Finite Group Databasebeta

← #10698051

Subgroups — #10698051

Order 768.

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

normal subgroups only

This group has 726 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
1#78611C211yesmaximal, characteristic
2#73999C212yesmaximal, complemented
3#69910C211yesmaximal, characteristic, complemented
4#69509C212yesmaximal, complemented
5C3×#1289C211yesmaximal, characteristic, complemented
6#10321—31nomaximal, Sylow
7C2×#930C2212yes—
8C22×C4×Dic3C2211yescharacteristic
9C2×#886—22no—
10C2×C3×C22⋊C8C2211yescharacteristic
11C3×#380C2212yes—
12C2×#830C2211yescharacteristic
13#5115—22no—
14#4988—22no—
15#4984—22no—
16#4958—22no—
17#4736—22no—
18#4732—22no—
19#4670—24no—
20#4550—22no—
21#4451—24no—
22#3186—31no—
23#2854—32no—
24#1817—31no—
25#1519—32no—
26#1289D311yescharacteristic, complemented
27C23×C3×C4C2311yescharacteristic
28C22×#67—22no—
29C23×Dic3D411yescharacteristic
30C23×Dic3D411yescharacteristic
31C22×D12D412yes—
32C22×C3×C8D412yes—
33C2×C3×C4⋊C4—22no—
34C2×C3×C22⋊C4—22no—
35#930—22no—
36#930—22no—
37#930—22no—
38#930—22no—
39C2×(C2×D3)⋊C4—24no—
40C2×(C2×D3)⋊C4—24no—
41C2×(C2×D3)⋊C4—22no—
42C2×C4⋊Dic3—24no—
43C2×C4⋊Dic3D412yes—
44C2×C4×Dic3—22no—
45C2×C4×Dic3—22no—
46C2×C4×Dic3—21no—
47C2×C4×Dic3—21no—
48C22×C3⋊C8—21no—
49C2×D24—22no—
50#886—44no—
51#886—44no—
52C3×C22⋊C8—22no—
53C3×#95D414yes—
54C3⋊#95—24no—
55#830—22no—
56#821—24no—
57#818—22no—
58C2×#123—62no—
59C2×#121—32no—
60C22×C42—31no—
61#480—62no—
62#466—62no—
63#447—62no—
64C2×C22⋊C8—31no—
65C2×C22⋊C8C2×D311yescharacteristic
66#390—62no—
67#389—62no—
68#388—64no—
69#386—62no—
70#380C2×D312yes—
71#336—62no—
72#327—64no—
73C24×C3C2×D411yescharacteristic
74C23×D3—22no—
75C22×C3×C4—24no—
76C22×C3×C4—42no—
77C22×C3×C4C2×D411yescharacteristic
78C22×C3×C4C2×D412yes—
79C22×C3×C4C2×D412yes—
80C22×C3×C4#4111yescharacteristic
81C2×#67—42no—
82C2×#67—42no—
83C2×#67—42no—
84C2×#67—42no—
85C2×#67—42no—
86C2×#67—42no—
87C22×Dic3—22no—
88C22×Dic3—22no—
89C22×Dic3—21no—
90C22×Dic3—21no—
91C22×Dic3—21no—
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C22×Dic3D812yescomplemented
96C2×D12—22no—
97C2×D12—22no—
98C2×D12—44no—
99C2×C3×C8—22no—
100C2×C3×C8—42no—
101C3×C4⋊C4—44no—
102C3×C22⋊C4—42no—
103C3×C22⋊C4—42no—
104(C2×D3)⋊C4—48no—
105(C2×D3)⋊C4—48no—
106(C2×D3)⋊C4—42no—
107(C2×D3)⋊C4—42no—
108C4⋊Dic3—22no—
109C4⋊Dic3—22no—
110C4⋊Dic3—48no—
111C4×Dic3—41no—
112C4×Dic3—42no—
113C4×Dic3—41no—
114C2×C3⋊C8—42no—
115C2×C3⋊C8—41no—
116D24—82no—
117C22×D4—32no—
118C22×D4—62no—
119C23×C4—31no—
120C23×C4—31no—
121C23×C4C22×D311yescharacteristic
122C2×D8—62no—
123C22×C8—61no—
124C22×C8D1212yes—
125#123—124no—
126#123—124no—
127#121—62no—
128#121—62no—
129#121—62no—
130#121—62no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—32no—
133C2×C4⋊C4—64no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—64no—
136C2×C22⋊C4—64no—
137C2×C22⋊C4—62no—
138C2×C42—61no—
139C2×C42—62no—
140C2×C42—62no—
141C2×C42—61no—
142#107—62no—
143#102—64no—
144C22⋊C8—22no—
145C22⋊C8—62no—
146#95—64no—
147#95D1214yes—
148C23×C3—41no—
149C23×C3—41no—
150C23×C3#12012yes—
151C23×C3#12111yescharacteristic
152C23×C3C2×D811yescharacteristic
153C23×C3#13511yescharacteristic
154C23×C3AGL(1,ℤ/8)12yes—
155C22×D3—44no—
156C22×D3—44no—
157C22×D3—42no—
158C22×D3—42no—
159C22×D3—42no—
160C2×C3×C4—22no—
161C2×C3×C4—24no—
162C2×C3×C4—42no—
163C2×C3×C4—41no—
164C2×C3×C4—44no—
165C2×C3×C4—44no—
166C2×C3×C4—44no—
167C2×C3×C4#12011yescharacteristic
168C2×C3×C4#12112yes—
169C2×C3×C4#12711yescharacteristic
170#67—82no—
171#67—82no—
172#67—82no—
173#67—82no—
174C2×Dic3—24no—
175C2×Dic3—22no—
176C2×Dic3—24no—
177C2×Dic3—22no—
178C2×Dic3—42no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181C2×Dic3—41no—
182C2×Dic3—42no—
183C2×Dic3—41no—
184C2×Dic3—42no—
185C2×Dic3—42no—
186C2×Dic3—42no—
187D12—44no—
188D12—82no—
189C3×C8—42no—
190C3⋊C8—81no—
191C24—62no—
192C24D3×D411yescharacteristic
193C2×D4—62no—
194C2×D4—62no—
195C2×D4—124no—
196C2×D4—122no—
197C2×D4—122no—
198C2×D4—122no—
199C2×D4—122no—
200C2×D4—122no—
201C2×D4—122no—
202C22×C4—24no—
203C22×C4—32no—
204C22×C4—42no—
205C22×C4—62no—
206C22×C4—62no—
207C22×C4—61no—
208C22×C4—61no—
209C22×C4—62no—
210C22×C4—61no—
211C22×C4—62no—
212C22×C4—122no—
213C22×C4C2×D1211yescharacteristic
214C22×C4C2×D1212yes—
215C22×C4D3×D412yes—
216C22×C4#23911yescharacteristic
217D8—242no—
218C2×C8—22no—
219C2×C8—42no—
220C2×C8—122no—
221C2×C8—121no—
222C4⋊C4—44no—
223C4⋊C4—62no—
224C4⋊C4—62no—
225C4⋊C4—128no—
226C22⋊C4—42no—
227C22⋊C4—42no—
228C22⋊C4—128no—
229C22⋊C4—128no—
230C22⋊C4—122no—
231C22⋊C4—122no—
232C42—121no—
233C42—121no—
234C42—122no—
235C22×C3—22no—
236C22×C3—21no—
237C22×C3—22no—
238C22×C3—21no—
239C22×C3—41no—
240C22×C3—41no—
241C22×C3—42no—
242C22×C3#39211yescharacteristic
243C22×C3#44712yes—
244C22×C3#44912yes—
245C22×C3#46611yescharacteristic
246C22×C3#46911yescharacteristic
247C2×D3—84no—
248C2×D3—84no—
249C2×D3—82no—
250C2×D3—82no—
251C2×D3—82no—
252C3×C4—21no—
253C3×C4—22no—
254C3×C4—84no—
255C3×C4#45812yes—
256Dic3—42no—
257Dic3—42no—
258Dic3—82no—
259C23—41no—
260C23—41no—
261C23—124no—
262C23—124no—
263C23—122no—
264C23—122no—
265C23—122no—
266C23#88212yes—
267C23#89511yescharacteristic
268C23D3×D811yescharacteristic
269C23#91412yes—
270C23#91911yescharacteristic
271D4—124no—
272D4—242no—
273D4—242no—
274D4—242no—
275D4—242no—
276D4—242no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—64no—
285C2×C4—62no—
286C2×C4—62no—
287C2×C4—64no—
288C2×C4—122no—
289C2×C4—121no—
290C2×C4—121no—
291C2×C4—121no—
292C2×C4—121no—
293C2×C4—122no—
294C2×C4—122no—
295C2×C4—122no—
296C2×C4—122no—
297C2×C4#87411yescharacteristic
298C2×C4#88211yescharacteristic
299C2×C4#89512yes—
300C8—42no—
301C8—241no—
302C2×C3—22no—
303C2×C3—22no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3#199911yescharacteristic
307C2×C3#201011yescharacteristic
308C2×C3#202211yescharacteristic
309D3—162no—
310C22—22no—
311C22—21no—
312C22—22no—
313C22—21no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—244no—
318C22—244no—
319C22—242no—
320C22—242no—
321C22—242no—
322C22#467111yescharacteristic
323C22#477312yes—
324C22#481012yes—
325C22#486511yescharacteristic
326C22#488311yescharacteristic
327C4—21no—
328C4—22no—
329C4—84no—
330C4—122no—
331C4—122no—
332C4—242no—
333C4#474212yes—
334C3#1032111yescharacteristic, complemented, Sylow
335C2—22no—
336C2—22no—
337C2—21no—
338C2—21no—
339C2—482no—
340C2#6870311yescharacteristic
341C2#6888611yescharacteristic
342C2#6896611yescharacteristic
subgroupG/H
centerC22#4671
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1289C2
socleC22×C3#392
prime radical#78611C2
hypercenter#1289D3
hypocenterC3#10321
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1289

Complements

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

NH ≅ G/Ntype
2. #73999C2semidirect
3. #69910C2semidirect
4. #69509C2semidirect
5. C3×#1289C2semidirect
26. #1289D3semidirect
95. C22×Dic3D8semidirect
334. C3#10321semidirect

Maximal subgroups

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

subgroupindex
#786112
#739992
#699102
#695092
C3×#12892
#103213