Finite Group Databasebeta

← #7542942

Subgroups — #7542942

Order 512.

1595 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#59864C211yesmaximal, characteristic
2#36594C211yesmaximal, characteristic
3#32269C214yesmaximal, complemented
4#29206C212yesmaximal, complemented
5#26297C214yesmaximal, complemented
6#25349C214yesmaximal, complemented
7#25014C214yesmaximal, complemented
8#24108C214yesmaximal, complemented
9#23176C211yesmaximal, characteristic
10#22663C214yesmaximal, complemented
11C4×#477C212yesmaximal, complemented
12C2×#556C2212yes—
13C2×#483C2214yes—
14C2×#477C2212yes—
15C2×#436C2214yes—
16C2×#428C2214yes—
17#2868C2211yescharacteristic
18#2858—21no—
19#2804—22no—
20#2580—24no—
21#2571—21no—
22#2563C2214yes—
23#2547C2212yes—
24#2545—22no—
25#2538C2214yes—
26#2512—24no—
27#2487C2214yes—
28#2366—21no—
29#2354C2214yes—
30D4×C4⋊C4C2214yes—
31C4×C4⋊Q8—21no—
32C4×C4⋊Q8C2212yes—
33C4×#127C2211yescharacteristic
34C42×Q8C2211yescharacteristic
35#2077C2218yes—
36#2075C2218yes—
37#2058C2218yes—
38#2035C2218yes—
39#1990C2214yes—
40#1987C2212yes—
41#1954C2214yes—
42#1948C2214yes—
43#1948C2214yes—
44#1947C2212yes—
45#1921C2214yes—
46#1894C2214yes—
47#1870C2214yes—
48#1863C2214yes—
49#1854C2218yes—
50#1853C2218yes—
51#1844C2214yes—
52#1825C2214yes—
53#1809C2218yes—
54#1803C2212yes—
55C4×#107C2214yes—
56C4×C4⋊C8C2212yes—
57C4×#103C2214yes—
58C22×M16C2314yes—
59#556—44no—
60#556—44no—
61C2×C4⋊Q8—21no—
62C2×C4⋊Q8—21no—
63C2×C4⋊Q8C2312yes—
64C2×C4⋊Q8C2312yes—
65C2×#127D411yescharacteristic
66C2×#126—24no—
67C2×#124—22no—
68C2×#124—22no—
69C2×C22⋊Q8C2314yes—
70C2×C4×Q8—22no—
71C2×C4×Q8—22no—
72C2×C4×Q8C2312yes—
73C2×C4×Q8C2311yescharacteristic
74C2×C4×D4—24no—
75C2×C4×D4C2314yes—
76C2×C4×D4C2312yes—
77C2×#117C2314yes—
78C22×C4⋊C4C2314yes—
79#483—48no—
80#477—28no—
81#436—216no—
82#428—48no—
83C2×#107C2314yes—
84C2×#107C2314yes—
85C2×#106C2318yes—
86C2×#106C2318yes—
87C2×C4⋊C8C2312yes—
88C2×C4⋊C8C2312yes—
89C2×#103C2314yes—
90C2×#103C2314yes—
91C2×C22⋊C8C2318yes—
92C2×C22⋊C8C2314yes—
93C2×C4×C8C2314yes—
94#400—24no—
95#399—24no—
96#397—24no—
97#396—28no—
98#393—28no—
99#391—22no—
100#391—22no—
101#390—24no—
102#390—24no—
103#390—24no—
104#390D414yes—
105#390D412yes—
106#389—24no—
107#389—24no—
108#389—22no—
109#389C2314yes—
110#388—28no—
111#386—24no—
112#385—24no—
113#385—28no—
114#384—22no—
115#384D411yescharacteristic
116#384C2312yes—
117#384C2314yes—
118#382—24no—
119#380C2314yes—
120#380C2318yes—
121#380C2318yes—
122C4×C4⋊C4—24no—
123C4×C4⋊C4—22no—
124C4×C4⋊C4—22no—
125C4×C4⋊C4—21no—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2312yes—
128C4×C4⋊C4C2311yescharacteristic
129C43—21no—
130C43C2311yescharacteristic
131#340C2318yes—
132#340C2318yes—
133#340C2314yes—
134#340C2314yes—
135#340C2314yes—
136#340C2314yes—
137#336C2314yes—
138C22×Q8—22no—
139C22×Q8—22no—
140C22×Q8C2411yescharacteristic
141C22×D4C2×D414yes—
142C22×D4C2×D412yes—
143C23×C4—24no—
144C23×C4C2414yes—
145C2×M16—216no—
146C2×M16—44no—
147C22×C8C2414yes—
148C22×C8C2414yes—
149C4⋊Q8—24no—
150C4⋊Q8—24no—
151C4⋊Q8—42no—
152C4⋊Q8—42no—
153C4⋊Q8—44no—
154C4⋊Q8—82no—
155#127—24no—
156#126—416no—
157#124—48no—
158#124—84no—
159C22⋊Q8—28no—
160C22⋊Q8—44no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—24no—
164C4×Q8—24no—
165C4×Q8—48no—
166C4×Q8—84no—
167C4×D4—28no—
168C4×D4—28no—
169C4×D4—42no—
170C4×D4—42no—
171C4×D4—88no—
172#117—28no—
173#117—44no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—24no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—22no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4C2×D414yes—
188C2×C4⋊C4C2×D412yes—
189C2×C4⋊C4C2414yes—
190C2×C4⋊C4C2414yes—
191C2×C4⋊C4C2412yes—
192C2×C4⋊C4C2414yes—
193C2×C4⋊C4C2411yescharacteristic
194C2×C22⋊C4—24no—
195C2×C22⋊C4—24no—
196C2×C22⋊C4—28no—
197C2×C22⋊C4—28no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4C2×D418yes—
200C2×C22⋊C4C2×D414yes—
201C2×C42—21no—
202C2×C42—24no—
203C2×C42—24no—
204C2×C42—22no—
205C2×C42—21no—
206C2×C42—22no—
207C2×C42C2×D412yes—
208C2×C42C2×D411yescharacteristic
209C2×C42C2×D411yescharacteristic
210C2×C42C2411yescharacteristic
211C2×C42C2412yes—
212#107—28no—
213#107—28no—
214#106—48no—
215#106—48no—
216C4⋊C8—24no—
217C4⋊C8#4118yes—
218#103—28no—
219#103—28no—
220C22⋊C8—216no—
221C22⋊C8—44no—
222C4×C8—28no—
223#95—22no—
224#95—21no—
225#95—28no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—28no—
230C24C22×D414yes—
231C2×Q8—21no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—44no—
235C2×Q8—44no—
236C2×Q8—44no—
237C2×Q8—44no—
238C2×Q8—82no—
239C2×Q8—82no—
240C2×Q8H(5,𝔽2)12yes—
241C2×D4—28no—
242C2×D4—28no—
243C2×D4—24no—
244C2×D4—28no—
245C2×D4—42no—
246C2×D4—42no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—28no—
250C22×C4—24no—
251C22×C4—24no—
252C22×C4—28no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—84no—
258C22×C4—88no—
259C22×C4C22×D414yes—
260C22×C4C22×D411yescharacteristic
261C22×C4C22×D412yes—
262C22×C4C22×D411yescharacteristic
263C22×C4C22×D412yes—
264C22×C4C2511yescharacteristic
265M16—416no—
266C2×C8—24no—
267C2×C8—24no—
268C2×C8—24no—
269C2×C8—24no—
270C2×C8—24no—
271C2×C8C2×#4118yes—
272C4⋊C4—24no—
273C4⋊C4—28no—
274C4⋊C4—21no—
275C4⋊C4—24no—
276C4⋊C4—22no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—24no—
281C4⋊C4—22no—
282C4⋊C4—24no—
283C4⋊C4—42no—
284C4⋊C4—48no—
285C4⋊C4—44no—
286C4⋊C4—44no—
287C4⋊C4—44no—
288C4⋊C4—48no—
289C4⋊C4—44no—
290C4⋊C4—84no—
291C4⋊C4—82no—
292C4⋊C4—84no—
293C4⋊C4—84no—
294C4⋊C4H(5,𝔽2)12yes—
295C22⋊C4—216no—
296C22⋊C4—44no—
297C22⋊C4—48no—
298C22⋊C4—48no—
299C22⋊C4—84no—
300C22⋊C4—88no—
301C22⋊C4—88no—
302C42—24no—
303C42—24no—
304C42—21no—
305C42—21no—
306C42—41no—
307C42—48no—
308C42—42no—
309C42—44no—
310C42—44no—
311C42—48no—
312C42#13714yes—
313C42C2×#4114yes—
314C422−1+414yes—
315C23—24no—
316C23—28no—
317C23—24no—
318C23—24no—
319C23—28no—
320C23C23×D411yescharacteristic
321Q8—24no—
322Q8—41no—
323Q8—41no—
324Q8—88no—
325Q8—88no—
326D4—416no—
327D4—84no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—28no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—44no—
343C2×C4—42no—
344C2×C4—44no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—44no—
350C2×C4—44no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—42no—
354C2×C4—44no—
355C2×C4—168no—
356C2×C4#54714yes—
357C2×C4#54714yes—
358C2×C4C2×#13712yes—
359C2×C4C2×#13714yes—
360C2×C4#57714yes—
361C2×C4C22×#4112yes—
362C2×C4C2×H(5,𝔽2)11yescharacteristic
363C2×C4C2×H(5,𝔽2)11yescharacteristic
364C2×C4C2×2−1+412yes—
365C8—28no—
366C8—44no—
367C22—48no—
368C22—48no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22C2×#54712yes—
373C22#348411yescharacteristic
374C22C22×#13711yescharacteristic
375C22C2×#57712yes—
376C22#358511yescharacteristic
377C4—24no—
378C4—22no—
379C4—44no—
380C4—42no—
381C4—41no—
382C4—42no—
383C4—82no—
384C4—84no—
385C4—84no—
386C4—84no—
387C4#335618yes—
388C4#348111yescharacteristic
389C4#352212yes—
390C4#355011yescharacteristic
391C2—84no—
392C2#5907011yescharacteristic
393C2C2×#335612yes—
394C2#6127912yes—
395C2#6158811yescharacteristic
396C2#6166411yescharacteristic
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC2×#127D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×#127 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #32269C2semidirect
4. #29206C2semidirect
5. #26297C2semidirect
6. #25349C2semidirect
7. #25014C2semidirect
8. #24108C2semidirect
10. #22663C2semidirect
11. C4×#477C2semidirect

Maximal subgroups

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

subgroupindex
#598642
#365942
#322692
#292062
#262972
#253492
#250142
#241082
#231762
#226632
C4×#4772