Finite Group Databasebeta

← #10690713

Subgroups — #10690713

Order 768.

602 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 602 conjugacy classes of subgroups — too many to draw. Only its 127 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×#2328C211yesmaximal, characteristic
2#82304C211yesmaximal, characteristic, complemented
3#71601C211yesmaximal, characteristic, complemented
4#71597C212yesmaximal, complemented
5#67614C212yesmaximal, complemented
6#9105—31nomaximal, Sylow
7C2×C3×#127C411yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C2×C3×C4×D4C2211yescharacteristic
10C3×#390C411yescharacteristic
11C3×#390C411yescharacteristic
12C3×#386—22no—
13C3×#386—22no—
14C3×#382C411yescharacteristic
15C3×#376C2211yescharacteristic
16#5207—22no—
17#5012—24no—
18#4925C2212yes—
19#4925C2212yes—
20D4⋊(C3⋊C8)—24no—
21#4525—22no—
22#2975—31no—
23#2328D311yescharacteristic
24#1478—32no—
25#1460—31no—
26#1274—32no—
27C22×C3×D4—22no—
28C22×C3×D4C2×C411yescharacteristic
29C22×C3×D4C2×C411yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4D412yes—
32C3×#127—22no—
33C3×#127—22no—
34C3×C4×D4—22no—
35C3×C4×D4—22no—
36C3×C4×D4—22no—
37C3×C4×D4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C411yescharacteristic
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4D412yes—
44C2×C3×C42C2×C411yescharacteristic
45C2×C3×C42C2×C411yescharacteristic
46C2×C3×C42C2311yescharacteristic
47C2×#208—24no—
48C3×#95—22no—
49C3×#95—21no—
50C3×#95—21no—
51#830—24no—
52#830—22no—
53#830—22no—
54C4⋊(C3⋊C8)—22no—
55C4⋊(C3⋊C8)—22no—
56C4×C3⋊C8—22no—
57C4×C3⋊C8D414yescomplemented
58C2×#127Dic311yescharacteristic
59C2×C4×D4C2×D311yescharacteristic
60C2×C4×D4C2×D311yescharacteristic
61#436—64no—
62#431—32no—
63#431—32no—
64#407—62no—
65#390Dic311yescharacteristic
66#390Dic311yescharacteristic
67#386—22no—
68#386—22no—
69#382Dic311yescharacteristic
70#376C2×D311yescharacteristic
71D4⋊C8—64no—
72#323—62no—
73C24×C3—21no—
74C24×C3C22⋊C412yes—
75C2×C3×D4—28no—
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—21no—
79C2×C3×D4—21no—
80C2×C3×D4—21no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—82no—
84C2×C3×D4—82no—
85C2×C3×D4M1612yescomplemented
86C22×C3×C4—21no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—42no—
92C22×C3×C4—82no—
93C22×C3×C4—82no—
94C22×C3×C4C22⋊C412yes—
95C22×C3×C4C22×C411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×C3×C4C2×D411yescharacteristic
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C22⋊C4—24no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—41no—
103C3×C22⋊C4—82no—
104C3×C22⋊C4—82no—
105C3×C42—21no—
106C3×C42—22no—
107C3×C42M1612yescomplemented
108C3×C42C2×D412yes—
109C3×C42#4112yes—
110#208—84no—
111C2×C3⋊C8—24no—
112C2×C3⋊C8—42no—
113C22×D4—22no—
114C22×D4C2×Dic311yescharacteristic
115C22×D4C2×Dic311yescharacteristic
116C23×C4—21no—
117C23×C4#6712yes—
118C2×M16—64no—
119#127—22no—
120#127—22no—
121C4×D4—22no—
122C4×D4—22no—
123C4×D4—22no—
124C4×D4—22no—
125C2×C4⋊C4C2×Dic311yescharacteristic
126C2×C4⋊C4C2×Dic311yescharacteristic
127C2×C22⋊C4—21no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—22no—
130C2×C22⋊C4#6712yes—
131C2×C42C2×Dic311yescharacteristic
132C2×C42C2×Dic311yescharacteristic
133C2×C42C22×D311yescharacteristic
134C4⋊C8—62no—
135C4⋊C8—62no—
136C22⋊C8—64no—
137C22⋊C8—62no—
138C22⋊C8—62no—
139C4×C8—34no—
140C4×C8—62no—
141#95—22no—
142#95—21no—
143#95—21no—
144C23×C3—22no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—42no—
149C23×C3—42no—
150C23×C3—42no—
151C23×C3—42no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3C2×C22⋊C411yescharacteristic
155C3×D4—24no—
156C3×D4—42no—
157C3×D4—42no—
158C3×D4—42no—
159C3×D4—88no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—44no—
165C2×C3×C4—41no—
166C2×C3×C4—41no—
167C2×C3×C4—42no—
168C2×C3×C4—41no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—41no—
173C2×C3×C4—82no—
174C2×C3×C4—84no—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12011yescharacteristic
177C2×C3×C4#12112yes—
178C2×C3×C4#12311yescharacteristic
179C2×C3×C4C2×M1611yescharacteristic
180C2×C3×C4C2×M1611yescharacteristic
181C2×C3×C4#13112yes—
182C3⋊C8—44no—
183C3⋊C8—82no—
184C24—21no—
185C24#21712yes—
186C2×D4—28no—
187C2×D4—21no—
188C2×D4—21no—
189C2×D4—21no—
190C2×D4—22no—
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—22no—
194C2×D4—82no—
195C2×D4—82no—
196C2×D4#20812yescomplemented
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4#21712yes—
206C22×C4C22×Dic311yescharacteristic
207C22×C4C2×#6711yescharacteristic
208C22×C4C2×#6711yescharacteristic
209M16—244no—
210C2×C8—64no—
211C2×C8—122no—
212C4⋊C4—22no—
213C4⋊C4—22no—
214C22⋊C4—24no—
215C22⋊C4—41no—
216C22⋊C4—41no—
217C22⋊C4—82no—
218C22⋊C4—82no—
219C42—21no—
220C42—22no—
221C42#20812yescomplemented
222C42D3×D412yes—
223C42#23712yes—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—44no—
227C22×C3—44no—
228C22×C3—42no—
229C22×C3—42no—
230C22×C3—42no—
231C22×C3—42no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3#38611yescharacteristic
237C22×C3#40711yescharacteristic
238C22×C3#40811yescharacteristic
239C3×C4—22no—
240C3×C4—21no—
241C3×C4—21no—
242C3×C4—42no—
243C3×C4—42no—
244C3×C4—81no—
245C3×C4—81no—
246C3×C4#43512yes—
247C3×C4#43612yes—
248C23—22no—
249C23—22no—
250C23—22no—
251C23—22no—
252C23—42no—
253C23—42no—
254C23—42no—
255C23—42no—
256C23—41no—
257C23—41no—
258C23C2×#21711yescharacteristic
259D4—24no—
260D4—42no—
261D4—42no—
262D4—42no—
263D4—88no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—44no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—42no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—82no—
278C2×C4—84no—
279C2×C4C2×#20811yescharacteristic
280C2×C4C2×#20811yescharacteristic
281C2×C4D4×Dic312yes—
282C2×C4#93511yescharacteristic
283C2×C4#93711yescharacteristic
284C2×C4#93912yes—
285C2×C4#94812yes—
286C8—124no—
287C8—242no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—82no—
291C2×C3—82no—
292C2×C3#187011yescharacteristic
293C2×C3#191011yescharacteristic
294C2×C3#191111yescharacteristic
295C22—21no—
296C22—21no—
297C22—44no—
298C22—44no—
299C22—42no—
300C22—42no—
301C22—42no—
302C22—42no—
303C22—82no—
304C22—82no—
305C22—81no—
306C22—81no—
307C22#520711yescharacteristic
308C22#522011yescharacteristic
309C22#523411yescharacteristic
310C4—21no—
311C4—21no—
312C4—22no—
313C4—42no—
314C4—42no—
315C4—81no—
316C4—81no—
317C4#501112yes—
318C4#501212yes—
319C3#910511yescharacteristic, complemented, Sylow
320C2—21no—
321C2—21no—
322C2—82no—
323C2—82no—
324C2#7161111yescharacteristic
325C2#7185311yescharacteristic
326C2#7185411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2328C2
socleC22×C3#386
prime radicalC2×C3×#127C4
cosocleC2×C3×C42C23
hypercenter#2328D3
hypocenterC3#9105
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2328

Complements

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

NH ≅ G/Ntype
2. #82304C2semidirect
3. #71601C2semidirect
4. #71597C2semidirect
5. #67614C2semidirect
57. C4×C3⋊C8D4semidirect
85. C2×C3×D4M16semidirect
107. C3×C42M16semidirect
196. C2×D4#208semidirect
221. C42#208semidirect
319. C3#9105semidirect

Maximal subgroups

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

subgroupindex
C3×#23282
#823042
#716012
#715972
#676142
#91053