Finite Group Databasebeta

← #11863923

Subgroups — #11863923

Order 1152.

2982 conjugacy classes of subgroups; only the 417 normal subgroups are shown.

normal subgroups only

This group has 2982 conjugacy classes of subgroups — too many to draw. Only its 417 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×C32×D4C211yesmaximal, characteristic, complemented
2C2×#64141C212yesmaximal, complemented
3#10596262C211yesmaximal, characteristic, complemented
4#10592917C214yesmaximal, complemented
5#10592916C214yesmaximal, complemented
6#10592914C212yesmaximal, complemented
7#10592906C211yesmaximal, characteristic, complemented
8#84038—34nomaximal
9C25×C32—21no—
10C22×C32×D4—28no—
11C22×C32×D4—22no—
12C22×C32×D4—21no—
13C22×C32×D4C2214yes—
14C22×C32×D4C2212yes—
15C23×C32×C4C2211yescharacteristic
16C2×#3828C2211yescharacteristic, complemented
17#64162—21no—
18C2×#3754C2214yescomplemented
19#64141—28no—
20C2×#3753C2214yescomplemented
21#64139C2212yescomplemented
22#64138—21no—
23C4×#638—22no—
24#64135C2211yescharacteristic, complemented
25#64133C2212yescomplemented
26C2×#3748C2212yescomplemented
27#63659C2218yescomplemented
28#63657C2214yescomplemented
29C23×C3×D4D314yescomplemented
30C2×#932—38no—
31#5791—34no—
32#5218—316no—
33#5217—316no—
34#5215—38no—
35#5207—34no—
36C24×C32—28no—
37C24×C32—24no—
38C24×C32—22no—
39C24×C32—48no—
40C24×C32D411yescharacteristic
41C2×C32×D4—28no—
42C2×C32×D4—216no—
43C2×C32×D4—216no—
44C2×C32×D4—24no—
45C2×C32×D4—22no—
46C2×C32×D4—24no—
47C2×C32×D4D418yescomplemented
48C2×C32×D4C2314yes—
49C22×C32×C4—24no—
50C22×C32×C4D412yes—
51C22×C32×C4D411yescharacteristic
52C22×C32×C4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C2×#638—21no—
55C2×#638—22no—
56#3828—24no—
57C2×C4⋊(C3⋊D3)C2311yescharacteristic, complemented
58C2×C4×C3⋊D3—21no—
59C2×#634C2311yescharacteristic, complemented
60#3757—22no—
61#3757—21no—
62#3754—28no—
63#3753—28no—
64#3752—22no—
65C4⋊(C3⋊Dic3)C2312yescomplemented
66#3750—22no—
67C4×C3⋊Dic3—21no—
68#3748—24no—
69C2×C3⋊(C3⋊C8)C2314yescomplemented
70#3003—91noSylow
71C25×C3—24no—
72C22×C3×D4—232no—
73C22×C3×D4—28no—
74C22×C3×D4—24no—
75C22×C3×D4C2×D3116yes—
76C22×C3×D4C2×D318yes—
77C23×C3×C4C2×D314yes—
78C2×#235—34no—
79#953—64no—
80C2×#214—316no—
81#932—632no—
82C2×#213—316no—
83#930—38no—
84#929—64no—
85C4×#67—68no—
86#926—34no—
87#924—38no—
88C2×#208—38no—
89#832—332no—
90#830—316no—
91C23×C32—22no—
92C23×C32—24no—
93C23×C32—216no—
94C23×C32—28no—
95C23×C32—24no—
96C23×C32—432no—
97C23×C32—48no—
98C23×C32—48no—
99C23×C32—48no—
100C23×C32C2×D411yescharacteristic
101C23×C32C2×D412yes—
102C22×C3⋊D3—21no—
103C32×D4—216no—
104C32×D4—28no—
105C32×D4—24no—
106C32×D4C2×D418yes—
107C2×C32×C4—22no—
108C2×C32×C4—24no—
109C2×C32×C4—24no—
110C2×C32×C4C2×D412yes—
111C2×C32×C4C2×D411yescharacteristic
112C2×C32×C4C2×D414yes—
113C2×C32×C4C2411yescharacteristic
114#638—42no—
115#638—44no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—21no—
118C4⋊(C3⋊D3)—22no—
119C4×C3⋊D3—42no—
120#634—22no—
121C3⋊(C3⋊C8)—24no—
122C23×D4C3⋊D311yescharacteristic, complemented
123C2×AGL(1,ℤ/8)—92no—
124#525—91no—
125#450—94no—
126#447—94no—
127#418—92no—
128#407—91no—
129C24×C3—232no—
130C24×C3—216no—
131C24×C3—28no—
132C24×C3—432no—
133C24×C3#6714yes—
134C2×C3×D4—232no—
135C2×C3×D4—264no—
136C2×C3×D4—264no—
137C2×C3×D4—216no—
138C2×C3×D4—28no—
139C2×C3×D4—216no—
140C2×C3×D4#67132yescomplemented
141C2×C3×D4C22×D3116yes—
142C22×C3×C4—216no—
143C22×C3×C4#6718yes—
144C22×C3×C4#6714yes—
145C22×C3×C4C22×D318yes—
146C22×C3×C4C22×D314yes—
147C2×#67—68no—
148C2×#67—64no—
149#235—616no—
150C2×D12—34no—
151C2×C4×D3—64no—
152C2×Dic6—34no—
153#217—68no—
154#217—64no—
155#214—632no—
156#213—632no—
157(C2×D3)⋊C4—68no—
158C4⋊Dic3—38no—
159Dic3⋊C4—68no—
160C4×Dic3—64no—
161#208—616no—
162C2×C3⋊C8—316no—
163C22×C32—28no—
164C22×C32—22no—
165C22×C32—22no—
166C22×C32—22no—
167C22×C32—28no—
168C22×C32—416no—
169C22×C32—432no—
170C22×C32—48no—
171C22×C32#12012yes—
172C22×C32AGL(1,ℤ/8)14yescomplemented
173C22×C32C22×D411yescharacteristic
174C2×C3⋊D3—41no—
175C2×C3⋊D3—42no—
176C32×C4—22no—
177C32×C4#12012yes—
178C32×C4C22×D412yes—
179C3⋊Dic3—41no—
180C3⋊Dic3—42no—
181C25—21no—
182C2×#41—91no—
183C22×D4—28no—
184C22×D4—22no—
185C22×D4—21no—
186C22×D4C2×C3⋊D314yes—
187C22×D4C2×C3⋊D312yes—
188C23×C4C2×C3⋊D311yescharacteristic
189AGL(1,ℤ/8)—188no—
190C2×SD16—94no—
191C2×D8—94no—
192C2×M16—92no—
193#123—181no—
194C22⋊Q8—92no—
195#121—92no—
196#120—181no—
197C4×D4—182no—
198#117—91no—
199#102—98no—
200C22⋊C8—94no—
201C23×C3—216no—
202C23×C3—28no—
203C23×C3—264no—
204C23×C3—232no—
205C23×C3—216no—
206C23×C3—4128no—
207C23×C3—432no—
208C23×C3—432no—
209C23×C3—432no—
210C23×C3C2×#6718yes—
211C23×C3C2×#6714yes—
212C22×D3—64no—
213C3×D4—264no—
214C3×D4—232no—
215C3×D4—216no—
216C3×D4C2×#67132yes—
217C2×C3×C4—28no—
218C2×C3×C4—216no—
219C2×C3×C4—216no—
220C2×C3×C4C2×#6718yes—
221C2×C3×C4C2×#6714yes—
222C2×C3×C4C2×#67116yes—
223C2×C3×C4C23×D314yes—
224#67—1216no—
225#67—128no—
226C2×Dic3—68no—
227C2×Dic3—64no—
228D12—68no—
229C4×D3—128no—
230Dic6—68no—
231C3⋊C8—616no—
232C2×C32—22no—
233C2×C32—24no—
234C2×C32—48no—
235C2×C32C2×#12011yescharacteristic
236C2×C32C2×AGL(1,ℤ/8)12yes—
237C3⋊D3—81no—
238C24—28no—
239C24—24no—
240C24—22no—
241C24—48no—
242C24#63811yescharacteristic
243#41—184no—
244C2×Q8—91no—
245C2×D4—28no—
246C2×D4—216no—
247C2×D4—216no—
248C2×D4—24no—
249C2×D4—24no—
250C2×D4—22no—
251C2×D4—91no—
252C2×D4—182no—
253C2×D4—181no—
254C2×D4#63818yescomplemented
255C2×D4C22×C3⋊D314yes—
256C22×C4—24no—
257C22×C4—181no—
258C22×C4#63812yes—
259C22×C4#63811yescharacteristic
260C22×C4C22×C3⋊D312yes—
261C22×C4C22×C3⋊D311yescharacteristic
262SD16—188no—
263D8—188no—
264M16—184no—
265C2×C8—94no—
266C4⋊C4—92no—
267C4⋊C4—182no—
268C22⋊C4—181no—
269C22⋊C4—182no—
270C22⋊C4—182no—
271C42—181no—
272C22×C3—232no—
273C22×C3—28no—
274C22×C3—28no—
275C22×C3—28no—
276C22×C3—232no—
277C22×C3—4128no—
278C22×C3—464no—
279C22×C3—432no—
280C22×C3#932116yescomplemented
281C22×C3#95318yes—
282C22×C3C22×#6714yes—
283C2×D3—128no—
284C2×D3—124no—
285C3×C4—28no—
286C3×C4#95318yes—
287C3×C4C22×#6718yes—
288Dic3—128no—
289Dic3—124no—
290C32#300311yescharacteristic, complemented, Sylow
291C23—24no—
292C23—22no—
293C23—216no—
294C23—24no—
295C23—28no—
296C23—432no—
297C23—48no—
298C23—48no—
299C23—48no—
300C23—181no—
301C23C2×#63812yes—
302C23C2×#63811yescharacteristic
303Q8—182no—
304D4—216no—
305D4—24no—
306D4—28no—
307D4—182no—
308D4—364no—
309D4—362no—
310D4C2×#63818yes—
311C2×C4—22no—
312C2×C4—24no—
313C2×C4—24no—
314C2×C4—182no—
315C2×C4—181no—
316C2×C4—362no—
317C2×C4C2×#63812yes—
318C2×C4C2×#63814yes—
319C2×C4C2×#63811yescharacteristic
320C2×C4C23×C3⋊D311yescharacteristic
321C8—184no—
322C2×C3—216no—
323C2×C3—28no—
324C2×C3—432no—
325C2×C3C2×#93218yes—
326C2×C3C2×#95314yes—
327D3—244no—
328C22—28no—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—28no—
333C22—432no—
334C22—416no—
335C22—48no—
336C22—362no—
337C22—361no—
338C22#6414114yescomplemented
339C22#6416212yes—
340C22C22×#63811yescharacteristic
341C4—22no—
342C4—362no—
343C4—361no—
344C4#6416212yes—
345C4C22×#63812yes—
346C3#8403814yescomplemented
347C2—24no—
348C2—22no—
349C2—48no—
350C2—721no—
351C2C2×#6414112yes—
352C2C2×#6416211yescharacteristic
subgroupG/H
centerC22C22×#638
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C23×C3⋊D3
FittingC23×C32×D4C2
socleC22×C32C22×D4
hypercenterC23×D4C3⋊D3
hypocenterC32#3003
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C23×C32×D4C2semidirect
2. C2×#64141C2semidirect
3. #10596262C2semidirect
4. #10592917C2semidirect
5. #10592916C2semidirect
6. #10592914C2semidirect
7. #10592906C2semidirect
16. C2×#3828C22semidirect
18. C2×#3754C22semidirect
20. C2×#3753C22semidirect
21. #64139C22semidirect
24. #64135C22semidirect
25. #64133C22semidirect
26. C2×#3748C22semidirect
27. #63659C22semidirect
28. #63657C22semidirect
29. C23×C3×D4D3semidirect
47. C2×C32×D4D4semidirect
57. C2×C4⋊(C3⋊D3)C23semidirect
59. C2×#634C23semidirect
65. C4⋊(C3⋊Dic3)C23semidirect
69. C2×C3⋊(C3⋊C8)C23semidirect
122. C23×D4C3⋊D3semidirect
140. C2×C3×D4#67semidirect
172. C22×C32AGL(1,ℤ/8)semidirect
254. C2×D4#638semidirect
280. C22×C3#932semidirect
290. C32#3003semidirect
338. C22#64141semidirect
346. C3#84038semidirect

Maximal subgroups

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

subgroupindex
C23×C32×D42
C2×#641412
#105962622
#105929172
#105929162
#105929142
#105929062
#840383