Finite Group Databasebeta

← #10690933

Subgroups — #10690933

Order 768.

756 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 143 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×#2854C211yesmaximal, characteristic
2C2×#4932C211yesmaximal, characteristic, complemented
3C2×#4537C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#67624C212yesmaximal, complemented
6#9149—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C22×C4×Dic3—21no—
11C3×#390C412yescomplemented
12C3×#389C412yescomplemented
13C3×#388—22no—
14C3×#386—21no—
15C3×#380C2211yescharacteristic
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19#4932—24no—
20#4537—24no—
21#4524—22no—
22#2854D311yescharacteristic
23C2×#376—31no—
24#1817—31no—
25C2×#327—32no—
26#1294—32no—
27C22×C3×D4—21no—
28C22×C3×D4C2×C411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4D411yescharacteristic
31C23×C3×C4C2311yescharacteristic
32C23×Dic3—22no—
33C3×#123—44no—
34C3×#121—22no—
35C3×#121—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4C2×C411yescharacteristic
38C2×C3×C4⋊C4C2×C412yes—
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—21no—
43C2×C3×C42—22no—
44C2×C3×C42—22no—
45C2×C3×C42C2×C412yes—
46C2×C3×C42D412yes—
47C2×C4×Dic3—24no—
48C2×C4×Dic3—28no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—21no—
52C3⋊#95—24no—
53C3⋊#95—22no—
54C3⋊#95D414yes—
55#830—22no—
56#830—22no—
57C2×#123—21no—
58C2×#121C2×D311yescharacteristic
59C22×C42—61no—
60C22×C42C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#390Dic312yescomplemented
63#389Dic312yescomplemented
64#388—22no—
65#386—21no—
66#380C2×D311yescharacteristic
67#376—64no—
68C2×#95—32no—
69C2×#95—61no—
70#336—62no—
71#327—64no—
72C24×C3—21no—
73C24×C3C2×D411yescharacteristic
74C2×C3×D4—22no—
75C2×C3×D4—42no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—24no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—21no—
89C22×C3×C4—42no—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22⋊C412yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4112yes—
96C22×Dic3—28no—
97C22×Dic3—24no—
98C22×Dic3—44no—
99C22×Dic3—42no—
100C22×Dic3—42no—
101C3×C4⋊C4—22no—
102C3×C4⋊C4—42no—
103C3×C4⋊C4—44no—
104C3×C22⋊C4—44no—
105C3×C22⋊C4—42no—
106C3×C22⋊C4—44no—
107C3×C22⋊C4—42no—
108C3×C42—24no—
109C3×C42—24no—
110C4×Dic3—216no—
111C2×C3⋊C8—44no—
112C2×C3⋊C8—42no—
113C22×D4—21no—
114C22×D4C2×Dic311yescharacteristic
115C23×C4—62no—
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120#123—44no—
121#121—22no—
122#121—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4C2×Dic311yescharacteristic
125C2×C4⋊C4C2×Dic312yes—
126C2×C22⋊C4—21no—
127C2×C22⋊C4—22no—
128C2×C22⋊C4—22no—
129C2×C22⋊C4—21no—
130C2×C42—22no—
131C2×C42—22no—
132C2×C42—64no—
133C2×C42—68no—
134C2×C42C2×Dic312yes—
135C2×C42#6712yes—
136C22⋊C8—62no—
137C22⋊C8—62no—
138#95—22no—
139#95—21no—
140#95—34no—
141#95—64no—
142#95—62no—
143C23×C3—22no—
144C23×C3—22no—
145C23×C3—41no—
146C23×C3—42no—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3—42no—
150C23×C3C2×C22⋊C411yescharacteristic
151C23×C3#12011yescharacteristic
152C23×C3#12311yescharacteristic
153C23×C3#13512yes—
154C23×C3AGL(1,ℤ/8)12yes—
155C3×D4—82no—
156C3×D4—82no—
157C3×D4—82no—
158C2×C3×C4—24no—
159C2×C3×C4—24no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—22no—
164C2×C3×C4—22no—
165C2×C3×C4—22no—
166C2×C3×C4—22no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4#10414yes—
178C2×C3×C4C4×D412yes—
179C2×C3×C4#12112yes—
180C2×Dic3—28no—
181C2×Dic3—44no—
182C2×Dic3—48no—
183C2×Dic3—48no—
184C2×Dic3—44no—
185C3⋊C8—82no—
186C24—21no—
187C24C2×#6711yescharacteristic
188C2×D4—22no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4—41no—
192C2×D4—41no—
193C2×D4—42no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—24no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—22no—
201C22×C4—22no—
202C22×C4—21no—
203C22×C4—42no—
204C22×C4—68no—
205C22×C4—64no—
206C22×C4—124no—
207C22×C4—122no—
208C22×C4—122no—
209C22×C4#21712yes—
210C22×C4#21712yes—
211C22×C4D3×D412yes—
212C22×C4#23712yes—
213C22×C4C22×Dic311yescharacteristic
214C22×C4C2×#6711yescharacteristic
215C2×C8—124no—
216C2×C8—122no—
217C4⋊C4—22no—
218C4⋊C4—42no—
219C4⋊C4—44no—
220C22⋊C4—44no—
221C22⋊C4—42no—
222C22⋊C4—44no—
223C22⋊C4—42no—
224C42—24no—
225C42—24no—
226C42—616no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—24no—
232C22×C3—24no—
233C22×C3—81no—
234C22×C3—81no—
235C22×C3—81no—
236C22×C3—82no—
237C22×C3—82no—
238C22×C3#38611yescharacteristic
239C22×C3#41012yes—
240C22×C3#41611yescharacteristic
241C22×C3#41811yescharacteristic
242C22×C3C2×#10412yes—
243C22×C3#44912yes—
244C22×C3#48212yes—
245C3×C4—24no—
246C3×C4—22no—
247C3×C4—22no—
248C3×C4—42no—
249C3×C4—81no—
250C3×C4—82no—
251Dic3—48no—
252C23—22no—
253C23—22no—
254C23—41no—
255C23—41no—
256C23—41no—
257C23—42no—
258C23—42no—
259C23#93212yes—
260C23#93511yescharacteristic
261C23#93711yescharacteristic
262C23#95012yes—
263C23C2×#21711yescharacteristic
264D4—82no—
265D4—82no—
266D4—82no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—68no—
287C2×C4—124no—
288C2×C4—128no—
289C2×C4—128no—
290C2×C4—124no—
291C2×C4#83714yes—
292C2×C4D4×Dic312yes—
293C2×C4#93912yes—
294C8—242no—
295C2×C3—22no—
296C2×C3—22no—
297C2×C3—161no—
298C2×C3#150012yes—
299C2×C3#150512yes—
300C2×C3#187511yescharacteristic
301C2×C3#189611yescharacteristic
302C2×C3#189911yescharacteristic
303C22—21no—
304C22—21no—
305C22—21no—
306C22—21no—
307C22—24no—
308C22—24no—
309C22—81no—
310C22—81no—
311C22—82no—
312C22—82no—
313C22—81no—
314C22#503512yes—
315C22#503712yes—
316C22#521511yescharacteristic
317C22#522011yescharacteristic
318C22#523311yescharacteristic
319C22C2×#83712yes—
320C22#524012yes—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—42no—
325C4—82no—
326C4—81no—
327C4—128no—
328C3#914911yescharacteristic, complemented, Sylow
329C2—22no—
330C2—22no—
331C2—161no—
332C2#7167811yescharacteristic
333C2#7170511yescharacteristic
334C2#7171111yescharacteristic
335C2#7175412yes—
336C2#7181212yes—
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2854C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2854D3
hypocenterC3#9149
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2854

Complements

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

NH ≅ G/Ntype
2. C2×#4932C2semidirect
4. #71531C2semidirect
5. #67624C2semidirect
11. C3×#390C4semidirect
12. C3×#389C4semidirect
62. #390Dic3semidirect
63. #389Dic3semidirect
328. C3#9149semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
C2×#49322
C2×#45372
#715312
#676242
#91493