Finite Group Databasebeta

← #10698071

Subgroups — #10698071

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
1C3×#3189C211yesmaximal, characteristic
2#82810C212yesmaximal
3#71690C212yesmaximal
4#69910C211yesmaximal, characteristic
5#67676C211yesmaximal, characteristic
6#10325—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2212yes—
8C22×C4×Dic3C2211yescharacteristic
9C3×#484—22no—
10C3×#467—22no—
11C3×#451—22no—
12C2×C3×C22⋊C8C2211yescharacteristic
13#5229—22no—
14#5212C2212yes—
15C2×#830C2211yescharacteristic
16#4978—24no—
17#4972—22no—
18#4946—22no—
19#4943—22no—
20#4550—22no—
21#4538—24no—
22#3189D311yescharacteristic
23#2855—32no—
24#1817—31no—
25#1520—32no—
26#1289—31no—
27C22×C3×Q8D412yes—
28C23×C3×C4C2311yescharacteristic
29C23×Dic3D411yescharacteristic
30C23×Dic3D411yescharacteristic
31C2×C3×Q16—22no—
32C22×C3×C8D412yes—
33C3×C22⋊Q8—22no—
34C3×C22⋊Q8—22no—
35C3×C22⋊Q8—22no—
36C3×C22⋊Q8—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C22⋊C4—22no—
40C2×#217—22no—
41C2×C4⋊Dic3—22no—
42C2×Dic3⋊C4—24no—
43C2×C4×Dic3—22no—
44C2×C4×Dic3—22no—
45C2×C4×Dic3—21no—
46C2×C4×Dic3—21no—
47C22×C3⋊C8—21no—
48C3×#107—22no—
49C3×#103—24no—
50C3×C22⋊C8—22no—
51C3⋊#95—24no—
52C3⋊#95—24no—
53C3⋊#95—24no—
54C3⋊#95D414yes—
55#830—22no—
56C2×C22⋊Q8C2×D312yes—
57C22×C42—31no—
58#484—22no—
59#467—22no—
60#451—22no—
61C2×C22⋊C8—31no—
62C2×C22⋊C8C2×D311yescharacteristic
63#391—62no—
64#389—62no—
65#387—64no—
66#385—62no—
67#381—62no—
68#380—32no—
69#336—62no—
70#328—64no—
71C24×C3C2×D411yescharacteristic
72C2×C3×Q8—22no—
73C2×C3×Q8—22no—
74C2×C3×Q8—44no—
75C22×C3×C4—22no—
76C22×C3×C4—22no—
77C22×C3×C4—42no—
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D412yes—
80C22×C3×C4C2×D411yescharacteristic
81C22×C3×C4#4111yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—21no—
85C22×Dic3—21no—
86C22×Dic3—21no—
87C22×Dic3—22no—
88C22×Dic3—24no—
89C22×Dic3—42no—
90C22×Dic3Q1612yescomplemented
91C3×Q16—82no—
92C2×C3×C8—22no—
93C2×C3×C8—42no—
94C3×C4⋊C4—22no—
95C3×C4⋊C4—22no—
96C3×C4⋊C4—42no—
97C3×C4⋊C4—42no—
98C3×C22⋊C4—42no—
99C3×C22⋊C4—42no—
100#217—42no—
101#217—42no—
102C4⋊Dic3—44no—
103Dic3⋊C4—48no—
104C4×Dic3—41no—
105C4×Dic3—42no—
106C4×Dic3—41no—
107C2×C3⋊C8—42no—
108C2×C3⋊C8—41no—
109C22×Q8#6712yes—
110C23×C4—31no—
111C23×C4—31no—
112C23×C4C22×D311yescharacteristic
113C2×Q16—22no—
114C22×C8—61no—
115C22×C8#6712yes—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C2×C4⋊C4—22no—
121C2×C4⋊C4—64no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4#6712yes—
124C2×C22⋊C4—22no—
125C2×C22⋊C4—62no—
126C2×C42—62no—
127C2×C42—61no—
128C2×C42—61no—
129C2×C42—62no—
130#107—22no—
131#103—24no—
132C22⋊C8—22no—
133C22⋊C8—62no—
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—
145C3×Q8—44no—
146C3×Q8—82no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—44no—
152C2×C3×C4—44no—
153C2×C3×C4—42no—
154C2×C3×C4—42no—
155C2×C3×C4#12011yescharacteristic
156C2×C3×C4#12112yes—
157C2×C3×C4#12711yescharacteristic
158C2×Dic3—24no—
159C2×Dic3—22no—
160C2×Dic3—24no—
161C2×Dic3—22no—
162C2×Dic3—42no—
163C2×Dic3—41no—
164C2×Dic3—41no—
165C2×Dic3—41no—
166C2×Dic3—42no—
167C2×Dic3—41no—
168C2×Dic3—44no—
169C2×Dic3—44no—
170C2×Dic3—44no—
171C3×C8—42no—
172C3⋊C8—81no—
173C24D3×D411yescharacteristic
174C2×Q8—22no—
175C2×Q8—22no—
176C2×Q8—44no—
177C22×C4—22no—
178C22×C4—22no—
179C22×C4—32no—
180C22×C4—42no—
181C22×C4—62no—
182C22×C4—61no—
183C22×C4—62no—
184C22×C4—62no—
185C22×C4—61no—
186C22×C4—61no—
187C22×C4—64no—
188C22×C4—122no—
189C22×C4D3×D412yes—
190C22×C4#23711yescharacteristic
191C22×C4C2×#6712yes—
192C22×C4C2×#6711yescharacteristic
193Q16—82no—
194C2×C8—22no—
195C2×C8—42no—
196C2×C8—122no—
197C2×C8—121no—
198C4⋊C4—22no—
199C4⋊C4—22no—
200C4⋊C4—42no—
201C4⋊C4—42no—
202C4⋊C4—124no—
203C4⋊C4—128no—
204C22⋊C4—42no—
205C22⋊C4—42no—
206C22⋊C4—122no—
207C22⋊C4—122no—
208C42—122no—
209C42—121no—
210C42—121no—
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—82no—
226C3×C4—82no—
227C3×C4#45812yes—
228Dic3—42no—
229Dic3—42no—
230Dic3—84no—
231C23—41no—
232C23—41no—
233C23D3×Q1611yescharacteristic
234C23#91812yes—
235C23#91911yescharacteristic
236C23#93712yes—
237C23#93811yescharacteristic
238Q8—44no—
239Q8—82no—
240C2×C4—24no—
241C2×C4—22no—
242C2×C4—42no—
243C2×C4—41no—
244C2×C4—44no—
245C2×C4—44no—
246C2×C4—42no—
247C2×C4—42no—
248C2×C4—64no—
249C2×C4—62no—
250C2×C4—64no—
251C2×C4—62no—
252C2×C4—122no—
253C2×C4—121no—
254C2×C4—122no—
255C2×C4—121no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—124no—
259C2×C4—124no—
260C2×C4—124no—
261C2×C4#93912yes—
262C2×C4#94011yescharacteristic
263C2×C4#95311yescharacteristic
264C8—42no—
265C8—241no—
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—42no—
279C22—41no—
280C22#518612yes—
281C22#518712yes—
282C22#518811yescharacteristic
283C22#518911yescharacteristic
284C22#522311yescharacteristic
285C4—21no—
286C4—22no—
287C4—82no—
288C4—82no—
289C4—122no—
290C4—122no—
291C4—244no—
292C4#524312yes—
293C3#1032511yescharacteristic, complemented, Sylow
294C2—22no—
295C2—22no—
296C2—21no—
297C2—21no—
298C2#7184911yescharacteristic
299C2#7207111yescharacteristic
300C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3189C2
socleC22×C3#392
prime radicalC24×C3C2×D4
hypercenter#3189D3
hypocenterC3#10325
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
90. C22×Dic3Q16semidirect
293. C3#10325semidirect

Maximal subgroups

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

subgroupindex
C3×#31892
#828102
#716902
#699102
#676762
#103253