Finite Group Databasebeta

← #10690973

Subgroups — #10690973

Order 768.

648 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 135 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×#4937C211yesmaximal, characteristic, complemented
3C2×#4537C212yesmaximal
4#71531C211yesmaximal, characteristic, complemented
5#67624C212yesmaximal, complemented
6#9157—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C22×C3×C42C2211yescharacteristic
10C3×#390C412yescomplemented
11C3×#389C412yescomplemented
12C3×#388—22no—
13C3×#386—21no—
14C3×#380C2211yescharacteristic
15C2×C3⋊#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19#4937—24no—
20#4537—24no—
21#4524—22no—
22#2854D311yescharacteristic
23C2×#383—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—
47C22×C3⋊C8—22no—
48C3×#95—22no—
49C3×#95—21no—
50C3⋊#95—24no—
51C3⋊#95—24no—
52C3⋊#95—22no—
53C3⋊#95D414yes—
54#830—22no—
55#830—22no—
56C2×#123—21no—
57C2×#121C2×D311yescharacteristic
58C22×C42C2×D311yescharacteristic
59C2×C22⋊C8—32no—
60#390Dic312yescomplemented
61#389Dic312yescomplemented
62#388—22no—
63#386—21no—
64#383—64no—
65#380C2×D311yescharacteristic
66C2×#95—32no—
67C2×#95—61no—
68C2×#95—61no—
69#336—62no—
70#327—64no—
71C24×C3—21no—
72C24×C3C2×D411yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—42no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—21no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4C22⋊C412yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D412yes—
93C22×C3×C4C2×D411yescharacteristic
94C22×C3×C4#4112yes—
95C22×Dic3—24no—
96C22×Dic3—44no—
97C22×Dic3—44no—
98C22×Dic3—42no—
99C22×Dic3—42no—
100C3×C4⋊C4—22no—
101C3×C4⋊C4—42no—
102C3×C4⋊C4—44no—
103C3×C22⋊C4—44no—
104C3×C22⋊C4—44no—
105C3×C22⋊C4—42no—
106C3×C22⋊C4—42no—
107C3×C42—24no—
108C3×C42—24no—
109C2×C3⋊C8—44no—
110C2×C3⋊C8—42no—
111C22×D4—21no—
112C22×D4C2×Dic311yescharacteristic
113C23×C4—62no—
114C23×C4#6711yescharacteristic
115C23×C4#6711yescharacteristic
116C23×C4C22×D311yescharacteristic
117C22×C8—62no—
118#123—44no—
119#121—22no—
120#121—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4C2×Dic311yescharacteristic
123C2×C4⋊C4C2×Dic312yes—
124C2×C22⋊C4—21no—
125C2×C22⋊C4—22no—
126C2×C22⋊C4—22no—
127C2×C22⋊C4—21no—
128C2×C42—22no—
129C2×C42—22no—
130C2×C42C2×Dic312yes—
131C2×C42#6712yes—
132C22⋊C8—62no—
133C22⋊C8—62no—
134#95—22no—
135#95—21no—
136#95—34no—
137#95—64no—
138#95—64no—
139#95—62no—
140C23×C3—22no—
141C23×C3—22no—
142C23×C3—41no—
143C23×C3—42no—
144C23×C3—41no—
145C23×C3—41no—
146C23×C3—42no—
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3#12011yescharacteristic
149C23×C3#12311yescharacteristic
150C23×C3#13512yes—
151C23×C3AGL(1,ℤ/8)12yes—
152C3×D4—82no—
153C3×D4—82no—
154C3×D4—82no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—22no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—22no—
163C2×C3×C4—42no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—42no—
168C2×C3×C4—42no—
169C2×C3×C4—41no—
170C2×C3×C4—42no—
171C2×C3×C4—41no—
172C2×C3×C4—42no—
173C2×C3×C4—41no—
174C2×C3×C4—42no—
175C2×C3×C4C4×D412yes—
176C2×C3×C4#12112yes—
177C2×Dic3—44no—
178C2×Dic3—44no—
179C2×Dic3—44no—
180C2×Dic3—84no—
181C2×Dic3—84no—
182C3⋊C8—82no—
183C24—21no—
184C24C2×#6711yescharacteristic
185C2×D4—22no—
186C2×D4—42no—
187C2×D4—42no—
188C2×D4—41no—
189C2×D4—41no—
190C2×D4—42no—
191C22×C4—22no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—21no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—64no—
202C22×C4—124no—
203C22×C4—124no—
204C22×C4—122no—
205C22×C4—122no—
206C22×C4#21712yes—
207C22×C4#21712yes—
208C22×C4D3×D412yes—
209C22×C4#23712yes—
210C22×C4C22×Dic311yescharacteristic
211C22×C4C2×#6711yescharacteristic
212C2×C8—124no—
213C2×C8—122no—
214C4⋊C4—22no—
215C4⋊C4—42no—
216C4⋊C4—44no—
217C22⋊C4—44no—
218C22⋊C4—44no—
219C22⋊C4—42no—
220C22⋊C4—42no—
221C42—24no—
222C42—24no—
223C22×C3—21no—
224C22×C3—21no—
225C22×C3—21no—
226C22×C3—21no—
227C22×C3—24no—
228C22×C3—24no—
229C22×C3—81no—
230C22×C3—81no—
231C22×C3—81no—
232C22×C3—82no—
233C22×C3—82no—
234C22×C3#38611yescharacteristic
235C22×C3#41012yes—
236C22×C3#41611yescharacteristic
237C22×C3#41811yescharacteristic
238C22×C3#42112yes—
239C22×C3#44912yes—
240C22×C3#48212yes—
241C3×C4—22no—
242C3×C4—22no—
243C3×C4—42no—
244C3×C4—42no—
245C3×C4—81no—
246C3×C4—82no—
247Dic3—84no—
248C23—22no—
249C23—22no—
250C23—41no—
251C23—41no—
252C23—41no—
253C23—42no—
254C23—42no—
255C23#93212yes—
256C23#93511yescharacteristic
257C23#93711yescharacteristic
258C23#95012yes—
259C23C2×#21711yescharacteristic
260D4—82no—
261D4—82no—
262D4—82no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—124no—
284C2×C4—124no—
285C2×C4—124no—
286C2×C4—244no—
287C2×C4—244no—
288C2×C4D4×Dic312yes—
289C2×C4#93912yes—
290C8—242no—
291C2×C3—22no—
292C2×C3—22no—
293C2×C3—161no—
294C2×C3#151412yes—
295C2×C3#152112yes—
296C2×C3#187511yescharacteristic
297C2×C3#189711yescharacteristic
298C2×C3#189911yescharacteristic
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—24no—
304C22—24no—
305C22—81no—
306C22—81no—
307C22—82no—
308C22—82no—
309C22—81no—
310C22#503512yes—
311C22#503712yes—
312C22#521511yescharacteristic
313C22#522011yescharacteristic
314C22#523311yescharacteristic
315C22#523612yes—
316C22#524012yes—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—42no—
321C4—82no—
322C4—81no—
323C4—244no—
324C3#915711yescharacteristic, complemented, Sylow
325C2—22no—
326C2—22no—
327C2—161no—
328C2#7167811yescharacteristic
329C2#7170511yescharacteristic
330C2#7171211yescharacteristic
331C2#7175712yes—
332C2#7181312yes—
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#9157
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×#4937C2semidirect
4. #71531C2semidirect
5. #67624C2semidirect
10. C3×#390C4semidirect
11. C3×#389C4semidirect
60. #390Dic3semidirect
61. #389Dic3semidirect
324. C3#9157semidirect

Maximal subgroups

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

subgroupindex
C3×#28542
C2×#49372
C2×#45372
#715312
#676242
#91573