Finite Group Databasebeta

← #10695787

Subgroups — #10695787

Order 768.

768 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 83 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#84082C212yesmaximal, complemented
2#78611C211yesmaximal, characteristic, complemented
3C3×#1780C211yesmaximal, characteristic, complemented
4#69921C212yesmaximal, complemented
5#67679C211yesmaximal, characteristic, complemented
6#9974—31nomaximal, Sylow
7C22×C3×#41C2211yescharacteristic
8C2×#949—22no—
9C2×#930C2212yes—
10C2×C3×C22⋊C8—22no—
11C2×C3×C22⋊C8C2211yescharacteristic
12C2×C3×C22⋊C8C2211yescharacteristic
13#5238—24no—
14#5232—22no—
15#5227—22no—
16#5217—22no—
17#5207C2212yes—
18#5115—22no—
19#4736—22no—
20#4732—22no—
21#4468—24no—
22#3186—31no—
23#3006—32no—
24#1780D311yescharacteristic, complemented
25#1471—32no—
26#1293—31no—
27C2×C3×#41—22no—
28C2×C3×#41—22no—
29C2×C3×#41—44no—
30C22×C3×Q8D411yescharacteristic
31C22×C3×D4—21no—
32C22×C3×D4D411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4C2311yescharacteristic
35C22×#67—22no—
36C22×D12D412yes—
37C22×C3×C8—21no—
38C22×C3×C8D412yes—
39#949—48no—
40C2×#215—24no—
41C2×#213—24no—
42#930—22no—
43#930—22no—
44#930—22no—
45#930—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3D412yes—
48C2×#208—22no—
49C2×D24—22no—
50C3×C22⋊C8—22no—
51C3×C22⋊C8—22no—
52C3×C22⋊C8—44no—
53#835—24no—
54#832—24no—
55#830—22no—
56#830D414yes—
57#821—24no—
58#818—22no—
59C22×#41C2×D311yescharacteristic
60C2×AGL(1,ℤ/8)—62no—
61C2×#121—32no—
62#480—62no—
63#466—62no—
64#449—64no—
65#448—62no—
66#447—62no—
67#447—62no—
68#417—62no—
69#407—32no—
70C2×C22⋊C8—22no—
71C2×C22⋊C8C2×D311yescharacteristic
72C2×C22⋊C8C2×D311yescharacteristic
73#323—64no—
74C24×C3—21no—
75C24×C3#4111yescharacteristic
76C23×D3—22no—
77C3×#41—28no—
78C3×#41—44no—
79C3×#41—44no—
80C3×#41—44no—
81C2×C3×Q8—22no—
82C2×C3×Q8—42no—
83C2×C3×D4—22no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—42no—
87C2×C3×D4—42no—
88C22×C3×C4—24no—
89C22×C3×C4—21no—
90C22×C3×C4—42no—
91C22×C3×C4—42no—
92C22×C3×C4—41no—
93C22×C3×C4—41no—
94C22×C3×C4C2×D411yescharacteristic
95C22×C3×C4C2×D412yes—
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D411yescharacteristic
98C2×#67—42no—
99C2×#67—42no—
100C2×#67—84no—
101C22×Dic3—22no—
102C2×D12—22no—
103C2×D12—22no—
104C2×D12—44no—
105C2×C3×C8—22no—
106C2×C3×C8—42no—
107C2×C3×C8—42no—
108C2×C3×C8—41no—
109#215—48no—
110#213—48no—
111(C2×D3)⋊C4—42no—
112(C2×D3)⋊C4—42no—
113C4⋊Dic3—22no—
114C4⋊Dic3—22no—
115#208—44no—
116C2×C3⋊C8—24no—
117D24—82no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—44no—
121C22×Q8D1211yescharacteristic
122C22×D4—21no—
123C22×D4—32no—
124C22×D4—62no—
125C22×D4D1211yescharacteristic
126C23×C4—21no—
127C23×C4C22×D311yescharacteristic
128AGL(1,ℤ/8)—128no—
129C2×SD16—64no—
130C2×D8—62no—
131C2×D8—64no—
132C2×M16—62no—
133C22×C8—21no—
134C22×C8#6712yes—
135#121—62no—
136#121—62no—
137#121—62no—
138#121—62no—
139C2×C4⋊C4—32no—
140C2×C22⋊C4—62no—
141#107—62no—
142#103—64no—
143#102—64no—
144#102—64no—
145C22⋊C8—22no—
146C22⋊C8—22no—
147C22⋊C8—34no—
148C22⋊C8—44no—
149C22⋊C8—62no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3—41no—
153C23×C3—41no—
154C23×C3—41no—
155C23×C3—42no—
156C23×C3—41no—
157C23×C3—42no—
158C23×C3#12112yes—
159C23×C3#12111yescharacteristic
160C22×D3—42no—
161C22×D3—44no—
162C22×D3—82no—
163C22×D3—82no—
164C3×Q8—24no—
165C3×Q8—42no—
166C3×D4—24no—
167C3×D4—42no—
168C3×D4—48no—
169C2×C3×C4—24no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—44no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—44no—
178C2×C3×C4#12012yes—
179C2×C3×C4#12011yescharacteristic
180C2×C3×C4#12711yescharacteristic
181#67—84no—
182#67—84no—
183C2×Dic3—42no—
184C2×Dic3—82no—
185D12—44no—
186D12—44no—
187C3×C8—42no—
188C3×C8—81no—
189C3⋊C8—44no—
190C24—21no—
191C24—62no—
192C24#23511yescharacteristic
193#41—28no—
194#41—44no—
195#41—44no—
196#41—44no—
197C2×Q8—22no—
198C2×Q8—42no—
199C2×D4—22no—
200C2×D4—42no—
201C2×D4—42no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—62no—
205C2×D4—62no—
206C2×D4—124no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—244no—
210C22×C4—24no—
211C22×C4—21no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—62no—
217C22×C4C2×D1211yescharacteristic
218C22×C4D3×D412yes—
219C22×C4D3×D412yes—
220C22×C4C2×#6711yescharacteristic
221SD16—128no—
222D8—128no—
223D8—242no—
224M16—124no—
225C2×C8—22no—
226C2×C8—42no—
227C2×C8—42no—
228C2×C8—41no—
229C2×C8—64no—
230C4⋊C4—62no—
231C4⋊C4—62no—
232C22⋊C4—122no—
233C22⋊C4—122no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—44no—
240C22×C3—42no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—81no—
244C22×C3—81no—
245C22×C3#39211yescharacteristic
246C22×C3#47212yes—
247C2×D3—82no—
248C2×D3—82no—
249C2×D3—82no—
250C2×D3—164no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—42no—
254C3×C4B(4,𝔽2)12yes—
255Dic3—82no—
256C23—22no—
257C23—22no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23—42no—
262C23—42no—
263C23—41no—
264C23—124no—
265C23—122no—
266C23—242no—
267C23—242no—
268C23#88412yes—
269C23#93011yescharacteristic
270Q8—24no—
271Q8—42no—
272D4—24no—
273D4—42no—
274D4—48no—
275D4—124no—
276D4—124no—
277D4—244no—
278D4—244no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—44no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—44no—
288C2×C4—122no—
289C2×C4—242no—
290C2×C4#88212yes—
291C2×C4#93711yescharacteristic
292C2×C4#94011yescharacteristic
293C8—42no—
294C8—81no—
295C8—124no—
296C2×C3—22no—
297C2×C3—42no—
298C2×C3—82no—
299C2×C3#201711yescharacteristic
300C2×C3#201811yescharacteristic
301C2×C3#202111yescharacteristic
302D3—162no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—44no—
309C22—42no—
310C22—82no—
311C22—82no—
312C22—81no—
313C22—81no—
314C22—242no—
315C22—242no—
316C22—242no—
317C22—484no—
318C22#496011yescharacteristic
319C22#514212yes—
320C4—21no—
321C4—22no—
322C4—42no—
323C4—242no—
324C4#503212yes—
325C3#997411yescharacteristic, complemented, Sylow
326C2—22no—
327C2—42no—
328C2—82no—
329C2—482no—
330C2#6950011yescharacteristic
331C2#7000611yescharacteristic
332C2#7001311yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1780C2
socleC22×C3#392
hypercenter#1780D3
hypocenterC3#9974
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1780

Complements

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

NH ≅ G/Ntype
1. #84082C2semidirect
2. #78611C2semidirect
3. C3×#1780C2semidirect
4. #69921C2semidirect
5. #67679C2semidirect
24. #1780D3semidirect
325. C3#9974semidirect

Maximal subgroups

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

subgroupindex
#840822
#786112
C3×#17802
#699212
#676792
#99743