Finite Group Databasebeta

← #7500216

Subgroups — #7500216

Order 512.

1312 conjugacy classes of subgroups.

The subgroup lattice has 391 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#39379C211yesmaximal, characteristic
3#32598C214yesmaximal
4#32166C214yesmaximal
5#32082C214yesmaximal, complemented
6#30078C212yesmaximal, complemented
7#29368C212yesmaximal
8#27490C214yesmaximal
9#27362C214yesmaximal, complemented
10#26320C214yesmaximal, complemented
11#23363C211yesmaximal, characteristic, complemented
12C2×#541—21no—
13#2859C2211yescharacteristic
14#2809—22no—
15#2725—24no—
16#2629—21no—
17#2571—21no—
18#2568C2214yes—
19#2546C2214yes—
20#2538C2212yes—
21#2512C2214yes—
22#2508—24no—
23#2506C2214yes—
24#2421—22no—
25#2369—21no—
26#2355C2214yes—
27Q8×C4⋊C4C2212yes—
28C4×C4⋊Q8C2212yes—
29C4×C4⋊Q8C2211yescharacteristic
30#2293C2211yescharacteristic
31#2100C2212yes—
32#2096C2214yes—
33#2096C2214yes—
34#2094C2212yes—
35#2084C2214yes—
36#2072C2214yes—
37#2072C2214yes—
38#2066C2214yes—
39#2055C2214yes—
40#2043C2214yes—
41#2043C2212yes—
42#2043C2214yes—
43#2013C2214yes—
44#2013C2212yes—
45#1984C2214yes—
46#1981C2214yes—
47#1968C2214yes—
48#1956C2214yes—
49#1948C2214yes—
50#1933C2214yes—
51#1927C2214yes—
52#1927C2214yes—
53#1927C2214yes—
54#1926C2214yes—
55#1919C2214yes—
56#1916C2214yes—
57#1916C2214yes—
58#1880C2214yes—
59#1877C2214yes—
60#1865C2214yes—
61#1865C2214yes—
62#1865C2214yes—
63#1839C2212yes—
64C4×#106C2214yes—
65C22×SD16C2314yes—
66#541—48no—
67C2×C4⋊Q8D411yescharacteristic
68C2×C4⋊Q8C2312yes—
69C2×C4⋊Q8C2314yes—
70C2×#126—24no—
71C2×#125—24no—
72C2×#125—21no—
73C2×C22⋊Q8—22no—
74C2×C4×Q8—22no—
75C2×C4×Q8C2314yes—
76C2×C4×Q8C2312yes—
77C2×C4×D4C2311yescharacteristic
78C2×#117—21no—
79C2×#106C2314yes—
80C2×#106C2314yes—
81C2×#106C2314yes—
82C2×C4⋊C8C2314yes—
83C2×#103C2314yes—
84C2×#103C2314yes—
85C2×#103C2314yes—
86C2×#103C2314yes—
87C2×#103C2314yes—
88C2×#103C2314yes—
89C2×#103C2314yes—
90C2×#102C2314yes—
91C2×C4×C8C2314yes—
92#400—24no—
93#400—24no—
94#400—24no—
95#398—24no—
96#398—24no—
97#398—24no—
98#397—24no—
99#396—24no—
100#396—24no—
101#391—24no—
102#391D412yes—
103#391C2314yes—
104#391C2314yes—
105#391C2314yes—
106#390C2314yes—
107#390C2312yes—
108#389—24no—
109#389—22no—
110#389—24no—
111#389—24no—
112#389—22no—
113#389—22no—
114#389—22no—
115#389D414yes—
116#389C2314yes—
117#389C2314yes—
118#389C2312yes—
119#388—22no—
120#388—24no—
121#387—24no—
122#387—24no—
123#387—24no—
124#384D411yescharacteristic
125#384C2312yes—
126#384C2312yes—
127#384C2314yes—
128#382—24no—
129#382—24no—
130#382—21no—
131#380—22no—
132C4×C4⋊C4—24no—
133C4×C4⋊C4—22no—
134C4×C4⋊C4—22no—
135C4×C4⋊C4C2314yes—
136C4×C4⋊C4C2311yescharacteristic
137C4×C4⋊C4C2314yes—
138C4×C4⋊C4C2314yes—
139C4×C22⋊C4—21no—
140C43C2311yescharacteristic
141#340C2314yes—
142#340C2314yes—
143#340C2314yes—
144#340C2314yes—
145#336C2314yes—
146#336C2314yes—
147#336C2314yes—
148#336C2312yes—
149#336C2312yes—
150#336C2314yes—
151C22×Q8C2×D412yes—
152C22×Q8C2414yes—
153C22×D4C2411yescharacteristic
154C23×C4—21no—
155C2×SD16—44no—
156C2×SD16—44no—
157C2×SD16—44no—
158C22×C8C2414yes—
159C22×C8C2414yes—
160C4⋊Q8—44no—
161C4⋊Q8—44no—
162C4⋊Q8—44no—
163C4⋊Q8SD1618yescomplemented
164#126—416no—
165#125—416no—
166#125—44no—
167C22⋊Q8—48no—
168C4×Q8—42no—
169C4×Q8—42no—
170C4×Q8—44no—
171C4×Q8—44no—
172C4×Q8—48no—
173C4×D4—42no—
174#117—44no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—24no—
178C2×C4⋊C4—24no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—24no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—24no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2×D414yes—
192C2×C4⋊C4C2×D414yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2412yes—
196C2×C4⋊C4C2414yes—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—24no—
202C2×C42—21no—
203C2×C42—24no—
204C2×C42—24no—
205C2×C42—24no—
206C2×C42C2×D411yescharacteristic
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2411yescharacteristic
210C2×C42C2412yes—
211C2×C42C2414yes—
212#106—44no—
213#106—44no—
214#106—44no—
215C4⋊C8—44no—
216#103—44no—
217#103—44no—
218#103—44no—
219#103—44no—
220#103—44no—
221#103—44no—
222#103—44no—
223#102—44no—
224C4×C8—44no—
225#95—22no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95—21no—
232#95—24no—
233#95—24no—
234#95—24no—
235#95—24no—
236#95—24no—
237#95C2×D414yes—
238#95C2×D414yes—
239C24—21no—
240C2×Q8—24no—
241C2×Q8—24no—
242C2×Q8—24no—
243C2×Q8—24no—
244C2×Q8—24no—
245C2×Q8—24no—
246C2×Q8—42no—
247C2×Q8—42no—
248C2×Q8C2×SD1618yes—
249C2×D4—24no—
250C2×D4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—21no—
254C22×C4—22no—
255C22×C4—24no—
256C22×C4—42no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4C22×D414yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D414yes—
264C22×C4C22×D412yes—
265C22×C4C2511yescharacteristic
266SD16—88no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C4⋊C4—24no—
274C4⋊C4—24no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—44no—
280C4⋊C4—42no—
281C4⋊C4—44no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—44no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—48no—
290C4⋊C4—48no—
291C4⋊C4—82no—
292C4⋊C4—82no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4C2×SD16116yes—
297C22⋊C4—42no—
298C22⋊C4—44no—
299C22⋊C4—48no—
300C22⋊C4—82no—
301C22⋊C4—84no—
302C42—24no—
303C42—24no—
304C42—22no—
305C42—24no—
306C42—42no—
307C42—41no—
308C42—48no—
309C42—42no—
310C42—48no—
311C42—84no—
312C42C2×SD1614yes—
313C23—42no—
314C23—41no—
315C23—41no—
316C23—42no—
317C23—41no—
318C23C23×D411yescharacteristic
319Q8—48no—
320Q8—44no—
321Q8—44no—
322Q8—44no—
323Q8—44no—
324D4—44no—
325C2×C4—22no—
326C2×C4—21no—
327C2×C4—24no—
328C2×C4—21no—
329C2×C4—24no—
330C2×C4—22no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—24no—
335C2×C4—24no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—44no—
340C2×C4—42no—
341C2×C4—44no—
342C2×C4—44no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—84no—
355C2×C4C22×SD1612yes—
356C2×C4C22×SD1618yes—
357C2×C4C22×SD1614yes—
358C2×C4#58512yes—
359C2×C4#58514yes—
360C8—44no—
361C8—44no—
362C22—81no—
363C22—81no—
364C22—82no—
365C22—82no—
366C22—81no—
367C22#348511yescharacteristic
368C22#348712yes—
369C22C23×SD1611yescharacteristic
370C22#358612yes—
371C22#358611yescharacteristic
372C4—24no—
373C4—22no—
374C4—28no—
375C4—41no—
376C4—44no—
377C4—42no—
378C4—84no—
379C4—82no—
380C4—84no—
381C4—84no—
382C4—81no—
383C4#356614yes—
384C2—161no—
385C2#5208411yescharacteristic
386C2#6169511yescharacteristic
387C2#6179811yescharacteristic
388C2#6188312yes—
389C2#6194112yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC22×D4C24
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C4⋊Q8 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
1. #59869C2semidirect
5. #32082C2semidirect
6. #30078C2semidirect
9. #27362C2semidirect
10. #26320C2semidirect
11. #23363C2semidirect
163. C4⋊Q8SD16semidirect

Maximal subgroups

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

subgroupindex
#598692
#393792
#325982
#321662
#320822
#300782
#293682
#274902
#273622
#263202
#233632