Finite Group Databasebeta

← #391843

Subgroups — #391843

Order 512.

978 conjugacy classes of subgroups.

The subgroup lattice has 367 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35639C211yesmaximal, characteristic
2#20695C211yesmaximal, characteristic
3#20640C211yesmaximal, characteristic
4#11383C212yesmaximal, complemented
5#11352C212yesmaximal, complemented
6#11223C212yesmaximal, complemented
7#11206C212yesmaximal, complemented
8#11175C212yesmaximal
9#11173C212yesmaximal
10C22×#125C2211yescharacteristic
11C2×#520C2211yescharacteristic
12#2905C2212yes—
13C2×#413C2212yes—
14#2879C2212yes—
15#2579C414yes—
16#2487C414yes—
17#2454—22no—
18#2453—22no—
19#2442—22no—
20#2441—22no—
21#2317C2211yescharacteristic
22#2316C2211yescharacteristic
23#2311C2211yescharacteristic
24#2297C2211yescharacteristic
25#2296C2211yescharacteristic
26#1822—22no—
27#1821—22no—
28#1801—22no—
29#1800—22no—
30C2×#344C2214yes—
31C2×#344C2214yes—
32#1742C2212yes—
33C2×#343C2212yes—
34C2×#343C2212yes—
35C2×#343C2214yes—
36#1728C2214yes—
37C22×#41C2311yescharacteristic
38C22×M16C2314yes—
39C22×M16C2314yes—
40C2×#125—22no—
41C2×#125—22no—
42C2×#125—44no—
43#520—44no—
44C2×C4×Q8D412yes—
45C2×C4×D4C2×C414yes—
46C2×C4×D4D412yes—
47C2×#117D412yes—
48C2×#117C2311yescharacteristic
49C2×#117C2311yescharacteristic
50C2×#117C2311yescharacteristic
51C22×C4⋊C4C2×C412yes—
52C22×C4⋊C4C2×C412yes—
53C22×C4⋊C4D412yes—
54C22×C4⋊C4C2311yescharacteristic
55C22×C4⋊C4C2311yescharacteristic
56C22×C42C2311yescharacteristic
57#424—44no—
58#423—24no—
59#413—44no—
60C2×#101—22no—
61C2×#100—22no—
62#408—44no—
63#407—24no—
64#400—24no—
65#399—24no—
66#398—24no—
67#396—24no—
68#391—21no—
69#391—21no—
70#390—21no—
71#390—21no—
72#389—22no—
73#389—22no—
74#389—21no—
75#389—21no—
76#389—24no—
77#389—22no—
78#387—24no—
79#386C2×C414yes—
80#386C2×C412yes—
81#386C2×C412yes—
82#385—24no—
83#385C2×C414yes—
84#385C2×C412yes—
85#385C2×C412yes—
86#384—21no—
87#382—21no—
88#380—24no—
89#380—22no—
90#380C2×C414yes—
91C4×C4⋊C4—22no—
92C4×C22⋊C4—22no—
93#344—44no—
94#344—44no—
95#343—24no—
96#343—22no—
97#343—24no—
98#343—22no—
99#343—44no—
100#336—44no—
101C2×#41—24no—
102C2×#41—22no—
103C2×#41—24no—
104C2×#41—22no—
105C22×Q8C2×D411yescharacteristic
106C22×D4C22×C412yes—
107C22×D4C2×D411yescharacteristic
108C23×C4C22⋊C414yes—
109C23×C4C22×C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C22×C411yescharacteristic
112C23×C4C22×C412yes—
113C23×C4C2×D412yes—
114C23×C4C2411yescharacteristic
115C2×M16—24no—
116C2×M16—24no—
117C2×M16—24no—
118C2×M16—24no—
119C2×M16—24no—
120C2×M16—24no—
121C2×M16—24no—
122C2×M16—24no—
123C2×M16—24no—
124C2×M16—24no—
125C2×M16—24no—
126C2×M16—24no—
127C22×C8—24no—
128C22×C8—24no—
129#125—48no—
130#125—48no—
131C4×Q8—44no—
132C4×D4—44no—
133C4×D4—48no—
134#117—24no—
135#117—22no—
136#117—22no—
137#117—22no—
138#117—22no—
139#117—44no—
140C2×C4⋊C4—24no—
141C2×C4⋊C4—24no—
142C2×C4⋊C4—24no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—24no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—24no—
152C2×C4⋊C4—44no—
153C2×C4⋊C4—44no—
154C2×C4⋊C4—44no—
155C2×C4⋊C4C22⋊C414yes—
156C2×C4⋊C4C2×D411yescharacteristic
157C2×C4⋊C4C2×D412yes—
158C2×C4⋊C4C2×D412yes—
159C2×C4⋊C4C2×D411yescharacteristic
160C2×C22⋊C4—22no—
161C2×C22⋊C4—22no—
162C2×C22⋊C4—24no—
163C2×C22⋊C4C22⋊C414yes—
164C2×C22⋊C4C22×C412yes—
165C2×C22⋊C4C22×C412yes—
166C2×C22⋊C4C22×C412yes—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—24no—
172C2×C42C22⋊C414yes—
173C2×C42C2×D412yes—
174C4⋊C8—48no—
175#101—44no—
176#100—44no—
177C22⋊C8—48no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—24no—
183#95—24no—
184#95—24no—
185C24C2×C22⋊C412yes—
186C24C23×C411yescharacteristic
187#41—44no—
188#41—44no—
189#41—44no—
190#41—44no—
191C2×Q8—22no—
192C2×Q8—21no—
193C2×Q8—22no—
194C2×Q8—21no—
195C2×D4—24no—
196C2×D4—24no—
197C2×D4—24no—
198C2×D4—22no—
199C2×D4—21no—
200C2×D4—22no—
201C2×D4—21no—
202C22×C4—24no—
203C22×C4—24no—
204C22×C4—24no—
205C22×C4—24no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—21no—
209C22×C4—21no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—24no—
214C22×C4—44no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—84no—
220C22×C4C2×C22⋊C412yes—
221C22×C4C2×C22⋊C412yes—
222C22×C4C2×C22⋊C412yes—
223C22×C4C2×C22⋊C414yes—
224C22×C4C22×D411yescharacteristic
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—
234M16—44no—
235M16—44no—
236M16—44no—
237M16—44no—
238C2×C8—44no—
239C2×C8—44no—
240C2×C8—44no—
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—44no—
244C4⋊C4—24no—
245C4⋊C4—24no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—44no—
249C4⋊C4—48no—
250C4⋊C4—48no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—82no—
257C4⋊C4—84no—
258C4⋊C4—84no—
259C4⋊C4—82no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C22⋊C4—24no—
263C22⋊C4—44no—
264C22⋊C4—84no—
265C22⋊C4—82no—
266C22⋊C4—82no—
267C42—24no—
268C42—44no—
269C42—42no—
270C42—42no—
271C42—41no—
272C42—41no—
273C42—42no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—41no—
279C23—42no—
280C23—42no—
281C23C22×C22⋊C411yescharacteristic
282C23#51812yes—
283C23#51912yes—
284C23#52012yes—
285Q8—42no—
286Q8—42no—
287D4—48no—
288D4—42no—
289D4—42no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—24no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—48no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—44no—
310C2×C4—44no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—84no—
317C2×C4—84no—
318C2×C4—164no—
319C2×C4#53412yes—
320C2×C4#53412yes—
321C2×C4#53511yescharacteristic
322C2×C4#53612yes—
323C2×C4#53611yescharacteristic
324C2×C4#53612yes—
325C2×C4#53712yes—
326C2×C4#56214yes—
327C2×C4#56214yes—
328C2×C4#56314yes—
329C2×C4#56314yes—
330C8—84no—
331C8—84no—
332C22—21no—
333C22—22no—
334C22—21no—
335C22—22no—
336C22—42no—
337C22—42no—
338C22—44no—
339C22—42no—
340C22—42no—
341C22—42no—
342C22—82no—
343C22#230811yescharacteristic
344C22#231011yescharacteristic
345C22#231111yescharacteristic
346C22#237814yes—
347C4—44no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—82no—
352C4—82no—
353C4—81no—
354C4—81no—
355C4—82no—
356C4#281414yes—
357C4#281612yes—
358C4#281712yes—
359C2—21no—
360C2—21no—
361C2—24no—
362C2—82no—
363C2#1466011yescharacteristic
364C2#2177911yescharacteristic
365C2#2178111yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC23×C4C24
prime radicalC22×#41C23
LCSG ▷ C24 ▷ 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. #11383C2semidirect
5. #11352C2semidirect
6. #11223C2semidirect
7. #11206C2semidirect

Maximal subgroups

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

subgroupindex
#356392
#206952
#206402
#113832
#113522
#112232
#112062
#111752
#111732