Finite Group Databasebeta

← #10692992

Subgroups — #10692992

Order 768.

818 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2406C211yesmaximal, characteristic
2#82144C211yesmaximal, characteristic, complemented
3#72138C212yesmaximal
4#71552C211yesmaximal, characteristic, complemented
5#67628C212yesmaximal, complemented
6#9476—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×C3×#123—21no—
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#926—21no—
11C3×#395C412yescomplemented
12C3×#394C412yescomplemented
13C3×#386—21no—
14C3×#385—22no—
15C3×#381C2211yescharacteristic
16C2×#837—24no—
17C4×#217—21no—
18C4×#217C2212yes—
19#5207C2212yes—
20Dic3×C4⋊C4—24no—
21#4528—22no—
22#2406D311yescharacteristic
23#2274—31no—
24#1789—32no—
25#1463—31no—
26#1284—32no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8C2×C411yescharacteristic
31C22×C3×D4—21no—
32C22×C3×D4C2×C411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C3×#123—44no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4C2×C412yes—
42C2×C3×C4⋊C4D412yes—
43C2×C3×C22⋊C4—21no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×#217—22no—
47#926—28no—
48C2×C4⋊Dic3—22no—
49C2×C4×Dic3—22no—
50C2×C4×Dic3—22no—
51C2×C4×Dic3D414yes—
52C2×#208—22no—
53C3×#95—22no—
54C3×#95—21no—
55C3×#95C2×C412yes—
56#837—48no—
57C3⋊#95—22no—
58C3⋊#95—22no—
59#830—22no—
60#830—22no—
61C22×#41C2×D311yescharacteristic
62C2×#123—21no—
63C2×#117—61no—
64C22×C4⋊C4C2×D311yescharacteristic
65C2×#104—64no—
66#407—32no—
67#395Dic312yescomplemented
68#394Dic312yescomplemented
69#386—21no—
70#385—22no—
71#381C2×D311yescharacteristic
72C4×C4⋊C4—64no—
73C4×C22⋊C4—32no—
74C4×C22⋊C4—61no—
75#324—62no—
76C24×C3—21no—
77C24×C3C2×D411yescharacteristic
78C3×#41—28no—
79C3×#41—44no—
80C3×#41—44no—
81C3×#41—44no—
82C2×C3×Q8—22no—
83C2×C3×Q8—42no—
84C2×C3×D4—24no—
85C2×C3×D4—22no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—24no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—42no—
102C22×C3×C4C22⋊C412yes—
103C22×C3×C4C22⋊C412yes—
104C22×C3×C4C22×C411yescharacteristic
105C22×C3×C4C2×D412yes—
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4112yes—
108C22×Dic3—24no—
109C3×C4⋊C4—24no—
110C3×C4⋊C4—24no—
111C3×C4⋊C4—44no—
112C3×C22⋊C4—44no—
113C3×C22⋊C4—42no—
114C3×C22⋊C4—44no—
115#217—28no—
116C4⋊Dic3—28no—
117C4×Dic3—28no—
118C4×Dic3—44no—
119C4×Dic3—82no—
120#208—44no—
121C2×C3⋊C8—42no—
122C2×#41—22no—
123C2×#41—22no—
124C2×#41—44no—
125C22×Q8C2×Dic311yescharacteristic
126C22×D4—21no—
127C22×D4C2×Dic311yescharacteristic
128C23×C4—21no—
129C23×C4#6711yescharacteristic
130C23×C4#6711yescharacteristic
131C23×C4C22×D311yescharacteristic
132C2×M16—62no—
133#123—44no—
134#117—68no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4C2×Dic312yes—
140C2×C4⋊C4#6712yes—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—22no—
144C2×C22⋊C4—62no—
145C2×C42—34no—
146C2×C42—62no—
147C2×C42—62no—
148#104—128no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151#95—22no—
152#95—21no—
153#95—62no—
154#95—62no—
155#95C2×Dic312yes—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—41no—
160C23×C3—41no—
161C23×C3—42no—
162C23×C3—41no—
163C23×C3—42no—
164C23×C3C2×C22⋊C411yescharacteristic
165C23×C3#12011yescharacteristic
166C23×C3#12311yescharacteristic
167C3×Q8—24no—
168C3×Q8—42no—
169C3×D4—24no—
170C3×D4—42no—
171C3×D4—44no—
172C3×D4—44no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—24no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
183C2×C3×C4—42no—
184C2×C3×C4—48no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—42no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—81no—
193C2×C3×C4#10414yescomplemented
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12112yes—
196C2×Dic3—28no—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C3⋊C8—82no—
200C24—21no—
201C24C2×#6711yescharacteristic
202#41—28no—
203#41—44no—
204#41—44no—
205#41—44no—
206C2×Q8—22no—
207C2×Q8—42no—
208C2×D4—24no—
209C2×D4—22no—
210C2×D4—42no—
211C2×D4—42no—
212C2×D4—41no—
213C2×D4—41no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—22no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—64no—
227C22×C4#21712yes—
228C22×C4#21712yes—
229C22×C4D3×D412yes—
230C22×C4#23712yes—
231C22×C4C22×Dic311yescharacteristic
232C22×C4C2×#6711yescharacteristic
233M16—124no—
234C2×C8—122no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—68no—
239C22⋊C4—44no—
240C22⋊C4—42no—
241C22⋊C4—44no—
242C22⋊C4—68no—
243C42—68no—
244C42—124no—
245C42—242no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—44no—
254C22×C3—44no—
255C22×C3—42no—
256C22×C3—82no—
257C22×C3—81no—
258C22×C3—81no—
259C22×C3#38611yescharacteristic
260C22×C3C2×#10412yes—
261C22×C3B(4,𝔽2)12yes—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—24no—
265C3×C4—42no—
266C3×C4—42no—
267C3×C4—82no—
268Dic3—48no—
269C23—22no—
270C23—22no—
271C23—22no—
272C23—41no—
273C23—41no—
274C23—42no—
275C23—41no—
276C23—42no—
277C23#93511yescharacteristic
278C23#93711yescharacteristic
279C23C2×#21711yescharacteristic
280Q8—24no—
281Q8—42no—
282D4—24no—
283D4—42no—
284D4—44no—
285D4—44no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—42no—
296C2×C4—41no—
297C2×C4—48no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—68no—
306C2×C4—81no—
307C2×C4—124no—
308C2×C4—124no—
309C2×C4#83714yescomplemented
310C2×C4D4×Dic312yes—
311C2×C4#93912yes—
312C8—242no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—41no—
316C2×C3—82no—
317C2×C3#188911yescharacteristic
318C2×C3#189611yescharacteristic
319C2×C3#191211yescharacteristic
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—44no—
328C22—44no—
329C22—42no—
330C22—82no—
331C22—81no—
332C22—81no—
333C22#503212yes—
334C22#522011yescharacteristic
335C22C2×#83712yes—
336C4—24no—
337C4—22no—
338C4—22no—
339C4—42no—
340C4—42no—
341C4—82no—
342C4—128no—
343C3#947611yescharacteristic, complemented, Sylow
344C2—22no—
345C2—22no—
346C2—41no—
347C2—82no—
348C2#7166911yescharacteristic
349C2#7171111yescharacteristic
350C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2406C2
socleC22×C3#386
prime radicalC22×C3×#41C22
cosocleC23×C3×C4C23
hypercenter#2406D3
hypocenterC3#9476
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
2. #82144C2semidirect
4. #71552C2semidirect
5. #67628C2semidirect
11. C3×#395C4semidirect
12. C3×#394C4semidirect
67. #395Dic3semidirect
68. #394Dic3semidirect
193. C2×C3×C4#104semidirect
309. C2×C4#837semidirect
343. C3#9476semidirect

Maximal subgroups

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

subgroupindex
C3×#24062
#821442
#721382
#715522
#676282
#94763