Finite Group Databasebeta

← #10692613

Subgroups — #10692613

Order 768.

706 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 706 conjugacy classes of subgroups — too many to draw. Only its 123 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×#2541C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3#71759C212yesmaximal, complemented
4#71617C211yesmaximal, characteristic, complemented
5#67620C212yesmaximal, complemented
6#9405—31nomaximal, Sylow
7C2×C3×#127—21no—
8C2×C3×#127C2211yescharacteristic
9C2×C3×#121—22no—
10C2×C3×C4×D4C2211yescharacteristic
11C3×#392C414yes—
12C3×#390—21no—
13C3×C4×C4⋊C4C2211yescharacteristic
14C2×#833—21no—
15#5207—22no—
16#5012—24no—
17#4925C2212yes—
18#4924C2212yes—
19#4544—24no—
20#4530—21no—
21#2975—31no—
22#2541D311yescharacteristic
23#1800—31no—
24#1545—32no—
25#1303—32no—
26C22×C3×D4—24no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C22×C3×D4D411yescharacteristic
31C23×C3×C4D412yes—
32C3×#127—22no—
33C3×#127—22no—
34C3×#127—44no—
35C3×#121—48no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4D411yescharacteristic
40C2×C3×C22⋊C4—24no—
41C2×C3×C22⋊C4C2×C412yes—
42C2×C3×C42—21no—
43C2×C3×C42C2311yescharacteristic
44C2×#208—24no—
45C2×#208—22no—
46C3×#95C2×C412yes—
47#833—42no—
48#830—24no—
49#830—22no—
50C4⋊(C3⋊C8)—22no—
51C4⋊(C3⋊C8)—22no—
52C4⋊(C3⋊C8)D414yescomplemented
53C4×C3⋊C8—22no—
54C2×#127—21no—
55C2×#127C2×D311yescharacteristic
56C2×#121—22no—
57C2×C4×D4C2×D311yescharacteristic
58#436—64no—
59#431—32no—
60#423—32no—
61C2×#100—61no—
62#407—62no—
63#392Dic314yes—
64#390—21no—
65C4×C4⋊C4C2×D311yescharacteristic
66#343—61no—
67#331—64no—
68C24×C3—22no—
69C24×C3C22⋊C412yes—
70C2×C3×D4—24no—
71C2×C3×D4—22no—
72C2×C3×D4—21no—
73C2×C3×D4—22no—
74C2×C3×D4—21no—
75C2×C3×D4—44no—
76C2×C3×D4—48no—
77C2×C3×D4—48no—
78C2×C3×D4—41no—
79C2×C3×D4—42no—
80C2×C3×D4—41no—
81C2×C3×D4—42no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
84C2×C3×D4—84no—
85C2×C3×D4—82no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—44no—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C3×C4⋊C4—24no—
95C3×C4⋊C4—22no—
96C3×C22⋊C4—24no—
97C3×C22⋊C4—48no—
98C3×C42—42no—
99C3×C42C2×D412yes—
100C3×C42#4112yes—
101#208—42no—
102#208—42no—
103#208—84no—
104C2×C3⋊C8—24no—
105C2×C3⋊C8—42no—
106C22×D4—24no—
107C22×D4—21no—
108C22×D4—21no—
109C22×D4C2×Dic312yes—
110C22×D4#6711yescharacteristic
111C23×C4#6712yes—
112C2×M16—64no—
113C2×M16—62no—
114#127—22no—
115#127—22no—
116#127—44no—
117#121—48no—
118C4×D4—22no—
119C4×D4—22no—
120C2×C4⋊C4—21no—
121C2×C4⋊C4#6711yescharacteristic
122C2×C22⋊C4—24no—
123C2×C22⋊C4C2×Dic312yes—
124C2×C42—21no—
125C2×C42C22×D311yescharacteristic
126C4⋊C8—34no—
127C4⋊C8—62no—
128C4⋊C8—62no—
129#100—122no—
130C22⋊C8—64no—
131C22⋊C8—62no—
132C4×C8—62no—
133#95C2×Dic312yes—
134C23×C3—22no—
135C23×C3—22no—
136C23×C3—22no—
137C23×C3—42no—
138C23×C3—42no—
139C23×C3—44no—
140C23×C3—44no—
141C23×C3—42no—
142C23×C3—42no—
143C23×C3—42no—
144C23×C3C2×C22⋊C411yescharacteristic
145C3×D4—42no—
146C3×D4—42no—
147C3×D4—84no—
148C3×D4—82no—
149C3×D4—816no—
150C3×D4—82no—
151C3×D4—82no—
152C3×D4—82no—
153C2×C3×C4—22no—
154C2×C3×C4—24no—
155C2×C3×C4—22no—
156C2×C3×C4—41no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—44no—
160C2×C3×C4—42no—
161C2×C3×C4—82no—
162C2×C3×C4C4×D412yes—
163C2×C3×C4#12011yescharacteristic
164C2×C3×C4#12112yes—
165C2×C3×C4#12311yescharacteristic
166C2×C3×C4AGL(1,ℤ/8)12yes—
167C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
168C2×C3×C4#13711yescharacteristic
169C3⋊C8—44no—
170C3⋊C8—82no—
171C24—22no—
172C24#21712yes—
173C2×D4—22no—
174C2×D4—24no—
175C2×D4—22no—
176C2×D4—21no—
177C2×D4—21no—
178C2×D4—44no—
179C2×D4—48no—
180C2×D4—48no—
181C2×D4—42no—
182C2×D4—41no—
183C2×D4—41no—
184C2×D4—42no—
185C2×D4—41no—
186C2×D4—41no—
187C2×D4—84no—
188C2×D4—82no—
189C22×C4—22no—
190C22×C4—22no—
191C22×C4—22no—
192C22×C4—44no—
193C22×C4#21712yes—
194C22×C4C22×Dic311yescharacteristic
195C22×C4C2×#6711yescharacteristic
196C22×C4C2×#6711yescharacteristic
197M16—122no—
198M16—122no—
199M16—244no—
200C2×C8—64no—
201C2×C8—122no—
202C4⋊C4—24no—
203C4⋊C4—22no—
204C22⋊C4—24no—
205C22⋊C4—48no—
206C42—42no—
207C42D3×D412yes—
208C42#23712yes—
209C22×C3—21no—
210C22×C3—21no—
211C22×C3—44no—
212C22×C3—42no—
213C22×C3—42no—
214C22×C3—42no—
215C22×C3—82no—
216C22×C3—84no—
217C22×C3—84no—
218C22×C3—82no—
219C22×C3—82no—
220C22×C3—82no—
221C22×C3#38611yescharacteristic
222C22×C3#41711yescharacteristic
223C22×C3#41811yescharacteristic
224C3×C4—21no—
225C3×C4—22no—
226C3×C4—44no—
227C3×C4—42no—
228C3×C4#45312yes—
229C3×C4#46812yes—
230C3×C4#46912yes—
231C23—22no—
232C23—22no—
233C23—22no—
234C23—42no—
235C23—42no—
236C23—44no—
237C23—44no—
238C23—42no—
239C23—42no—
240C23—42no—
241C23C2×#21711yescharacteristic
242D4—42no—
243D4—42no—
244D4—84no—
245D4—82no—
246D4—816no—
247D4—82no—
248D4—82no—
249D4—82no—
250C2×C4—22no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—41no—
254C2×C4—42no—
255C2×C4—42no—
256C2×C4—44no—
257C2×C4—42no—
258C2×C4—82no—
259C2×C4#93212yes—
260C2×C4D4×Dic312yes—
261C2×C4#93511yescharacteristic
262C2×C4#93711yescharacteristic
263C2×C4#93912yes—
264C2×C4#94911yescharacteristic
265C2×C4#95111yescharacteristic
266C8—124no—
267C8—242no—
268C2×C3—21no—
269C2×C3—21no—
270C2×C3—82no—
271C2×C3—162no—
272C2×C3#188711yescharacteristic
273C2×C3#189511yescharacteristic
274C2×C3#191411yescharacteristic
275C22—21no—
276C22—21no—
277C22—44no—
278C22—42no—
279C22—42no—
280C22—42no—
281C22—82no—
282C22—84no—
283C22—84no—
284C22—82no—
285C22—82no—
286C22—82no—
287C22#521511yescharacteristic
288C22#522011yescharacteristic
289C22#523211yescharacteristic
290C4—21no—
291C4—22no—
292C4—44no—
293C4—42no—
294C4#503912yes—
295C4#504212yes—
296C4#507712yes—
297C3#940511yescharacteristic, complemented, Sylow
298C2—21no—
299C2—21no—
300C2—82no—
301C2—162no—
302C2#7168211yescharacteristic
303C2#7176311yescharacteristic
304C2#7185511yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#2541C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#2541D3
hypocenterC3#9405
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2541

Complements

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

NH ≅ G/Ntype
2. #82304C2semidirect
3. #71759C2semidirect
4. #71617C2semidirect
5. #67620C2semidirect
52. C4⋊(C3⋊C8)D4semidirect
297. C3#9405semidirect

Maximal subgroups

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

subgroupindex
C3×#25412
#823042
#717592
#716172
#676202
#94053