Finite Group Databasebeta

← #10698074

Subgroups — #10698074

Order 768.

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

normal subgroups only

This group has 602 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#82075C212yesmaximal
2#78617C211yesmaximal, characteristic
3C3×#1817C211yesmaximal, characteristic
4#69946C212yesmaximal
5#67676C211yesmaximal, characteristic
6#10325—31nomaximal, Sylow
7C22×C3×C42C2211yescharacteristic
8C2×#924C2212yes—
9C2×C3×C22⋊C8C2211yescharacteristic
10C2×C3×C22⋊C8C2211yescharacteristic
11#5212C2212yes—
12#5211—22no—
13#5210—22no—
14#5116—22no—
15#4933—24no—
16#4929—22no—
17#4928—22no—
18#4739—22no—
19#4722—22no—
20C3×#336—22no—
21#4467—24no—
22#3189—31no—
23#2855—32no—
24#1817D311yescharacteristic
25#1520—32no—
26#1289—31no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×Dic6D412yes—
31C22×C3×C8—21no—
32C22×C3×C8D412yes—
33C2×C3×C42—22no—
34C2×C3×C42—22no—
35C2×C3×C42—21no—
36C2×C3×C42—21no—
37C2×#217—22no—
38C2×#217—22no—
39C2×C4⋊Dic3—22no—
40C2×C4⋊Dic3D412yes—
41#924—22no—
42#924—22no—
43#924—22no—
44#924—22no—
45C2×Dic3⋊C4—24no—
46C2×Dic3⋊C4—22no—
47C2×Dic12—22no—
48C3×C22⋊C8—22no—
49C3×C22⋊C8—22no—
50C3⋊#95—24no—
51C3⋊#95—24no—
52C3⋊#95—24no—
53C3⋊#95D414yes—
54#818—22no—
55#816—24no—
56C2×C22⋊Q8—32no—
57C22×C42C2×D311yescharacteristic
58#484—62no—
59#467—62no—
60#451—62no—
61C2×C22⋊C8C2×D311yescharacteristic
62C2×C22⋊C8C2×D311yescharacteristic
63#391—62no—
64#389—62no—
65#387—64no—
66#385—62no—
67#381—62no—
68#380—32no—
69#336—22no—
70#328—64no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—22no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—42no—
79C22×C3×C4—42no—
80C22×C3×C4Q1612yescomplemented
81C22×C3×C4C2×D412yes—
82C22×C3×C4C2×D412yes—
83C22×C3×C4C2×D411yescharacteristic
84C22×C3×C4#4111yescharacteristic
85C22×Dic3—22no—
86C22×Dic3—24no—
87C22×Dic3—22no—
88C2×Dic6—22no—
89C2×Dic6—22no—
90C2×Dic6—44no—
91C2×C3×C8—22no—
92C2×C3×C8—42no—
93C2×C3×C8—42no—
94C2×C3×C8—41no—
95C3×C42—41no—
96C3×C42—42no—
97C3×C42—41no—
98#217—42no—
99#217—42no—
100#217—42no—
101#217—42no—
102C4⋊Dic3—22no—
103C4⋊Dic3—22no—
104C4⋊Dic3—44no—
105Dic3⋊C4—48no—
106Dic3⋊C4—42no—
107Dic3⋊C4—42no—
108Dic12—82no—
109C22×Q8—32no—
110C23×C4D1211yescharacteristic
111C23×C4D1211yescharacteristic
112C23×C4C22×D311yescharacteristic
113C2×Q16—62no—
114C22×C8—21no—
115C22×C8#6712yes—
116C22⋊Q8—62no—
117C22⋊Q8—62no—
118C22⋊Q8—62no—
119C22⋊Q8—62no—
120C2×C4⋊C4—32no—
121C2×C4⋊C4—64no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—62no—
124C2×C22⋊C4—62no—
125C2×C22⋊C4—62no—
126C2×C42—21no—
127C2×C42—22no—
128C2×C42—22no—
129C2×C42—21no—
130#107—62no—
131#103—64no—
132C22⋊C8—22no—
133C22⋊C8—22no—
134#95—34no—
135#95—64no—
136#95—64no—
137#95—64no—
138C23×C3—41no—
139C23×C3—41no—
140C23×C3#12012yes—
141C23×C3#12111yescharacteristic
142C23×C3C2×Q1611yescharacteristic
143C23×C3#13511yescharacteristic
144C23×C3#13712yes—
145C2×C3×C4—24no—
146C2×C3×C4—22no—
147C2×C3×C4—24no—
148C2×C3×C4—22no—
149C2×C3×C4—24no—
150C2×C3×C4—22no—
151C2×C3×C4—42no—
152C2×C3×C4—41no—
153C2×C3×C4—42no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—41no—
159C2×C3×C4#12011yescharacteristic
160C2×C3×C4#12112yes—
161C2×C3×C4#12711yescharacteristic
162C2×Dic3—44no—
163C2×Dic3—44no—
164C2×Dic3—44no—
165C2×Dic3—44no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—44no—
169Dic6—44no—
170Dic6—82no—
171C3×C8—42no—
172C3×C8—81no—
173C24C2×D1211yescharacteristic
174C2×Q8—62no—
175C2×Q8—62no—
176C2×Q8—124no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—64no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4Dic1212yescomplemented
189C22×C4#23511yescharacteristic
190C22×C4D3×D412yes—
191C22×C4D3×D412yes—
192C22×C4C2×#6711yescharacteristic
193Q16—242no—
194C2×C8—22no—
195C2×C8—42no—
196C2×C8—42no—
197C2×C8—41no—
198C4⋊C4—62no—
199C4⋊C4—62no—
200C4⋊C4—128no—
201C4⋊C4—124no—
202C4⋊C4—122no—
203C4⋊C4—122no—
204C22⋊C4—122no—
205C22⋊C4—122no—
206C22⋊C4—122no—
207C22⋊C4—122no—
208C42—42no—
209C42—41no—
210C42—41no—
211C22×C3—22no—
212C22×C3—21no—
213C22×C3—22no—
214C22×C3—21no—
215C22×C3—41no—
216C22×C3—41no—
217C22×C3—42no—
218C22×C3#39211yescharacteristic
219C22×C3#45112yes—
220C22×C3#45212yes—
221C22×C3#46711yescharacteristic
222C22×C3#47011yescharacteristic
223C3×C4—21no—
224C3×C4—22no—
225C3×C4—42no—
226C3×C4—42no—
227C3×C4#45812yes—
228Dic3—84no—
229Dic3—82no—
230Dic3—82no—
231C23—41no—
232C23—41no—
233C23#88212yes—
234C23#90411yescharacteristic
235C23C2×Dic1211yescharacteristic
236C23#90912yes—
237C23#93011yescharacteristic
238Q8—124no—
239Q8—242no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—24no—
243C2×C4—22no—
244C2×C4—24no—
245C2×C4—22no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—41no—
250C2×C4—41no—
251C2×C4—41no—
252C2×C4—41no—
253C2×C4—42no—
254C2×C4—124no—
255C2×C4—124no—
256C2×C4—124no—
257C2×C4—124no—
258C2×C4—122no—
259C2×C4—122no—
260C2×C4—124no—
261C2×C4#88412yes—
262C2×C4#93711yescharacteristic
263C2×C4#94011yescharacteristic
264C8—42no—
265C8—81no—
266C2×C3—22no—
267C2×C3—22no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3#200111yescharacteristic
271C2×C3#201011yescharacteristic
272C2×C3#202211yescharacteristic
273C22—22no—
274C22—21no—
275C22—22no—
276C22—21no—
277C22—41no—
278C22—41no—
279C22—42no—
280C22#473312yes—
281C22#473912yes—
282C22#496011yescharacteristic
283C22#511611yescharacteristic
284C22#513711yescharacteristic
285C4—21no—
286C4—22no—
287C4—42no—
288C4—42no—
289C4—244no—
290C4—242no—
291C4—242no—
292C4#505312yes—
293C3#1032511yescharacteristic, complemented, Sylow
294C2—22no—
295C2—21no—
296C2—22no—
297C2—21no—
298C2#6953911yescharacteristic
299C2#6996411yescharacteristic
300C2#6999311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1817C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#1817D3
hypocenterC3#10325
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1817

Complements

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

NH ≅ G/Ntype
80. C22×C3×C4Q16semidirect
188. C22×C4Dic12semidirect
293. C3#10325semidirect

Maximal subgroups

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

subgroupindex
#820752
#786172
C3×#18172
#699462
#676762
#103253