C2×C4#51812yes—
320C2×C4#51912yes—
321C2×C4#52012yes—
322C2×C4#53412yes—
323C2×C4#53511yescharacteristic
324C2×C4#53612yes—
325C2×C4#53611yescharacteristic
326C2×C4#56214yes—
327C2×C4#56214yes—
328C2×C4#56314yes—
329C2×C4#56314yes—
330C8—84no—
331C22—21no—
332C22—22no—
333C22—21no—
334C22—22no—
335C22—42no—
336C22—42no—
337C22#230811yescharacteristic
338C22#231011yescharacteristic
339C22#231111yescharacteristic
340C22#281414yes—
341C4—42no—
342C4—42no—
343C4—42no—
344C4—42no—
345C4—42no—
346C4—44no—
347C4—82no—
348C4—81no—
349C4—81no—
350C4—82no—
351C4—81no—
352C4—164no—
353C4#237814yes—
354C4#281612yes—
355C4#281712yes—
356C2—21no—
357C2—24no—
358C2—21no—
359C2#1466011yescharacteristic
360C2#2178311yescharacteristic
361C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C23×C4 ◁ G
LECSG ▷ C23×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
14. #2568C4semidirect
15. #2567C4semidirect
16. #2565C4semidirect
17. #2565C4semidirect

Maximal subgroups

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

subgroupindex
#356682
#206912
#206392
#113882
#113472
#112252
#112122
#111622
#111562