Finite Group Databasebeta

← #4308297

Subgroups — #4308297

Order 512.

3124 conjugacy classes of subgroups.

The subgroup lattice has 447 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2854C211yesmaximal, characteristic
2C2×#2853C212yesmaximal, complemented
3C22×#376C211yesmaximal, characteristic, complemented
4#15122C214yesmaximal, complemented
5#15120C214yesmaximal, complemented
6#15117C218yesmaximal, complemented
7#14706C212yesmaximal, complemented
8#14685C211yesmaximal, characteristic, complemented
9#14275C212yesmaximal, complemented
10#14271C212yesmaximal
11#14259C214yesmaximal, complemented
12C22×#123C2212yes—
13C22×#121C2211yescharacteristic
14C23×C42C2211yescharacteristic
15C23×C42C2211yescharacteristic
16C23×C42C2211yescharacteristic
17#2854—28no—
18#2853—216no—
19C2×#390C2212yes—
20C2×#389C2212yes—
21C2×#388C2218yes—
22C2×#388C2214yes—
23C2×#387C2218yes—
24C2×#386C2214yes—
25C2×#386C2212yes—
26C2×#386C2214yes—
27C2×#385C2214yes—
28C2×#381C2212yes—
29C2×#380C2211yescharacteristic
30C2×#380C2212yes—
31C2×#376—28no—
32C2×#376C2214yes—
33C2×#376C2214yes—
34C22×#95C2212yes—
35C22×#95C2212yes—
36#1445C2214yes—
37#1444C2214yes—
38#1443C2218yes—
39#1442C2218yes—
40#1442C2214yes—
41#1441C2218yes—
42#1440C2218yes—
43#1439C2218yes—
44#1439C2218yes—
45#1438C2214yes—
46#1437C2218yes—
47#1436C2218yes—
48C4⋊#95C2214yes—
49#1434C2214yes—
50C22⋊#95C2212yes—
51C22⋊#95C2214yes—
52C23×D4C2312yes—
53C23×D4C2311yescharacteristic
54C24×C4C2311yescharacteristic
55C24×C4C2311yescharacteristic
56C24×C4C2312yes—
57C24×C4C2312yes—
58C24×C4C2311yescharacteristic
59C2×#123—28no—
60C2×#123—28no—
61C2×#123—28no—
62C2×#121—24no—
63C2×#121—28no—
64C22×C4⋊C4C2311yescharacteristic
65C22×C4⋊C4C2314yes—
66C22×C4⋊C4C2314yes—
67C22×C4⋊C4C2318yes—
68C22×C4⋊C4C2312yes—
69C22×C22⋊C4C2312yes—
70C22×C22⋊C4C2312yes—
71C22×C22⋊C4C2314yes—
72C22×C22⋊C4C2314yes—
73C22×C22⋊C4C2318yes—
74C22×C22⋊C4C2318yes—
75C22×C22⋊C4C2314yes—
76C22×C42—24no—
77C22×C42—24no—
78C22×C42—28no—
79C22×C42—24no—
80C22×C42—28no—
81C22×C42D418yes—
82C22×C42D418yescomplemented
83C22×C42C2314yes—
84C22×C42C2312yes—
85C22×C42C2312yes—
86C22×C42C2314yes—
87#390—28no—
88#389—28no—
89#388—232no—
90#388—216no—
91#387—232no—
92#386—216no—
93#386—28no—
94#386—216no—
95#385—216no—
96#381—28no—
97#380—24no—
98#380—28no—
99#376—216no—
100#376—216no—
101C2×#95—28no—
102C2×#95—28no—
103C2×#95C2314yes—
104C2×#95C2318yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2314yes—
108C2×#95C2314yes—
109C2×#95C2318yes—
110C2×#95C2314yes—
111C2×#95C2318yes—
112C2×#95C2312yes—
113C2×#95C2314yes—
114C2×#95C2318yes—
115C2×#95C2318yes—
116C2×#95C2318yes—
117C25C2412yes—
118C25C2411yescharacteristic
119C22×D4—24no—
120C22×D4—24no—
121C22×D4—28no—
122C22×D4—24no—
123C22×D4—28no—
124C22×D4—22no—
125C22×D4—24no—
126C22×D4—44no—
127C23×C4—24no—
128C23×C4—24no—
129C23×C4—24no—
130C23×C4—24no—
131C23×C4—24no—
132C23×C4—24no—
133C23×C4—24no—
134C23×C4—44no—
135C23×C4—44no—
136C23×C4—44no—
137C23×C4—44no—
138C23×C4—44no—
139C23×C4—44no—
140C23×C4C2×D414yes—
141C23×C4C2×D414yes—
142C23×C4C2×D414yes—
143C23×C4C2×D418yes—
144C23×C4C2×D414yes—
145C23×C4#4114yes—
146C23×C4#4118yes—
147C23×C4#4118yes—
148C23×C4C2414yes—
149C23×C4C2418yes—
150C23×C4C2412yes—
151C23×C4C2414yes—
152C23×C4C2412yes—
153C23×C4C2414yes—
154C23×C4C2412yes—
155C23×C4C2411yescharacteristic
156C23×C4C2411yescharacteristic
157#123—432no—
158#121—416no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—28no—
161C2×C4⋊C4—28no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—216no—
164C2×C4⋊C4—24no—
165C2×C4⋊C4—28no—
166C2×C4⋊C4—28no—
167C2×C4⋊C4—44no—
168C2×C4⋊C4—48no—
169C2×C4⋊C4—48no—
170C2×C4⋊C4—44no—
171C2×C4⋊C4—44no—
172C2×C22⋊C4—24no—
173C2×C22⋊C4—24no—
174C2×C22⋊C4—28no—
175C2×C22⋊C4—24no—
176C2×C22⋊C4—28no—
177C2×C22⋊C4—28no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—28no—
180C2×C22⋊C4—28no—
181C2×C22⋊C4—216no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—28no—
184C2×C22⋊C4—28no—
185C2×C22⋊C4—216no—
186C2×C22⋊C4—48no—
187C2×C22⋊C4—48no—
188C2×C22⋊C4—48no—
189C2×C22⋊C4—48no—
190C2×C22⋊C4—48no—
191C2×C42—232no—
192C2×C42—232no—
193C2×C42—232no—
194C2×C42—28no—
195C2×C42—28no—
196C2×C42—216no—
197C2×C42—28no—
198C2×C42—28no—
199C2×C42—28no—
200C2×C42C2×D418yes—
201C2×C42C2×D418yes—
202C2×C42C2×D4116yes—
203#95—28no—
204#95—28no—
205#95—28no—
206#95—28no—
207#95—24no—
208#95—216no—
209#95—216no—
210#95—216no—
211#95—28no—
212#95—28no—
213#95—216no—
214#95—48no—
215#95—48no—
216#95—48no—
217C24—24no—
218C24—24no—
219C24—22no—
220C24—22no—
221C24—22no—
222C24—24no—
223C24—24no—
224C24—28no—
225C24—28no—
226C24C22×D412yes—
227C24C22×D412yes—
228C24C2×#4112yes—
229C24C2×#4114yes—
230C24C2×#4114yes—
231C24C2511yescharacteristic
232C2×D4—48no—
233C2×D4—48no—
234C2×D4—48no—
235C2×D4—48no—
236C2×D4—416no—
237C2×D4—416no—
238C2×D4—44no—
239C2×D4—88no—
240C22×C4—28no—
241C22×C4—28no—
242C22×C4—216no—
243C22×C4—216no—
244C22×C4—24no—
245C22×C4—24no—
246C22×C4—24no—
247C22×C4—24no—
248C22×C4—216no—
249C22×C4—28no—
250C22×C4—216no—
251C22×C4—24no—
252C22×C4—24no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—24no—
256C22×C4—24no—
257C22×C4—28no—
258C22×C4—28no—
259C22×C4—24no—
260C22×C4—28no—
261C22×C4—28no—
262C22×C4—28no—
263C22×C4—22no—
264C22×C4—24no—
265C22×C4—24no—
266C22×C4—22no—
267C22×C4—24no—
268C22×C4—28no—
269C22×C4—28no—
270C22×C4—24no—
271C22×C4—24no—
272C22×C4—22no—
273C22×C4—28no—
274C22×C4—28no—
275C22×C4—28no—
276C22×C4—28no—
277C22×C4—28no—
278C22×C4—28no—
279C22×C4—48no—
280C22×C4—48no—
281C22×C4—416no—
282C22×C4—416no—
283C22×C4—416no—
284C22×C4—48no—
285C22×C4—48no—
286C22×C4—48no—
287C22×C4—48no—
288C22×C4—416no—
289C22×C4—416no—
290C22×C4—416no—
291C22×C4#121116yes—
292C22×C4#123116yes—
293C22×C4C22×D414yes—
294C22×C4C22×D418yes—
295C22×C4C22×D414yes—
296C22×C4C22×D414yes—
297C22×C4C22×D412yes—
298C22×C4C22×D412yes—
299C22×C4C2×#4118yes—
300C22×C4C2×#4114yes—
301C22×C4C2×#4114yes—
302C22×C4C2×#4114yes—
303C4⋊C4—48no—
304C4⋊C4—432no—
305C4⋊C4—416no—
306C4⋊C4—44no—
307C4⋊C4—88no—
308C22⋊C4—416no—
309C22⋊C4—432no—
310C22⋊C4—48no—
311C22⋊C4—416no—
312C22⋊C4—416no—
313C22⋊C4—48no—
314C22⋊C4—816no—
315C42—232no—
316C42—232no—
317C42—216no—
318C42—216no—
319C23—24no—
320C23—24no—
321C23—28no—
322C23—28no—
323C23—28no—
324C23—28no—
325C23—28no—
326C23—28no—
327C23—42no—
328C23—42no—
329C23—42no—
330C23—44no—
331C23—44no—
332C23—44no—
333C23—48no—
334C23—44no—
335C23—48no—
336C23—48no—
337C23—48no—
338C23—48no—
339C23—48no—
340C23C2×#12114yes—
341C23C2×#12314yes—
342C23#52514yes—
343C23#52918yes—
344C23#53218yes—
345C23C23×D411yescharacteristic
346C23C23×D411yescharacteristic
347C23C22×#4111yescharacteristic
348C23C22×#4112yes—
349C23C22×#4112yes—
350C23#58514yes—
351C23#58514yes—
352D4—88no—
353D4—816no—
354C2×C4—232no—
355C2×C4—24no—
356C2×C4—24no—
357C2×C4—28no—
358C2×C4—28no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—28no—
362C2×C4—28no—
363C2×C4—28no—
364C2×C4—24no—
365C2×C4—28no—
366C2×C4—24no—
367C2×C4—216no—
368C2×C4—24no—
369C2×C4—216no—
370C2×C4—416no—
371C2×C4—432no—
372C2×C4—432no—
373C2×C4—416no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—48no—
379C2×C4—48no—
380C2×C4—48no—
381C2×C4—44no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—44no—
386C2×C4—48no—
387C2×C4—44no—
388C2×C4—48no—
389C2×C4—44no—
390C2×C4—48no—
391C2×C4—44no—
392C2×C4—48no—
393C2×C4—48no—
394C2×C4—48no—
395C2×C4—48no—
396C2×C4—48no—
397C2×C4—48no—
398C2×C4C2×#12118yes—
399C2×C4C2×#12314yes—
400C2×C4C2×#12314yes—
401C22—24no—
402C22—216no—
403C22—28no—
404C22—216no—
405C22—28no—
406C22—24no—
407C22—82no—
408C22—82no—
409C22—88no—
410C22—84no—
411C22—88no—
412C22—84no—
413C22—82no—
414C22#285418yes—
415C22C22×#12111yescharacteristic
416C22C22×#12311yescharacteristic
417C22C2×#52511yescharacteristic
418C22C2×#52912yes—
419C22C2×#53212yes—
420C22#344912yes—
421C22#344912yes—
422C22#344914yes—
423C22#345514yes—
424C22#349114yes—
425C22#349114yes—
426C22#349314yes—
427C22#349314yes—
428C4—28no—
429C4—24no—
430C4—24no—
431C4—48no—
432C4—48no—
433C4—44no—
434C4—82no—
435C4—88no—
436C4—84no—
437C2—28no—
438C2—162no—
439C2#3463312yes—
440C2#3509711yescharacteristic
441C2#3509914yes—
442C2#3511812yes—
443C2#3511914yes—
444C2#3515711yescharacteristic
445C2C2×#285411yescharacteristic
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC22×#121C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2853C2semidirect
3. C22×#376C2semidirect
4. #15122C2semidirect
5. #15120C2semidirect
6. #15117C2semidirect
7. #14706C2semidirect
8. #14685C2semidirect
9. #14275C2semidirect
11. #14259C2semidirect
82. C22×C42D4semidirect

Maximal subgroups

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

subgroupindex
C2×#28542
C2×#28532
C22×#3762
#151222
#151202
#151172
#147062
#146852
#142752
#142712
#142592