Finite Group Databasebeta

← #10714854

Subgroups — #10714854

Order 768.

540 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 81 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×#2683C211yesmaximal, characteristic
2#69642C211yesmaximal, characteristic
3#69597C212yesmaximal
4#67641C211yesmaximal, characteristic
5#67637C212yesmaximal
6#13461—31nomaximal, Sylow
7C2×C3×C22⋊Q8—22no—
8#5798C2211yescharacteristic
9C3×#396—22no—
10C3×#395—21no—
11C3×#390C2211yescharacteristic
12C3×#389—21no—
13C3×#386C2212yes—
14C2×#834C2212yes—
15C2×#833C2211yescharacteristic
16#5210—22no—
17#4964—21no—
18#4527—22no—
19#4523—21no—
20#2683D311yescharacteristic
21#1867—32no—
22#1856—31no—
23#1393—31no—
24#1390—32no—
25C22×C3×Q8—21no—
26C22×C3×D4C2311yescharacteristic
27C23×C3×C4D412yes—
28C3×C22⋊Q8—48no—
29C2×C3×C4⋊C4—21no—
30C2×C3×C4⋊C4—22no—
31C2×C3×C4⋊C4—22no—
32C2×C3×C4⋊C4—21no—
33C2×C3×C4⋊C4D411yescharacteristic
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4D412yes—
37C2×C3×C42D411yescharacteristic
38C2×#217—22no—
39C2×#217—22no—
40D4×Dic3—41no—
41D4×Dic3—41no—
42#926—21no—
43C2×Dic3⋊C4—42no—
44C2×#208—21no—
45C3×#95—21no—
46C3×#95—22no—
47#834—22no—
48#834Q814yescomplemented
49#833—21no—
50#833Q812yescomplemented
51C3⋊#95—42no—
52#808—41no—
53#807—41no—
54C2×C22⋊Q8—22no—
55#518—31no—
56#428—61no—
57C2×#100—31no—
58C2×#99—32no—
59#396—22no—
60#395—21no—
61#390C2×D311yescharacteristic
62#389—21no—
63#386C2×D312yes—
64#385—62no—
65#342—62no—
66#339—61no—
67C24×C3C2×D412yes—
68C2×C3×Q8—44no—
69C2×C3×Q8—81no—
70C2×C3×D4—41no—
71C2×C3×D4—41no—
72C2×C3×D4C2×Q811yescharacteristic
73C2×C3×D4C2×Q812yes—
74C2×C3×D4#4111yescharacteristic
75C22×C3×C4—22no—
76C22×C3×C4—21no—
77C22×C3×C4—22no—
78C22×C3×C4—21no—
79C22×C3×C4—44no—
80C22×C3×C4—42no—
81C22×C3×C4C2×D411yescharacteristic
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—42no—
85C3×C4⋊C4—24no—
86C3×C4⋊C4—41no—
87C3×C4⋊C4—44no—
88C3×C4⋊C4—44no—
89C3×C4⋊C4—81no—
90C3×C22⋊C4—44no—
91C3×C22⋊C4—44no—
92C3×C22⋊C4—42no—
93C3×C42—41no—
94#217—24no—
95#217—42no—
96#217—42no—
97#217—41no—
98C4⋊Dic3—21no—
99C4⋊Dic3—21no—
100Dic3⋊C4—84no—
101C4×Dic3—41no—
102#208—22no—
103#208—41no—
104C2×C3⋊C8—41no—
105C22×Q8—21no—
106C22×D4C22×D311yescharacteristic
107C23×C4#6712yes—
108C2×M16—61no—
109C22⋊Q8—48no—
110C4×D4—121no—
111C4×D4—121no—
112#117—61no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—22no—
115C2×C4⋊C4—22no—
116C2×C4⋊C4—21no—
117C2×C4⋊C4—122no—
118C2×C4⋊C4#6711yescharacteristic
119C2×C22⋊C4—22no—
120C2×C22⋊C4—22no—
121C2×C22⋊C4—62no—
122C2×C22⋊C4—62no—
123C2×C22⋊C4#6712yes—
124C2×C42#6711yescharacteristic
125#107—121no—
126#106—121no—
127#100—32no—
128#100—61no—
129#99—34no—
130#99—62no—
131#95—21no—
132#95—22no—
133#95—122no—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—42no—
137C23×C3—42no—
138C23×C3#12011yescharacteristic
139C23×C3C22⋊Q812yes—
140C23×C3C4⋊Q812yes—
141C3×Q8—84no—
142C3×D4—41no—
143C3×D4—42no—
144C3×D4—41no—
145C2×C3×C4—21no—
146C2×C3×C4—21no—
147C2×C3×C4—22no—
148C2×C3×C4—24no—
149C2×C3×C4—42no—
150C2×C3×C4—41no—
151C2×C3×C4—42no—
152C2×C3×C4—41no—
153C2×C3×C4—44no—
154C2×C3×C4—42no—
155C2×C3×C4—42no—
156C2×C3×C4—42no—
157C2×C3×C4—41no—
158C2×C3×C4—42no—
159C2×C3×C4—82no—
160C2×C3×C4C22⋊Q811yescharacteristic
161C2×C3×C4C4⋊Q811yescharacteristic
162C2×Dic3—44no—
163C2×Dic3—41no—
164C2×Dic3—41no—
165C2×Dic3—82no—
166C2×Dic3—82no—
167C2×Dic3—82no—
168C2×Dic3—82no—
169C3⋊C8—42no—
170C24C2×#6712yes—
171C2×Q8—44no—
172C2×Q8—81no—
173C2×D4—41no—
174C2×D4—41no—
175C2×D4D3×Q812yes—
176C2×D4D3×Q811yescharacteristic
177C2×D4#23911yescharacteristic
178C22×C4—22no—
179C22×C4—21no—
180C22×C4—22no—
181C22×C4—21no—
182C22×C4—44no—
183C22×C4—42no—
184C22×C4—62no—
185C22×C4—62no—
186C22×C4—122no—
187C22×C4C2×#6711yescharacteristic
188M16—62no—
189M16—121no—
190C2×C8—121no—
191C4⋊C4—24no—
192C4⋊C4—41no—
193C4⋊C4—44no—
194C4⋊C4—44no—
195C4⋊C4—61no—
196C4⋊C4—61no—
197C4⋊C4—81no—
198C4⋊C4—244no—
199C22⋊C4—44no—
200C22⋊C4—44no—
201C22⋊C4—42no—
202C22⋊C4—64no—
203C22⋊C4—122no—
204C22⋊C4—122no—
205C22⋊C4—121no—
206C42—41no—
207C42—121no—
208C22×C3—21no—
209C22×C3—21no—
210C22×C3—22no—
211C22×C3—22no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—42no—
216C22×C3—82no—
217C22×C3#39511yescharacteristic
218C22×C3B(4,𝔽2)11yescharacteristic
219C22×C3#45811yescharacteristic
220C3×C4—21no—
221C3×C4—21no—
222C3×C4—42no—
223C3×C4—44no—
224C3×C4—82no—
225C3×C4—81no—
226Dic3—84no—
227Dic3—81no—
228Dic3—81no—
229C23—22no—
230C23—22no—
231C23—42no—
232C23—42no—
233C23#94412yes—
234C23#94612yes—
235C23#95311yescharacteristic
236Q8—84no—
237D4—41no—
238D4—42no—
239D4—41no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—24no—
244C2×C4—42no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—41no—
254C2×C4—82no—
255C2×C4—124no—
256C2×C4—121no—
257C2×C4—121no—
258C2×C4—242no—
259C2×C4—242no—
260C2×C4—242no—
261C2×C4—242no—
262C2×C4#94411yescharacteristic
263C2×C4#94611yescharacteristic
264C8—122no—
265C2×C3—21no—
266C2×C3—21no—
267C2×C3—42no—
268C2×C3—42no—
269C2×C3#206211yescharacteristic
270C2×C3#219611yescharacteristic
271C2×C3#220111yescharacteristic
272C22—21no—
273C22—21no—
274C22—22no—
275C22—22no—
276C22—42no—
277C22—42no—
278C22—42no—
279C22—42no—
280C22—82no—
281C22#523011yescharacteristic
282C22#524311yescharacteristic
283C22#524411yescharacteristic
284C4—21no—
285C4—21no—
286C4—42no—
287C4—44no—
288C4—82no—
289C4—81no—
290C4—244no—
291C4—241no—
292C4—241no—
293C3#1346111yescharacteristic, complemented, Sylow
294C2—21no—
295C2—21no—
296C2—42no—
297C2—42no—
298C2#7189511yescharacteristic
299C2#7190911yescharacteristic
300C2#7195811yescharacteristic
subgroupG/H
centerC22#5244
commutator, prime radical, cosocleC22×C3×D4C23
FrattiniC22×D4C22×D3
FittingC3×#2683C2
socleC22×C3B(4,𝔽2)
hypercenter#2683D3
hypocenterC3#13461
LCSG ▷ C22×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×D4 ◁ #2683

Complements

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

NH ≅ G/Ntype
48. #834Q8semidirect
50. #833Q8semidirect
293. C3#13461semidirect

Maximal subgroups

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

subgroupindex
C3×#26832
#696422
#695972
#676412
#676372
#134613