Finite Group Databasebeta

← #10697657

Subgroups — #10697657

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
1C3×#3245C211yesmaximal, characteristic, complemented
2#83297C212yesmaximal, complemented
3#71802C212yesmaximal, complemented
4#70470C211yesmaximal, characteristic, complemented
5#67705C211yesmaximal, characteristic, complemented
6#10254—31nomaximal, Sylow
7C3×#532C2212yes—
8C2×#949—22no—
9C2×C4×D12C2211yescharacteristic
10C3×#474—22no—
11C3×#466—22no—
12C3×#459—22no—
13C2×C3×C4⋊C8C2211yescharacteristic
14#5232—22no—
15C2×(C2×D3)⋊C8—22no—
16#5083—22no—
17#5072—22no—
18#5057—24no—
19#4924C2212yes—
20C2×C4×C3⋊C8C2211yescharacteristic
21D4⋊(C3⋊C8)—24no—
22#3245D311yescharacteristic, complemented
23#3036—32no—
24#1954—31no—
25#1492—32no—
26#1292—31no—
27C2×C3×#41—22no—
28C22×D12D411yescharacteristic
29C22×C4×D3—21no—
30C2×C3×D8—22no—
31C22×C3×C8D412yes—
32C3×C4⋊Q8D412yes—
33C3×#127D412yes—
34C3×#124—22no—
35C3×#121—22no—
36C3×#121—22no—
37C3×C4×D4—22no—
38C3×C4×D4—22no—
39C2×C3×C42C2311yescharacteristic
40#949—44no—
41#949—44no—
42C2×#215—24no—
43C2×#213—24no—
44C2×(C2×D3)⋊C4—21no—
45C2×C4⋊Dic3D411yescharacteristic
46C2×#208—22no—
47C22×C3⋊C8—21no—
48C4×D12—22no—
49C4×D12—22no—
50C3×#107—22no—
51C3×C4⋊C8—22no—
52C3×#102—24no—
53#835—24no—
54#832—24no—
55(C2×D3)⋊C8—44no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yes—
58C4×C3⋊C8—22no—
59C2×AGL(1,ℤ/8)—62no—
60#532C2×D312yes—
61C2×C4×D4—31no—
62#474—22no—
63#466—22no—
64#463—64no—
65#461—62no—
66#459—22no—
67#459—62no—
68#423—32no—
69C2×C4⋊C8C2×D311yescharacteristic
70#417—62no—
71C2×C22⋊C8—62no—
72C2×C4×C8—31no—
73D4⋊C8—64no—
74C23×D3—21no—
75C3×#41—42no—
76C3×#41—42no—
77C3×#41—42no—
78C3×#41—42no—
79C2×C3×Q8—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—42no—
84C22×C3×C4—42no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4#4111yescharacteristic
88C22×Dic3—21no—
89C2×D12—22no—
90C2×D12—22no—
91C2×D12—21no—
92C2×D12D812yescomplemented
93C2×C4×D3—42no—
94C2×C4×D3—42no—
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C3×D8—82no—
98C2×C3×C8—22no—
99C2×C3×C8—42no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—42no—
104C3×C42C2×D412yes—
105C3×C42C2×D412yes—
106#215—48no—
107#213—48no—
108(C2×D3)⋊C4—42no—
109C4⋊Dic3—22no—
110#208—42no—
111#208—42no—
112C2×C3⋊C8—22no—
113C2×C3⋊C8—24no—
114C2×C3⋊C8—41no—
115C2×C3⋊C8—41no—
116C2×#41—22no—
117C22×D4—31no—
118C23×C4—61no—
119AGL(1,ℤ/8)—124no—
120AGL(1,ℤ/8)—124no—
121C2×SD16—64no—
122C2×D8—22no—
123C2×D8—64no—
124C2×M16—62no—
125C22×C8—61no—
126C22×C8#6712yes—
127C4⋊Q8#6712yes—
128#127#6712yes—
129#124—22no—
130#121—22no—
131#121—22no—
132C4×D4—22no—
133C4×D4—22no—
134C4×D4—62no—
135C4×D4—62no—
136C2×C4⋊C4—31no—
137C2×C22⋊C4—61no—
138C2×C42C22×D311yescharacteristic
139#107—22no—
140C4⋊C8—22no—
141C4⋊C8—34no—
142C4⋊C8—62no—
143#103—64no—
144#102—24no—
145#102—64no—
146C22⋊C8—124no—
147C4×C8—62no—
148C23×C3—42no—
149C23×C3#12111yescharacteristic
150C22×D3—22no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—41no—
154C22×D3—41no—
155C3×Q8—42no—
156C3×Q8—42no—
157C3×D4—44no—
158C3×D4—42no—
159C3×D4—42no—
160C3×D4—42no—
161C3×D4—42no—
162C3×D4—82no—
163C3×D4—82no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—82no—
169C2×C3×C4—82no—
170C2×C3×C4#12012yes—
171C2×C3×C4#12011yescharacteristic
172C2×C3×C4#12112yes—
173C2×C3×C4#12711yescharacteristic
174C2×C3×C4C2×D811yescharacteristic
175C2×C3×C4#13511yescharacteristic
176C2×C3×C4AGL(1,ℤ/8)12yes—
177C2×Dic3—42no—
178C2×Dic3—41no—
179D12—24no—
180D12—42no—
181C4×D3—82no—
182C4×D3—81no—
183C4×D3—81no—
184C3×C8—42no—
185C3⋊C8—42no—
186C3⋊C8—44no—
187C24—61no—
188#41—42no—
189#41—42no—
190#41—42no—
191#41—42no—
192C2×Q8—22no—
193C2×D4—22no—
194C2×D4—22no—
195C2×D4—22no—
196C2×D4—32no—
197C2×D4—42no—
198C2×D4—62no—
199C2×D4—61no—
200C2×D4—62no—
201C22×C4—42no—
202C22×C4—61no—
203C22×C4—122no—
204C22×C4—122no—
205C22×C4—121no—
206C22×C4—121no—
207C22×C4D3×D411yescharacteristic
208C22×C4#23711yescharacteristic
209C22×C4C2×#6711yescharacteristic
210SD16—128no—
211D8—82no—
212D8—128no—
213M16—122no—
214M16—122no—
215C2×C8—22no—
216C2×C8—42no—
217C2×C8—62no—
218C2×C8—64no—
219C2×C8—121no—
220C2×C8—121no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—62no—
224C22⋊C4—42no—
225C22⋊C4—42no—
226C22⋊C4—122no—
227C42D3×D412yes—
228C42C2×#6712yes—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—82no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3#39211yescharacteristic
235C22×C3#46611yescharacteristic
236C22×C3#46911yescharacteristic
237C2×D3—44no—
238C2×D3—42no—
239C2×D3—82no—
240C2×D3—81no—
241C2×D3—81no—
242C3×C4—21no—
243C3×C4—22no—
244C3×C4—82no—
245C3×C4#44712yes—
246C3×C4#44912yes—
247C3×C4#45311yescharacteristic
248C3×C4#45411yescharacteristic
249Dic3—81no—
250C23—42no—
251C23—62no—
252C23—122no—
253C23—122no—
254C23—121no—
255C23—121no—
256C23#93811yescharacteristic
257Q8—42no—
258Q8—42no—
259D4—44no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—42no—
264D4—64no—
265D4—82no—
266D4—82no—
267D4—122no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—82no—
273C2×C4—82no—
274C2×C4—122no—
275C2×C4—121no—
276C2×C4—242no—
277C2×C4—241no—
278C2×C4—241no—
279C2×C4D3×D811yescharacteristic
280C2×C4#91112yes—
281C2×C4#91211yescharacteristic
282C2×C4#93712yes—
283C2×C4#93912yes—
284C2×C4#94011yescharacteristic
285C2×C4#95311yescharacteristic
286C8—42no—
287C8—122no—
288C8—124no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—162no—
292C2×C3#199911yescharacteristic
293C2×C3#200611yescharacteristic
294C2×C3#201711yescharacteristic
295D3—82no—
296C22—21no—
297C22—21no—
298C22—82no—
299C22—82no—
300C22—82no—
301C22—124no—
302C22—122no—
303C22—242no—
304C22—241no—
305C22—241no—
306C22#515711yescharacteristic
307C22#515911yescharacteristic
308C22#522311yescharacteristic
309C4—22no—
310C4—21no—
311C4—82no—
312C4—241no—
313C4#515612yes—
314C4#515812yes—
315C4#524111yescharacteristic
316C4#524511yescharacteristic
317C3#1025411yescharacteristic, complemented, Sylow
318C2—21no—
319C2—21no—
320C2—162no—
321C2—242no—
322C2#7187911yescharacteristic
323C2#7204311yescharacteristic
324C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3245C2
socleC22×C3#392
hypercenter#3245D3
hypocenterC3#10254
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3245

Complements

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

NH ≅ G/Ntype
1. C3×#3245C2semidirect
2. #83297C2semidirect
3. #71802C2semidirect
4. #70470C2semidirect
5. #67705C2semidirect
22. #3245D3semidirect
92. 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
C3×#32452
#832972
#718022
#704702
#677052
#102543