Finite Group Databasebeta

← #607613

Subgroups — #607613

Order 512.

3402 conjugacy classes of subgroups.

The subgroup lattice has 419 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2298C212yesmaximal, complemented
2C2×#2293C211yesmaximal, characteristic
3#14701C218yesmaximal, complemented
4#14700C214yesmaximal
5#14699C214yesmaximal, complemented
6C4×#389C212yesmaximal, complemented
7C4×#390C212yesmaximal
8C4×#388C214yesmaximal, complemented
9#13761C212yesmaximal, complemented
10#13755C211yesmaximal, characteristic, complemented
11C44C211yesmaximal, characteristic, complemented
12C22×C4×D4C2211yescharacteristic
13C22×#117C2212yes—
14#2298—216no—
15#2293C4116yescomplemented
16C2×#390C2214yes—
17C2×#390C2212yes—
18C2×#389C2214yes—
19C2×#389C2212yes—
20C2×#388C2218yes—
21C2×#388C2218yes—
22C2×#388C2214yes—
23C2×#387C2218yes—
24C2×C4×C4⋊C4C2212yes—
25C2×C4×C4⋊C4C2214yes—
26C2×C4×C4⋊C4C2211yescharacteristic
27C2×C4×C22⋊C4C2214yes—
28C2×C4×C22⋊C4C2214yes—
29C2×C4×C22⋊C4C2212yes—
30C2×#376C2212yes—
31C2×C43C2214yes—
32C2×C43C2214yes—
33C2×C43C2211yescharacteristic
34C2×C43C2212yes—
35C2×C43C2212yes—
36C2×C43C2211yescharacteristic
37C2×C43C2211yescharacteristic
38#1437C2218yes—
39#1437C2218yes—
40#1437C2218yes—
41#1436C2218yes—
42#1436C2218yes—
43#1436C2218yes—
44C4⋊#95C2214yes—
45C4⋊#95C2214yes—
46C4⋊#95C2214yes—
47#1434C2218yes—
48#1434C2214yes—
49C4×#95C2212yes—
50C4×#95C2214yes—
51C23×D4C2311yescharacteristic
52C24×C4C2312yes—
53C2×C4×D4—28no—
54C2×C4×D4C2×C418yes—
55C2×#117—28no—
56C2×#117—28no—
57C2×#117C2×C4116yes—
58C22×C4⋊C4C2314yes—
59C22×C4⋊C4C2318yes—
60C22×C4⋊C4C2314yes—
61C22×C4⋊C4C2311yescharacteristic
62C22×C4⋊C4C2312yes—
63C22×C22⋊C4C2318yes—
64C22×C22⋊C4C2318yes—
65C22×C22⋊C4C2314yes—
66C22×C22⋊C4C2312yes—
67C22×C22⋊C4C2314yes—
68C22×C42C2314yes—
69C22×C42C2314yes—
70C22×C42C2314yes—
71C22×C42C2314yes—
72C22×C42C2312yes—
73C22×C42C2312yes—
74C22×C42C2312yes—
75C22×C42C2311yescharacteristic
76C22×C42C2314yes—
77C22×C42C2312yes—
78C22×C42C2314yes—
79C22×C42C2314yes—
80C22×C42C2312yes—
81C22×C42C2311yescharacteristic
82C22×C42C2311yescharacteristic
83#390—216no—
84#390C2×C4116yes—
85#389—216no—
86#389C2×C4116yes—
87#388—232no—
88#388—232no—
89#388C2×C4132yes—
90#387—232no—
91C4×C4⋊C4—216no—
92C4×C4⋊C4—28no—
93C4×C4⋊C4C2×C418yes—
94C4×C22⋊C4—216no—
95C4×C22⋊C4—216no—
96C4×C22⋊C4C2×C4116yes—
97#376—28no—
98C2×#95C2318yes—
99C2×#95C2318yes—
100C2×#95C2318yes—
101C2×#95C2314yes—
102C2×#95C2318yes—
103C2×#95C2318yes—
104C2×#95C2318yes—
105C2×#95C2318yes—
106C2×#95C2314yes—
107C2×#95C2312yes—
108C43—28no—
109C43—216no—
110C43—216no—
111C43—28no—
112C43—24no—
113C43C2×C418yes—
114C43D418yescomplemented
115C25C2412yes—
116C22×D4—24no—
117C22×D4—28no—
118C22×D4C22×C414yes—
119C23×C4—24no—
120C23×C4—24no—
121C23×C4—28no—
122C23×C4—28no—
123C23×C4C22×C418yes—
124C23×C4C2418yes—
125C23×C4C2414yes—
126C23×C4C2412yes—
127C23×C4C2412yes—
128C23×C4C2411yescharacteristic
129C23×C4C2412yes—
130C23×C4C2414yes—
131C23×C4C2414yes—
132C23×C4C2411yescharacteristic
133C23×C4C2411yescharacteristic
134C4×D4—232no—
135#117—264no—
136C2×C4⋊C4—28no—
137C2×C4⋊C4—28no—
138C2×C4⋊C4—24no—
139C2×C4⋊C4—28no—
140C2×C4⋊C4—216no—
141C2×C4⋊C4—28no—
142C2×C4⋊C4—216no—
143C2×C4⋊C4—28no—
144C2×C4⋊C4—216no—
145C2×C4⋊C4—28no—
146C2×C4⋊C4C22×C4116yes—
147C2×C4⋊C4C22×C414yes—
148C2×C4⋊C4C22×C418yes—
149C2×C22⋊C4—28no—
150C2×C22⋊C4—28no—
151C2×C22⋊C4—28no—
152C2×C22⋊C4—216no—
153C2×C22⋊C4—216no—
154C2×C22⋊C4—216no—
155C2×C22⋊C4—216no—
156C2×C22⋊C4—216no—
157C2×C22⋊C4—216no—
158C2×C22⋊C4—216no—
159C2×C22⋊C4C22×C4116yes—
160C2×C22⋊C4C22×C418yes—
161C2×C22⋊C4C22×C4116yes—
162C2×C42—24no—
163C2×C42—24no—
164C2×C42—28no—
165C2×C42—28no—
166C2×C42—28no—
167C2×C42—28no—
168C2×C42—28no—
169C2×C42—28no—
170C2×C42—28no—
171C2×C42—28no—
172C2×C42—28no—
173C2×C42—28no—
174C2×C42—28no—
175C2×C42—24no—
176C2×C42—28no—
177C2×C42—28no—
178C2×C42—28no—
179C2×C42—24no—
180C2×C42—24no—
181C2×C42—28no—
182C2×C42—28no—
183C2×C42—28no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42—28no—
187C2×C42—28no—
188C2×C42—28no—
189C2×C42C22×C418yes—
190C2×C42C22×C414yes—
191C2×C42C22×C418yes—
192C2×C42C22×C418yes—
193C2×C42C22×C414yes—
194C2×C42C22×C414yes—
195C2×C42C2×D418yes—
196C2×C42C2×D4116yes—
197C2×C42C2×D414yes—
198C2×C42#4118yes—
199C2×C42#41116yes—
200C2×C42#41116yes—
201C2×C42#4118yes—
202C2×C42#4114yes—
203#95—216no—
204#95—216no—
205#95—216no—
206#95—216no—
207#95—28no—
208#95—216no—
209#95—216no—
210#95—216no—
211#95C22×C4116yes—
212#95C22×C418yes—
213C24—22no—
214C24—28no—
215C24—22no—
216C24—24no—
217C24—28no—
218C24—24no—
219C24C23×C414yes—
220C24C2511yescharacteristic
221C2×D4—28no—
222C2×D4—216no—
223C2×D4—416no—
224C22×C4—22no—
225C22×C4—24no—
226C22×C4—24no—
227C22×C4—24no—
228C22×C4—22no—
229C22×C4—24no—
230C22×C4—24no—
231C22×C4—24no—
232C22×C4—24no—
233C22×C4—24no—
234C22×C4—24no—
235C22×C4—24no—
236C22×C4—24no—
237C22×C4—24no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—24no—
241C22×C4—28no—
242C22×C4—28no—
243C22×C4—24no—
244C22×C4—24no—
245C22×C4—22no—
246C22×C4—216no—
247C22×C4—216no—
248C22×C4—216no—
249C22×C4—28no—
250C22×C4—28no—
251C22×C4—28no—
252C22×C4—28no—
253C22×C4—28no—
254C22×C4—28no—
255C22×C4—28no—
256C22×C4—416no—
257C22×C4—416no—
258C22×C4C23×C418yes—
259C22×C4C23×C414yes—
260C22×C4C23×C412yes—
261C22×C4C23×C414yes—
262C22×C4C23×C414yes—
263C22×C4C23×C412yes—
264C22×C4C23×C412yes—
265C22×C4C22×D414yes—
266C22×C4C22×D418yes—
267C22×C4C22×D412yes—
268C22×C4C2×#4114yes—
269C22×C4C2×#4114yes—
270C22×C4C2×#4118yes—
271C22×C4C2×#4118yes—
272C22×C4C2×#4118yes—
273C22×C4C2×#4118yes—
274C22×C4C2×#4118yes—
275C22×C4C2×#4118yes—
276C22×C4C2×#4114yes—
277C22×C4C2×#4114yes—
278C22×C4C2×#4112yes—
279C22×C4C2×#4118yes—
280C22×C4C2×#4114yes—
281C4⋊C4—28no—
282C4⋊C4—232no—
283C4⋊C4—216no—
284C4⋊C4—416no—
285C4⋊C4—432no—
286C22⋊C4—216no—
287C22⋊C4—232no—
288C22⋊C4—232no—
289C22⋊C4—432no—
290C22⋊C4—432no—
291C42—232no—
292C42—232no—
293C42—232no—
294C42—216no—
295C42—216no—
296C42—28no—
297C42—216no—
298C42—232no—
299C42—216no—
300C42—232no—
301C42—232no—
302C42—216no—
303C42—416no—
304C42#117116yes—
305C42C4×D4116yes—
306C23—24no—
307C23—24no—
308C23—28no—
309C23—24no—
310C23—28no—
311C23—48no—
312C23—44no—
313C23—48no—
314C23—48no—
315C23—44no—
316C23—48no—
317C23—48no—
318C23—48no—
319C23C24×C411yescharacteristic
320C23C23×D411yescharacteristic
321C23C22×#4112yes—
322C23C22×#4111yescharacteristic
323C23C22×#4112yes—
324C23C22×#4114yes—
325C23C22×#4114yes—
326D4—416no—
327C2×C4—24no—
328C2×C4—24no—
329C2×C4—24no—
330C2×C4—24no—
331C2×C4—24no—
332C2×C4—28no—
333C2×C4—28no—
334C2×C4—28no—
335C2×C4—28no—
336C2×C4—28no—
337C2×C4—28no—
338C2×C4—28no—
339C2×C4—28no—
340C2×C4—28no—
341C2×C4—28no—
342C2×C4—28no—
343C2×C4—28no—
344C2×C4—28no—
345C2×C4—28no—
346C2×C4—28no—
347C2×C4—28no—
348C2×C4—28no—
349C2×C4—24no—
350C2×C4—28no—
351C2×C4—24no—
352C2×C4—28no—
353C2×C4—28no—
354C2×C4—24no—
355C2×C4—44no—
356C2×C4—48no—
357C2×C4—44no—
358C2×C4—48no—
359C2×C4—432no—
360C2×C4—48no—
361C2×C4—48no—
362C2×C4—48no—
363C2×C4—48no—
364C2×C4—48no—
365C2×C4—48no—
366C2×C4—48no—
367C2×C4C2×#11714yes—
368C2×C4C2×#11714yes—
369C2×C4C2×#11714yes—
370C2×C4C2×C4×D418yes—
371C2×C4C2×C4×D414yes—
372C2×C4C4×#4118yes—
373C2×C4C4×#4114yes—
374C2×C4C4×#4118yes—
375C2×C4#529116yes—
376C2×C4#52918yes—
377C2×C4#529116yes—
378C2×C4#532116yes—
379C2×C4#53218yes—
380C22—44no—
381C22—44no—
382C22—44no—
383C22—48no—
384C22—48no—
385C22—88no—
386C22—88no—
387C22C22×#11711yescharacteristic
388C22C22×C4×D411yescharacteristic
389C22C2×C4×#4111yescharacteristic
390C22C2×C4×#4112yes—
391C22C2×C4×#4112yes—
392C22C2×#52914yes—
393C22C2×#52914yes—
394C22C2×#52912yes—
395C22C2×#53212yes—
396C22C2×#53214yes—
397C22#344414yes—
398C22#344414yes—
399C22#344414yes—
400C4—216no—
401C4—216no—
402C4—24no—
403C4—28no—
404C4—28no—
405C4—44no—
406C4—48no—
407C4—88no—
408C4#229318yes—
409C4#229818yescomplemented
410C2—84no—
411C2C2×#229311yescharacteristic
412C2C2×#229811yescharacteristic
413C2#3402611yescharacteristic
414C2C4×#52912yes—
415C2C4×#53212yes—
416C2#3663114yes—
417C2#3663814yes—
subgroupG/H
centerC23×C4C24
commutatorC23C24×C4
Frattini, socle, cosocleC24C25
prime radicalC23×D4C23
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C4 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2298C2semidirect
3. #14701C2semidirect
5. #14699C2semidirect
6. C4×#389C2semidirect
8. C4×#388C2semidirect
9. #13761C2semidirect
10. #13755C2semidirect
11. C44C2semidirect
15. #2293C4semidirect
114. C43D4semidirect
409. C4#2298semidirect

Maximal subgroups

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

subgroupindex
C2×#22982
C2×#22932
#147012
#147002
#146992
C4×#3892
C4×#3902
C4×#3882
#137612
#137552
C442