Finite Group Databasebeta

← #391844

Subgroups — #391844

Order 512.

1081 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35529C211yesmaximal, characteristic
2#20690C211yesmaximal, characteristic
3#20638C211yesmaximal, characteristic
4#11388C212yesmaximal, complemented
5#11347C212yesmaximal, complemented
6#11225C212yesmaximal, complemented
7#11212C212yesmaximal, complemented
8#11161C212yesmaximal
9#11155C212yesmaximal
10C2×#544C2211yescharacteristic
11C2×#518C2211yescharacteristic
12#2905C2212yes—
13C22×#100C2212yes—
14#2566C412yescomplemented
15#2566C412yescomplemented
16#2563C412yescomplemented
17#2542C412yescomplemented
18#2454—22no—
19#2452—22no—
20#2437—24no—
21#2317C2211yescharacteristic
22#2309C2211yescharacteristic
23#2309C2212yes—
24#2296C2211yescharacteristic
25C2×#390C2211yescharacteristic
26#2285C2211yescharacteristic
27#1819—24no—
28#1800—22no—
29#1800—22no—
30#1742C2212yes—
31#1740C2212yes—
32#1740C2212yes—
33#1736C2214yes—
34#1733C2214yes—
35C2×#339C2214yes—
36C2×H(3,ℤ/4)C2214yes—
37C23×D4C2311yescharacteristic
38C22×M16C2314yes—
39#544—44no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8C2×C412yes—
42C2×#125—21no—
43C2×#125C2×C412yes—
44C2×C22⋊Q8—24no—
45C2×C22⋊Q8—22no—
46#518—44no—
47C2×C4×D4C2×C412yes—
48C2×C4×D4C2×C412yes—
49C2×C4×D4D412yes—
50C2×C4×D4D412yes—
51C2×#117C2314yes—
52C2×#117C2311yescharacteristic
53C2×#117C2311yescharacteristic
54C2×#117C2311yescharacteristic
55C2×#117C2311yescharacteristic
56C22×C4⋊C4C2311yescharacteristic
57C22×C22⋊C4—22no—
58C22×C22⋊C4D414yes—
59C22×C42C2311yescharacteristic
60#424—44no—
61#423—24no—
62C2×#100—24no—
63C2×#100—44no—
64#399—24no—
65#396—24no—
66#396—24no—
67#393—24no—
68#390—22no—
69#390—22no—
70#390—44no—
71#390C2×C412yes—
72#390C2×C412yes—
73#390C2×C412yes—
74#390C2×C412yes—
75#389—21no—
76#389—21no—
77#389—22no—
78#385—24no—
79#385—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#380—24no—
89#380—22no—
90#379—22no—
91#379—24no—
92C4×C4⋊C4—22no—
93C4×C4⋊C4C2×C412yes—
94C4×C4⋊C4C2×C412yes—
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—
103C25C2×D412yes—
104C22×Q8—22no—
105C22×D4—24no—
106C22×D4—22no—
107C22×D4—24no—
108C22×D4—22no—
109C22×D4C22×C412yes—
110C22×D4C2×D412yes—
111C23×C4—21no—
112C23×C4—21no—
113C23×C4C22⋊C414yes—
114C23×C4C22⋊C414yes—
115C23×C4C2411yescharacteristic
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C22×C8—24no—
123C4⋊Q8—44no—
124C4⋊Q8—44no—
125#125—44no—
126#125—44no—
127C22⋊Q8—48no—
128C22⋊Q8—88no—
129C4×D4—44no—
130C4×D4—44no—
131C4×D4—44no—
132C4×D4—44no—
133#117—24no—
134#117—24no—
135#117—22no—
136#117—22no—
137#117—24no—
138#117—24no—
139#117—24no—
140#117—24no—
141#117—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
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—24no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—24no—
165C2×C22⋊C4—24no—
166C2×C22⋊C4—24no—
167C2×C22⋊C4—24no—
168C2×C22⋊C4—48no—
169C2×C22⋊C4—44no—
170C2×C22⋊C4—44no—
171C2×C22⋊C4—84no—
172C2×C22⋊C4C22⋊C414yes—
173C2×C22⋊C4C22⋊C414yes—
174C2×C22⋊C4C2×D412yes—
175C2×C22⋊C4C2×D412yes—
176C2×C42—22no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—21no—
180C2×C42—24no—
181C2×C42C22×C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C412yes—
185C2×C42C2×D412yes—
186C4⋊C8—48no—
187#100—48no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—24no—
193C24—24no—
194C24—24no—
195C24—24no—
196C24—82no—
197C24—82no—
198C24C2×C22⋊C414yes—
199C24C22×D411yescharacteristic
200C2×Q8—44no—
201C2×Q8—84no—
202C2×D4—24no—
203C2×D4—24no—
204C2×D4—24no—
205C2×D4—24no—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—44no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—22no—
216C22×C4—21no—
217C22×C4—21no—
218C22×C4—22no—
219C22×C4—44no—
220C22×C4—44no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—44no—
225C22×C4—81no—
226C22×C4—82no—
227C22×C4—84no—
228C22×C4—84no—
229C22×C4C2×C22⋊C412yes—
230C22×C4C2×C22⋊C412yes—
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—22no—
247C4⋊C4—22no—
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—44no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C22⋊C4—24no—
267C22⋊C4—24no—
268C22⋊C4—44no—
269C22⋊C4—44no—
270C22⋊C4—48no—
271C22⋊C4—42no—
272C22⋊C4—42no—
273C22⋊C4—44no—
274C22⋊C4—44no—
275C22⋊C4—84no—
276C22⋊C4—84no—
277C22⋊C4—84no—
278C22⋊C4—164no—
279C22⋊C4—164no—
280C42—24no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—24no—
285C42—42no—
286C42—41no—
287C42—41no—
288C42—44no—
289C42—44no—
290C23—24no—
291C23—24no—
292C23—24no—
293C23—41no—
294C23—41no—
295C23—44no—
296C23—44no—
297C23—48no—
298C23—84no—
299C23—84no—
300C23—88no—
301C23—164no—
302C23C22×C22⋊C411yescharacteristic
303C23#53412yes—
304C23#53612yes—
305C23#53712yes—
306Q8—164no—
307D4—44no—
308D4—44no—
309D4—48no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—44no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—42no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—84no—
334C2×C4—84no—
335C2×C4—84no—
336C2×C4—84no—
337C2×C4—164no—
338C2×C4—164no—
339C2×C4—162no—
340C2×C4#51812yes—
341C2×C4#51912yes—
342C2×C4#52012yes—
343C2×C4#53412yes—
344C2×C4#53511yescharacteristic
345C2×C4#53612yes—
346C2×C4#53611yescharacteristic
347C2×C4#56214yes—
348C2×C4#56214yes—
349C2×C4#56314yes—
350C2×C4#56314yes—
351C8—84no—
352C22—21no—
353C22—22no—
354C22—21no—
355C22—22no—
356C22—42no—
357C22—42no—
358C22—48no—
359C22—44no—
360C22—44no—
361C22—44no—
362C22—84no—
363C22—168no—
364C22#230811yescharacteristic
365C22#231011yescharacteristic
366C22#231111yescharacteristic
367C22#281414yes—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—42no—
372C4—42no—
373C4—82no—
374C4—81no—
375C4—82no—
376C4—81no—
377C4—81no—
378C4—164no—
379C4#237814yes—
380C4#281612yes—
381C4#281712yes—
382C2—21no—
383C2—24no—
384C2—21no—
385C2—84no—
386C2#1466011yescharacteristic
387C2#2178211yescharacteristic
388C2#2178411yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC23×C4C24
prime radicalC23×D4C23
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. #11347C2semidirect
6. #11225C2semidirect
7. #11212C2semidirect
14. #2566C4semidirect
15. #2566C4semidirect
16. #2563C4semidirect
17. #2542C4semidirect

Maximal subgroups

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

subgroupindex
#355292
#206902
#206382
#113882
#113472
#112252
#112122
#111612
#111552