Finite Group Databasebeta

← #10695729

Subgroups — #10695729

Order 768.

944 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 121 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×#2656C211yesmaximal, characteristic
2C2×#5040C212yesmaximal, complemented
3C2×#4537C212yesmaximal, complemented
4#69424C211yesmaximal, characteristic, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9964—31nomaximal, Sylow
7C23×Dic6C2211yescharacteristic
8C2×C3×#121—22no—
9C2×C3×#121C2212yes—
10C22×#214—22no—
11C3×#399—22no—
12C3×#392—22no—
13C2×C3×#95C2211yescharacteristic
14C2×C3⋊#95—22no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5040—24no—
18#5040—24no—
19C2×#810—22no—
20#4537—24no—
21C2×#448—32no—
22#2656D311yescharacteristic
23C2×#327—32no—
24#1446—31no—
25#1280—31no—
26C22×C3×D4—22no—
27C22×C3×D4—22no—
28C22×C3×D4D412yes—
29C23×C3×C4D412yes—
30C23×C3×C4C2311yescharacteristic
31C23×Dic3—21no—
32C23×Dic3—21no—
33C22×Dic6—22no—
34C22×Dic6—22no—
35C22×Dic6—21no—
36C22×Dic6—44no—
37C22×Dic6D412yescomplemented
38C3×#121—22no—
39C3×#121—22no—
40C3×#121—22no—
41C3×#121—22no—
42C3×#121—44no—
43C3×#121—44no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4D412yes—
46C2×C3×C22⋊C4—24no—
47C2×C3×C22⋊C4—22no—
48C2×#214—28no—
49C2×#214—48no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3⋊#95—22no—
53C3⋊#95—44no—
54#830—22no—
55#830D414yescomplemented
56#810—28no—
57C23×Q8—31no—
58C22×SD16—62no—
59C2×#121—22no—
60C2×#121C2×D312yes—
61#448—64no—
62#448—64no—
63C2×#103—62no—
64C2×C22⋊C8—32no—
65#399—22no—
66#392—22no—
67C2×#95—31no—
68C2×#95—62no—
69C2×#95C2×D311yescharacteristic
70#327—64no—
71C24×C3—22no—
72C24×C3C2×D411yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—44no—
76C2×C3×D4—44no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—44no—
80C2×C3×D4—44no—
81C2×C3×D4—82no—
82C2×C3×D4—82no—
83C22×C3×C4—22no—
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—42no—
87C22×C3×C4—42no—
88C22×C3×C4—42no—
89C22×C3×C4SD1614yescomplemented
90C22×C3×C4C2×D411yescharacteristic
91C22×C3×C4C2×D412yes—
92C22×C3×C4C2×D412yes—
93C22×C3×C4#4111yescharacteristic
94C22×Dic3—24no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—42no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—42no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3—42no—
104C2×Dic6—28no—
105C2×Dic6—24no—
106C2×Dic6—44no—
107C2×Dic6—48no—
108C2×Dic6—44no—
109C2×Dic6—44no—
110C2×Dic6—42no—
111C3×C4⋊C4—22no—
112C3×C4⋊C4—22no—
113C3×C4⋊C4—44no—
114C3×C22⋊C4—48no—
115C3×C22⋊C4—42no—
116C3×C22⋊C4—42no—
117#214—416no—
118C2×C3⋊C8—24no—
119C2×C3⋊C8—44no—
120C22×Q8—32no—
121C22×Q8—61no—
122C22×Q8—62no—
123C22×Q8—62no—
124C22×Q8—124no—
125C22×D4—22no—
126C22×D4—22no—
127C22×D4#6712yes—
128C23×C4—61no—
129C23×C4—61no—
130C23×C4#6712yes—
131C23×C4C22×D311yescharacteristic
132C2×SD16—68no—
133C2×SD16—128no—
134C22×C8—62no—
135#121—22no—
136#121—22no—
137#121—22no—
138#121—22no—
139#121—44no—
140#121—44no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4#6712yes—
143C2×C22⋊C4—24no—
144C2×C22⋊C4—22no—
145#103—68no—
146C22⋊C8—34no—
147C22⋊C8—62no—
148#95—22no—
149#95—62no—
150#95—124no—
151C23×C3—21no—
152C23×C3—21no—
153C23×C3—22no—
154C23×C3—42no—
155C23×C3—44no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—44no—
159C23×C3#12011yescharacteristic
160C23×C3#12111yescharacteristic
161C23×C3#12711yescharacteristic
162C23×C3C2×SD1612yes—
163C23×C3AGL(1,ℤ/8)12yes—
164C3×D4—44no—
165C3×D4—82no—
166C3×D4—88no—
167C3×D4—84no—
168C3×D4—82no—
169C3×D4—82no—
170C2×C3×C4—24no—
171C2×C3×C4—24no—
172C2×C3×C4—41no—
173C2×C3×C4—42no—
174C2×C3×C4—41no—
175C2×C3×C4—42no—
176C2×C3×C4—42no—
177C2×C3×C4—44no—
178C2×C3×C4—42no—
179C2×C3×C4—44no—
180C2×C3×C4—82no—
181C2×C3×C4#12012yes—
182C2×C3×C4#12112yes—
183C2×Dic3—24no—
184C2×Dic3—42no—
185C2×Dic3—44no—
186C2×Dic3—42no—
187C2×Dic3—44no—
188C2×Dic3—44no—
189C2×Dic3—42no—
190C2×Dic3—82no—
191C2×Dic3—81no—
192C2×Dic3—81no—
193Dic6—28no—
194Dic6—44no—
195Dic6—48no—
196C3⋊C8—44no—
197C24—22no—
198C24C2×#6711yescharacteristic
199C2×Q8—68no—
200C2×Q8—64no—
201C2×Q8—124no—
202C2×Q8—124no—
203C2×Q8—124no—
204C2×Q8—128no—
205C2×Q8—122no—
206C2×D4—22no—
207C2×D4—22no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—44no—
211C2×D4—44no—
212C2×D4—42no—
213C2×D4—42no—
214C2×D4—82no—
215C2×D4—82no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—64no—
223C22×C4—62no—
224C22×C4—62no—
225C22×C4—122no—
226C22×C4—121no—
227C22×C4—121no—
228C22×C4—122no—
229C22×C4—121no—
230C22×C4—122no—
231C22×C4—121no—
232C22×C4#21414yescomplemented
233C22×C4D3×D411yescharacteristic
234C22×C4D3×D412yes—
235C22×C4#23711yescharacteristic
236C22×C4C2×#6712yes—
237SD16—1216no—
238C2×C8—64no—
239C2×C8—124no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—44no—
243C22⋊C4—48no—
244C22⋊C4—42no—
245C22⋊C4—42no—
246C22×C3—22no—
247C22×C3—22no—
248C22×C3—22no—
249C22×C3—22no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3—84no—
255C22×C3—84no—
256C22×C3—82no—
257C22×C3—82no—
258C22×C3—82no—
259C22×C3#39211yescharacteristic
260C22×C3#45012yes—
261C22×C3#45511yescharacteristic
262C22×C3#45611yescharacteristic
263C22×C3B(4,𝔽2)12yes—
264C22×C3#46012yes—
265C22×C3#49211yescharacteristic
266C22×C3#49611yescharacteristic
267C3×C4—22no—
268C3×C4—22no—
269C3×C4—44no—
270C3×C4—82no—
271Dic3—44no—
272Dic3—82no—
273C23—21no—
274C23—22no—
275C23—21no—
276C23—44no—
277C23—42no—
278C23—42no—
279C23—44no—
280C23—42no—
281C23#93212yes—
282C23C2×#21412yes—
283C23#93811yescharacteristic
284C23#94011yescharacteristic
285C23#95311yescharacteristic
286Q8—68no—
287Q8—124no—
288Q8—128no—
289D4—44no—
290D4—82no—
291D4—88no—
292D4—82no—
293D4—84no—
294D4—82no—
295C2×C4—24no—
296C2×C4—24no—
297C2×C4—41no—
298C2×C4—42no—
299C2×C4—41no—
300C2×C4—42no—
301C2×C4—42no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—42no—
305C2×C4—64no—
306C2×C4—82no—
307C2×C4—122no—
308C2×C4—124no—
309C2×C4—122no—
310C2×C4—124no—
311C2×C4—124no—
312C2×C4—122no—
313C2×C4—242no—
314C2×C4—241no—
315C2×C4—241no—
316C2×C4#93712yes—
317C2×C4#93912yes—
318C8—124no—
319C2×C3—21no—
320C2×C3—22no—
321C2×C3—21no—
322C2×C3—162no—
323C2×C3#160112yes—
324C2×C3#160412yes—
325C2×C3#200711yescharacteristic
326C2×C3#201411yescharacteristic
327C2×C3#202111yescharacteristic
328C22—22no—
329C22—22no—
330C22—21no—
331C22—21no—
332C22—22no—
333C22—22no—
334C22—21no—
335C22—21no—
336C22—82no—
337C22—84no—
338C22—82no—
339C22—84no—
340C22—82no—
341C22#507411yescharacteristic
342C22#507512yes—
343C22#507611yescharacteristic
344C22#521812yes—
345C22#522311yescharacteristic
346C22#524211yescharacteristic
347C22#524412yes—
348C22#524611yescharacteristic
349C4—22no—
350C4—22no—
351C4—44no—
352C4—82no—
353C4—124no—
354C4—242no—
355C3#996411yescharacteristic, complemented, Sylow
356C2—21no—
357C2—22no—
358C2—21no—
359C2—162no—
360C2#7176811yescharacteristic
361C2#7184612yes—
362C2#7185111yescharacteristic
363C2#7188012yes—
364C2#7188111yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2656C2
socleC23×C3#120
hypercenter#2656D3
hypocenterC3#9964
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2656

Complements

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

NH ≅ G/Ntype
2. C2×#5040C2semidirect
3. C2×#4537C2semidirect
4. #69424C2semidirect
5. #67613C2semidirect
37. C22×Dic6D4semidirect
55. #830D4semidirect
89. C22×C3×C4SD16semidirect
232. C22×C4#214semidirect
355. C3#9964semidirect

Maximal subgroups

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

subgroupindex
C3×#26562
C2×#50402
C2×#45372
#694242
#676132
#99643