Finite Group Databasebeta

← #4414340

Subgroups — #4414340

Order 512.

2506 conjugacy classes of subgroups.

The subgroup lattice has 445 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2822C211yesmaximal, characteristic, complemented
2C2×#2851C212yesmaximal
3C2×#2285C211yesmaximal, characteristic
4#16563C214yesmaximal, complemented
5#16400C214yesmaximal
6#16399C214yesmaximal
7#14907C214yesmaximal
8#14869C214yesmaximal, complemented
9#14818C214yesmaximal, complemented
10#14677C211yesmaximal, characteristic, complemented
11#14664C212yesmaximal, complemented
12C22×#125C2212yes—
13C22×#117C2212yes—
14C23×C42C2211yescharacteristic
15#2851—216no—
16#2822—28no—
17C2×#400C2214yes—
18C2×#398C2214yes—
19C2×#398C2214yes—
20C2×#397C2214yes—
21C2×#389C2212yes—
22C2×#389C2212yes—
23C2×#389C2214yes—
24C2×#387C2214yes—
25C2×#387C2214yes—
26C2×#385C2214yes—
27C2×#385C2214yes—
28#2285—28no—
29C2×#384C2212yes—
30C2×#382C2212yes—
31C2×#382C2211yescharacteristic
32C2×#382C2211yescharacteristic
33C2×#380C2212yes—
34C2×#380C2212yes—
35C2×#380C2214yes—
36C2×C4×C4⋊C4C2214yes—
37#1445C2214yes—
38#1445C2214yes—
39#1444C2214yes—
40#1442C2214yes—
41#1441C2214yes—
42#1441C2218yes—
43#1441C2218yes—
44#1441C2218yes—
45#1441C2214yes—
46#1441C2218yes—
47#1441C2218yes—
48#1441C2214yes—
49#1440C2218yes—
50#1439C2214yes—
51#1436C2218yes—
52#1434C2214yes—
53C24×C4C2311yescharacteristic
54C24×C4C2312yes—
55C2×#125—28no—
56C2×#125—28no—
57C2×#125—28no—
58C2×#117—28no—
59C2×#117—44no—
60C2×#117—44no—
61C22×C4⋊C4C2314yes—
62C22×C4⋊C4C2312yes—
63C22×C4⋊C4C2314yes—
64C22×C4⋊C4C2312yes—
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2318yes—
68C22×C4⋊C4C2314yes—
69C22×C4⋊C4C2312yes—
70C22×C4⋊C4C2312yes—
71C22×C4⋊C4C2312yes—
72C22×C4⋊C4C2314yes—
73C22×C4⋊C4C2312yes—
74C22×C22⋊C4C2314yes—
75C22×C22⋊C4C2314yes—
76C22×C22⋊C4C2314yes—
77C22×C42—24no—
78C22×C42—24no—
79C22×C42D418yes—
80C22×C42C2314yes—
81C22×C42C2312yes—
82C22×C42C2311yescharacteristic
83C22×C42C2311yescharacteristic
84#400—216no—
85#398—216no—
86#398—216no—
87#397—48no—
88#389—28no—
89#389—28no—
90#389—48no—
91#387—216no—
92#387—216no—
93#385—216no—
94#385—216no—
95#384—28no—
96#382—28no—
97#382—24no—
98#382D418yes—
99#380—28no—
100#380—28no—
101#380—216no—
102C4×C4⋊C4—216no—
103C2×#95C2318yes—
104C2×#95C2314yes—
105C2×#95C2314yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2314yes—
110C2×#95C2314yes—
111C2×#95C2314yes—
112C2×#95C2314yes—
113C2×#95C2318yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C2×#95C2318yes—
118C2×#95C2314yes—
119C25C2411yescharacteristic
120C23×C4—22no—
121C23×C4—22no—
122C23×C4—22no—
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—22no—
126C23×C4—24no—
127C23×C4—24no—
128C23×C4—28no—
129C23×C4—44no—
130C23×C4C2×D414yes—
131C23×C4C2×D418yes—
132C23×C4C2414yes—
133C23×C4C2414yes—
134C23×C4C2414yes—
135C23×C4C2414yes—
136C23×C4C2414yes—
137C23×C4C2414yes—
138C23×C4C2411yescharacteristic
139C23×C4C2412yes—
140C23×C4C2411yescharacteristic
141C23×C4C2412yes—
142#125—432no—
143#117—432no—
144C2×C4⋊C4—24no—
145C2×C4⋊C4—24no—
146C2×C4⋊C4—24no—
147C2×C4⋊C4—28no—
148C2×C4⋊C4—28no—
149C2×C4⋊C4—28no—
150C2×C4⋊C4—24no—
151C2×C4⋊C4—28no—
152C2×C4⋊C4—24no—
153C2×C4⋊C4—216no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—28no—
157C2×C4⋊C4—28no—
158C2×C4⋊C4—28no—
159C2×C4⋊C4—24no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—24no—
162C2×C4⋊C4—28no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—216no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—48no—
168C2×C4⋊C4—44no—
169C2×C4⋊C4—44no—
170C2×C4⋊C4—48no—
171C2×C4⋊C4—42no—
172C2×C4⋊C4—42no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—44no—
176C2×C4⋊C4—44no—
177C2×C4⋊C4—44no—
178C2×C4⋊C4C2×D418yes—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—28no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—48no—
186C2×C22⋊C4—44no—
187C2×C42—24no—
188C2×C42—22no—
189C2×C42—24no—
190C2×C42—22no—
191C2×C42—232no—
192C2×C42—28no—
193C2×C42—28no—
194C2×C42—44no—
195C2×C42C2×D414yes—
196C2×C42C2×D418yes—
197C2×C42C2×D414yes—
198C2×C42C2×D414yes—
199C2×C42#4114yes—
200#95—28no—
201#95—28no—
202#95—28no—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—216no—
208#95—28no—
209#95—48no—
210#95—48no—
211#95—48no—
212#95—48no—
213#95—44no—
214#95—48no—
215#95C2×D4116yes—
216C24—21no—
217C24—21no—
218C24—21no—
219C24—24no—
220C24—21no—
221C24—24no—
222C24C22×D412yes—
223C24H(5,𝔽2)12yes—
224C242−1+412yes—
225C24C2511yescharacteristic
226C22×C4—28no—
227C22×C4—28no—
228C22×C4—28no—
229C22×C4—28no—
230C22×C4—28no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—24no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—24no—
244C22×C4—28no—
245C22×C4—28no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—28no—
249C22×C4—28no—
250C22×C4—216no—
251C22×C4—216no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—24no—
258C22×C4—24no—
259C22×C4—24no—
260C22×C4—24no—
261C22×C4—24no—
262C22×C4—24no—
263C22×C4—24no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—24no—
269C22×C4—28no—
270C22×C4—28no—
271C22×C4—24no—
272C22×C4—24no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—48no—
280C22×C4—416no—
281C22×C4—416no—
282C22×C4C22×D418yes—
283C22×C4C22×D412yes—
284C22×C4C22×D414yes—
285C22×C4C22×D414yes—
286C22×C4C22×D412yes—
287C22×C4C22×D414yes—
288C22×C4C22×D412yes—
289C22×C4C2×#4112yes—
290C22×C4C2×#4114yes—
291C22×C4H(5,𝔽2)12yes—
292C22×C42−1+418yes—
293C22×C42−1+412yes—
294C4⋊C4—48no—
295C4⋊C4—432no—
296C4⋊C4—48no—
297C4⋊C4—48no—
298C4⋊C4—48no—
299C4⋊C4—416no—
300C4⋊C4—48no—
301C4⋊C4—48no—
302C4⋊C4—416no—
303C4⋊C4—416no—
304C4⋊C4—416no—
305C4⋊C4—416no—
306C4⋊C4—88no—
307C22⋊C4—416no—
308C22⋊C4—416no—
309C22⋊C4—416no—
310C42—216no—
311C42—28no—
312C42—416no—
313C42#121116yes—
314C23—22no—
315C23—24no—
316C23—22no—
317C23—24no—
318C23—22no—
319C23—22no—
320C23—22no—
321C23—22no—
322C23—28no—
323C23—28no—
324C23—44no—
325C23—44no—
326C23—42no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23#53714yes—
331C23C23×D411yescharacteristic
332C23C23×D411yescharacteristic
333C23C22×#4111yescharacteristic
334C23C2×H(5,𝔽2)11yescharacteristic
335C23C2×H(5,𝔽2)11yescharacteristic
336C23C2×2−1+414yes—
337C23C2×2−1+411yescharacteristic
338C23C2×2−1+411yescharacteristic
339C23#58514yes—
340C2×C4—232no—
341C2×C4—22no—
342C2×C4—24no—
343C2×C4—24no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—24no—
348C2×C4—24no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—28no—
354C2×C4—28no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—22no—
359C2×C4—28no—
360C2×C4—28no—
361C2×C4—28no—
362C2×C4—216no—
363C2×C4—432no—
364C2×C4—44no—
365C2×C4—44no—
366C2×C4—48no—
367C2×C4—44no—
368C2×C4—44no—
369C2×C4—44no—
370C2×C4—44no—
371C2×C4—44no—
372C2×C4—44no—
373C2×C4—44no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—44no—
380C2×C4—44no—
381C2×C4—44no—
382C2×C4—48no—
383C2×C4—48no—
384C2×C4—44no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—44no—
389C2×C4—48no—
390C2×C4—44no—
391C2×C4—48no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4C2×#12114yes—
395C2×C4C2×#12118yes—
396C2×C4#53618yes—
397C2×C4#53714yes—
398C22—24no—
399C22—28no—
400C22—42no—
401C22—42no—
402C22—44no—
403C22—44no—
404C22—44no—
405C22—44no—
406C22—44no—
407C22C22×#12111yescharacteristic
408C22C2×#53612yes—
409C22C2×#53711yescharacteristic
410C22C2×#53711yescharacteristic
411C22#344912yes—
412C22#348411yescharacteristic
413C22#348611yescharacteristic
414C22#348914yes—
415C22#349211yescharacteristic
416C22#349211yescharacteristic
417C22#349514yes—
418C22#352912yes—
419C22#353112yes—
420C22#355012yes—
421C22#355612yes—
422C22#355914yes—
423C22#356314yes—
424C4—24no—
425C4—24no—
426C4—28no—
427C4—28no—
428C4—48no—
429C4—84no—
430C4—84no—
431C4—84no—
432C4—84no—
433C4—84no—
434C2—44no—
435C2#3522311yescharacteristic
436C2#3530612yes—
437C2#3668411yescharacteristic
438C2#3754912yes—
439C2#3764111yescharacteristic
440C2#4622812yes—
441C2#4622912yes—
442C2#4623412yes—
443C2#4623512yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC25C24
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2822C2semidirect
4. #16563C2semidirect
8. #14869C2semidirect
9. #14818C2semidirect
10. #14677C2semidirect
11. #14664C2semidirect

Maximal subgroups

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

subgroupindex
C2×#28222
C2×#28512
C2×#22852
#165632
#164002
#163992
#149072
#148692
#148182
#146772
#146642