Finite Group Databasebeta

← #10690262

Subgroups — #10690262

Order 768.

484 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 111 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×#2399C211yesmaximal, characteristic
2#82436C211yesmaximal, characteristic
3C4×#836C211yesmaximal, characteristic
4#71623C212yesmaximal
5#67615C212yesmaximal
6#9030—31nomaximal, Sylow
7C2×C3×C4×Q8C2211yescharacteristic
8C2×C3×C4×Q8C2211yescharacteristic
9C3×#398—21no—
10C3×#398C411yescharacteristic
11C3×#398C411yescharacteristic
12C3×#395—21no—
13C3×#395C411yescharacteristic
14C3×#395C411yescharacteristic
15C3×C4×C4⋊C4—21no—
16C3×C4×C4⋊C4—21no—
17C3×#376C2211yescharacteristic
18C2×#836—21no—
19#5024—24no—
20#4926—21no—
21#4926C2212yes—
22#4925—21no—
23C4×#208C2212yes—
24#4535—24no—
25#4530—21no—
26#2983—31no—
27#2399D311yescharacteristic
28C4×#101—31no—
29#1491—32no—
30#1300—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×Q8D411yescharacteristic
33C3×C4×Q8—22no—
34C3×C4×Q8—22no—
35C3×C4×Q8—22no—
36C3×C4×Q8—22no—
37C2×C3×C4⋊C4—21no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C42—21no—
47C2×C3×C42—21no—
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51C2×#208—22no—
52C3×#95—21no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95C2×C411yescharacteristic
58C3×#95C2×C411yescharacteristic
59#836—24no—
60#830—22no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)—22no—
63C4⋊(C3⋊C8)D414yes—
64#803—22no—
65#803—22no—
66#803—22no—
67C4×C3⋊C8—22no—
68C4×C3⋊C8—22no—
69C2×C4×Q8C2×D311yescharacteristic
70C2×C4×Q8C2×D311yescharacteristic
71#446—64no—
72#432—32no—
73#432—61no—
74#431—61no—
75C2×#101—61no—
76C4×M16—32no—
77#398—21no—
78#398Dic311yescharacteristic
79#398Dic311yescharacteristic
80#395—21no—
81#395Dic311yescharacteristic
82#395Dic311yescharacteristic
83C4×C4⋊C4—21no—
84C4×C4⋊C4—21no—
85#376C2×D311yescharacteristic
86#343—61no—
87Q8⋊C8—64no—
88C2×C3×Q8—22no—
89C2×C3×Q8—22no—
90C2×C3×Q8—22no—
91C2×C3×Q8—41no—
92C2×C3×Q8—42no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—21no—
96C22×C3×C4—21no—
97C22×C3×C4C22⋊C411yescharacteristic
98C22×C3×C4C22⋊C411yescharacteristic
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4C2×D411yescharacteristic
104C3×C4⋊C4—22no—
105C3×C4⋊C4—22no—
106C3×C4⋊C4—22no—
107C3×C4⋊C4—22no—
108C3×C4⋊C4—42no—
109C3×C4⋊C4—42no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C42—24no—
114C3×C42—21no—
115C3×C42—21no—
116C3×C42—21no—
117C3×C42—21no—
118C3×C42—41no—
119C3×C42—41no—
120C3×C42C2×D412yes—
121C3×C42#4112yes—
122#208—24no—
123#208—42no—
124C2×C3⋊C8—24no—
125C2×C3⋊C8—42no—
126C22×Q8C2×Dic311yescharacteristic
127C22×Q8#6711yescharacteristic
128C2×M16—62no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4C2×Dic311yescharacteristic
139C2×C4⋊C4C2×Dic311yescharacteristic
140C2×C4⋊C4C2×Dic311yescharacteristic
141C2×C4⋊C4#6711yescharacteristic
142C2×C42—21no—
143C2×C42—21no—
144C2×C42#6711yescharacteristic
145C2×C42#6711yescharacteristic
146C2×C42C22×D311yescharacteristic
147C4⋊C8—34no—
148C4⋊C8—62no—
149C4⋊C8—62no—
150#101—64no—
151C22⋊C8—62no—
152#97—62no—
153#97—62no—
154#97—62no—
155C4×C8—62no—
156C4×C8—62no—
157#95—21no—
158#95—21no—
159#95—21no—
160#95—21no—
161#95—21no—
162#95C2×Dic311yescharacteristic
163#95C2×Dic311yescharacteristic
164C23×C3C2×C22⋊C411yescharacteristic
165C3×Q8—24no—
166C3×Q8—44no—
167C3×Q8—42no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—22no—
173C2×C3×C4—22no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—41no—
182C2×C3×C4—41no—
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#10412yes—
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12011yescharacteristic
196C2×C3×C4#12112yes—
197C2×C3×C4#12311yescharacteristic
198C3⋊C8—44no—
199C3⋊C8—44no—
200C2×Q8—22no—
201C2×Q8—22no—
202C2×Q8—22no—
203C2×Q8—41no—
204C2×Q8—42no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—21no—
208C22×C4—21no—
209C22×C4#21711yescharacteristic
210C22×C4#21711yescharacteristic
211C22×C4#21711yescharacteristic
212C22×C4#21711yescharacteristic
213C22×C4C22×Dic311yescharacteristic
214C22×C4C2×#6711yescharacteristic
215C22×C4C2×#6711yescharacteristic
216M16—64no—
217M16—122no—
218C2×C8—64no—
219C2×C8—122no—
220C4⋊C4—22no—
221C4⋊C4—22no—
222C4⋊C4—22no—
223C4⋊C4—22no—
224C4⋊C4—42no—
225C4⋊C4—42no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—42no—
229C42—24no—
230C42—21no—
231C42—21no—
232C42—21no—
233C42—21no—
234C42—41no—
235C42—41no—
236C42D3×D412yes—
237C42#23712yes—
238C22×C3—21no—
239C22×C3—21no—
240C22×C3#38611yescharacteristic
241C22×C3C2×#10411yescharacteristic
242C22×C3#42111yescharacteristic
243C3×C4—22no—
244C3×C4—22no—
245C3×C4—21no—
246C3×C4—22no—
247C3×C4—42no—
248C3×C4—42no—
249C3×C4—41no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—42no—
253C3×C4—81no—
254C3×C4#45612yes—
255C23C2×#21711yescharacteristic
256Q8—24no—
257Q8—44no—
258Q8—42no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#83712yes—
285C2×C4D4×Dic312yes—
286C2×C4#93511yescharacteristic
287C2×C4#93711yescharacteristic
288C2×C4#93912yes—
289C8—124no—
290C8—124no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3#187111yescharacteristic
294C2×C3#188611yescharacteristic
295C2×C3#189111yescharacteristic
296C22—21no—
297C22—21no—
298C22#522011yescharacteristic
299C22C2×#83711yescharacteristic
300C22#523611yescharacteristic
301C4—22no—
302C4—22no—
303C4—22no—
304C4—21no—
305C4—42no—
306C4—42no—
307C4—41no—
308C4—42no—
309C4—41no—
310C4—41no—
311C4—81no—
312C4#509512yes—
313C3#903011yescharacteristic, complemented, Sylow
314C2—21no—
315C2—21no—
316C2#7171311yescharacteristic
317C2#7171411yescharacteristic
318C2#7178411yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2399C2
socleC22×C3#386
prime radicalC23×C3C2×C22⋊C4
cosocleC2×C3×C42C23
hypercenter#2399D3
hypocenterC3#9030
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
313. C3#9030semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824362
C4×#8362
#716232
#676152
#90303