Finite Group Databasebeta

← #6437303

Subgroups — #6437303

Order 512.

1452 conjugacy classes of subgroups.

The subgroup lattice has 393 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36635C212yesmaximal
2#30378C214yesmaximal, complemented
3#30358C214yesmaximal
4#30338C214yesmaximal, complemented
5#30318C214yesmaximal
6#23835C211yesmaximal, characteristic, complemented
7#23816C212yesmaximal, complemented
8#23801C211yesmaximal, characteristic, complemented
9#23698C214yesmaximal
10#21276C214yesmaximal, complemented
11C4×#431C211yesmaximal, characteristic
12C2×C4×Q16C2214yes—
13C2×#431C2212yes—
14#2864—22no—
15#2863—22no—
16#2863C2214yes—
17#2862C2212yes—
18#2859C2212yes—
19#2857C2212yes—
20#2857C2212yes—
21#2855—22no—
22#2855C2214yes—
23#2851—22no—
24C4×C4⋊Q8C2212yes—
25C4×C4⋊Q8C2212yes—
26C4×#125—22no—
27C4×C22⋊Q8—22no—
28C4×C22⋊Q8C2214yes—
29#2295—22no—
30#2295C2214yes—
31#2283—22no—
32C2×C43C2211yescharacteristic
33#2100C2214yes—
34#2095C2214yes—
35#2087C2214yes—
36#2083C2214yes—
37#2076C2214yes—
38#2072C2214yes—
39#2068C2218yes—
40#2057C2214yes—
41#2056C2214yes—
42#2043C2214yes—
43#2036C2218yes—
44#2024C2214yes—
45#2016C2214yes—
46#2001C2214yes—
47#1969C2218yes—
48#1942C2218yes—
49#1933C2218yes—
50#1933C2218yes—
51C4×#107C2214yes—
52C4×C4⋊C8C2214yes—
53C4×#103C2218yes—
54C4×C22⋊C8C2214yes—
55#1744C2211yescharacteristic
56#1744C2211yescharacteristic
57C42×C8C2212yes—
58C22×Q16C2314yes—
59C2×C4⋊Q8C2312yes—
60C2×C4⋊Q8C2312yes—
61C2×#125—22no—
62C2×C22⋊Q8—22no—
63C2×C22⋊Q8C2314yes—
64C2×C4×Q8—22no—
65C2×C4×Q8C2314yes—
66C22×C42D411yescharacteristic
67C22×C42D412yes—
68C22×C42D411yescharacteristic
69C22×C42C2311yescharacteristic
70C22×C42C2312yes—
71C4×Q16—216no—
72#431—28no—
73C2×#107C2314yes—
74C2×#107C2318yes—
75C2×C4⋊C8C2314yes—
76C2×#103C2318yes—
77C2×#103C2318yes—
78C2×#103C2318yes—
79C2×C22⋊C8C2314yes—
80C2×C4×C8C2312yes—
81C2×C4×C8C2318yes—
82C2×C4×C8C2312yes—
83#391—22no—
84#391—22no—
85#391—22no—
86#391—22no—
87#391C2314yes—
88#391C2314yes—
89#391C2314yes—
90#391C2314yes—
91#389—22no—
92#389—22no—
93#389—22no—
94#389—22no—
95#389—22no—
96#389—22no—
97#389—22no—
98#389—22no—
99#389C2314yes—
100#389C2314yes—
101#389C2314yes—
102#389C2314yes—
103#387—24no—
104#387—24no—
105#387—24no—
106#387—24no—
107#387—24no—
108#387—24no—
109#387—24no—
110#387—24no—
111#385—24no—
112#385—24no—
113#384C2312yes—
114#384C2312yes—
115#383—24no—
116#382—22no—
117#382—22no—
118#382—22no—
119#381—24no—
120#380—22no—
121#380—22no—
122#380—22no—
123#380C2314yes—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—22no—
126C4×C4⋊C4—24no—
127C4×C4⋊C4—24no—
128C4×C4⋊C4—22no—
129C4×C4⋊C4C2314yes—
130C4×C22⋊C4—24no—
131#376—24no—
132C43D411yescharacteristic
133C43D412yes—
134C43D411yescharacteristic
135C43C2311yescharacteristic
136C43C2312yes—
137C43C2311yescharacteristic
138#340C2314yes—
139#340C2318yes—
140#340C2318yes—
141#340C2314yes—
142#336C2314yes—
143#336C2314yes—
144C22×Q8—22no—
145C22×Q8C2414yes—
146C23×C4C2×D412yes—
147C23×C4C2×D411yescharacteristic
148C23×C4C2×D411yescharacteristic
149C23×C4C2×D412yes—
150C23×C4C2411yescharacteristic
151C2×Q16—28no—
152C2×Q16—48no—
153C22×C8C2414yes—
154C22×C8C2414yes—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157#125—48no—
158C22⋊Q8—48no—
159C22⋊Q8—48no—
160C4×Q8—216no—
161C4×Q8—48no—
162C2×C4⋊C4—22no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4C2414yes—
176C2×C4⋊C4C2414yes—
177C2×C4⋊C4C2414yes—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C42—24no—
181C2×C42—24no—
182C2×C42—24no—
183C2×C42—24no—
184C2×C42—44no—
185C2×C42—44no—
186C2×C42Q1618yescomplemented
187C2×C42C2×D412yes—
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D412yes—
190C2×C42C2×D412yes—
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D412yes—
194C2×C42C2×D412yes—
195C2×C42C2×D412yes—
196C2×C42C2×D412yes—
197C2×C42C2×D412yes—
198C2×C42C2×D411yescharacteristic
199C2×C42C2×D411yescharacteristic
200C2×C42C2×D411yescharacteristic
201C2×C42C2412yes—
202C2×C42C2412yes—
203C2×C42C2411yescharacteristic
204C2×C42C2411yescharacteristic
205#107—28no—
206#107—48no—
207C4⋊C8—28no—
208#103—216no—
209#103—48no—
210#103—48no—
211C22⋊C8—28no—
212C4×C8—24no—
213C4×C8—48no—
214C4×C8#4118yes—
215#95—24no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—24no—
224#95—24no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229C24C22×D411yescharacteristic
230C2×Q8—28no—
231C2×Q8—24no—
232C2×Q8—28no—
233C2×Q8—24no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—44no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—24no—
241C22×C4—24no—
242C22×C4—22no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—42no—
249C22×C4—42no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4C2×Q1614yes—
253C22×C4C2×Q1618yes—
254C22×C4C22×D411yescharacteristic
255C22×C4C22×D411yescharacteristic
256C22×C4C22×D411yescharacteristic
257C22×C4C22×D411yescharacteristic
258C22×C4C22×D412yes—
259C22×C4C22×D412yes—
260C22×C4C22×D412yes—
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C2511yescharacteristic
264Q16—416no—
265C2×C8—28no—
266C2×C8—24no—
267C2×C8—28no—
268C2×C8C2×#4118yes—
269C4⋊C4—28no—
270C4⋊C4—24no—
271C4⋊C4—28no—
272C4⋊C4—24no—
273C4⋊C4—44no—
274C4⋊C4—48no—
275C4⋊C4—44no—
276C4⋊C4—44no—
277C4⋊C4—48no—
278C4⋊C4—44no—
279C4⋊C4—44no—
280C4⋊C4—44no—
281C4⋊C4—48no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—48no—
286C22⋊C4—48no—
287C22⋊C4—48no—
288C42—22no—
289C42—22no—
290C42—24no—
291C42—22no—
292C42—24no—
293C42—24no—
294C42—22no—
295C42—24no—
296C42—24no—
297C42—42no—
298C42—42no—
299C42—42no—
300C42—42no—
301C42—48no—
302C42—48no—
303C42C2×Q1618yes—
304C42C2×Q1614yes—
305C42C2×Q1614yes—
306C42#13514yes—
307C42C2×#4114yes—
308C23—21no—
309C23—22no—
310C23—21no—
311C23—22no—
312C23C22×Q1612yes—
313C23C23×D411yescharacteristic
314Q8—48no—
315Q8—48no—
316Q8—88no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—28no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—48no—
334C2×C4—48no—
335C2×C4—44no—
336C2×C4—44no—
337C2×C4—44no—
338C2×C4—44no—
339C2×C4—44no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4#53212yes—
350C2×C4#53212yes—
351C2×C4C22×Q1612yes—
352C2×C4C22×Q1614yes—
353C2×C4C22×Q1614yes—
354C2×C4C22×Q1612yes—
355C2×C4C2×#13514yes—
356C2×C4C2×#13512yes—
357C2×C4#57514yes—
358C2×C4C22×#4112yes—
359C2×C4#58514yes—
360C8—28no—
361C8—44no—
362C22—22no—
363C22—24no—
364C22—41no—
365C22—42no—
366C22—41no—
367C22C2×#53211yescharacteristic
368C22#344912yes—
369C22C23×Q1611yescharacteristic
370C22C22×#13511yescharacteristic
371C22C2×#57512yes—
372C4—22no—
373C4—22no—
374C4—24no—
375C4—24no—
376C4—24no—
377C4—24no—
378C4—41no—
379C4—41no—
380C4—84no—
381C4—84no—
382C4—84no—
383C4—84no—
384C4#317014yes—
385C4#345514yes—
386C2—42no—
387C2#3464311yescharacteristic
388C2C2×#317011yescharacteristic
389C2#6067611yescharacteristic
390C2#6094212yes—
391C2#6094312yes—
subgroupG/H
centerC22×C4C22×D4
commutator, Frattini, cosocleC22×C4C25
socleC23C23×D4
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4 ◁ C2×C43 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #30378C2semidirect
4. #30338C2semidirect
6. #23835C2semidirect
7. #23816C2semidirect
8. #23801C2semidirect
10. #21276C2semidirect
186. C2×C42Q16semidirect

Maximal subgroups

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

subgroupindex
#366352
#303782
#303582
#303382
#303182
#238352
#238162
#238012
#236982
#212762
C4×#4312