Finite Group Databasebeta

← #10691236

Subgroups — #10691236

Order 768.

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

normal subgroups only

This group has 560 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×#2627C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3#71690C212yesmaximal
4C2×#4528C211yesmaximal, characteristic
5#67625C212yesmaximal
6#9200—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C2×#926—21no—
10C3×#400C412yescomplemented
11C3×#398—22no—
12C3×#395C412yescomplemented
13C3×#385—21no—
14C3×#381—21no—
15C3×#380C2211yescharacteristic
16#5212—21no—
17#5212C2212yes—
18C2×#830C2212yes—
19#4984—24no—
20#4538—24no—
21#4528—22no—
22#2627D311yescharacteristic
23#2316—31no—
24#1520—32no—
25C2×#324—31no—
26#1289—32no—
27C22×C3×Q8C2×C411yescharacteristic
28C23×C3×C4D411yescharacteristic
29C23×C3×C4D411yescharacteristic
30C23×C3×C4C2311yescharacteristic
31C3×C22⋊Q8—22no—
32C3×C22⋊Q8—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—21no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4C2×C411yescharacteristic
39C2×C3×C4⋊C4C2×C412yes—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—21no—
42C2×C3×C22⋊C4—21no—
43C2×#217—22no—
44#926—42no—
45#926—42no—
46C2×C4⋊Dic3—22no—
47C2×C4⋊Dic3—22no—
48C2×C4⋊Dic3—22no—
49C2×C4×Dic3—22no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3×#95—22no—
54C3×#95C2×C412yes—
55C3⋊#95—22no—
56C3⋊#95D414yes—
57#830—22no—
58#830—22no—
59C2×C22⋊Q8C2×D311yescharacteristic
60C2×#117—61no—
61C22×C4⋊C4C2×D311yescharacteristic
62C2×C22⋊C8—32no—
63#400Dic312yescomplemented
64#398—22no—
65#395Dic312yescomplemented
66#389—64no—
67#385—21no—
68#381—21no—
69#380—32no—
70#380—61no—
71#380C2×D311yescharacteristic
72#328—64no—
73#324—62no—
74C24×C3C2×D411yescharacteristic
75C2×C3×Q8—22no—
76C2×C3×Q8—81no—
77C22×C3×C4—21no—
78C22×C3×C4—21no—
79C22×C3×C4—21no—
80C22×C3×C4—21no—
81C22×C3×C4—21no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—22no—
85C22×C3×C4—21no—
86C22×C3×C4—81no—
87C22×C3×C4—81no—
88C22×C3×C4C22⋊C412yes—
89C22×C3×C4C22⋊C412yes—
90C22×C3×C4SD1612yes—
91C22×C3×C4Q1612yes—
92C22×C3×C4C22×C411yescharacteristic
93C22×C3×C4C2×D412yes—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4#4112yes—
96C22×Dic3—24no—
97C3×C4⋊C4—24no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—42no—
101C3×C4⋊C4—81no—
102C3×C4⋊C4—82no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4—41no—
107#217—42no—
108#217—42no—
109C4⋊Dic3—44no—
110C4⋊Dic3—82no—
111C4⋊Dic3—82no—
112C4×Dic3—44no—
113C2×C3⋊C8—42no—
114C2×C3⋊C8—82no—
115C22×Q8C2×Dic311yescharacteristic
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122#117—122no—
123#117—122no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—62no—
131C2×C4⋊C4—62no—
132C2×C4⋊C4C2×Dic311yescharacteristic
133C2×C4⋊C4C2×Dic312yes—
134C2×C4⋊C4#6712yes—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—62no—
138C2×C42—62no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#95—22no—
142#95—22no—
143#95—22no—
144#95—34no—
145#95—62no—
146#95C2×Dic312yes—
147C23×C3—41no—
148C23×C3—41no—
149C23×C3C2×C22⋊C411yescharacteristic
150C23×C3#12011yescharacteristic
151C23×C3#12311yescharacteristic
152C23×C3C2×SD1611yescharacteristic
153C23×C3C2×Q1611yescharacteristic
154C23×C3AGL(1,ℤ/8)11yescharacteristic
155C23×C3#13711yescharacteristic
156C3×Q8—82no—
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—22no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—41no—
167C2×C3×C4—41no—
168C2×C3×C4—42no—
169C2×C3×C4—42no—
170C2×C3×C4—42no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—41no—
176C2×C3×C4—42no—
177C2×C3×C4—81no—
178C2×C3×C4#10314yes—
179C2×C3×C4C4×D412yes—
180C2×C3×C4#12112yes—
181C2×Dic3—44no—
182C2×Dic3—44no—
183C2×Dic3—44no—
184C3⋊C8—82no—
185C24C2×#6711yescharacteristic
186C2×Q8—22no—
187C2×Q8—81no—
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—22no—
196C22×C4—21no—
197C22×C4—64no—
198C22×C4—81no—
199C22×C4—81no—
200C22×C4#21512yes—
201C22×C4#21612yes—
202C22×C4#21712yes—
203C22×C4#21712yes—
204C22×C4D3×D412yes—
205C22×C4#23712yes—
206C22×C4C22×Dic311yescharacteristic
207C22×C4C2×#6711yescharacteristic
208C2×C8—122no—
209C2×C8—242no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—42no—
214C4⋊C4—81no—
215C4⋊C4—82no—
216C4⋊C4—124no—
217C4⋊C4—242no—
218C4⋊C4—242no—
219C22⋊C4—41no—
220C22⋊C4—41no—
221C22⋊C4—41no—
222C22⋊C4—41no—
223C22⋊C4—122no—
224C22⋊C4—122no—
225C42—124no—
226C22×C3—21no—
227C22×C3—21no—
228C22×C3—21no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—21no—
232C22×C3—41no—
233C22×C3—41no—
234C22×C3—41no—
235C22×C3—41no—
236C22×C3#38611yescharacteristic
237C22×C3C2×#10311yescharacteristic
238C22×C3#41711yescharacteristic
239C22×C3#45011yescharacteristic
240C22×C3#45111yescharacteristic
241C22×C3#48311yescharacteristic
242C22×C3#48411yescharacteristic
243C3×C4—24no—
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—42no—
247C3×C4—82no—
248C3×C4—81no—
249C3×C4—81no—
250Dic3—84no—
251C23—41no—
252C23—41no—
253C23#93511yescharacteristic
254C23#93711yescharacteristic
255C23C2×#21511yescharacteristic
256C23C2×#21611yescharacteristic
257C23#94911yescharacteristic
258C23#95111yescharacteristic
259C23C2×#21711yescharacteristic
260Q8—82no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—41no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—41no—
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—42no—
281C2×C4—81no—
282C2×C4—124no—
283C2×C4—124no—
284C2×C4—124no—
285C2×C4#83514yes—
286C2×C4D4×Dic312yes—
287C2×C4#93912yes—
288C8—242no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—21no—
294C2×C3—21no—
295C2×C3#187211yescharacteristic
296C2×C3#191211yescharacteristic
297C2×C3#191311yescharacteristic
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22#504311yescharacteristic
309C22#504411yescharacteristic
310C22#504611yescharacteristic
311C22#505011yescharacteristic
312C22#522011yescharacteristic
313C22C2×#83511yescharacteristic
314C22#523211yescharacteristic
315C4—24no—
316C4—22no—
317C4—22no—
318C4—42no—
319C4—82no—
320C4—81no—
321C4—81no—
322C4—244no—
323C3#920011yescharacteristic, complemented, Sylow
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2—21no—
329C2—21no—
330C2#7169211yescharacteristic
331C2#7184811yescharacteristic
332C2#7186411yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2627C2
socleC22×C3#386
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2627D3
hypocenterC3#9200
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2627

Complements

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

NH ≅ G/Ntype
10. C3×#400C4semidirect
12. C3×#395C4semidirect
63. #400Dic3semidirect
65. #395Dic3semidirect
323. C3#9200semidirect

Maximal subgroups

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

subgroupindex
C3×#26272
#821562
#716902
C2×#45282
#676252
#92003