Finite Group Databasebeta

← #10692899

Subgroups — #10692899

Order 768.

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

normal subgroups only

This group has 536 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#82476C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic, complemented
3C3×#1467C211yesmaximal, characteristic, complemented
4#70818C212yesmaximal
5#67749C212yesmaximal, complemented
6#9458—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×#947C412yes—
9C2×#874C2211yescharacteristic
10C3×#432C2212yes—
11#5024—24no—
12#4991—21no—
13#4991—21no—
14#4988C412yes—
15#4939—21no—
16#4939—21no—
17C4×C4⋊Dic3C2211yescharacteristic
18#4926—21no—
19#4926C2212yes—
20#4925—21no—
21C3×#324—21no—
22C3×#324—21no—
23#4485—24no—
24#2983—31no—
25#2548—31no—
26#1555—32no—
27#1467D311yescharacteristic, complemented
28#1285—32no—
29C22×C3×Q8C2×C411yescharacteristic
30C22×D12—21no—
31C22×D12—21no—
32C22×D12C2×C412yes—
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C2×C3×C4⋊C4C2×C411yescharacteristic
36C2×C3×C4⋊C4D411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C42D411yescharacteristic
39C2×C3×C42D411yescharacteristic
40C2×C3×C42C2311yescharacteristic
41#947—44no—
42C2×(C2×D3)⋊C4—22no—
43C2×(C2×D3)⋊C4—22no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×C4⋊Dic3—21no—
47C2×C4×Dic3C2×C412yes—
48#874—22no—
49#874—22no—
50C3×C4⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×#97—22no—
53C3⋊#95—21no—
54#830—22no—
55C4⋊(C3⋊C8)—22no—
56C4⋊(C3⋊C8)—22no—
57C4⋊(C3⋊C8)D414yes—
58#803—22no—
59#803—22no—
60C4×C3⋊C8—22no—
61C2×#127—31no—
62C2×#124—32no—
63C2×C4×Q8C2×D311yescharacteristic
64#446—64no—
65#432—32no—
66#432—61no—
67#432C2×D312yes—
68#431—61no—
69#394—61no—
70#394—61no—
71#390—32no—
72#390—61no—
73#390—61no—
74C4×C4⋊C4—31no—
75#331—64no—
76#324—21no—
77#324—21no—
78C23×D3—22no—
79C2×C3×Q8—22no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C22×C3×C4C22⋊C411yescharacteristic
83C22×C3×C4C22⋊C411yescharacteristic
84C22×C3×C4C22⋊C411yescharacteristic
85C22×C3×C4C22⋊C411yescharacteristic
86C22×C3×C4C22×C411yescharacteristic
87C22×C3×C4C2×D411yescharacteristic
88C22×C3×C4C2×D411yescharacteristic
89C22×Dic3—22no—
90C2×D12—24no—
91C2×D12—42no—
92C2×D12—42no—
93C2×D12—42no—
94C2×D12—42no—
95C2×D12—41no—
96C2×D12—42no—
97C2×D12—41no—
98C2×D12—82no—
99C2×C3×C8—42no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—22no—
103C3×C42—21no—
104C3×C42—21no—
105C3×C42—21no—
106C3×C42—21no—
107C3×C42C2×D412yes—
108C3×C42#4112yes—
109(C2×D3)⋊C4—44no—
110(C2×D3)⋊C4—44no—
111(C2×D3)⋊C4—82no—
112(C2×D3)⋊C4—82no—
113C4⋊Dic3—42no—
114C4×Dic3—42no—
115C2×C3⋊C8—24no—
116C22×Q8C4×D311yescharacteristic
117C22×D4—32no—
118C22×D4—61no—
119C22×D4—61no—
120#127—62no—
121#127—62no—
122#124—124no—
123C4×Q8—22no—
124C4×Q8—22no—
125C2×C4⋊C4—61no—
126C2×C4⋊C4C4×D311yescharacteristic
127C2×C4⋊C4#6711yescharacteristic
128C2×C4⋊C4#6711yescharacteristic
129C2×C22⋊C4—62no—
130C2×C22⋊C4—62no—
131C2×C22⋊C4—62no—
132C2×C22⋊C4—62no—
133C2×C42—32no—
134C2×C42D1211yescharacteristic
135C2×C42D1211yescharacteristic
136C2×C42C22×D311yescharacteristic
137C4⋊C8—22no—
138C4⋊C8—34no—
139C4⋊C8—62no—
140C4⋊C8—62no—
141C22⋊C8—22no—
142C22⋊C8—62no—
143#97—22no—
144#97—62no—
145#97—62no—
146C4×C8—62no—
147#95—61no—
148C23×C3C2×C22⋊C411yescharacteristic
149C22×D3—42no—
150C22×D3—42no—
151C22×D3—42no—
152C22×D3—42no—
153C22×D3—42no—
154C22×D3—42no—
155C22×D3—42no—
156C3×Q8—44no—
157C2×C3×C4—21no—
158C2×C3×C4—21no—
159C2×C3×C4—21no—
160C2×C3×C4—21no—
161C2×C3×C4—22no—
162C2×C3×C4—21no—
163C2×C3×C4—21no—
164C2×C3×C4—21no—
165C2×C3×C4—21no—
166C2×C3×C4—21no—
167C2×C3×C4—41no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4#10012yes—
171C2×C3×C4C4×D412yes—
172C2×C3×C4#12011yescharacteristic
173C2×C3×C4#12112yes—
174C2×C3×C4#12311yescharacteristic
175C2×Dic3—42no—
176C2×Dic3—42no—
177C2×Dic3—42no—
178D12—84no—
179D12—82no—
180D12—84no—
181D12—82no—
182C3×C8—82no—
183C3⋊C8—44no—
184C24—62no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—41no—
188C2×D4—64no—
189C2×D4—122no—
190C2×D4—122no—
191C2×D4—122no—
192C2×D4—122no—
193C2×D4—121no—
194C2×D4—121no—
195C2×D4—122no—
196C2×D4—242no—
197C22×C4—62no—
198C22×C4(C2×D3)⋊C411yescharacteristic
199C22×C4(C2×D3)⋊C411yescharacteristic
200C22×C4(C2×D3)⋊C411yescharacteristic
201C22×C4(C2×D3)⋊C411yescharacteristic
202C22×C4C2×C4×D311yescharacteristic
203C22×C4C2×D1211yescharacteristic
204C22×C4C2×#6711yescharacteristic
205C2×C8—42no—
206C2×C8—64no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—22no—
210C4⋊C4—122no—
211C22⋊C4—124no—
212C22⋊C4—124no—
213C22⋊C4—242no—
214C22⋊C4—242no—
215C42—21no—
216C42—21no—
217C42—21no—
218C42—21no—
219C42—122no—
220C42D3×D412yes—
221C42#23712yes—
222C22×C3—21no—
223C22×C3—21no—
224C22×C3#38611yescharacteristic
225C22×C3C2×#10011yescharacteristic
226C22×C3#41311yescharacteristic
227C2×D3—82no—
228C2×D3—82no—
229C2×D3—82no—
230C2×D3—82no—
231C2×D3—82no—
232C2×D3—82no—
233C2×D3—82no—
234C3×C4—21no—
235C3×C4—22no—
236C3×C4—22no—
237C3×C4—22no—
238C3×C4—41no—
239C3×C4—41no—
240C3×C4—41no—
241C3×C4#45412yes—
242Dic3—82no—
243C23—122no—
244C23—122no—
245C23—122no—
246C23—122no—
247C23—122no—
248C23—122no—
249C23—122no—
250C23C2×(C2×D3)⋊C411yescharacteristic
251Q8—44no—
252D4—244no—
253D4—242no—
254D4—244no—
255D4—242no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—122no—
270C2×C4—122no—
271C2×C4—122no—
272C2×C4#82312yes—
273C2×C4#67⋊C412yes—
274C2×C4#88211yescharacteristic
275C2×C4#88611yescharacteristic
276C2×C4#93912yes—
277C8—82no—
278C8—124no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3#188811yescharacteristic
282C2×C3#190011yescharacteristic
283C2×C3#191011yescharacteristic
284D3—162no—
285C22—21no—
286C22—21no—
287C22—242no—
288C22—242no—
289C22—242no—
290C22—242no—
291C22—242no—
292C22—242no—
293C22—242no—
294C22#495811yescharacteristic
295C22C2×#82311yescharacteristic
296C22#513211yescharacteristic
297C4—22no—
298C4—22no—
299C4—22no—
300C4—21no—
301C4—41no—
302C4—41no—
303C4—41no—
304C4—242no—
305C4#475312yes—
306C3#945811yescharacteristic, complemented, Sylow
307C2—21no—
308C2—21no—
309C2—482no—
310C2#6998511yescharacteristic
311C2#7004711yescharacteristic
312C2#7010411yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1467C2
socleC22×C3#386
prime radicalC2×#874C22
cosocleC2×C3×C42C23
hypercenter#1467D3
hypocenterC3#9458
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1467

Complements

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

NH ≅ G/Ntype
2. #82436C2semidirect
3. C3×#1467C2semidirect
5. #67749C2semidirect
27. #1467D3semidirect
306. C3#9458semidirect

Maximal subgroups

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

subgroupindex
#824762
#824362
C3×#14672
#708182
#677492
#94583