Finite Group Databasebeta

← #7567416

Subgroups — #7567416

Order 512.

1545 conjugacy classes of subgroups.

The subgroup lattice has 399 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59869C211yesmaximal, characteristic, complemented
2#36180C211yesmaximal, characteristic
3#32645C214yesmaximal, complemented
4#29141C212yesmaximal, complemented
5#27777C214yesmaximal, complemented
6#26027C214yesmaximal, complemented
7#25945C214yesmaximal
8#24730C214yesmaximal
9#23250C211yesmaximal, characteristic, complemented
10#22762C214yesmaximal
11C4×#461C212yesmaximal
12C2×#554C2212yes—
13C2×#491C2214yes—
14C2×#461C2212yes—
15C2×#461C2214yes—
16C2×C4×SD16C2214yes—
17#2868C2211yescharacteristic
18#2857—21no—
19#2809—22no—
20#2571—21no—
21#2568C2214yes—
22#2567C2212yes—
23#2552C2214yes—
24#2546C2214yes—
25#2508—24no—
26#2502—22no—
27#2440—24no—
28#2369—21no—
29#2355C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8C2212yes—
32C4×C4⋊Q8C2211yescharacteristic
33C2×#384—21no—
34C42×D4C2211yescharacteristic
35#2066C2218yes—
36#2043C2218yes—
37#2026C2218yes—
38#2024C2218yes—
39#1989C2214yes—
40#1985C2214yes—
41#1981C2212yes—
42#1956C2214yes—
43#1948C2212yes—
44#1939C2214yes—
45#1935C2218yes—
46#1927C2214yes—
47#1920C2218yes—
48#1877C2214yes—
49#1877C2218yes—
50#1834C2214yes—
51#1803C2214yes—
52#1803C2214yes—
53#1802C2212yes—
54#1770C2214yes—
55C4×C4⋊C8C2212yes—
56#1762C2214yes—
57C4×#103C2214yes—
58C22×SD16C2314yes—
59#554—44no—
60#554—44no—
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8D411yescharacteristic
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2314yes—
65C2×C4⋊Q8C2312yes—
66C2×#125C2314yes—
67C2×C22⋊Q8—22no—
68C2×C22⋊Q8—24no—
69C2×C4×Q8C2314yes—
70C2×C4×Q8C2312yes—
71C2×C4×D4C2311yescharacteristic
72C2×C4×D4C2314yes—
73C2×C4×D4C2312yes—
74C22×C4⋊C4—22no—
75C22×C4⋊C4—24no—
76C22×C42—21no—
77#491—48no—
78#461—28no—
79#461—48no—
80C4×SD16—216no—
81C2×#106C2314yes—
82C2×C4⋊C8C2312yes—
83C2×C4⋊C8C2312yes—
84C2×C4⋊C8C2314yes—
85C2×#103C2314yes—
86C2×#103C2314yes—
87C2×#103C2314yes—
88C2×#103C2318yes—
89C2×#103C2318yes—
90C2×#102C2314yes—
91C2×#97C2314yes—
92C2×C4×C8C2314yes—
93#400—24no—
94#400—24no—
95#398—28no—
96#398—28no—
97#395—24no—
98#391—22no—
99#391—24no—
100#391D412yes—
101#391C2314yes—
102#391C2318yes—
103#389—24no—
104#389—24no—
105#389—24no—
106#389—22no—
107#389—22no—
108#389D414yes—
109#389C2318yes—
110#387—28no—
111#385—28no—
112#384—22no—
113#384—24no—
114#384—21no—
115#384—21no—
116#384D411yescharacteristic
117#384C2312yes—
118#384C2314yes—
119#382—24no—
120#382—24no—
121#380—24no—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4C2311yescharacteristic
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2314yes—
129C4×C22⋊C4—22no—
130C4×C22⋊C4—21no—
131C43C2311yescharacteristic
132#340C2314yes—
133#340C2318yes—
134#340C2318yes—
135#340C2314yes—
136#336C2318yes—
137#336C2314yes—
138C22×Q8—22no—
139C22×Q8C2×D412yes—
140C22×Q8C2414yes—
141C22×D4C2411yescharacteristic
142C23×C4—21no—
143C23×C4—22no—
144C23×C4—24no—
145C2×SD16—28no—
146C2×SD16—28no—
147C2×SD16—28no—
148C22×C8C2414yes—
149C22×C8C2414yes—
150C4⋊Q8—24no—
151C4⋊Q8—24no—
152C4⋊Q8—24no—
153C4⋊Q8—42no—
154C4⋊Q8—42no—
155C4⋊Q8—44no—
156C4⋊Q8—44no—
157C4⋊Q8—84no—
158#125—28no—
159#125—44no—
160C22⋊Q8—416no—
161C22⋊Q8—48no—
162C4×Q8—28no—
163C4×Q8—28no—
164C4×Q8—42no—
165C4×Q8—42no—
166C4×D4—22no—
167C4×D4—22no—
168C4×D4—28no—
169C4×D4—24no—
170C4×D4—44no—
171C4×D4#4118yes—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—28no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—24no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—88no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D412yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2414yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2412yes—
194C2×C4⋊C4C2411yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—24no—
198C2×C42—24no—
199C2×C42—22no—
200C2×C42—21no—
201C2×C42—24no—
202C2×C42—44no—
203C2×C42C2×D412yes—
204C2×C42C2×D411yescharacteristic
205C2×C42C2×D411yescharacteristic
206C2×C42C2411yescharacteristic
207C2×C42C2414yes—
208C2×C42C2412yes—
209#106—28no—
210C4⋊C8—24no—
211C4⋊C8—24no—
212C4⋊C8—44no—
213#103—28no—
214#103—28no—
215#103—28no—
216#103—48no—
217#103—48no—
218#102—28no—
219#97—44no—
220C4×C8—28no—
221#95—21no—
222#95—22no—
223#95—24no—
224#95—28no—
225#95—24no—
226#95—24no—
227#95—28no—
228#95—28no—
229#95C2×D418yes—
230C24—21no—
231C2×Q8—24no—
232C2×Q8—24no—
233C2×Q8—24no—
234C2×Q8—24no—
235C2×Q8—24no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—42no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—84no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4C2×#4114yes—
245C2×D4H(5,𝔽2)12yes—
246C22×C4—24no—
247C22×C4—22no—
248C22×C4—21no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—42no—
254C22×C4—48no—
255C22×C4—42no—
256C22×C4—44no—
257C22×C4—44no—
258C22×C4—88no—
259C22×C4C22×D411yescharacteristic
260C22×C4C22×D414yes—
261C22×C4C22×D414yes—
262C22×C4C22×D412yes—
263C22×C4C22×D411yescharacteristic
264C22×C4C22×D412yes—
265C22×C4C2511yescharacteristic
266SD16—416no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8—24no—
272C2×C8—24no—
273C4⋊C4—28no—
274C4⋊C4—28no—
275C4⋊C4—21no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—48no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—48no—
287C4⋊C4—44no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—48no—
291C4⋊C4—82no—
292C4⋊C4—88no—
293C4⋊C4—84no—
294C4⋊C4—84no—
295C4⋊C4—84no—
296C4⋊C4C2×#4114yes—
297C4⋊C4H(5,𝔽2)12yes—
298C22⋊C4—24no—
299C22⋊C4—28no—
300C22⋊C4—48no—
301C42—24no—
302C42—24no—
303C42—24no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—48no—
308C42—42no—
309C42—44no—
310C42—84no—
311C42#13714yes—
312C42C2×#4114yes—
313C422−1+414yes—
314C23—24no—
315C23—22no—
316C23—41no—
317C23—41no—
318C23—42no—
319C23C23×D411yescharacteristic
320Q8—44no—
321Q8—44no—
322Q8—44no—
323Q8—44no—
324Q8—84no—
325Q8—164no—
326D4—24no—
327D4—22no—
328D4#55514yes—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—28no—
342C2×C4—48no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—416no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—48no—
355C2×C4—44no—
356C2×C4—168no—
357C2×C4#54814yes—
358C2×C4#54814yes—
359C2×C4C2×#13712yes—
360C2×C4C2×#13714yes—
361C2×C4#57714yes—
362C2×C4C22×#4112yes—
363C2×C4C2×H(5,𝔽2)11yescharacteristic
364C2×C4C2×H(5,𝔽2)11yescharacteristic
365C2×C4C2×2−1+412yes—
366C8—44no—
367C8—44no—
368C22—24no—
369C22—44no—
370C22—42no—
371C22—44no—
372C22—42no—
373C22C2×#54812yes—
374C22#348411yescharacteristic
375C22C22×#13711yescharacteristic
376C22C2×#57712yes—
377C22#358611yescharacteristic
378C4—22no—
379C4—24no—
380C4—44no—
381C4—44no—
382C4—42no—
383C4—44no—
384C4—41no—
385C4—84no—
386C4—84no—
387C4—84no—
388C4#337218yes—
389C4C2×#55511yescharacteristic
390C4#351112yes—
391C4#354411yescharacteristic
392C2—44no—
393C2#5913511yescharacteristic
394C2C2×#337212yes—
395C2#6134612yes—
396C2#6158611yescharacteristic
397C2#6166211yescharacteristic
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
3. #32645C2semidirect
4. #29141C2semidirect
5. #27777C2semidirect
6. #26027C2semidirect
9. #23250C2semidirect

Maximal subgroups

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

subgroupindex
#598692
#361802
#326452
#291412
#277772
#260272
#259452
#247302
#232502
#227622
C4×#4612