Finite Group Databasebeta

← #6528516

Subgroups — #6528516

Order 512.

1444 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#35104C211yesmaximal, characteristic
2C2×#2565C211yesmaximal, characteristic
3#29112C212yesmaximal
4#29111C212yesmaximal
5#27969C212yesmaximal
6#27966C212yesmaximal
7#27820C218yesmaximal, complemented
8#27734C218yesmaximal, complemented
9#21836C212yesmaximal
10#20735C212yesmaximal
11#20536C211yesmaximal, characteristic
12C22×#125C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#431C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2627—22no—
18#2606—22no—
19#2605—22no—
20#2570C2214yes—
21#2565—22no—
22#2565—22no—
23#2565C2218yes—
24#2550C2214yes—
25C2×#398—22no—
26#2329—22no—
27C2×#391C2211yescharacteristic
28C2×#391C2211yescharacteristic
29C2×#387—22no—
30C2×#384C2211yescharacteristic
31#2283C2212yes—
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2101C2218yes—
35#2100C2218yes—
36#2074C2214yes—
37#2072C2214yes—
38#2026C2214yes—
39#2024C2214yes—
40#1993C2214yes—
41#1992C2214yes—
42#1991C2214yes—
43#1991C2214yes—
44#1989C2218yes—
45#1935C2218yes—
46#1935C2218yes—
47#1836C2214yes—
48#1836C2214yes—
49#1824C2214yes—
50#1823C2214yes—
51#1823C2214yes—
52#1805C2214yes—
53#1805C2214yes—
54#1787C2214yes—
55#1770C2218yes—
56C4×#103C2218yes—
57#1749C2212yes—
58#1749C2212yes—
59C23×Q8C2311yescharacteristic
60C2×C4⋊Q8C2312yes—
61C2×#125—22no—
62C2×#125—28no—
63C2×#125C2314yes—
64C2×#125C2312yes—
65C2×C22⋊Q8—22no—
66C2×C4×Q8—22no—
67C2×C4×Q8—28no—
68C2×C4×Q8C2314yes—
69C22×C4⋊C4—21no—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—22no—
72C22×C4⋊C4—21no—
73C22×C4⋊C4D412yes—
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C4⋊C4C2311yescharacteristic
77C22×C42—21no—
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80C22×C42C2311yescharacteristic
81#431—28no—
82C2×C4⋊C8C2314yes—
83C2×#103C2318yes—
84C2×#103C2318yes—
85C2×C22⋊C8C2314yes—
86C2×#97C2314yes—
87C2×#97C2314yes—
88C2×C4×C8C2314yes—
89#400—24no—
90#398—28no—
91#398—28no—
92#398—24no—
93#398—24no—
94#395—24no—
95#391—21no—
96#391—21no—
97#391—22no—
98#391—22no—
99#391—24no—
100#391C2314yes—
101#391C2314yes—
102#389—22no—
103#389—24no—
104#389C2314yes—
105#389C2314yes—
106#387—24no—
107#387—28no—
108#387—28no—
109#387—24no—
110#387—24no—
111#385—24no—
112#384—22no—
113#384C2314yes—
114#384C2312yes—
115#383—22no—
116#382—21no—
117#382—21no—
118#382C2314yes—
119#382C2312yes—
120#382C2312yes—
121#382C2312yes—
122#381—22no—
123#380—22no—
124#380C2314yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4D412yes—
127C4×C4⋊C4C2314yes—
128C4×C4⋊C4C2314yes—
129C4×C22⋊C4—22no—
130#376—22no—
131C2×#95—22no—
132C2×#95—22no—
133C2×#95—22no—
134C2×#95D412yes—
135#340C2318yes—
136#340C2318yes—
137#340C2318yes—
138#340C2318yes—
139#340C2318yes—
140#340C2318yes—
141#336C2314yes—
142#336C2314yes—
143C22×Q8—24no—
144C22×Q8—21no—
145C22×Q8—22no—
146C22×Q8—24no—
147C22×Q8—42no—
148C22×Q8C2412yes—
149C23×C4—21no—
150C23×C4—21no—
151C23×C4—21no—
152C23×C4—21no—
153C23×C4C2×D411yescharacteristic
154C23×C4C2×D412yes—
155C23×C4C2×D412yes—
156C23×C4C2×D411yescharacteristic
157C23×C4C2411yescharacteristic
158C22×C8C2418yes—
159C4⋊Q8—44no—
160#125—28no—
161#125—28no—
162#125—44no—
163#125—48no—
164C22⋊Q8—48no—
165C4×Q8—28no—
166C4×Q8—28no—
167C4×Q8—48no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—24no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—28no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—24no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—22no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—24no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—48no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4—44no—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2×D414yes—
191C2×C4⋊C4C2412yes—
192C2×C4⋊C4C2412yes—
193C2×C4⋊C4C2412yes—
194C2×C4⋊C4C2414yes—
195C2×C4⋊C4C2414yes—
196C2×C22⋊C4—24no—
197C2×C42—24no—
198C2×C42—24no—
199C2×C42—42no—
200C2×C42—42no—
201C2×C42—44no—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42C2×D412yes—
206C2×C42#4118yes—
207C2×C42C2412yes—
208C2×C42C2412yes—
209C2×C42C2412yes—
210C4⋊C8—28no—
211#103—216no—
212#103—48no—
213C22⋊C8—28no—
214#97—44no—
215#97—44no—
216C4×C8—28no—
217#95—24no—
218#95—24no—
219#95—28no—
220#95—24no—
221#95—24no—
222#95—24no—
223#95—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—24no—
228#95—24no—
229#95—24no—
230#95—24no—
231#95C2×D414yes—
232#95C2×D414yes—
233C24C22×D411yescharacteristic
234C2×Q8—24no—
235C2×Q8—22no—
236C2×Q8—24no—
237C2×Q8—24no—
238C2×Q8—416no—
239C2×Q8—42no—
240C2×Q8—44no—
241C2×Q8—42no—
242C2×Q8—82no—
243C2×Q8—82no—
244C2×Q82−1+414yes—
245C22×C4—24no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—24no—
249C22×C4—24no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—22no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—42no—
258C22×C4—42no—
259C22×C4—42no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—44no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—44no—
269C22×C4—44no—
270C22×C4#12414yes—
271C22×C4#12414yes—
272C22×C4C22×D412yes—
273C22×C4C22×D412yes—
274C22×C4C22×D414yes—
275C22×C4C22×D411yescharacteristic
276C22×C4C22×D412yes—
277C22×C4C22×D412yes—
278C22×C4C2×#4114yes—
279C22×C4C2511yescharacteristic
280C2×C8—28no—
281C2×C8—28no—
282C2×C8—28no—
283C4⋊C4—24no—
284C4⋊C4—24no—
285C4⋊C4—24no—
286C4⋊C4—28no—
287C4⋊C4—28no—
288C4⋊C4—44no—
289C4⋊C4—44no—
290C4⋊C4—416no—
291C4⋊C4—48no—
292C4⋊C4—44no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4—82no—
303C22⋊C4—48no—
304C42—24no—
305C42—24no—
306C42—41no—
307C42—42no—
308C42—41no—
309C42—48no—
310C42—84no—
311C42#12418yes—
312C42C2×#4114yes—
313C42C2×#4114yes—
314C23—22no—
315C23—21no—
316C23—21no—
317C23C2×#12412yes—
318C23#57514yes—
319C23C23×D411yescharacteristic
320Q8—28no—
321Q8—48no—
322Q8—44no—
323Q8—164no—
324C2×C4—24no—
325C2×C4—28no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—24no—
329C2×C4—28no—
330C2×C4—24no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—24no—
337C2×C4—24no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—44no—
341C2×C4—48no—
342C2×C4—48no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4C2×#12414yes—
362C2×C4C2×#12412yes—
363C2×C4C2×#12414yes—
364C2×C4C22×#4112yes—
365C2×C4C2×2−1+412yes—
366C2×C4#58512yes—
367C8—48no—
368C22—24no—
369C22—22no—
370C22—24no—
371C22—22no—
372C22#313514yes—
373C22C22×#12411yescharacteristic
374C22#348511yescharacteristic
375C22#348611yescharacteristic
376C22C2×#57512yes—
377C22#358612yes—
378C4—22no—
379C4—24no—
380C4—24no—
381C4—44no—
382C4—42no—
383C4—42no—
384C4—44no—
385C4—84no—
386C4—84no—
387C4—82no—
388C4—82no—
389C4—82no—
390C4#349514yes—
391C2—24no—
392C2#3564411yescharacteristic
393C2C2×#313511yescharacteristic
394C2#6062011yescharacteristic
395C2#6171612yes—
396C2#6183212yes—
subgroupG/H
center, socleC23C23×D4
commutator, Frattini, cosocleC22×C4C25
prime radicalC24C22×D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×C4⋊C4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
7. #27820C2semidirect
8. #27734C2semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25652
#291122
#291112
#279692
#279662
#278202
#277342
#218362
#207352
#205362