Finite Group Databasebeta

← #10695753

Subgroups — #10695753

Order 768.

772 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×#4968C212yesmaximal, complemented
2#73381C211yesmaximal, characteristic
3C3×#1445C211yesmaximal, characteristic, complemented
4C2×#4451C212yesmaximal, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9968—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#930—22no—
9C2×#930C2212yes—
10C2×C3×#95—22no—
11C2×C3×#95C2211yescharacteristic
12C2×C3×#95C2211yescharacteristic
13C2×#830C2212yes—
14#4968—24no—
15#4968—24no—
16C2×#809—22no—
17C2×#808—22no—
18#4677—22no—
19#4671—22no—
20#4451—24no—
21C2×#481—32no—
22#2656—31no—
23C2×#327—32no—
24#1445D311yescharacteristic, complemented
25#1280—31no—
26C23×C3×C4—21no—
27C23×C3×C4—21no—
28C23×C3×C4D412yes—
29C23×C3×C4C2311yescharacteristic
30C22×#67—22no—
31C22×D12—22no—
32C22×D12D412yes—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4D412yescomplemented
37#930—22no—
38#930—22no—
39#930—22no—
40#930—22no—
41#930—44no—
42#930—44no—
43C2×(C2×D3)⋊C4—24no—
44C2×(C2×D3)⋊C4—22no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3D412yes—
47C22×C3⋊C8—22no—
48C3×#95—22no—
49C3×#95—22no—
50C3×#95—44no—
51#830—22no—
52#830D414yescomplemented
53#809—28no—
54#808—28no—
55C2×#121—32no—
56C2×#121—62no—
57C22×C4⋊C4C2×D311yescharacteristic
58#481—64no—
59#481—64no—
60C2×#106—62no—
61C2×#102—62no—
62C2×C22⋊C8—32no—
63#399—62no—
64#392—62no—
65C2×#95—22no—
66C2×#95C2×D311yescharacteristic
67C2×#95C2×D311yescharacteristic
68#327—64no—
69C24×C3C2×D411yescharacteristic
70C23×D3—22no—
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—22no—
75C22×C3×C4—42no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4—42no—
79C22×C3×C4—41no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4—41no—
85C22×C3×C4—42no—
86C22×C3×C4SD1614yes—
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D412yes—
89C22×C3×C4C2×D412yes—
90C22×C3×C4#4111yescharacteristic
91C2×#67—42no—
92C2×#67—42no—
93C2×#67—44no—
94C2×#67—44no—
95C22×Dic3—22no—
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—44no—
99C2×D12—44no—
100C2×D12—82no—
101C2×D12—82no—
102C3×C4⋊C4—24no—
103C3×C4⋊C4—42no—
104(C2×D3)⋊C4—48no—
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C4⋊Dic3—44no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—44no—
112C22×D4—32no—
113C22×D4—62no—
114C22×D4—62no—
115C23×C4—21no—
116C23×C4—21no—
117C23×C4#6712yes—
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120#121—62no—
121#121—62no—
122#121—62no—
123#121—62no—
124#121—124no—
125#121—124no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4D1212yescomplemented
132C2×C22⋊C4—64no—
133C2×C22⋊C4—62no—
134#106—68no—
135#102—68no—
136C22⋊C8—34no—
137C22⋊C8—62no—
138#95—22no—
139#95—22no—
140#95—44no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—22no—
144C23×C3#12011yescharacteristic
145C23×C3#12111yescharacteristic
146C23×C3#12711yescharacteristic
147C23×C3C2×SD1612yes—
148C23×C3AGL(1,ℤ/8)12yes—
149C22×D3—42no—
150C22×D3—44no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—44no—
154C2×C3×C4—24no—
155C2×C3×C4—24no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4—42no—
160C2×C3×C4—42no—
161C2×C3×C4—44no—
162C2×C3×C4—44no—
163C2×C3×C4—42no—
164C2×C3×C4—44no—
165C2×C3×C4—42no—
166C2×C3×C4—82no—
167C2×C3×C4—82no—
168C2×C3×C4—81no—
169C2×C3×C4—82no—
170C2×C3×C4—81no—
171C2×C3×C4—81no—
172C2×C3×C4—81no—
173C2×C3×C4#12012yes—
174C2×C3×C4#12112yes—
175#67—84no—
176#67—82no—
177#67—82no—
178C2×Dic3—42no—
179C2×Dic3—44no—
180D12—44no—
181D12—82no—
182D12—88no—
183C3⋊C8—44no—
184C24—62no—
185C24C2×#6711yescharacteristic
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—124no—
189C2×D4—124no—
190C2×D4—122no—
191C2×D4—122no—
192C2×D4—124no—
193C2×D4—124no—
194C2×D4—242no—
195C2×D4—242no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—41no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—42no—
211C22×C4—62no—
212C22×C4#21514yes—
213C22×C4C2×D1211yescharacteristic
214C22×C4#23511yescharacteristic
215C22×C4D3×D412yes—
216C22×C4D3×D412yes—
217C2×C8—64no—
218C2×C8—124no—
219C4⋊C4—24no—
220C4⋊C4—42no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—124no—
224C22⋊C4—128no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3#39211yescharacteristic
236C22×C3#45012yes—
237C22×C3#45311yescharacteristic
238C22×C3#45411yescharacteristic
239C22×C3#45812yes—
240C22×C3#46012yes—
241C22×C3#49211yescharacteristic
242C22×C3#49611yescharacteristic
243C2×D3—84no—
244C2×D3—84no—
245C2×D3—82no—
246C2×D3—82no—
247C2×D3—82no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—44no—
251C3×C4—82no—
252C3×C4—82no—
253Dic3—82no—
254C23—22no—
255C23—21no—
256C23—21no—
257C23—122no—
258C23—124no—
259C23—124no—
260C23—122no—
261C23—122no—
262C23#93011yescharacteristic
263C23#93711yescharacteristic
264C23#94011yescharacteristic
265C23C2×#21512yes—
266C23#94912yes—
267D4—124no—
268D4—242no—
269D4—248no—
270D4—244no—
271D4—242no—
272D4—242no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—42no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—42no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—81no—
288C2×C4—82no—
289C2×C4—81no—
290C2×C4—81no—
291C2×C4—81no—
292C2×C4—122no—
293C2×C4—124no—
294C2×C4#88212yes—
295C2×C4#88412yes—
296C8—124no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—21no—
300C2×C3#161312yes—
301C2×C3#161612yes—
302C2×C3#200611yescharacteristic
303C2×C3#201411yescharacteristic
304C2×C3#202211yescharacteristic
305D3—162no—
306C22—21no—
307C22—21no—
308C22—22no—
309C22—22no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—242no—
315C22—242no—
316C22—244no—
317C22—244no—
318C22—242no—
319C22#496011yescharacteristic
320C22#502712yes—
321C22#504612yes—
322C22#505312yes—
323C22#505811yescharacteristic
324C22#506211yescharacteristic
325C22#507711yescharacteristic
326C22#508511yescharacteristic
327C4—22no—
328C4—22no—
329C4—44no—
330C4—82no—
331C4—82no—
332C4—242no—
333C3#996811yescharacteristic, complemented, Sylow
334C2—21no—
335C2—22no—
336C2—21no—
337C2—482no—
338C2#6953911yescharacteristic
339C2#6966411yescharacteristic
340C2#6968811yescharacteristic
341C2#6971412yes—
342C2#6971812yes—
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1445C2
socleC23×C3#120
prime radical#73381C2
hypercenter#1445D3
hypocenterC3#9968
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1445

Complements

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

NH ≅ G/Ntype
1. C2×#4968C2semidirect
3. C3×#1445C2semidirect
4. C2×#4451C2semidirect
5. #67613C2semidirect
24. #1445D3semidirect
36. C2×C3×C4⋊C4D4semidirect
52. #830D4semidirect
131. C2×C4⋊C4D12semidirect
333. C3#9968semidirect

Maximal subgroups

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

subgroupindex
C2×#49682
#733812
C3×#14452
C2×#44512
#676132
#99683