Finite Group Databasebeta

← #391847

Subgroups — #391847

Order 512.

992 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35620C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20634C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11348C212yesmaximal, complemented
6#11224C212yesmaximal, complemented
7#11211C212yesmaximal, complemented
8#11160C212yesmaximal
9#11154C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#520C2211yescharacteristic
12C2×#410C2212yes—
13#2879C2212yes—
14#2568C412yescomplemented
15#2566C412yescomplemented
16#2565C412yescomplemented
17#2563C412yescomplemented
18#2450—24no—
19#2442—22no—
20#2439—22no—
21#2317C2211yescharacteristic
22#2316C2212yes—
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26C2×#382C2211yescharacteristic
27#1822—22no—
28#1821—22no—
29#1799—24no—
30#1742C2212yes—
31C2×#343C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C22×#41C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—22no—
45#520—44no—
46C2×C4×Q8C2×C412yes—
47C2×C4×D4C2×C412yes—
48C2×C4×D4D414yes—
49C2×#117D412yes—
50C2×#117C2314yes—
51C2×#117C2311yescharacteristic
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C22×C4⋊C4D412yes—
55C22×C4⋊C4C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C42C2311yescharacteristic
58#410—44no—
59C2×#99—24no—
60#408—44no—
61#407—24no—
62#400—24no—
63#398—24no—
64#396—24no—
65#393—24no—
66#391—24no—
67#391C2×C411yescharacteristic
68#391C2×C412yes—
69#391C2×C411yescharacteristic
70#390C2×C411yescharacteristic
71#390C2×C412yes—
72#390C2×C411yescharacteristic
73#389—22no—
74#389—21no—
75#389—21no—
76#389—44no—
77#387—24no—
78#386—22no—
79#386—22no—
80#385—24no—
81#385—22no—
82#385—22no—
83#384C2×C412yes—
84#382—22no—
85#382C2×C412yes—
86#382C2×C412yes—
87#382C2×C412yes—
88#380—24no—
89#380—22no—
90#380—24no—
91C4×C4⋊C4C2×C412yes—
92C4×C4⋊C4C2×C412yes—
93C4×C22⋊C4—22no—
94C2×#95—22no—
95#343—22no—
96#343—24no—
97#343—22no—
98#342—24no—
99#340—44no—
100#339—44no—
101#339—44no—
102H(3,ℤ/4)—44no—
103C2×#41—24no—
104C2×#41—22no—
105C2×#41—24no—
106C2×#41—22no—
107C22×Q8—22no—
108C22×Q8C22×C411yescharacteristic
109C22×D4C22×C411yescharacteristic
110C22×D4C2×D412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C22⋊C414yes—
114C23×C4C2×D412yes—
115C23×C4C2×D412yes—
116C23×C4C2411yescharacteristic
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C22×C8—24no—
124C4⋊Q8—44no—
125C4⋊Q8—44no—
126#125—44no—
127#125—44no—
128C22⋊Q8—48no—
129C4×Q8—44no—
130C4×D4—44no—
131C4×D4—48no—
132#117—24no—
133#117—22no—
134#117—22no—
135#117—24no—
136#117—24no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—44no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
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⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4C22⋊C414yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C411yescharacteristic
157C2×C4⋊C4C22×C411yescharacteristic
158C2×C4⋊C4C22×C412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C4⋊C4C2×D412yes—
161C2×C4⋊C4C2×D411yescharacteristic
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4—22no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4C22⋊C414yes—
169C2×C22⋊C4C2×D412yes—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42—21no—
173C2×C42—21no—
174C2×C42—24no—
175C2×C42C22⋊C414yes—
176C2×C42C22×C412yes—
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C22×C412yes—
180#99—48no—
181C22⋊C8—48no—
182#95—24no—
183#95—24no—
184#95—22no—
185#95—22no—
186#95—22no—
187#95—22no—
188#95—24no—
189#95—24no—
190#95—44no—
191C24C2×C22⋊C412yes—
192C24C22×D411yescharacteristic
193#41—44no—
194#41—44no—
195#41—44no—
196#41—44no—
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—44no—
201C2×Q8—84no—
202C2×D4—24no—
203C2×D4—22no—
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—22no—
209C22×C4—24no—
210C22×C4—24no—
211C22×C4—24no—
212C22×C4—24no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—44no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—42no—
223C22×C4—44no—
224C22×C4—81no—
225C22×C4—81no—
226C22×C4—82no—
227C22×C4—82no—
228C22×C4—84no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C414yes—
231C22×C4C2×C22⋊C412yes—
232C22×C4C2×C22⋊C412yes—
233C22×C4C23×C411yescharacteristic
234C22×C4C22×D411yescharacteristic
235C22×C4H(5,𝔽2)14yes—
236C22×C4H(5,𝔽2)12yes—
237C22×C42−1+414yes—
238C22×C42−1+412yes—
239M16—44no—
240M16—44no—
241M16—44no—
242M16—44no—
243C2×C8—44no—
244C2×C8—44no—
245C2×C8—44no—
246C4⋊C4—24no—
247C4⋊C4—24no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—24no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—44no—
254C4⋊C4—44no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—44no—
261C4⋊C4—84no—
262C4⋊C4—82no—
263C4⋊C4—82no—
264C4⋊C4—164no—
265C22⋊C4—24no—
266C22⋊C4—44no—
267C22⋊C4—42no—
268C22⋊C4—41no—
269C22⋊C4—41no—
270C22⋊C4—44no—
271C22⋊C4—44no—
272C22⋊C4—84no—
273C22⋊C4—82no—
274C22⋊C4—82no—
275C42—24no—
276C42—22no—
277C42—22no—
278C42—24no—
279C42—44no—
280C42—42no—
281C42—42no—
282C42—44no—
283C42—44no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—41no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23C22×C22⋊C411yescharacteristic
292C23#53412yes—
293C23#53511yescharacteristic
294C23#53612yes—
295C23#53611yescharacteristic
296Q8—44no—
297Q8—164no—
298D4—44no—
299D4—44no—
300D4—44no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—24no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—48no—
308C2×C4—44no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—44no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—84no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—84no—
330C2×C4—84no—
331C2×C4—84no—
332C2×C4—164no—
333C2×C4—162no—
334C2×C4#51812yes—
335C2×C4#51912yes—
336C2×C4#52012yes—
337C2×C4#53412yes—
338C2×C4#53612yes—
339C2×C4#53712yes—
340C2×C4#56214yes—
341C2×C4#56214yes—
342C2×C4#56314yes—
343C2×C4#56314yes—
344C8—84no—
345C22—21no—
346C22—22no—
347C22—21no—
348C22—22no—
349C22—41no—
350C22—41no—
351C22—42no—
352C22—44no—
353C22—42no—
354C22—42no—
355C22—42no—
356C22—82no—
357C22#230811yescharacteristic
358C22#231011yescharacteristic
359C22#231111yescharacteristic
360C22#281612yes—
361C22#281712yes—
362C4—44no—
363C4—42no—
364C4—42no—
365C4—42no—
366C4—82no—
367C4—81no—
368C4—81no—
369C4—81no—
370C4—82no—
371C4—81no—
372C4—164no—
373C4#237814yes—
374C4#281414yes—
375C2—21no—
376C2—22no—
377C2—21no—
378C2—22no—
379C2—82no—
380C2#1466011yescharacteristic
381C2#2178411yescharacteristic
382C2#2178511yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
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
4. #11388C2semidirect
5. #11348C2semidirect
6. #11224C2semidirect
7. #11211C2semidirect
14. #2568C4semidirect
15. #2566C4semidirect
16. #2565C4semidirect
17. #2563C4semidirect

Maximal subgroups

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

subgroupindex
#356202
#206952
#206342
#113882
#113482
#112242
#112112
#111602
#111542