Finite Group Databasebeta

← #10697660

Subgroups — #10697660

Order 768.

634 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 634 conjugacy classes of subgroups — too many to draw. Only its 103 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
1#79055C212yesmaximal, complemented
2#79026C211yesmaximal, characteristic, complemented
3#70498C212yesmaximal, complemented
4#70470C211yesmaximal, characteristic, complemented
5C3×#1292C211yesmaximal, characteristic, complemented
6#10254—31nomaximal, Sylow
7C2×#908—22no—
8#5477C2212yes—
9C2×C4×D12C2211yescharacteristic
10C3×#423C2212yes—
11C2×C3×C4⋊C8C2211yescharacteristic
12#5133—22no—
13#5115—22no—
14C2×(C2×D3)⋊C8—22no—
15C2×C4×C3⋊C8C2211yescharacteristic
16#4846—22no—
17#4844—24no—
18#4843—22no—
19#4843—22no—
20#4841—22no—
21#4483—24no—
22#3245—31no—
23#3036—32no—
24#1954—31no—
25#1492—32no—
26#1292D311yescharacteristic, complemented
27C2×#235—22no—
28C22×D12D411yescharacteristic
29C22×C4×D3—21no—
30C2×C3×M16—22no—
31C22×C3×C8D412yes—
32C2×C3×C42C2311yescharacteristic
33#930—22no—
34#930—22no—
35C2×(C2×D3)⋊C4—21no—
36C2×C4⋊Dic3D411yescharacteristic
37C22×C3⋊C8—21no—
38#908—44no—
39#908—44no—
40C2×D24—24no—
41C2×D24—22no—
42C2×#204—24no—
43#875—22no—
44#874D412yes—
45C4×D12—22no—
46C4×D12—22no—
47C4×D12—22no—
48C4×D12—22no—
49#869D412yes—
50C3×C4⋊C8—22no—
51C3×C4⋊C8—22no—
52C3×C4⋊C8D414yes—
53#821—24no—
54#821—24no—
55(C2×D3)⋊C8—44no—
56#818—22no—
57#816—24no—
58C4×C3⋊C8—22no—
59C2×AGL(1,ℤ/8)—62no—
60#532—32no—
61C2×C4×D4—31no—
62#474—62no—
63#466—62no—
64#463—64no—
65#461—62no—
66#459—62no—
67#459—62no—
68#423C2×D312yes—
69C2×C4⋊C8C2×D311yescharacteristic
70#417—62no—
71C2×C22⋊C8—62no—
72C2×C4×C8—31no—
73D4⋊C8—64no—
74C23×D3—21no—
75C22×C3×C4C2×D411yescharacteristic
76C22×C3×C4C2×D411yescharacteristic
77C22×C3×C4#4111yescharacteristic
78C2×#67—42no—
79C22×Dic3—21no—
80#235—42no—
81#235—42no—
82#235—42no—
83#235—42no—
84C2×D12—22no—
85C2×D12—22no—
86C2×D12—22no—
87C2×D12—22no—
88C2×D12—22no—
89C2×D12—21no—
90C2×D12D812yescomplemented
91C2×C4×D3—42no—
92C2×C4×D3—42no—
93C2×C4×D3—41no—
94C2×C4×D3—42no—
95C2×C4×D3—41no—
96C2×Dic6—22no—
97C3×M16—42no—
98C3×M16—42no—
99C2×C3×C8—24no—
100C2×C3×C8—22no—
101C2×C3×C8—42no—
102C3×C42C2×D412yes—
103C3×C42C2×D412yes—
104(C2×D3)⋊C4—42no—
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110C2×C3⋊C8—22no—
111C2×C3⋊C8—41no—
112C2×C3⋊C8—41no—
113D24—48no—
114D24—82no—
115#204—48no—
116C2×#41—62no—
117C22×D4—31no—
118C23×C4—61no—
119AGL(1,ℤ/8)—124no—
120AGL(1,ℤ/8)—124no—
121C2×SD16—64no—
122C2×D8—64no—
123C2×D8—62no—
124C2×M16—22no—
125C22×C8—61no—
126C22×C8D1212yes—
127C4⋊Q8—32no—
128#127—32no—
129#124—62no—
130#121—62no—
131#121—62no—
132C4×D4—62no—
133C4×D4—62no—
134C4×D4—62no—
135C4×D4—62no—
136C2×C4⋊C4—31no—
137C2×C22⋊C4—61no—
138C2×C42C22×D311yescharacteristic
139#107—62no—
140C4⋊C8—22no—
141C4⋊C8—22no—
142C4⋊C8D1214yes—
143#103—64no—
144#102—64no—
145#102—64no—
146C22⋊C8—124no—
147C4×C8—62no—
148C23×C3#12111yescharacteristic
149C22×D3—22no—
150C22×D3—42no—
151C22×D3—42no—
152C22×D3—41no—
153C22×D3—41no—
154C22×D3—42no—
155C2×C3×C4—21no—
156C2×C3×C4—21no—
157C2×C3×C4—41no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4#12112yes—
161C2×C3×C4#12711yescharacteristic
162C2×C3×C4C2×D811yescharacteristic
163C2×C3×C4#13511yescharacteristic
164C2×C3×C4AGL(1,ℤ/8)12yes—
165#67—82no—
166#67—82no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169C2×Dic3—41no—
170D12—24no—
171D12—44no—
172D12—42no—
173D12—42no—
174D12—42no—
175D12—42no—
176D12—42no—
177C4×D3—82no—
178C4×D3—82no—
179C4×D3—82no—
180C4×D3—81no—
181C4×D3—81no—
182Dic6—42no—
183Dic6—42no—
184C3×C8—44no—
185C3×C8—42no—
186C3⋊C8—42no—
187C24—61no—
188#41—122no—
189#41—122no—
190#41—122no—
191#41—122no—
192C2×Q8—62no—
193C2×D4—32no—
194C2×D4—62no—
195C2×D4—62no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—61no—
200C2×D4—122no—
201C22×C4—61no—
202C22×C4—122no—
203C22×C4—122no—
204C22×C4—121no—
205C22×C4—121no—
206C22×C4—122no—
207C22×C4C2×D1211yescharacteristic
208C22×C4D3×D411yescharacteristic
209C22×C4#23911yescharacteristic
210SD16—128no—
211D8—128no—
212D8—242no—
213M16—42no—
214M16—42no—
215C2×C8—22no—
216C2×C8—24no—
217C2×C8—42no—
218C2×C8—62no—
219C2×C8—121no—
220C2×C8—121no—
221C4⋊C4—62no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C22⋊C4—122no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C42C2×D1212yes—
228C42D3×D412yes—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3#39211yescharacteristic
232C22×C3#46611yescharacteristic
233C22×C3#46911yescharacteristic
234C2×D3—44no—
235C2×D3—42no—
236C2×D3—82no—
237C2×D3—81no—
238C2×D3—81no—
239C2×D3—82no—
240C2×D3—82no—
241C2×D3—82no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4#44712yes—
245C3×C4#44912yes—
246C3×C4#45311yescharacteristic
247C3×C4#45411yescharacteristic
248Dic3—82no—
249Dic3—81no—
250C23—62no—
251C23—122no—
252C23—122no—
253C23—121no—
254C23—121no—
255C23—122no—
256C23#89511yescharacteristic
257Q8—122no—
258Q8—122no—
259D4—64no—
260D4—124no—
261D4—122no—
262D4—122no—
263D4—122no—
264D4—122no—
265D4—122no—
266D4—242no—
267D4—242no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—122no—
272C2×C4—121no—
273C2×C4—122no—
274C2×C4—242no—
275C2×C4—241no—
276C2×C4—241no—
277C2×C4—242no—
278C2×C4—242no—
279C2×C4#87411yescharacteristic
280C2×C4#88211yescharacteristic
281C2×C4#88212yes—
282C2×C4#89512yes—
283C2×C4D3×D811yescharacteristic
284C2×C4#91412yes—
285C2×C4#91911yescharacteristic
286C8—42no—
287C8—44no—
288C8—122no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3#199911yescharacteristic
292C2×C3#200611yescharacteristic
293C2×C3#201711yescharacteristic
294D3—82no—
295D3—162no—
296C22—21no—
297C22—21no—
298C22—124no—
299C22—122no—
300C22—242no—
301C22—241no—
302C22—241no—
303C22—242no—
304C22—242no—
305C22—242no—
306C22#467111yescharacteristic
307C22#486511yescharacteristic
308C22#488311yescharacteristic
309C4—22no—
310C4—21no—
311C4—242no—
312C4—241no—
313C4#474711yescharacteristic
314C4#475311yescharacteristic
315C4#477312yes—
316C4#481012yes—
317C3#1025411yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—242no—
321C2—482no—
322C2#6872711yescharacteristic
323C2#6888611yescharacteristic
324C2#6900811yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#392
hypercenter#1292D3
hypocenterC3#10254
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
1. #79055C2semidirect
2. #79026C2semidirect
3. #70498C2semidirect
4. #70470C2semidirect
5. C3×#1292C2semidirect
26. #1292D3semidirect
90. C2×D12D8semidirect
317. C3#10254semidirect

Maximal subgroups

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

subgroupindex
#790552
#790262
#704982
#704702
C3×#12922
#102543