Finite Group Databasebeta

← #10690127

Subgroups — #10690127

Order 768.

578 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 119 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
1C2×C3×#497C211yesmaximal, characteristic
2#82957C211yesmaximal, characteristic
3#71788C212yesmaximal, complemented
4C2×#4540C211yesmaximal, characteristic
5#67659C212yesmaximal, complemented
6#9003—31nomaximal, Sylow
7C22×C3×Q16—21no—
8C22×C3×SD16—21no—
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#497—22no—
12C3×#497C414yescomplemented
13C2×C3×#97C2211yescharacteristic
14#5229—21no—
15#5220—21no—
16#4951—22no—
17#4950—22no—
18#4936C2212yes—
19C2×#803C2212yes—
20#4540—22no—
21#4529—22no—
22#4529—22no—
23C2×#497D311yescharacteristic
24#2379—31no—
25#1993—32no—
26C2×#329—31no—
27#1295—32no—
28C22×C3×Q8—21no—
29C22×C3×Q8D411yescharacteristic
30C22×C3×D4D411yescharacteristic
31C23×Dic3—21no—
32C2×C3×Q16—24no—
33C2×C3×Q16—42no—
34C2×C3×Q16—42no—
35C2×C3×SD16—24no—
36C2×C3×SD16—42no—
37C2×C3×SD16—42no—
38C22×C3×C8—21no—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—22no—
41C3×C4⋊Q8C2×C412yescomplemented
42C3×#124—22no—
43C3×#124C2×C412yes—
44C3×#124D412yes—
45C2×C3×C4⋊C4—22no—
46C2×C3×C22⋊C4—22no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—22no—
49C2×#217—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3D412yes—
52C2×C4⋊Dic3D412yes—
53C2×C4×Dic3—21no—
54C22×C3⋊C8—22no—
55C3×#97—21no—
56C3×#97C2×C412yescomplemented
57C3⋊#95—22no—
58C3⋊#95—22no—
59C3⋊#95—22no—
60#803—22no—
61#803—22no—
62C22×Q16—21no—
63C22×SD16—21no—
64C2×C4⋊Q8C2×D311yescharacteristic
65C2×#124C2×D311yescharacteristic
66#497—22no—
67#497Dic314yescomplemented
68C2×#97—32no—
69C2×#97C2×D311yescharacteristic
70#391—61no—
71#388—62no—
72#386—61no—
73#385—62no—
74#382—32no—
75#340—62no—
76#340—62no—
77#329—62no—
78C24×C3—21no—
79C2×C3×Q8—21no—
80C2×C3×Q8—22no—
81C2×C3×Q8—41no—
82C2×C3×Q8—41no—
83C2×C3×Q8—41no—
84C2×C3×Q8—41no—
85C2×C3×Q8—41no—
86C2×C3×Q8—41no—
87C2×C3×Q8—41no—
88C2×C3×Q8C22⋊C412yes—
89C2×C3×D4—21no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C2×C3×D4C22⋊C412yes—
93C22×C3×C4—21no—
94C22×C3×C4—41no—
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—22no—
98C22×Dic3—22no—
99C22×Dic3—81no—
100C22×Dic3—82no—
101C3×Q16—48no—
102C3×SD16—48no—
103C2×C3×C8—22no—
104C2×C3×C8—22no—
105C2×C3×C8—22no—
106C3×C4⋊C4—24no—
107C3×C4⋊C4—42no—
108C3×C22⋊C4—24no—
109C3×C22⋊C4—24no—
110C3×C42C22×C411yescharacteristic
111C3×C42C2×D411yescharacteristic
112C3×C42#4112yes—
113#217—81no—
114#217—82no—
115C4⋊Dic3—42no—
116C4⋊Dic3—42no—
117C4⋊Dic3—81no—
118C4×Dic3—81no—
119C2×C3⋊C8—24no—
120C2×C3⋊C8—42no—
121C2×C3⋊C8—42no—
122C22×Q8—21no—
123C22×Q8#6711yescharacteristic
124C22×D4#6711yescharacteristic
125C23×C4—61no—
126C2×Q16—24no—
127C2×Q16—42no—
128C2×Q16—42no—
129C2×SD16—24no—
130C2×SD16—42no—
131C2×SD16—42no—
132C22×C8—21no—
133C22×C8—62no—
134C4⋊Q8—22no—
135C4⋊Q8—21no—
136C4⋊Q8C2×Dic312yescomplemented
137#124—22no—
138#124C2×Dic312yes—
139#124#6712yes—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—32no—
142C2×C4⋊C4—32no—
143C2×C4⋊C4—61no—
144C2×C22⋊C4—22no—
145C2×C22⋊C4—62no—
146C2×C22⋊C4—61no—
147C2×C42—61no—
148C2×C42C22×D311yescharacteristic
149#97—21no—
150#97—62no—
151#97—62no—
152#97C2×Dic312yescomplemented
153#95—62no—
154#95—62no—
155#95—62no—
156C23×C3—22no—
157C23×C3—41no—
158C23×C3—41no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3#12011yescharacteristic
162C3×Q8—42no—
163C3×Q8—42no—
164C3×Q8—42no—
165C3×Q8—81no—
166C3×Q8—81no—
167C3×Q8—81no—
168C3×D4—42no—
169C3×D4—81no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—81no—
179C2×C3×C4—81no—
180C2×C3×C4C2×C22⋊C411yescharacteristic
181C2×C3×C4C4×D412yes—
182C2×C3×C4#12112yes—
183C2×C3×C4#12311yescharacteristic
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187C2×Dic3—42no—
188C2×Dic3—42no—
189C2×Dic3—42no—
190C2×Dic3—84no—
191C3×C8—24no—
192C3⋊C8—44no—
193C24—21no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—41no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8#21712yes—
204C2×D4—21no—
205C2×D4—41no—
206C2×D4—41no—
207C2×D4#21712yes—
208C22×C4—21no—
209C22×C4—41no—
210C22×C4—62no—
211C22×C4—62no—
212C22×C4—242no—
213C22×C4—241no—
214C22×C4D3×D412yes—
215C22×C4C2×#6711yescharacteristic
216Q16—48no—
217SD16—48no—
218C2×C8—22no—
219C2×C8—22no—
220C2×C8—22no—
221C2×C8—64no—
222C2×C8—122no—
223C2×C8—122no—
224C4⋊C4—24no—
225C4⋊C4—42no—
226C4⋊C4—122no—
227C4⋊C4—122no—
228C4⋊C4—241no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—242no—
232C22⋊C4—241no—
233C42—241no—
234C42#23712yes—
235C42C22×Dic311yescharacteristic
236C42C2×#6711yescharacteristic
237C22×C3—44no—
238C22×C3—42no—
239C22×C3—81no—
240C22×C3—81no—
241C22×C3—82no—
242C22×C3#38611yescharacteristic
243C22×C3#45311yescharacteristic
244C22×C3#45311yescharacteristic
245C22×C3#45512yes—
246C22×C3#45612yes—
247C3×C4—22no—
248C3×C4—22no—
249C3×C4—21no—
250C3×C4—21no—
251C3×C4—42no—
252C3×C4—82no—
253Dic3—82no—
254Dic3—82no—
255C23—22no—
256C23—42no—
257C23—42no—
258C23—41no—
259C23—41no—
260C23#93711yescharacteristic
261Q8—42no—
262Q8—42no—
263Q8—42no—
264Q8—81no—
265Q8—81no—
266Q8—81no—
267D4—42no—
268D4—81no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—22no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—81no—
278C2×C4—81no—
279C2×C4—122no—
280C2×C4—122no—
281C2×C4—122no—
282C2×C4—122no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—244no—
286C2×C4D4×Dic312yes—
287C2×C4#93511yescharacteristic
288C2×C4#93912yes—
289C2×C4C2×#21711yescharacteristic
290C8—24no—
291C8—124no—
292C2×C3—82no—
293C2×C3#165911yescharacteristic
294C2×C3#166011yescharacteristic
295C2×C3#166112yes—
296C2×C3#188212yes—
297C2×C3#188711yescharacteristic
298C22—42no—
299C22—44no—
300C22—81no—
301C22—81no—
302C22—82no—
303C22#507711yescharacteristic
304C22#507711yescharacteristic
305C22#520212yes—
306C22#520512yes—
307C22#522011yescharacteristic
308C4—21no—
309C4—21no—
310C4—22no—
311C4—22no—
312C4—42no—
313C4—82no—
314C4—242no—
315C4—242no—
316C3#900311yescharacteristic, complemented, Sylow
317C2—82no—
318C2#7132911yescharacteristic
319C2#7133011yescharacteristic
320C2#7133112yes—
321C2#7176311yescharacteristic
322C2#7210012yes—
subgroupG/H
centerC23#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#497C2
socleC23×C3#120
prime radicalC22×C3×D4D4
cosocleC2×C3×C42C23
hypercenterC2×#497D3
hypocenterC3#9003
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#497

Complements

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

NH ≅ G/Ntype
3. #71788C2semidirect
5. #67659C2semidirect
12. C3×#497C4semidirect
41. C3×C4⋊Q8C2×C4semidirect
56. C3×#97C2×C4semidirect
67. #497Dic3semidirect
136. C4⋊Q8C2×Dic3semidirect
152. #97C2×Dic3semidirect
316. C3#9003semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4972
#829572
#717882
C2×#45402
#676592
#90033