Finite Group Databasebeta

← #10692652

Subgroups — #10692652

Order 768.

720 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2312C211yesmaximal, characteristic, complemented
2#82115C211yesmaximal, characteristic
3#71539C211yesmaximal, characteristic, complemented
4#69444C212yesmaximal, complemented
5#67618C212yesmaximal, complemented
6#9413—31nomaximal, Sylow
7C22×C3×C22⋊C4C2211yescharacteristic
8C2×#953—21no—
9C2×#930C2211yescharacteristic
10C2×#929—21no—
11C3×#388—24no—
12C3×#380—21no—
13C3×#379—21no—
14C3×C4×C22⋊C4C2212yes—
15C4×#217C2211yescharacteristic
16#5207C2212yes—
17#4960C412yescomplemented
18#4957—22no—
19#4949C412yescomplemented
20#4525—22no—
21#4451—24no—
22#2539—31no—
23#2312D311yescharacteristic, complemented
24#1507—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C22×#67—22no—
32C22×D12C2×C411yescharacteristic
33C2×C3×C4⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35C2×C3×C22⋊C4—22no—
36C2×C3×C22⋊C4—22no—
37C2×C3×C22⋊C4—22no—
38C2×C3×C22⋊C4—22no—
39C2×C3×C22⋊C4C2×C412yes—
40C2×C3×C22⋊C4D412yes—
41C2×C3×C42—22no—
42#953—44no—
43C2×#217—21no—
44C2×#217—22no—
45#930—22no—
46#930—22no—
47#929—44no—
48C2×(C2×D3)⋊C4—22no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3C2×C411yescharacteristic
51C2×Dic3⋊C4—22no—
52C2×C4×Dic3—21no—
53C2×#208—22no—
54C3×#95—22no—
55C3×#95D414yescomplemented
56C3⋊#95C2×C412yes—
57#830—22no—
58#830—22no—
59C2×#123—61no—
60C2×#121—31no—
61C2×#120—61no—
62C22×C22⋊C4C2×D311yescharacteristic
63#407—32no—
64#399—32no—
65#392—32no—
66#390—62no—
67#388—24no—
68#380—21no—
69#379—21no—
70C4×C22⋊C4—31no—
71C4×C22⋊C4C2×D312yes—
72#327—64no—
73#323—62no—
74C24×C3—21no—
75C24×C3—21no—
76C24×C3—42no—
77C24×C3—42no—
78C24×C3—81no—
79C24×C3C22⋊C412yes—
80C24×C3C2×D411yescharacteristic
81C23×D3—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—24no—
85C22×C3×C4—42no—
86C22×C3×C4—81no—
87C22×C3×C4C22⋊C412yes—
88C22×C3×C4C22×C411yescharacteristic
89C22×C3×C4C2×D411yescharacteristic
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4112yes—
92C2×#67—44no—
93C2×#67—42no—
94C2×#67—42no—
95C2×#67—44no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—82no—
99C22×Dic3—22no—
100C22×Dic3—21no—
101C2×D12—22no—
102C2×D12—81no—
103C3×C4⋊C4—82no—
104C3×C22⋊C4—24no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—44no—
107C3×C22⋊C4—82no—
108C3×C22⋊C4—82no—
109C3×C42—82no—
110#217—24no—
111#217—44no—
112(C2×D3)⋊C4—44no—
113(C2×D3)⋊C4—42no—
114C4⋊Dic3—22no—
115Dic3⋊C4—44no—
116C4×Dic3—81no—
117#208—82no—
118C2×C3⋊C8—42no—
119C25#6711yescharacteristic
120C22×D4—31no—
121C22×D4—61no—
122C22×D4—62no—
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C2×M16—62no—
126#123—124no—
127#121—62no—
128#121—62no—
129#120—124no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—62no—
133C2×C22⋊C4—22no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—61no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—61no—
142C2×C22⋊C4C4×D312yes—
143C2×C22⋊C4D1212yes—
144C2×C42—22no—
145C2×C42—61no—
146C22⋊C8—62no—
147C22⋊C8—62no—
148#95—22no—
149#95—32no—
150#95#6714yescomplemented
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—22no—
156C23×C3—41no—
157C23×C3—42no—
158C23×C3—42no—
159C23×C3—42no—
160C23×C3—42no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—81no—
164C23×C3—81no—
165C23×C3—88no—
166C23×C3C2×C22⋊C411yescharacteristic
167C23×C3#12011yescharacteristic
168C23×C3#12311yescharacteristic
169C22×D3—41no—
170C22×D3—42no—
171C22×D3—41no—
172C22×D3—41no—
173C22×D3—42no—
174C2×C3×C4—22no—
175C2×C3×C4—42no—
176C2×C3×C4—41no—
177C2×C3×C4—42no—
178C2×C3×C4—41no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—44no—
182C2×C3×C4—44no—
183C2×C3×C4—44no—
184C2×C3×C4—81no—
185C2×C3×C4C4×D412yes—
186C2×C3×C4#12112yes—
187#67—88no—
188#67—82no—
189#67—82no—
190C2×Dic3—24no—
191C2×Dic3—42no—
192C2×Dic3—42no—
193C2×Dic3—41no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196D12—82no—
197C3⋊C8—82no—
198C24—21no—
199C24—21no—
200C24—42no—
201C24—42no—
202C24—61no—
203C24—81no—
204C24(C2×D3)⋊C412yes—
205C24C2×#6711yescharacteristic
206C2×D4—62no—
207C2×D4—124no—
208C2×D4—122no—
209C2×D4—122no—
210C2×D4—124no—
211C2×D4—121no—
212C2×D4—121no—
213C2×D4—241no—
214C2×D4—242no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—42no—
219C22×C4—62no—
220C22×C4—61no—
221C22×C4—81no—
222C22×C4(C2×D3)⋊C412yes—
223C22×C4C2×C4×D311yescharacteristic
224C22×C4C2×D1211yescharacteristic
225C22×C4#23512yes—
226C22×C4C2×#6712yes—
227M16—242no—
228C2×C8—122no—
229C4⋊C4—62no—
230C4⋊C4—82no—
231C4⋊C4—124no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—44no—
235C22⋊C4—64no—
236C22⋊C4—82no—
237C22⋊C4—82no—
238C22⋊C4—124no—
239C22⋊C4—124no—
240C22⋊C4—122no—
241C42—82no—
242C42—241no—
243C22×C3—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—44no—
251C22×C3—44no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—42no—
255C22×C3—82no—
256C22×C3—84no—
257C22×C3—84no—
258C22×C3#38611yescharacteristic
259C22×C3#42112yes—
260C22×C3#45312yes—
261C2×D3—82no—
262C2×D3—82no—
263C2×D3—81no—
264C2×D3—81no—
265C2×D3—81no—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—42no—
269C3×C4—84no—
270Dic3—44no—
271Dic3—81no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—42no—
284C23—81no—
285C23—81no—
286C23—88no—
287C23—121no—
288C23—121no—
289C23—122no—
290C23—121no—
291C23—122no—
292C23C2×(C2×D3)⋊C411yescharacteristic
293C23#92911yescharacteristic
294C23#95311yescharacteristic
295D4—242no—
296D4—248no—
297D4—242no—
298D4—242no—
299C2×C4—22no—
300C2×C4—42no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—42no—
306C2×C4—44no—
307C2×C4—44no—
308C2×C4—44no—
309C2×C4—64no—
310C2×C4—81no—
311C2×C4—122no—
312C2×C4—122no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4C4×#6712yes—
317C2×C4#93012yes—
318C8—242no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—41no—
322C2×C3—82no—
323C2×C3#188711yescharacteristic
324C2×C3#189711yescharacteristic
325C2×C3#191311yescharacteristic
326D3—161no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22—44no—
336C22—42no—
337C22—42no—
338C22—42no—
339C22—82no—
340C22—84no—
341C22—84no—
342C22—242no—
343C22—242no—
344C22—241no—
345C22—241no—
346C22—241no—
347C22#500512yes—
348C22#521311yescharacteristic
349C22#524112yes—
350C4—22no—
351C4—22no—
352C4—42no—
353C4—84no—
354C4—124no—
355C4—241no—
356C3#941311yescharacteristic, complemented, Sylow
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—481no—
362C2#7159111yescharacteristic
363C2#7165011yescharacteristic
364C2#7165611yescharacteristic
subgroupG/H
centerC22#5213
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2312C2
socleC22×C3#386
prime radical#82115C2
cosocleC23×C3×C4C23
hypercenter#2312D3
hypocenterC3#9413
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2312

Complements

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

NH ≅ G/Ntype
1. C3×#2312C2semidirect
3. #71539C2semidirect
4. #69444C2semidirect
5. #67618C2semidirect
17. #4960C4semidirect
19. #4949C4semidirect
23. #2312D3semidirect
55. C3×#95D4semidirect
150. #95#67semidirect
356. C3#9413semidirect

Maximal subgroups

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

subgroupindex
C3×#23122
#821152
#715392
#694442
#676182
#94133