Finite Group Databasebeta

← #10690224

Subgroups — #10690224

Order 768.

616 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 616 conjugacy classes of subgroups — too many to draw. Only its 143 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
1C2×#4986C211yesmaximal, characteristic
2#82156C211yesmaximal, characteristic
3C2×C3×#324C211yesmaximal, characteristic
4#69925C212yesmaximal
5#67675C212yesmaximal
6#9023—31nomaximal, Sylow
7C22×C3×C4⋊C4C2211yescharacteristic
8C2×#926—21no—
9C22×C4⋊Dic3C2211yescharacteristic
10C2×C3×C22⋊C8C2212yes—
11#5212—21no—
12#5212C2212yes—
13C2×C3⋊#95—21no—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16#4986—22no—
17#4986C414yescomplemented
18#4984—24no—
19C3×#324—22no—
20#4551—24no—
21C2×#400—31no—
22#2316—31no—
23#1824—32no—
24C2×#324D311yescharacteristic
25#1294—32no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C22×C3×C8—22no—
32C2×C3×C4⋊C4—22no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4C2×C412yes—
35C2×C3×C4⋊C4D412yes—
36C2×#217—22no—
37#926—42no—
38#926—42no—
39C2×C4⋊Dic3—22no—
40C2×C4⋊Dic3—22no—
41C2×C4⋊Dic3—22no—
42C2×C4⋊Dic3—21no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3C2×C412yes—
45C2×C4⋊Dic3D414yes—
46C2×C4×Dic3—22no—
47C3×C22⋊C8—22no—
48C3×C22⋊C8—22no—
49C3⋊#95—22no—
50C3⋊#95—24no—
51C3⋊#95—24no—
52C3⋊#95—21no—
53C3⋊#95—22no—
54C3⋊#95C2×C412yes—
55C2×#117—61no—
56C22×C4⋊C4—31no—
57C22×C4⋊C4C2×D311yescharacteristic
58C2×C22⋊C8C2×D312yes—
59#400—34no—
60#400—62no—
61#389—64no—
62#380—32no—
63#380—61no—
64C2×#95—31no—
65C2×#95—61no—
66C2×#95—61no—
67#340—64no—
68#324—22no—
69C24×C3C2×D411yescharacteristic
70C22×C3×C4—21no—
71C22×C3×C4—21no—
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—42no—
77C22×C3×C4—42no—
78C22×C3×C4C22⋊C412yes—
79C22×C3×C4C22⋊C412yes—
80C22×C3×C4SD1612yes—
81C22×C3×C4Q1612yes—
82C22×C3×C4C22×C411yescharacteristic
83C22×C3×C4C2×D412yes—
84C22×C3×C4C2×D411yescharacteristic
85C22×C3×C4#4112yes—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—24no—
89C22×Dic3—42no—
90C22×Dic3—42no—
91C22×Dic3—44no—
92C22×Dic3—41no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C2×C3×C8—44no—
97C2×C3×C8—42no—
98C3×C4⋊C4—24no—
99C3×C4⋊C4—24no—
100#217—42no—
101#217—42no—
102C4⋊Dic3—24no—
103C4⋊Dic3—44no—
104C4⋊Dic3—42no—
105C4⋊Dic3—44no—
106C4⋊Dic3—82no—
107C4×Dic3—44no—
108C23×C4—61no—
109C23×C4—61no—
110C23×C4D1211yescharacteristic
111C23×C4D1211yescharacteristic
112C23×C4C22×D311yescharacteristic
113C22×C8—22no—
114#117—122no—
115#117—122no—
116C2×C4⋊C4—22no—
117C2×C4⋊C4—22no—
118C2×C4⋊C4—32no—
119C2×C4⋊C4—34no—
120C2×C4⋊C4—62no—
121C2×C4⋊C4—62no—
122C2×C4⋊C4—62no—
123C2×C4⋊C4—61no—
124C2×C4⋊C4—62no—
125C2×C4⋊C4C4×D312yes—
126C2×C4⋊C4#6712yes—
127C2×C22⋊C4—62no—
128C2×C42—62no—
129C22⋊C8—22no—
130C22⋊C8—22no—
131#95—32no—
132#95—62no—
133#95—64no—
134#95—64no—
135#95—62no—
136#95—61no—
137C23×C3—22no—
138C23×C3—22no—
139C23×C3C2×C22⋊C411yescharacteristic
140C23×C3#12011yescharacteristic
141C23×C3#12311yescharacteristic
142C23×C3C2×SD1611yescharacteristic
143C23×C3C2×Q1611yescharacteristic
144C23×C3AGL(1,ℤ/8)11yescharacteristic
145C23×C3#13711yescharacteristic
146C2×C3×C4—22no—
147C2×C3×C4—22no—
148C2×C3×C4—22no—
149C2×C3×C4—22no—
150C2×C3×C4—24no—
151C2×C3×C4—22no—
152C2×C3×C4—22no—
153C2×C3×C4—22no—
154C2×C3×C4—41no—
155C2×C3×C4—41no—
156C2×C3×C4—42no—
157C2×C3×C4—42no—
158C2×C3×C4—42no—
159C2×C3×C4—81no—
160C2×C3×C4#10314yes—
161C2×C3×C4C4×D412yes—
162C2×C3×C4#12112yes—
163C2×Dic3—42no—
164C2×Dic3—42no—
165C2×Dic3—44no—
166C2×Dic3—42no—
167C2×Dic3—42no—
168C2×Dic3—44no—
169C2×Dic3—44no—
170C2×Dic3—44no—
171C2×Dic3—82no—
172C2×Dic3—82no—
173C2×Dic3—82no—
174C2×Dic3—82no—
175C3×C8—82no—
176C24C2×D1211yescharacteristic
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—42no—
184C22×C4—42no—
185C22×C4—64no—
186C22×C4—62no—
187C22×C4—62no—
188C22×C4—122no—
189C22×C4—122no—
190C22×C4—124no—
191C22×C4—121no—
192C22×C4—121no—
193C22×C4—121no—
194C22×C4—121no—
195C22×C4#20412yes—
196C22×C4Dic1212yes—
197C22×C4(C2×D3)⋊C412yes—
198C22×C4(C2×D3)⋊C412yes—
199C22×C4C2×C4×D311yescharacteristic
200C22×C4D3×D412yes—
201C22×C4#23712yes—
202C22×C4C2×#6711yescharacteristic
203C2×C8—44no—
204C2×C8—42no—
205C4⋊C4—24no—
206C4⋊C4—24no—
207C4⋊C4—64no—
208C4⋊C4—124no—
209C4⋊C4—124no—
210C4⋊C4—122no—
211C4⋊C4—242no—
212C22⋊C4—122no—
213C22⋊C4—122no—
214C42—124no—
215C22×C3—22no—
216C22×C3—22no—
217C22×C3—22no—
218C22×C3—22no—
219C22×C3—21no—
220C22×C3—21no—
221C22×C3—21no—
222C22×C3—21no—
223C22×C3#38611yescharacteristic
224C22×C3C2×#10311yescharacteristic
225C22×C3#41711yescharacteristic
226C22×C3#45011yescharacteristic
227C22×C3#45111yescharacteristic
228C22×C3#45412yes—
229C22×C3#45812yes—
230C22×C3#48311yescharacteristic
231C22×C3#48411yescharacteristic
232C3×C4—24no—
233C3×C4—22no—
234C3×C4—22no—
235C3×C4—42no—
236Dic3—82no—
237Dic3—84no—
238Dic3—82no—
239C23—22no—
240C23—22no—
241C23#88211yescharacteristic
242C23#88611yescharacteristic
243C23C2×#20411yescharacteristic
244C23C2×Dic1211yescharacteristic
245C23#90811yescharacteristic
246C23#90911yescharacteristic
247C23C2×(C2×D3)⋊C411yescharacteristic
248C2×C4—22no—
249C2×C4—22no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—24no—
255C2×C4—22no—
256C2×C4—41no—
257C2×C4—42no—
258C2×C4—41no—
259C2×C4—42no—
260C2×C4—42no—
261C2×C4—81no—
262C2×C4—122no—
263C2×C4—122no—
264C2×C4—124no—
265C2×C4—122no—
266C2×C4—122no—
267C2×C4—124no—
268C2×C4—124no—
269C2×C4—124no—
270C2×C4—242no—
271C2×C4—242no—
272C2×C4—242no—
273C2×C4—242no—
274C2×C4#81614yes—
275C2×C4#67⋊C412yes—
276C2×C4#93912yes—
277C8—82no—
278C2×C3—21no—
279C2×C3—21no—
280C2×C3—21no—
281C2×C3—21no—
282C2×C3#161612yes—
283C2×C3#161712yes—
284C2×C3#187211yescharacteristic
285C2×C3#188811yescharacteristic
286C2×C3#189011yescharacteristic
287C22—22no—
288C22—22no—
289C22—22no—
290C22—22no—
291C22—21no—
292C22—21no—
293C22—21no—
294C22—21no—
295C22#472111yescharacteristic
296C22#472211yescharacteristic
297C22#473111yescharacteristic
298C22#473911yescharacteristic
299C22#474212yes—
300C22#475312yes—
301C22#495811yescharacteristic
302C22C2×#81611yescharacteristic
303C22#513311yescharacteristic
304C4—24no—
305C4—22no—
306C4—22no—
307C4—42no—
308C4—242no—
309C4—244no—
310C4—242no—
311C3#902311yescharacteristic, complemented, Sylow
312C2—21no—
313C2—21no—
314C2—21no—
315C2—21no—
316C2#6813712yes—
317C2#6814112yes—
318C2#6993011yescharacteristic
319C2#7002311yescharacteristic
320C2#7010411yescharacteristic
subgroupG/H
centerC23#882
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#324C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenterC2×#324D3
hypocenterC3#9023
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
17. #4986C4semidirect
311. C3#9023semidirect

Maximal subgroups

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

subgroupindex
C2×#49862
#821562
C2×C3×#3242
#699252
#676752
#90233