Finite Group Databasebeta

← #10693182

Subgroups — #10693182

Order 768.

478 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 103 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×#2484C211yesmaximal, characteristic
2#82209C211yesmaximal, characteristic
3#71589C211yesmaximal, characteristic
4#71552C212yesmaximal
5#67629C212yesmaximal
6#9514—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C3×#400—21no—
10C3×#398—21no—
11C3×#387—21no—
12C3×#387—21no—
13C3×#385C411yescharacteristic
14C3×#385C411yescharacteristic
15C3×#381C411yescharacteristic
16C3×#381C411yescharacteristic
17C3×#381C2211yescharacteristic
18#5207C2212yes—
19#5207C2212yes—
20#5000—24no—
21#4924—21no—
22C4×#208—21no—
23#4530—21no—
24#4530—21no—
25#4528—24no—
26#2921—31no—
27#2484D311yescharacteristic
28#1822—31no—
29#1463—32no—
30#1306—32no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4C2311yescharacteristic
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4—21no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—22no—
42C2×C3×C4⋊C4D411yescharacteristic
43C2×C3×C4⋊C4D411yescharacteristic
44C2×C3×C22⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4C2×C411yescharacteristic
47C2×C3×C22⋊C4C2×C411yescharacteristic
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×#208—22no—
51C2×#208—22no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60#830—22no—
61#830—22no—
62#830—22no—
63#830D414yes—
64C4⋊(C3⋊C8)—24no—
65#803—22no—
66C4×C3⋊C8—22no—
67C2×#117C2×D311yescharacteristic
68C22×C4⋊C4C2×D311yescharacteristic
69#433—64no—
70#423—61no—
71#407—32no—
72#407—32no—
73C4×M16—61no—
74#400—21no—
75#398—21no—
76#387—21no—
77#387—21no—
78#385Dic311yescharacteristic
79#385Dic311yescharacteristic
80#381Dic311yescharacteristic
81#381Dic311yescharacteristic
82#381C2×D311yescharacteristic
83#343—61no—
84#343—61no—
85#324—64no—
86C24×C3C22×C411yescharacteristic
87C22×C3×C4—21no—
88C22×C3×C4—22no—
89C22×C3×C4—21no—
90C22×C3×C4—21no—
91C22×C3×C4—21no—
92C22×C3×C4—22no—
93C22×C3×C4—42no—
94C22×C3×C4—41no—
95C22×C3×C4—41no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4#4112yes—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—41no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—41no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C42—21no—
118C3×C42—21no—
119C3×C42—21no—
120C3×C42—21no—
121#208—42no—
122#208—42no—
123#208—42no—
124#208—42no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—42no—
127C23×C4C2×Dic311yescharacteristic
128C23×C4C2×Dic311yescharacteristic
129C23×C4C22×D311yescharacteristic
130C2×M16—62no—
131C2×M16—62no—
132#117—22no—
133#117—22no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4#6711yescharacteristic
141C2×C4⋊C4#6711yescharacteristic
142C2×C22⋊C4C2×Dic311yescharacteristic
143C2×C22⋊C4C2×Dic311yescharacteristic
144C2×C22⋊C4C2×Dic311yescharacteristic
145C2×C22⋊C4C2×Dic311yescharacteristic
146C2×C42#6711yescharacteristic
147C2×C42#6711yescharacteristic
148C4⋊C8—64no—
149C22⋊C8—34no—
150C22⋊C8—62no—
151C22⋊C8—62no—
152C22⋊C8—62no—
153#97—62no—
154C4×C8—62no—
155#95—21no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95—21no—
163C23×C3—21no—
164C23×C3—21no—
165C23×C3—41no—
166C23×C3—41no—
167C23×C3C2×C22⋊C411yescharacteristic
168C23×C3C4×D412yes—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—22no—
174C2×C3×C4—24no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—42no—
180C2×C3×C4—42no—
181C2×C3×C4—41no—
182C2×C3×C4—42no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—82no—
198C2×C3×C4#12011yescharacteristic
199C2×C3×C4#12112yes—
200C2×C3×C4#12311yescharacteristic
201C3⋊C8—44no—
202C3⋊C8—82no—
203C24C22×Dic311yescharacteristic
204C22×C4—22no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4—22no—
210C22×C4—42no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—41no—
215C22×C4#21711yescharacteristic
216C22×C4#21711yescharacteristic
217C22×C4#21711yescharacteristic
218C22×C4#21711yescharacteristic
219C22×C4D3×D412yes—
220C22×C4#23712yes—
221C22×C4C2×#6711yescharacteristic
222C22×C4C2×#6711yescharacteristic
223M16—122no—
224M16—122no—
225M16—122no—
226M16—122no—
227C2×C8—64no—
228C2×C8—122no—
229C4⋊C4—22no—
230C4⋊C4—22no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—41no—
234C4⋊C4—41no—
235C4⋊C4—42no—
236C22⋊C4—22no—
237C22⋊C4—22no—
238C22⋊C4—41no—
239C22⋊C4—41no—
240C42—21no—
241C42—21no—
242C42—21no—
243C42—21no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—42no—
250C22×C3—42no—
251C22×C3#38611yescharacteristic
252C22×C3#41011yescharacteristic
253C22×C3#41011yescharacteristic
254C3×C4—21no—
255C3×C4—22no—
256C3×C4—21no—
257C3×C4—42no—
258C3×C4—41no—
259C3×C4—41no—
260C3×C4—41no—
261C3×C4—41no—
262C3×C4—81no—
263C3×C4—81no—
264C3×C4—81no—
265C23—21no—
266C23—21no—
267C23—41no—
268C23—41no—
269C23D4×Dic312yes—
270C23C2×#21711yescharacteristic
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—82no—
300C2×C4#93511yescharacteristic
301C2×C4#93711yescharacteristic
302C2×C4#93912yes—
303C8—124no—
304C8—242no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3—42no—
308C2×C3#189111yescharacteristic
309C2×C3#189911yescharacteristic
310C2×C3#191511yescharacteristic
311C22—21no—
312C22—21no—
313C22—22no—
314C22—22no—
315C22—22no—
316C22—42no—
317C22—42no—
318C22#522011yescharacteristic
319C22#524011yescharacteristic
320C22#524011yescharacteristic
321C4—21no—
322C4—22no—
323C4—21no—
324C4—42no—
325C4—41no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—81no—
330C4—81no—
331C4—81no—
332C3#951411yescharacteristic, complemented, Sylow
333C2—21no—
334C2—21no—
335C2—42no—
336C2#7170511yescharacteristic
337C2#7171411yescharacteristic
338C2#7186611yescharacteristic
subgroupG/H
centerC22#5220
commutator, prime radicalC24×C3C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2484C2
socleC22×C3#386
cosocleC23×C3×C4C23
hypercenter#2484D3
hypocenterC3#9514
LCSG ▷ C24×C3 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2484

Complements

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

NH ≅ G/Ntype
332. C3#9514semidirect

Maximal subgroups

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

subgroupindex
C3×#24842
#822092
#715892
#715522
#676292
#95143