Finite Group Databasebeta

← #6528500

Subgroups — #6528500

Order 512.

1472 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35104C211yesmaximal, characteristic
2C2×#2568C211yesmaximal, characteristic
3#29111C212yesmaximal
4#29106C212yesmaximal
5#27969C212yesmaximal
6#27964C212yesmaximal
7#27828C218yesmaximal, complemented
8#27742C218yesmaximal, complemented
9#21840C212yesmaximal
10#20741C212yesmaximal
11#20534C211yesmaximal, characteristic
12C22×C4⋊Q8C2211yescharacteristic
13C22×C4×Q8C2211yescharacteristic
14C2×#432C2212yes—
15#2857C2212yes—
16#2851C2212yes—
17#2627—22no—
18#2606—22no—
19#2605—22no—
20#2570C2214yes—
21#2568—22no—
22#2568—22no—
23#2568C2218yes—
24#2550C2214yes—
25C2×#400—22no—
26#2329—22no—
27C2×#391—22no—
28C2×#391C2211yescharacteristic
29C2×#391C2211yescharacteristic
30#2283C2212yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×C4×C4⋊C4C2211yescharacteristic
34#2101C2218yes—
35#2101C2218yes—
36#2074C2214yes—
37#2072C2214yes—
38#2026C2214yes—
39#2024C2214yes—
40#1993C2214yes—
41#1993C2214yes—
42#1993C2214yes—
43#1991C2214yes—
44#1987C2218yes—
45#1935C2218yes—
46#1933C2218yes—
47#1836C2214yes—
48#1835C2214yes—
49#1824C2214yes—
50#1823C2214yes—
51#1823C2214yes—
52#1805C2214yes—
53#1805C2214yes—
54#1787C2214yes—
55#1771C2218yes—
56#1763C2218yes—
57#1749C2212yes—
58C4×C22⋊C8C2212yes—
59C23×Q8—21no—
60C23×Q8C2311yescharacteristic
61C2×C4⋊Q8—22no—
62C2×C4⋊Q8—28no—
63C2×C4⋊Q8C2314yes—
64C2×C4⋊Q8C2312yes—
65C2×#125C2312yes—
66C2×C22⋊Q8—22no—
67C2×C4×Q8—22no—
68C2×C4×Q8—28no—
69C2×C4×Q8C2314yes—
70C22×C4⋊C4—21no—
71C22×C4⋊C4—21no—
72C22×C4⋊C4D412yes—
73C22×C4⋊C4C2311yescharacteristic
74C22×C4⋊C4C2311yescharacteristic
75C22×C4⋊C4C2311yescharacteristic
76C22×C42—21no—
77C22×C42C2311yescharacteristic
78C22×C42C2311yescharacteristic
79C22×C42C2311yescharacteristic
80#432—28no—
81C2×C4⋊C8C2314yes—
82C2×#103C2318yes—
83C2×#103C2318yes—
84C2×C22⋊C8C2314yes—
85C2×#97C2314yes—
86C2×#97C2314yes—
87C2×C4×C8C2314yes—
88#400—24no—
89#400—28no—
90#400—28no—
91#398—24no—
92#398—24no—
93#395—24no—
94#391—28no—
95#391—28no—
96#391—22no—
97#391—21no—
98#391—21no—
99#391—22no—
100#391—24no—
101#391C2314yes—
102#391C2314yes—
103#389—22no—
104#389—24no—
105#389C2314yes—
106#389C2314yes—
107#387—24no—
108#387—24no—
109#387—24no—
110#385—24no—
111#384C2312yes—
112#383—22no—
113#382—22no—
114#382—21no—
115#382—21no—
116#382C2314yes—
117#382C2314yes—
118#382C2312yes—
119#382C2312yes—
120#382C2312yes—
121#381—22no—
122#380—22no—
123#380C2314yes—
124C4×C4⋊C4D412yes—
125C4×C4⋊C4D412yes—
126C4×C4⋊C4C2314yes—
127C4×C4⋊C4C2314yes—
128C4×C22⋊C4—22no—
129#376—22no—
130C2×#95—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—24no—
146C22×Q8—22no—
147C22×Q8—24no—
148C22×Q8—42no—
149C22×Q8—48no—
150C22×Q8—44no—
151C22×Q8C2412yes—
152C23×C4—21no—
153C23×C4—21no—
154C23×C4—21no—
155C23×C4—21no—
156C23×C4C2×D411yescharacteristic
157C23×C4C2×D412yes—
158C23×C4C2×D412yes—
159C23×C4C2×D411yescharacteristic
160C23×C4C2411yescharacteristic
161C22×C8C2418yes—
162C4⋊Q8—28no—
163C4⋊Q8—28no—
164C4⋊Q8—44no—
165C4⋊Q8—48no—
166#125—44no—
167C22⋊Q8—48no—
168C4×Q8—28no—
169C4×Q8—28no—
170C4×Q8—48no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—24no—
173C2×C4⋊C4—24no—
174C2×C4⋊C4—24no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—24no—
181C2×C4⋊C4—24no—
182C2×C4⋊C4—22no—
183C2×C4⋊C4—22no—
184C2×C4⋊C4—24no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—44no—
187C2×C4⋊C4—44no—
188C2×C4⋊C4C2×D414yes—
189C2×C4⋊C4C2×D414yes—
190C2×C4⋊C4C2412yes—
191C2×C4⋊C4C2412yes—
192C2×C4⋊C4C2412yes—
193C2×C4⋊C4C2414yes—
194C2×C4⋊C4C2414yes—
195C2×C22⋊C4—24no—
196C2×C42—24no—
197C2×C42—24no—
198C2×C42—42no—
199C2×C42—42no—
200C2×C42—44no—
201C2×C42C2×D412yes—
202C2×C42C2×D411yescharacteristic
203C2×C42C2×D411yescharacteristic
204C2×C42C2×D412yes—
205C2×C42#4118yes—
206C2×C42C2412yes—
207C2×C42C2412yes—
208C2×C42C2412yes—
209C4⋊C8—28no—
210#103—216no—
211#103—48no—
212C22⋊C8—28no—
213#97—28no—
214#97—44no—
215C4×C8—44no—
216#95—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—416no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—48no—
244C2×Q8—82no—
245C2×Q8—82no—
246C2×Q8—816no—
247C2×Q82−1+414yes—
248C22×C4—24no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—24no—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—24no—
260C22×C4—42no—
261C22×C4—42no—
262C22×C4—42no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—44no—
268C22×C4—42no—
269C22×C4—42no—
270C22×C4—42no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4#12414yes—
274C22×C4#12414yes—
275C22×C4C22×D414yes—
276C22×C4C22×D412yes—
277C22×C4C22×D412yes—
278C22×C4C22×D412yes—
279C22×C4C22×D411yescharacteristic
280C22×C4C22×D412yes—
281C22×C4C2×#4114yes—
282C22×C4C2511yescharacteristic
283C2×C8—28no—
284C2×C8—28no—
285C2×C8—28no—
286C4⋊C4—24no—
287C4⋊C4—24no—
288C4⋊C4—24no—
289C4⋊C4—28no—
290C4⋊C4—28no—
291C4⋊C4—44no—
292C4⋊C4—44no—
293C4⋊C4—48no—
294C4⋊C4—44no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—48no—
299C4⋊C4—48no—
300C4⋊C4—82no—
301C4⋊C4—84no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C22⋊C4—48no—
305C42—24no—
306C42—24no—
307C42—42no—
308C42—41no—
309C42—41no—
310C42—48no—
311C42—84no—
312C42#12418yes—
313C42C2×#4114yes—
314C42C2×#4114yes—
315C23—22no—
316C23—21no—
317C23—21no—
318C23C2×#12412yes—
319C23#57514yes—
320C23C23×D411yescharacteristic
321Q8—28no—
322Q8—48no—
323Q8—44no—
324Q8—816no—
325Q8—164no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—24no—
331C2×C4—28no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—24no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—44no—
343C2×C4—48no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—44no—
353C2×C4—44no—
354C2×C4—44no—
355C2×C4—44no—
356C2×C4—44no—
357C2×C4—44no—
358C2×C4—84no—
359C2×C4—84no—
360C2×C4—84no—
361C2×C4—84no—
362C2×C4—84no—
363C2×C4C2×#12414yes—
364C2×C4C2×#12412yes—
365C2×C4C2×#12414yes—
366C2×C4C22×#4112yes—
367C2×C4C2×2−1+412yes—
368C2×C4#58512yes—
369C8—48no—
370C22—24no—
371C22—22no—
372C22—24no—
373C22—22no—
374C22#314014yes—
375C22C22×#12411yescharacteristic
376C22#348511yescharacteristic
377C22#348611yescharacteristic
378C22C2×#57512yes—
379C22#358612yes—
380C4—22no—
381C4—24no—
382C4—24no—
383C4—42no—
384C4—44no—
385C4—42no—
386C4—44no—
387C4—84no—
388C4—84no—
389C4—82no—
390C4—82no—
391C4—82no—
392C4#349514yes—
393C2—24no—
394C2#3564411yescharacteristic
395C2C2×#314011yescharacteristic
396C2#6062011yescharacteristic
397C2#6171212yes—
398C2#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. #27828C2semidirect
8. #27742C2semidirect

Maximal subgroups

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

subgroupindex
#351042
C2×#25682
#291112
#291062
#279692
#279642
#278282
#277422
#218402
#207412
#205342