Finite Group Databasebeta

← #10692633

Subgroups — #10692633

Order 768.

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

normal subgroups only

This group has 840 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×#2539C211yesmaximal, characteristic
2#82083C211yesmaximal, characteristic, complemented
3#71673C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic, complemented
5#67618C212yesmaximal, complemented
6#9409—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C2×C3×#120—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—21no—
12C3×#399C412yescomplemented
13C3×#392C412yescomplemented
14C3×#390—22no—
15C3×C4×C22⋊C4C2211yescharacteristic
16C4×#217—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19Dic3×C22⋊C4—24no—
20#4537—24no—
21#4525—22no—
22#2539D311yescharacteristic
23#2273—31no—
24#1507—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4C2×C411yescharacteristic, complemented
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3—22no—
34C3×#123—44no—
35C3×#121—22no—
36C3×#121—22no—
37C3×#120—44no—
38C2×C3×C4⋊C4—22no—
39C2×C3×C4⋊C4C2×C411yescharacteristic, complemented
40C2×C3×C22⋊C4—21no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4C2×C412yes—
47C2×C3×C22⋊C4D412yes—
48C2×C3×C42—21no—
49C2×#217—22no—
50C2×#217—28no—
51C2×#217—22no—
52C2×C4×Dic3—22no—
53C2×C4×Dic3—22no—
54C2×#208—22no—
55C3×#95C2×C412yescomplemented
56C3⋊#95—22no—
57C3⋊#95D414yescomplemented
58#830—22no—
59#830—22no—
60C2×#123—21no—
61C2×#121C2×D311yescharacteristic
62C2×#120—21no—
63C22×C22⋊C4—61no—
64C22×C22⋊C4C2×D311yescharacteristic
65#407—32no—
66#399Dic312yescomplemented
67#392Dic312yescomplemented
68#390—22no—
69C4×C22⋊C4—32no—
70C4×C22⋊C4—64no—
71C4×C22⋊C4—61no—
72C4×C22⋊C4C2×D311yescharacteristic
73#327—64no—
74#323—62no—
75C24×C3—21no—
76C24×C3—24no—
77C24×C3—21no—
78C24×C3—21no—
79C24×C3—42no—
80C24×C3—81no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C2×C3×D4—22no—
84C2×C3×D4—44no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—44no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—82no—
91C2×C3×D4—81no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—42no—
97C22×C3×C4—81no—
98C22×C3×C4C22⋊C412yes—
99C22×C3×C4C22×C411yescharacteristic
100C22×C3×C4C2×D412yes—
101C22×C3×C4C2×D411yescharacteristic
102C22×C3×C4#4112yes—
103C22×Dic3—24no—
104C22×Dic3—28no—
105C22×Dic3—82no—
106C22×Dic3—82no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—44no—
109C3×C22⋊C4—24no—
110C3×C22⋊C4—24no—
111C3×C22⋊C4—24no—
112C3×C22⋊C4—44no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—42no—
115C3×C42—81no—
116#217—28no—
117#217—28no—
118C4×Dic3—82no—
119C4×Dic3—82no—
120#208—82no—
121C2×C3⋊C8—42no—
122C25#6711yescharacteristic
123C22×D4—21no—
124C22×D4—22no—
125C22×D4C2×Dic311yescharacteristic, complemented
126C23×C4—62no—
127C23×C4#6711yescharacteristic
128C23×C4C22×D311yescharacteristic
129C2×M16—62no—
130#123—44no—
131#121—22no—
132#121—22no—
133#120—44no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4C2×Dic311yescharacteristic, complemented
136C2×C22⋊C4—21no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—22no—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—62no—
143C2×C22⋊C4—68no—
144C2×C22⋊C4—62no—
145C2×C22⋊C4C2×Dic312yes—
146C2×C22⋊C4#6712yes—
147C2×C42—21no—
148C2×C42—62no—
149C2×C42—62no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152#95—34no—
153#95—62no—
154#95C2×Dic312yescomplemented
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—24no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—41no—
163C23×C3—42no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—41no—
171C23×C3—42no—
172C23×C3—81no—
173C23×C3—81no—
174C23×C3—88no—
175C23×C3#10414yes—
176C23×C3C2×C22⋊C411yescharacteristic
177C23×C3#12011yescharacteristic
178C23×C3#12311yescharacteristic
179C3×D4—82no—
180C3×D4—88no—
181C3×D4—82no—
182C3×D4—82no—
183C2×C3×C4—24no—
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—81no—
197C2×C3×C4C4×D412yes—
198C2×C3×C4#12112yes—
199C2×Dic3—28no—
200C2×Dic3—44no—
201C2×Dic3—44no—
202C2×Dic3—88no—
203C3⋊C8—82no—
204C24—21no—
205C24—24no—
206C24—21no—
207C24—21no—
208C24—42no—
209C24—81no—
210C24#21712yes—
211C24C2×#6711yescharacteristic
212C2×D4—22no—
213C2×D4—44no—
214C2×D4—42no—
215C2×D4—42no—
216C2×D4—44no—
217C2×D4—41no—
218C2×D4—41no—
219C2×D4—82no—
220C2×D4—81no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—64no—
227C22×C4—68no—
228C22×C4—81no—
229C22×C4—242no—
230C22×C4—242no—
231C22×C4#21712yes—
232C22×C4D3×D412yes—
233C22×C4#23712yes—
234C22×C4C22×Dic311yescharacteristic
235C22×C4C2×#6711yescharacteristic
236M16—242no—
237C2×C8—122no—
238C4⋊C4—22no—
239C4⋊C4—44no—
240C22⋊C4—24no—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—42no—
246C22⋊C4—68no—
247C22⋊C4—68no—
248C42—81no—
249C42—242no—
250C42—242no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—24no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—44no—
260C22×C3—44no—
261C22×C3—44no—
262C22×C3—44no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—82no—
266C22×C3—82no—
267C22×C3—82no—
268C22×C3—84no—
269C22×C3—81no—
270C22×C3—81no—
271C22×C3—81no—
272C22×C3#38611yescharacteristic
273C22×C3C2×#10412yes—
274C22×C3#45312yes—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—44no—
278C3×C4—42no—
279C3×C4—81no—
280Dic3—48no—
281C23—22no—
282C23—22no—
283C23—24no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—41no—
289C23—42no—
290C23—42no—
291C23—42no—
292C23—41no—
293C23—41no—
294C23—42no—
295C23—42no—
296C23—41no—
297C23—42no—
298C23—81no—
299C23—81no—
300C23—88no—
301C23#83714yes—
302C23#93511yescharacteristic
303C23#93711yescharacteristic
304C23C2×#21711yescharacteristic
305D4—82no—
306D4—88no—
307D4—82no—
308D4—82no—
309C2×C4—24no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—68no—
323C2×C4—81no—
324C2×C4—124no—
325C2×C4—124no—
326C2×C4—248no—
327C2×C4D4×Dic312yes—
328C2×C4#93912yes—
329C8—242no—
330C2×C3—22no—
331C2×C3—22no—
332C2×C3—41no—
333C2×C3—44no—
334C2×C3—161no—
335C2×C3#188711yescharacteristic
336C2×C3#189611yescharacteristic
337C2×C3#191311yescharacteristic
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—24no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—44no—
347C22—44no—
348C22—44no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22—82no—
354C22—82no—
355C22—84no—
356C22—81no—
357C22—81no—
358C22—81no—
359C22#507712yes—
360C22#522011yescharacteristic
361C22C2×#83712yes—
362C4—22no—
363C4—22no—
364C4—44no—
365C4—42no—
366C4—81no—
367C4—128no—
368C3#940911yescharacteristic, complemented, Sylow
369C2—22no—
370C2—22no—
371C2—41no—
372C2—44no—
373C2—161no—
374C2#7171111yescharacteristic
375C2#7176311yescharacteristic
376C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2539C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2539D3
hypocenterC3#9409
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2539

Complements

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

NH ≅ G/Ntype
2. #82083C2semidirect
3. #71673C2semidirect
4. #71539C2semidirect
5. #67618C2semidirect
12. C3×#399C4semidirect
13. C3×#392C4semidirect
30. C22×C3×D4C2×C4semidirect
39. C2×C3×C4⋊C4C2×C4semidirect
55. C3×#95C2×C4semidirect
57. C3⋊#95D4semidirect
66. #399Dic3semidirect
67. #392Dic3semidirect
125. C22×D4C2×Dic3semidirect
135. C2×C4⋊C4C2×Dic3semidirect
154. #95C2×Dic3semidirect
368. C3#9409semidirect

Maximal subgroups

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

subgroupindex
C3×#25392
#820832
#716732
#715392
#676182
#94093