Finite Group Databasebeta

← #10697659

Subgroups — #10697659

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#82307C212yesmaximal, complemented
2#79026C211yesmaximal, characteristic, complemented
3C3×#1954C211yesmaximal, characteristic, complemented
4#69735C212yesmaximal, complemented
5#67705C211yesmaximal, characteristic, complemented
6#10254—31nomaximal, Sylow
7C2×C3×C4×D4C2211yescharacteristic
8C2×#932—22no—
9#5477C2212yes—
10C2×C3×C4⋊C8C2211yescharacteristic
11C2×C3×C22⋊C8—22no—
12C2×C3×C4×C8C2211yescharacteristic
13#5115—22no—
14#5019—22no—
15#5016—24no—
16#5015—22no—
17#4966—22no—
18#4924C2212yes—
19#4846—22no—
20#4843—22no—
21#4459—24no—
22#3245—31no—
23#3036—32no—
24#1954D311yescharacteristic, complemented
25#1492—32no—
26#1292—31no—
27C22×C3×D4D411yescharacteristic
28C23×C3×C4—21no—
29C2×#235—22no—
30C22×C3×C8—21no—
31C22×C3×C8D412yes—
32C3×C4×D4—22no—
33C3×C4×D4—22no—
34C2×C3×C4⋊C4D411yescharacteristic
35C2×C3×C22⋊C4—21no—
36C2×C3×C42C2311yescharacteristic
37C2×#214—24no—
38#932—44no—
39#932—44no—
40C2×#213—24no—
41#930—22no—
42#930—22no—
43C2×#208—22no—
44C2×D24—22no—
45#875—22no—
46#874D412yes—
47C4×D12—22no—
48C4×D12—22no—
49#869D412yes—
50C3×C4⋊C8—22no—
51C3×C22⋊C8—44no—
52C3×C4×C8—22no—
53#821—24no—
54#818—22no—
55#810—24no—
56#809—24no—
57C4⋊(C3⋊C8)—22no—
58C4⋊(C3⋊C8)D414yes—
59C2×AGL(1,ℤ/8)—62no—
60#532—32no—
61C2×C4×D4C2×D311yescharacteristic
62#474—62no—
63#466—62no—
64#463—64no—
65#461—62no—
66#459—62no—
67#459—62no—
68#423—32no—
69C2×C4⋊C8C2×D311yescharacteristic
70#417—62no—
71C2×C22⋊C8—22no—
72C2×C4×C8C2×D311yescharacteristic
73D4⋊C8—64no—
74C24×C3—21no—
75C2×C3×D4—22no—
76C2×C3×D4—21no—
77C2×C3×D4—22no—
78C2×C3×D4D812yescomplemented
79C22×C3×C4—21no—
80C22×C3×C4—42no—
81C22×C3×C4—42no—
82C22×C3×C4—41no—
83C22×C3×C4—41no—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4#4111yescharacteristic
87C2×#67—42no—
88#235—42no—
89#235—42no—
90#235—42no—
91#235—42no—
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—22no—
95C2×C4×D3—42no—
96C2×Dic6—22no—
97C2×C3×C8—22no—
98C2×C3×C8—22no—
99C2×C3×C8—41no—
100C2×C3×C8—41no—
101C2×C3×C8—42no—
102C3×C4⋊C4—22no—
103C3×C22⋊C4—42no—
104C3×C42C2×D412yes—
105C3×C42C2×D412yes—
106#214—48no—
107#213—48no—
108(C2×D3)⋊C4—42no—
109(C2×D3)⋊C4—42no—
110C4⋊Dic3—22no—
111C4⋊Dic3—22no—
112#208—42no—
113#208—42no—
114C2×C3⋊C8—24no—
115D24—82no—
116C2×#41—62no—
117C22×D4D1211yescharacteristic
118C23×C4—21no—
119AGL(1,ℤ/8)—124no—
120AGL(1,ℤ/8)—124no—
121C2×SD16—64no—
122C2×D8—62no—
123C2×D8—64no—
124C2×M16—62no—
125C22×C8—21no—
126C22×C8#6712yes—
127C4⋊Q8—32no—
128#127—32no—
129#124—62no—
130#121—62no—
131#121—62no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—62no—
135C4×D4—62no—
136C2×C4⋊C4D1211yescharacteristic
137C2×C22⋊C4—21no—
138C2×C42C22×D311yescharacteristic
139#107—62no—
140C4⋊C8—22no—
141C4⋊C8—34no—
142C4⋊C8—62no—
143#103—64no—
144#102—64no—
145#102—64no—
146C22⋊C8—44no—
147C4×C8—22no—
148C23×C3—22no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3#12111yescharacteristic
154C22×D3—42no—
155C3×D4—24no—
156C3×D4—42no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—41no—
160C2×C3×C4—42no—
161C2×C3×C4—41no—
162C2×C3×C4—82no—
163C2×C3×C4—81no—
164C2×C3×C4—81no—
165C2×C3×C4#12012yes—
166C2×C3×C4#12011yescharacteristic
167C2×C3×C4#12112yes—
168C2×C3×C4#12711yescharacteristic
169C2×C3×C4C2×D811yescharacteristic
170C2×C3×C4#13511yescharacteristic
171C2×C3×C4AGL(1,ℤ/8)12yes—
172#67—82no—
173#67—82no—
174C2×Dic3—42no—
175D12—44no—
176D12—42no—
177D12—42no—
178D12—42no—
179D12—42no—
180C4×D3—82no—
181C4×D3—82no—
182Dic6—42no—
183Dic6—42no—
184C3×C8—42no—
185C3×C8—42no—
186C3⋊C8—44no—
187C24—21no—
188#41—122no—
189#41—122no—
190#41—122no—
191#41—122no—
192C2×Q8—62no—
193C2×D4—22no—
194C2×D4—21no—
195C2×D4—22no—
196C2×D4—62no—
197C2×D4—62no—
198C2×D4—62no—
199C2×D4—122no—
200C2×D4D2412yescomplemented
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—41no—
205C22×C4—41no—
206C22×C4—122no—
207C22×C4C2×D1211yescharacteristic
208C22×C4#23511yescharacteristic
209C22×C4C2×#6711yescharacteristic
210SD16—128no—
211D8—128no—
212D8—242no—
213M16—122no—
214M16—122no—
215C2×C8—22no—
216C2×C8—22no—
217C2×C8—41no—
218C2×C8—41no—
219C2×C8—42no—
220C2×C8—64no—
221C4⋊C4—22no—
222C4⋊C4—62no—
223C4⋊C4—62no—
224C22⋊C4—42no—
225C22⋊C4—122no—
226C22⋊C4—122no—
227C42D3×D412yes—
228C42D3×D412yes—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—44no—
232C22×C3—42no—
233C22×C3—82no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3#39211yescharacteristic
237C22×C3#46611yescharacteristic
238C22×C3#46911yescharacteristic
239C2×D3—82no—
240C2×D3—82no—
241C2×D3—82no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—81no—
245C3×C4#44712yes—
246C3×C4#44912yes—
247C3×C4#45311yescharacteristic
248C3×C4#45411yescharacteristic
249Dic3—82no—
250C23—22no—
251C23—42no—
252C23—42no—
253C23—41no—
254C23—41no—
255C23—122no—
256C23#93011yescharacteristic
257Q8—122no—
258Q8—122no—
259D4—24no—
260D4—42no—
261D4—124no—
262D4—122no—
263D4—122no—
264D4—122no—
265D4—122no—
266D4—242no—
267D4—242no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—82no—
274C2×C4—81no—
275C2×C4—81no—
276C2×C4—122no—
277C2×C4—242no—
278C2×C4—242no—
279C2×C4#88212yes—
280C2×C4#88412yes—
281C2×C4C2×D2411yescharacteristic
282C2×C4#90411yescharacteristic
283C2×C4#90812yes—
284C2×C4#93711yescharacteristic
285C2×C4#94011yescharacteristic
286C8—42no—
287C8—42no—
288C8—124no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—82no—
292C2×C3#199911yescharacteristic
293C2×C3#200611yescharacteristic
294C2×C3#201711yescharacteristic
295D3—162no—
296C22—21no—
297C22—21no—
298C22—44no—
299C22—42no—
300C22—82no—
301C22—81no—
302C22—81no—
303C22—242no—
304C22—242no—
305C22—242no—
306C22#496011yescharacteristic
307C22#511511yescharacteristic
308C22#513511yescharacteristic
309C4—21no—
310C4—22no—
311C4—81no—
312C4—242no—
313C4#473212yes—
314C4#473412yes—
315C4#506211yescharacteristic
316C4#507711yescharacteristic
317C3#1025411yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—82no—
321C2—482no—
322C2#6966411yescharacteristic
323C2#6996111yescharacteristic
324C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1954C2
socleC22×C3#392
hypercenter#1954D3
hypocenterC3#10254
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1954

Complements

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

NH ≅ G/Ntype
1. #82307C2semidirect
2. #79026C2semidirect
3. C3×#1954C2semidirect
4. #69735C2semidirect
5. #67705C2semidirect
24. #1954D3semidirect
78. C2×C3×D4D8semidirect
200. C2×D4D24semidirect
317. C3#10254semidirect

Maximal subgroups

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

subgroupindex
#823072
#790262
C3×#19542
#697352
#677052
#102543