Finite Group Databasebeta

← #10697528

Subgroups — #10697528

Order 768.

712 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 131 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×#5084C212yesmaximal, complemented
2#74368C211yesmaximal, characteristic
3C3×#1956C211yesmaximal, characteristic, complemented
4C2×#4459C212yesmaximal, complemented
5#67634C211yesmaximal, characteristic, complemented
6#10235—31nomaximal, Sylow
7C2×C3×C4⋊Q8C2211yescharacteristic
8C2×#895—22no—
9C2×C4×D12C2212yes—
10C2×C3×#103—22no—
11C2×C3×C4×C8C2211yescharacteristic
12C3×#384C2211yescharacteristic
13#5084—24no—
14#5084—24no—
15#4991—22no—
16#4987—22no—
17C2×#809—22no—
18C2×#808—22no—
19C2×C4⋊(C3⋊C8)C2212yes—
20#4459—24no—
21C2×#476—32no—
22#2537—31no—
23#1956D311yescharacteristic, complemented
24C2×D4⋊C8—32no—
25#1296—31no—
26C22×C3×Q8—21no—
27C22×D12—22no—
28C22×D12D412yes—
29C22×C4×D3—22no—
30C22×C3×C8—21no—
31C3×C4⋊Q8—22no—
32C3×C4⋊Q8—21no—
33C3×C4⋊Q8D412yescomplemented
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4D412yes—
37C2×C3×C42C2311yescharacteristic
38C2×(C2×D3)⋊C4—24no—
39C2×(C2×D3)⋊C4—22no—
40C2×C4⋊Dic3D412yes—
41C22×C3⋊C8—22no—
42#895—44no—
43#895—44no—
44C4×D12—22no—
45C4×D12—22no—
46C4×D12—22no—
47C4×D12—22no—
48C3×#103—44no—
49C3×C4×C8—22no—
50C3⋊#95—22no—
51#809—28no—
52#808—28no—
53C4⋊(C3⋊C8)—22no—
54C4⋊(C3⋊C8)D414yescomplemented
55C2×C4⋊Q8C2×D311yescharacteristic
56C2×#121—62no—
57C2×C4×D4—32no—
58#476—64no—
59#476—64no—
60C2×#106—62no—
61C2×C4⋊C8—32no—
62C2×#103—22no—
63C2×#102—62no—
64C2×C4×C8C2×D311yescharacteristic
65#394—62no—
66#384C2×D311yescharacteristic
67#380—62no—
68D4⋊C8—64no—
69C23×D3—22no—
70C2×C3×Q8—22no—
71C2×C3×Q8—42no—
72C2×C3×Q8—42no—
73C2×C3×Q8—41no—
74C22×C3×C4—22no—
75C22×C3×C4—41no—
76C22×C3×C4C2×D412yes—
77C22×C3×C4C2×D411yescharacteristic
78C22×Dic3—22no—
79C2×D12—22no—
80C2×D12—22no—
81C2×D12—22no—
82C2×D12—22no—
83C2×D12—22no—
84C2×D12—48no—
85C2×D12—82no—
86C2×D12SD1614yes—
87C2×C4×D3—42no—
88C2×C4×D3—42no—
89C2×C4×D3—42no—
90C2×C4×D3—42no—
91C2×C4×D3—42no—
92C2×C4×D3—42no—
93C2×C3×C8—22no—
94C2×C3×C8—41no—
95C2×C3×C8—41no—
96C3×C4⋊C4—24no—
97C3×C4⋊C4—42no—
98C3×C4⋊C4—42no—
99C3×C4⋊C4—42no—
100C3×C4⋊C4—42no—
101C3×C42C2×D411yescharacteristic
102C3×C42C2×D412yes—
103C3×C42#4111yescharacteristic
104(C2×D3)⋊C4—48no—
105(C2×D3)⋊C4—42no—
106(C2×D3)⋊C4—42no—
107C4⋊Dic3—22no—
108C4⋊Dic3—22no—
109C2×C3⋊C8—24no—
110C2×C3⋊C8—42no—
111C2×C3⋊C8—42no—
112C22×Q8—21no—
113C22×D4—32no—
114C22×D4—62no—
115C23×C4—62no—
116C22×C8—21no—
117C22×C8—62no—
118C4⋊Q8—21no—
119C4⋊Q8—22no—
120C4⋊Q8D1212yescomplemented
121#121—124no—
122#121—124no—
123C4×D4—62no—
124C4×D4—62no—
125C4×D4—62no—
126C4×D4—62no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—32no—
130C2×C4⋊C4#6712yes—
131C2×C22⋊C4—64no—
132C2×C22⋊C4—62no—
133C2×C42C22×D311yescharacteristic
134#106—68no—
135C4⋊C8—34no—
136C4⋊C8—62no—
137#103—44no—
138#102—68no—
139C4×C8—22no—
140#95—62no—
141C23×C3#12011yescharacteristic
142C22×D3—24no—
143C22×D3—42no—
144C22×D3—42no—
145C22×D3—42no—
146C22×D3—42no—
147C22×D3—42no—
148C22×D3—42no—
149C3×Q8—44no—
150C3×Q8—82no—
151C2×C3×C4—21no—
152C2×C3×C4—21no—
153C2×C3×C4—42no—
154C2×C3×C4—44no—
155C2×C3×C4—42no—
156C2×C3×C4—81no—
157C2×C3×C4—81no—
158C2×C3×C4#12012yes—
159C2×C3×C4#12112yes—
160C2×C3×C4#12111yescharacteristic
161C2×C3×C4#12711yescharacteristic
162C2×C3×C4C2×SD1612yes—
163C2×C3×C4#13512yes—
164C2×C3×C4AGL(1,ℤ/8)12yes—
165C2×C3×C4#13712yes—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—42no—
169D12—24no—
170D12—24no—
171D12—42no—
172D12—42no—
173D12—88no—
174C4×D3—82no—
175C4×D3—82no—
176C4×D3—82no—
177C4×D3—82no—
178C3×C8—42no—
179C3⋊C8—44no—
180C24—62no—
181C2×Q8—22no—
182C2×Q8—42no—
183C2×Q8—42no—
184C2×Q8—41no—
185C2×D4—34no—
186C2×D4—62no—
187C2×D4—62no—
188C2×D4—62no—
189C2×D4—62no—
190C2×D4—62no—
191C2×D4—128no—
192C2×D4—242no—
193C22×C4—22no—
194C22×C4—41no—
195C22×C4—62no—
196C22×C4—122no—
197C22×C4—122no—
198C22×C4—122no—
199C22×C4—122no—
200C22×C4—122no—
201C22×C4—122no—
202C22×C4D3×D412yes—
203C22×C4C2×#6711yescharacteristic
204C2×C8—22no—
205C2×C8—41no—
206C2×C8—41no—
207C2×C8—64no—
208C2×C8—122no—
209C2×C8—122no—
210C4⋊C4—24no—
211C4⋊C4—42no—
212C4⋊C4—42no—
213C4⋊C4—42no—
214C4⋊C4—42no—
215C4⋊C4—62no—
216C4⋊C4—62no—
217C22⋊C4—128no—
218C22⋊C4—122no—
219C22⋊C4—122no—
220C42C2×D1211yescharacteristic
221C42#23511yescharacteristic
222C42D3×D412yes—
223C22×C3#39211yescharacteristic
224C22×C3#45012yes—
225C22×C3#45212yes—
226C22×C3#45311yescharacteristic
227C22×C3#45411yescharacteristic
228C2×D3—44no—
229C2×D3—44no—
230C2×D3—44no—
231C2×D3—82no—
232C2×D3—82no—
233C2×D3—82no—
234C2×D3—82no—
235C3×C4—21no—
236C3×C4—21no—
237C3×C4—82no—
238C3×C4—82no—
239C3×C4#44812yes—
240C3×C4#44912yes—
241C3×C4#46512yes—
242C3×C4#46812yes—
243Dic3—82no—
244C23—64no—
245C23—122no—
246C23—122no—
247C23—122no—
248C23—122no—
249C23—122no—
250C23—122no—
251C23#93711yescharacteristic
252Q8—44no—
253Q8—82no—
254D4—64no—
255D4—64no—
256D4—122no—
257D4—122no—
258D4—248no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—44no—
262C2×C4—42no—
263C2×C4—42no—
264C2×C4—81no—
265C2×C4—81no—
266C2×C4—122no—
267C2×C4—122no—
268C2×C4—122no—
269C2×C4—242no—
270C2×C4—242no—
271C2×C4—242no—
272C2×C4—242no—
273C2×C4#88212yes—
274C2×C4#88412yes—
275C2×C4D3×SD1612yes—
276C2×C4#91412yes—
277C2×C4#91812yes—
278C2×C4#91912yes—
279C2×C4#93011yescharacteristic
280C2×C4#94011yescharacteristic
281C8—42no—
282C8—124no—
283C2×C3#162911yescharacteristic
284C2×C3#163411yescharacteristic
285C2×C3#163512yes—
286C2×C3#200012yes—
287C2×C3#200611yescharacteristic
288D3—84no—
289C22—124no—
290C22—124no—
291C22—124no—
292C22—242no—
293C22—242no—
294C22—242no—
295C22—242no—
296C22#496011yescharacteristic
297C22#506211yescharacteristic
298C22#507711yescharacteristic
299C22#516912yes—
300C22#518712yes—
301C4—21no—
302C4—21no—
303C4—82no—
304C4—82no—
305C4—242no—
306C4#480612yes—
307C4#480712yes—
308C4#481012yes—
309C4#481212yes—
310C3#1023511yescharacteristic, complemented, Sylow
311C2—244no—
312C2#6966411yescharacteristic
313C2#7027212yes—
314C2#7122812yes—
315C2#7126611yescharacteristic
316C2#7130411yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1956C2
socleC23×C3#120
prime radical#74368C2
hypercenter#1956D3
hypocenterC3#10235
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1956

Complements

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

NH ≅ G/Ntype
1. C2×#5084C2semidirect
3. C3×#1956C2semidirect
4. C2×#4459C2semidirect
5. #67634C2semidirect
23. #1956D3semidirect
33. C3×C4⋊Q8D4semidirect
54. C4⋊(C3⋊C8)D4semidirect
120. C4⋊Q8D12semidirect
310. C3#10235semidirect

Maximal subgroups

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

subgroupindex
C2×#50842
#743682
C3×#19562
C2×#44592
#676342
#102353