Finite Group Databasebeta

← #10695785

Subgroups — #10695785

Order 768.

768 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 85 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×#3186C211yesmaximal, characteristic, complemented
2#82624C212yesmaximal, complemented
3#71676C212yesmaximal, complemented
4#69956C211yesmaximal, characteristic, complemented
5#67679C211yesmaximal, characteristic, complemented
6#9974—31nomaximal, Sylow
7C22×#235C2211yescharacteristic
8C2×C3×#121C2212yes—
9C2×#932—22no—
10C3×#480—22no—
11C3×#466—22no—
12C3×#447—22no—
13C2×C3×C22⋊C8C2211yescharacteristic
14#5207C2212yes—
15C2×#830C2211yescharacteristic
16C2×(C2×D3)⋊C8—22no—
17#5040—22no—
18#5037—24no—
19#5036—22no—
20#4966—22no—
21#4525—24no—
22#3186D311yescharacteristic, complemented
23#3006—32no—
24#1780—31no—
25#1471—32no—
26#1293—31no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4C2311yescharacteristic
30C22×#67—21no—
31C2×#235—22no—
32C2×#235—22no—
33C2×#235—44no—
34C22×D12D411yescharacteristic
35C22×C4×D3—21no—
36C22×Dic6D411yescharacteristic
37C2×C3×D8—22no—
38C22×C3×C8D412yes—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—22no—
42C3×#121—22no—
43C2×C3×C4⋊C4D412yes—
44C2×C3×C22⋊C4—22no—
45C2×#214—24no—
46#932—48no—
47C2×#213—24no—
48C2×#208—22no—
49C22×C3⋊C8—21no—
50C3×#107—22no—
51C3×#102—24no—
52C3×C22⋊C8—22no—
53#830—22no—
54#830—22no—
55#830D414yes—
56(C2×D3)⋊C8—44no—
57#810—24no—
58#809—24no—
59C22×#41—31no—
60C2×AGL(1,ℤ/8)—62no—
61C2×#121C2×D312yes—
62#480—22no—
63#466—22no—
64#449—64no—
65#448—62no—
66#447—22no—
67#447—62no—
68#417—62no—
69#407—32no—
70C2×C22⋊C8—31no—
71C2×C22⋊C8—62no—
72C2×C22⋊C8C2×D311yescharacteristic
73#323—64no—
74C24×C3—22no—
75C24×C3#4111yescharacteristic
76C23×D3—21no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—44no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—84no—
83C22×C3×C4—22no—
84C22×C3×C4—42no—
85C22×C3×C4C2×D411yescharacteristic
86C22×C3×C4C2×D412yes—
87C22×C3×C4C2×D412yes—
88C22×C3×C4C2×D411yescharacteristic
89C2×#67—42no—
90C2×#67—42no—
91C2×#67—42no—
92C22×Dic3—21no—
93#235—28no—
94#235—44no—
95#235—44no—
96#235—44no—
97C2×D12—22no—
98C2×D12—42no—
99C2×C4×D3—24no—
100C2×C4×D3—42no—
101C2×C4×D3—41no—
102C2×C4×D3—41no—
103C2×Dic6—22no—
104C2×Dic6—42no—
105C3×D8—82no—
106C2×C3×C8—22no—
107C2×C3×C8—42no—
108C3×C4⋊C4—22no—
109C3×C4⋊C4—22no—
110C3×C22⋊C4—42no—
111C3×C22⋊C4—42no—
112#214—48no—
113#213—48no—
114#208—44no—
115C2×C3⋊C8—24no—
116C2×C3⋊C8—42no—
117C2×C3⋊C8—41no—
118C2×#41—62no—
119C2×#41—62no—
120C2×#41—124no—
121C22×Q8—31no—
122C22×D4—22no—
123C22×D4—31no—
124C22×D4—61no—
125C22×D4#6712yes—
126C23×C4—61no—
127C23×C4C22×D311yescharacteristic
128AGL(1,ℤ/8)—128no—
129C2×SD16—64no—
130C2×D8—22no—
131C2×D8—64no—
132C2×M16—62no—
133C22×C8—61no—
134C22×C8#6712yes—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—22no—
139C2×C4⋊C4#6712yes—
140C2×C22⋊C4—22no—
141#107—22no—
142#103—64no—
143#102—24no—
144#102—64no—
145C22⋊C8—22no—
146C22⋊C8—34no—
147C22⋊C8—62no—
148C22⋊C8—62no—
149C22⋊C8—124no—
150C23×C3—22no—
151C23×C3—41no—
152C23×C3—41no—
153C23×C3—44no—
154C23×C3—42no—
155C23×C3—82no—
156C23×C3—82no—
157C23×C3#12111yescharacteristic
158C23×C3#12112yes—
159C22×D3—22no—
160C22×D3—41no—
161C22×D3—42no—
162C22×D3—41no—
163C22×D3—42no—
164C3×D4—44no—
165C3×D4—44no—
166C3×D4—84no—
167C3×D4—84no—
168C2×C3×C4—22no—
169C2×C3×C4—24no—
170C2×C3×C4—41no—
171C2×C3×C4—42no—
172C2×C3×C4—42no—
173C2×C3×C4—82no—
174C2×C3×C4#12012yes—
175C2×C3×C4#12011yescharacteristic
176C2×C3×C4#12711yescharacteristic
177#67—48no—
178C2×Dic3—22no—
179C2×Dic3—41no—
180C2×Dic3—41no—
181D12—24no—
182D12—42no—
183C4×D3—44no—
184C4×D3—44no—
185Dic6—24no—
186Dic6—42no—
187C3×C8—42no—
188C3⋊C8—44no—
189C3⋊C8—81no—
190C24—22no—
191C24—61no—
192C24#23711yescharacteristic
193#41—68no—
194#41—124no—
195#41—124no—
196#41—124no—
197C2×Q8—62no—
198C2×Q8—122no—
199C2×D4—22no—
200C2×D4—22no—
201C2×D4—44no—
202C2×D4—42no—
203C2×D4—42no—
204C2×D4—62no—
205C2×D4—84no—
206C2×D4—122no—
207C2×D4—122no—
208C2×D4—122no—
209C2×D4—122no—
210C22×C4—22no—
211C22×C4—42no—
212C22×C4—64no—
213C22×C4—61no—
214C22×C4—122no—
215C22×C4—121no—
216C22×C4—121no—
217C22×C4D3×D411yescharacteristic
218C22×C4D3×D412yes—
219C22×C4C2×#6712yes—
220C22×C4C2×#6711yescharacteristic
221SD16—128no—
222D8—82no—
223D8—128no—
224M16—124no—
225C2×C8—22no—
226C2×C8—42no—
227C2×C8—64no—
228C2×C8—122no—
229C2×C8—121no—
230C4⋊C4—22no—
231C4⋊C4—22no—
232C22⋊C4—42no—
233C22⋊C4—42no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—82no—
240C22×C3—82no—
241C22×C3—82no—
242C22×C3—82no—
243C22×C3—164no—
244C22×C3#39211yescharacteristic
245C22×C3#47212yes—
246C2×D3—44no—
247C2×D3—42no—
248C2×D3—82no—
249C2×D3—81no—
250C2×D3—81no—
251C3×C4—22no—
252C3×C4—21no—
253C3×C4—82no—
254C3×C4B(4,𝔽2)12yes—
255Dic3—42no—
256C23—22no—
257C23—41no—
258C23—41no—
259C23—44no—
260C23—42no—
261C23—62no—
262C23—82no—
263C23—82no—
264C23—121no—
265C23—122no—
266C23—122no—
267C23—121no—
268C23#93811yescharacteristic
269C23#93912yes—
270Q8—64no—
271Q8—122no—
272D4—44no—
273D4—44no—
274D4—64no—
275D4—84no—
276D4—84no—
277D4—122no—
278D4—128no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—41no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—62no—
285C2×C4—82no—
286C2×C4—124no—
287C2×C4—121no—
288C2×C4—121no—
289C2×C4—124no—
290C2×C4#93712yes—
291C2×C4#94011yescharacteristic
292C2×C4#95311yescharacteristic
293C8—42no—
294C8—124no—
295C8—241no—
296C2×C3—22no—
297C2×C3—42no—
298C2×C3—162no—
299C2×C3#201711yescharacteristic
300C2×C3#201811yescharacteristic
301C2×C3#202111yescharacteristic
302D3—82no—
303C22—21no—
304C22—21no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—82no—
309C22—82no—
310C22—82no—
311C22—82no—
312C22—124no—
313C22—122no—
314C22—164no—
315C22—242no—
316C22—241no—
317C22—241no—
318C22#516012yes—
319C22#522311yescharacteristic
320C4—22no—
321C4—21no—
322C4—82no—
323C4—122no—
324C4#524412yes—
325C3#997411yescharacteristic, complemented, Sylow
326C2—22no—
327C2—42no—
328C2—162no—
329C2—242no—
330C2#7185111yescharacteristic
331C2#7204611yescharacteristic
332C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3186C2
socleC22×C3#392
hypercenter#3186D3
hypocenterC3#9974
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. C3×#3186C2semidirect
2. #82624C2semidirect
3. #71676C2semidirect
4. #69956C2semidirect
5. #67679C2semidirect
22. #3186D3semidirect
325. C3#9974semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#826242
#716762
#699562
#676792
#99743