Finite Group Databasebeta

← #10692991

Subgroups — #10692991

Order 768.

818 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2274C211yesmaximal, characteristic, complemented
2#82192C211yesmaximal, characteristic
3#71552C211yesmaximal, characteristic, complemented
4#69454C212yesmaximal
5#67616C212yesmaximal, complemented
6#9476—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×#117—21no—
9C22×C3×C4⋊C4C2211yescharacteristic
10C2×#929—21no—
11C3×C4×C4⋊C4—24no—
12C3×C4×C22⋊C4—21no—
13C3×C4×C22⋊C4C2212yes—
14#5213—21no—
15#5211C2211yescharacteristic
16#5207C2212yes—
17#4991C412yescomplemented
18#4989—22no—
19#4979C412yescomplemented
20C2×#805—24no—
21#4528—22no—
22#2406—31no—
23#2274D311yescharacteristic, complemented
24#1789—32no—
25#1463—31no—
26#1284—32no—
27C23×C3×C4D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C2×#235—22no—
32C2×#235—22no—
33C2×#235—44no—
34C22×D12C2×C411yescharacteristic
35C22×C4×D3—21no—
36C22×Dic6C2×C411yescharacteristic
37C3×#117—28no—
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—22no—
44C2×C3×C42—22no—
45C2×C3×C42—22no—
46C2×C3×C42D414yes—
47C2×#217—21no—
48#929—44no—
49C2×(C2×D3)⋊C4—22no—
50C2×(C2×D3)⋊C4—22no—
51C2×Dic3⋊C4—22no—
52C2×#208—22no—
53C3×#95—22no—
54C3×#95—22no—
55C3⋊#95—22no—
56C3⋊#95—21no—
57C3⋊#95C2×C412yes—
58#830—22no—
59#830—22no—
60#805—48no—
61C22×#41—31no—
62C2×#123—61no—
63C2×#117—21no—
64C22×C4⋊C4C2×D311yescharacteristic
65C2×#104—64no—
66#407—32no—
67#395—32no—
68#394—32no—
69#386—61no—
70#385—62no—
71#381—31no—
72C4×C4⋊C4—24no—
73C4×C22⋊C4—21no—
74C4×C22⋊C4C2×D312yes—
75#324—62no—
76C24×C3C2×D411yescharacteristic
77C23×D3—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—24no—
81C22×C3×C4—24no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4C22⋊C412yes—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4112yes—
92C2×#67—24no—
93C2×#67—42no—
94C2×#67—41no—
95C2×#67—41no—
96C22×Dic3—21no—
97C22×Dic3—22no—
98#235—28no—
99#235—44no—
100#235—44no—
101#235—44no—
102C2×D12—22no—
103C2×D12—42no—
104C2×C4×D3—42no—
105C2×C4×D3—42no—
106C2×C4×D3—42no—
107C2×Dic6—22no—
108C2×Dic6—42no—
109C3×C4⋊C4—28no—
110C3×C4⋊C4—24no—
111C3×C4⋊C4—24no—
112C3×C22⋊C4—28no—
113C3×C42—28no—
114C3×C42—44no—
115C3×C42—82no—
116#217—42no—
117(C2×D3)⋊C4—44no—
118(C2×D3)⋊C4—44no—
119Dic3⋊C4—44no—
120#208—44no—
121C2×C3⋊C8—42no—
122C2×#41—62no—
123C2×#41—62no—
124C2×#41—124no—
125C22×Q8—31no—
126C22×D4—31no—
127C22×D4—61no—
128C23×C4—61no—
129C23×C4#6711yescharacteristic
130C23×C4#6711yescharacteristic
131C23×C4C22×D311yescharacteristic
132C2×M16—62no—
133#123—124no—
134#117—28no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—62no—
139C2×C4⋊C4C4×D312yes—
140C2×C4⋊C4D1212yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—61no—
143C2×C22⋊C4—62no—
144C2×C22⋊C4—62no—
145C2×C42—22no—
146C2×C42—22no—
147C2×C42#6714yes—
148#104—128no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151#95—22no—
152#95—22no—
153#95—32no—
154#95—62no—
155#95—61no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—41no—
159C23×C3C2×C22⋊C411yescharacteristic
160C23×C3#12011yescharacteristic
161C23×C3#12311yescharacteristic
162C22×D3—22no—
163C22×D3—41no—
164C22×D3—42no—
165C22×D3—41no—
166C22×D3—42no—
167C2×C3×C4—22no—
168C2×C3×C4—22no—
169C2×C3×C4—24no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—24no—
173C2×C3×C4—28no—
174C2×C3×C4—22no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—44no—
179C2×C3×C4—44no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—81no—
183C2×C3×C4#10414yescomplemented
184C2×C3×C4C4×D412yes—
185C2×C3×C4#12112yes—
186#67—44no—
187#67—44no—
188C2×Dic3—22no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194D12—24no—
195D12—42no—
196C4×D3—48no—
197Dic6—24no—
198Dic6—42no—
199C3⋊C8—82no—
200C24—61no—
201C24C2×#6711yescharacteristic
202#41—68no—
203#41—124no—
204#41—124no—
205#41—124no—
206C2×Q8—62no—
207C2×Q8—122no—
208C2×D4—64no—
209C2×D4—62no—
210C2×D4—122no—
211C2×D4—122no—
212C2×D4—121no—
213C2×D4—121no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—61no—
223C22×C4—62no—
224C22×C4—122no—
225C22×C4—122no—
226C22×C4—122no—
227C22×C4(C2×D3)⋊C412yes—
228C22×C4(C2×D3)⋊C412yes—
229C22×C4C2×C4×D311yescharacteristic
230C22×C4C2×D1211yescharacteristic
231C22×C4#23512yes—
232C22×C4C2×#6712yes—
233M16—124no—
234C2×C8—122no—
235C4⋊C4—28no—
236C4⋊C4—24no—
237C4⋊C4—24no—
238C4⋊C4—124no—
239C22⋊C4—28no—
240C22⋊C4—124no—
241C22⋊C4—122no—
242C22⋊C4—124no—
243C42—28no—
244C42—44no—
245C42—82no—
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#38611yescharacteristic
255C22×C3C2×#10412yes—
256C22×C3B(4,𝔽2)12yes—
257C2×D3—44no—
258C2×D3—42no—
259C2×D3—82no—
260C2×D3—81no—
261C2×D3—81no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—24no—
265C3×C4—48no—
266C3×C4—42no—
267Dic3—42no—
268Dic3—82no—
269C23—22no—
270C23—22no—
271C23—41no—
272C23—62no—
273C23—121no—
274C23—122no—
275C23—122no—
276C23—121no—
277C23C2×(C2×D3)⋊C411yescharacteristic
278C23#92911yescharacteristic
279C23#95311yescharacteristic
280Q8—64no—
281Q8—122no—
282D4—64no—
283D4—122no—
284D4—124no—
285D4—124no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—24no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—24no—
292C2×C4—28no—
293C2×C4—22no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—42no—
297C2×C4—44no—
298C2×C4—44no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—62no—
302C2×C4—81no—
303C2×C4—128no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—122no—
307C2×C4—122no—
308C2×C4—122no—
309C2×C4#80514yescomplemented
310C2×C4C4×#6712yes—
311C2×C4#93012yes—
312C8—242no—
313C2×C3—22no—
314C2×C3—22no—
315C2×C3—41no—
316C2×C3#188911yescharacteristic
317C2×C3#189611yescharacteristic
318C2×C3#191211yescharacteristic
319D3—82no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—44no—
328C22—124no—
329C22—122no—
330C22—242no—
331C22—241no—
332C22—241no—
333C22C2×#80512yes—
334C22#521311yescharacteristic
335C22#524412yes—
336C4—24no—
337C4—22no—
338C4—22no—
339C4—48no—
340C4—42no—
341C4—122no—
342C4—242no—
343C3#947611yescharacteristic, complemented, Sylow
344C2—22no—
345C2—22no—
346C2—41no—
347C2—242no—
348C2#7153211yescharacteristic
349C2#7165411yescharacteristic
350C2#7166311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2274C2
socleC22×C3#386
prime radicalC22×#235C22
cosocleC23×C3×C4C23
hypercenter#2274D3
hypocenterC3#9476
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
1. C3×#2274C2semidirect
3. #71552C2semidirect
5. #67616C2semidirect
17. #4991C4semidirect
19. #4979C4semidirect
23. #2274D3semidirect
183. C2×C3×C4#104semidirect
309. C2×C4#805semidirect
343. C3#9476semidirect

Maximal subgroups

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

subgroupindex
C3×#22742
#821922
#715522
#694542
#676162
#94763