Finite Group Databasebeta

← #10692653

Subgroups — #10692653

Order 768.

720 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 107 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#82093C211yesmaximal, characteristic, complemented
3#71673C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic, complemented
5#67618C212yesmaximal, complemented
6#9413—31nomaximal, Sylow
7C2×C3×#123—21no—
8C2×C3×#121C2211yescharacteristic
9C2×C3×#120—21no—
10C22×C3×C22⋊C4C2211yescharacteristic
11C3×#399C412yescomplemented
12C3×#392C412yescomplemented
13C3×#390—22no—
14C3×C4×C22⋊C4C2211yescharacteristic
15#5247—21no—
16#5212—21no—
17C4×#217C2212yes—
18#5207C2212yes—
19#4951—24no—
20#4537—24no—
21#4525—22no—
22#2539D311yescharacteristic
23#2312—31no—
24#1507—32no—
25#1462—31no—
26#1297—32no—
27C25×C3D411yescharacteristic
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4C2×C411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C3×#123—44no—
34C3×#121—22no—
35C3×#121—22no—
36C3×#120—44no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4—21no—
40C2×C3×C22⋊C4—22no—
41C2×C3×C22⋊C4—22no—
42C2×C3×C22⋊C4—22no—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4C2×C412yes—
46C2×C3×C22⋊C4D412yes—
47C2×C3×C42—21no—
48C2×#217—22no—
49C2×#217—22no—
50C2×#217—22no—
51C2×C4⋊Dic3—22no—
52C2×C4×Dic3—22no—
53C2×#208—22no—
54C3×#95C2×C412yes—
55C3⋊#95—22no—
56C3⋊#95D414yescomplemented
57#830—22no—
58#830—22no—
59C2×#123—21no—
60C2×#121C2×D311yescharacteristic
61C2×#120—21no—
62C22×C22⋊C4C2×D311yescharacteristic
63#407—32no—
64#399Dic312yescomplemented
65#392Dic312yescomplemented
66#390—22no—
67#388—64no—
68#380—61no—
69#379—61no—
70C4×C22⋊C4—32no—
71C4×C22⋊C4C2×D311yescharacteristic
72#327—64no—
73#323—62no—
74C24×C3—21no—
75C24×C3—21no—
76C24×C3—21no—
77C24×C3—42no—
78C24×C3—42no—
79C24×C3—81no—
80C24×C3C22⋊C412yes—
81C24×C3C2×D411yescharacteristic
82C2×C3×D4—22no—
83C2×C3×D4—44no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—44no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C2×C3×D4—82no—
90C2×C3×D4—81no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—81no—
97C22×C3×C4C22⋊C412yes—
98C22×C3×C4C22×C411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×C3×C4C2×D411yescharacteristic
101C22×C3×C4#4112yes—
102C22×Dic3—24no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—44no—
105C3×C22⋊C4—24no—
106C3×C22⋊C4—24no—
107C3×C22⋊C4—24no—
108C3×C22⋊C4—44no—
109C3×C22⋊C4—44no—
110C3×C22⋊C4—42no—
111C3×C42—81no—
112#217—44no—
113#217—82no—
114#217—82no—
115C4⋊Dic3—82no—
116C4×Dic3—82no—
117#208—82no—
118C2×C3⋊C8—42no—
119C25#6711yescharacteristic
120C22×D4—21no—
121C22×D4—22no—
122C22×D4C2×Dic311yescharacteristic
123C23×C4#6711yescharacteristic
124C23×C4C22×D311yescharacteristic
125C2×M16—62no—
126#123—44no—
127#121—22no—
128#121—22no—
129#120—44no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4C2×Dic311yescharacteristic
133C2×C22⋊C4—21no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—22no—
136C2×C22⋊C4—22no—
137C2×C22⋊C4—22no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—62no—
140C2×C22⋊C4—62no—
141C2×C22⋊C4—62no—
142C2×C22⋊C4C2×Dic312yes—
143C2×C22⋊C4#6712yes—
144C2×C42—21no—
145C2×C42—62no—
146C22⋊C8—62no—
147C22⋊C8—62no—
148#95—34no—
149#95—62no—
150#95C2×Dic312yes—
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—41no—
163C23×C3—41no—
164C23×C3—42no—
165C23×C3—42no—
166C23×C3—41no—
167C23×C3—42no—
168C23×C3—81no—
169C23×C3—81no—
170C23×C3—88no—
171C23×C3C2×C22⋊C411yescharacteristic
172C23×C3#12011yescharacteristic
173C23×C3#12311yescharacteristic
174C3×D4—82no—
175C3×D4—88no—
176C3×D4—82no—
177C3×D4—82no—
178C2×C3×C4—24no—
179C2×C3×C4—22no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—81no—
192C2×C3×C4C4×D412yes—
193C2×C3×C4#12112yes—
194C2×Dic3—44no—
195C2×Dic3—44no—
196C2×Dic3—44no—
197C3⋊C8—82no—
198C24—21no—
199C24—21no—
200C24—21no—
201C24—42no—
202C24—42no—
203C24—81no—
204C24#21712yes—
205C24C2×#6711yescharacteristic
206C2×D4—22no—
207C2×D4—44no—
208C2×D4—42no—
209C2×D4—42no—
210C2×D4—44no—
211C2×D4—41no—
212C2×D4—41no—
213C2×D4—82no—
214C2×D4—81no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—64no—
221C22×C4—81no—
222C22×C4#21712yes—
223C22×C4D3×D412yes—
224C22×C4#23712yes—
225C22×C4C22×Dic311yescharacteristic
226C22×C4C2×#6711yescharacteristic
227M16—242no—
228C2×C8—122no—
229C4⋊C4—22no—
230C4⋊C4—44no—
231C4⋊C4—242no—
232C22⋊C4—24no—
233C22⋊C4—24no—
234C22⋊C4—24no—
235C22⋊C4—44no—
236C22⋊C4—44no—
237C22⋊C4—42no—
238C22⋊C4—124no—
239C22⋊C4—242no—
240C22⋊C4—242no—
241C42—81no—
242C42—242no—
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—82no—
258C22×C3—82no—
259C22×C3—84no—
260C22×C3—81no—
261C22×C3—81no—
262C22×C3—81no—
263C22×C3#38611yescharacteristic
264C22×C3#42112yes—
265C22×C3#45312yes—
266C3×C4—22no—
267C3×C4—22no—
268C3×C4—44no—
269C3×C4—42no—
270C3×C4—81no—
271Dic3—84no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—41no—
278C23—42no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—42no—
283C23—41no—
284C23—41no—
285C23—42no—
286C23—42no—
287C23—41no—
288C23—42no—
289C23—81no—
290C23—81no—
291C23—88no—
292C23#93511yescharacteristic
293C23#93711yescharacteristic
294C23C2×#21711yescharacteristic
295D4—82no—
296D4—88no—
297D4—82no—
298D4—82no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—42no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—81no—
313C2×C4—124no—
314C2×C4—124no—
315C2×C4—124no—
316C2×C4D4×Dic312yes—
317C2×C4#93912yes—
318C8—242no—
319C2×C3—22no—
320C2×C3—22no—
321C2×C3—41no—
322C2×C3—82no—
323C2×C3—161no—
324C2×C3#188711yescharacteristic
325C2×C3#189711yescharacteristic
326C2×C3#191311yescharacteristic
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—82no—
342C22—82no—
343C22—84no—
344C22—81no—
345C22—81no—
346C22—81no—
347C22#507712yes—
348C22#522011yescharacteristic
349C22#523612yes—
350C4—22no—
351C4—22no—
352C4—44no—
353C4—42no—
354C4—81no—
355C4—244no—
356C3#941311yescharacteristic, complemented, Sylow
357C2—22no—
358C2—22no—
359C2—41no—
360C2—82no—
361C2—161no—
362C2#7171211yescharacteristic
363C2#7176311yescharacteristic
364C2#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#9413
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. #82093C2semidirect
3. #71673C2semidirect
4. #71539C2semidirect
5. #67618C2semidirect
11. C3×#399C4semidirect
12. C3×#392C4semidirect
56. C3⋊#95D4semidirect
64. #399Dic3semidirect
65. #392Dic3semidirect
356. C3#9413semidirect

Maximal subgroups

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

subgroupindex
C3×#25392
#820932
#716732
#715392
#676182
#94133