Finite Group Databasebeta

← #10695751

Subgroups — #10695751

Order 768.

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

normal subgroups only

This group has 772 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×#5034C212yesmaximal, complemented
3C2×#4537C212yesmaximal, complemented
4#69423C211yesmaximal, characteristic, complemented
5#67613C211yesmaximal, characteristic, complemented
6#9968—31nomaximal, Sylow
7C2×C3×#121—22no—
8C2×C3×#121C2212yes—
9C22×C4⋊Dic3C2211yescharacteristic
10C3×#399—22no—
11C3×#392—22no—
12C2×C3×#95C2211yescharacteristic
13C2×#832—22no—
14C2×C3⋊#95—22no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#5034—24no—
18#5034—24no—
19C2×#808—22no—
20#4537—24no—
21C2×#481—32no—
22#2656D311yescharacteristic
23C2×#327—32no—
24#1445—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—
33C3×#121—22no—
34C3×#121—22no—
35C3×#121—22no—
36C3×#121—22no—
37C3×#121—44no—
38C3×#121—44no—
39C2×C3×C4⋊C4—22no—
40C2×C3×C4⋊C4D412yes—
41C2×C3×C22⋊C4—24no—
42C2×C3×C22⋊C4—22no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3—22no—
45C2×C4⋊Dic3—21no—
46C2×C4⋊Dic3D412yescomplemented
47C22×C3⋊C8—22no—
48C3×#95—22no—
49#832—28no—
50C3⋊#95—22no—
51C3⋊#95—44no—
52#830—22no—
53#830D414yescomplemented
54#808—28no—
55C2×#121—22no—
56C2×#121C2×D312yes—
57C22×C4⋊C4—31no—
58#481—64no—
59#481—64no—
60C2×#106—62no—
61C2×#102—62no—
62C2×C22⋊C8—32no—
63#399—22no—
64#392—22no—
65C2×#95—31no—
66C2×#95—62no—
67C2×#95C2×D311yescharacteristic
68#327—64no—
69C24×C3—22no—
70C24×C3C2×D411yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—22no—
73C2×C3×D4—44no—
74C2×C3×D4—44no—
75C2×C3×D4—42no—
76C2×C3×D4—42no—
77C2×C3×D4—44no—
78C2×C3×D4—44no—
79C2×C3×D4—82no—
80C2×C3×D4—82no—
81C22×C3×C4—22no—
82C22×C3×C4—22no—
83C22×C3×C4—22no—
84C22×C3×C4—42no—
85C22×C3×C4—42no—
86C22×C3×C4—42no—
87C22×C3×C4SD1614yes—
88C22×C3×C4C2×D411yescharacteristic
89C22×C3×C4C2×D412yes—
90C22×C3×C4C2×D412yes—
91C22×C3×C4#4111yescharacteristic
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—42no—
95C22×Dic3—41no—
96C22×Dic3—42no—
97C22×Dic3—42no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C22×Dic3—42no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—22no—
104C3×C4⋊C4—44no—
105C3×C22⋊C4—48no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108C4⋊Dic3—24no—
109C4⋊Dic3—42no—
110C2×C3⋊C8—24no—
111C2×C3⋊C8—44no—
112C22×D4—22no—
113C22×D4—22no—
114C22×D4#6712yes—
115C23×C4—61no—
116C23×C4—61no—
117C23×C4#6712yes—
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120#121—22no—
121#121—22no—
122#121—22no—
123#121—22no—
124#121—44no—
125#121—44no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—32no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—62no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4#6712yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—22no—
134#106—68no—
135#102—68no—
136C22⋊C8—34no—
137C22⋊C8—62no—
138#95—22no—
139#95—62no—
140#95—124no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—22no—
144C23×C3—42no—
145C23×C3—44no—
146C23×C3—42no—
147C23×C3—42no—
148C23×C3—44no—
149C23×C3#12011yescharacteristic
150C23×C3#12111yescharacteristic
151C23×C3#12711yescharacteristic
152C23×C3C2×SD1612yes—
153C23×C3AGL(1,ℤ/8)12yes—
154C3×D4—44no—
155C3×D4—82no—
156C3×D4—88no—
157C3×D4—84no—
158C3×D4—82no—
159C3×D4—82no—
160C2×C3×C4—24no—
161C2×C3×C4—24no—
162C2×C3×C4—41no—
163C2×C3×C4—42no—
164C2×C3×C4—41no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—42no—
169C2×C3×C4—44no—
170C2×C3×C4—82no—
171C2×C3×C4#12012yes—
172C2×C3×C4#12112yes—
173C2×Dic3—44no—
174C2×Dic3—42no—
175C2×Dic3—44no—
176C2×Dic3—42no—
177C2×Dic3—82no—
178C2×Dic3—81no—
179C2×Dic3—82no—
180C2×Dic3—81no—
181C2×Dic3—81no—
182C2×Dic3—81no—
183C3⋊C8—44no—
184C24—22no—
185C24C2×#6711yescharacteristic
186C2×D4—22no—
187C2×D4—22no—
188C2×D4—44no—
189C2×D4—44no—
190C2×D4—44no—
191C2×D4—44no—
192C2×D4—42no—
193C2×D4—42no—
194C2×D4—82no—
195C2×D4—82no—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4—22no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—62no—
203C22×C4—62no—
204C22×C4—122no—
205C22×C4—122no—
206C22×C4—121no—
207C22×C4—122no—
208C22×C4—122no—
209C22×C4—121no—
210C22×C4—121no—
211C22×C4—121no—
212C22×C4#21414yes—
213C22×C4D3×D411yescharacteristic
214C22×C4D3×D412yes—
215C22×C4#23711yescharacteristic
216C22×C4C2×#6712yes—
217C2×C8—64no—
218C2×C8—124no—
219C4⋊C4—22no—
220C4⋊C4—22no—
221C4⋊C4—44no—
222C4⋊C4—64no—
223C4⋊C4—122no—
224C22⋊C4—48no—
225C22⋊C4—42no—
226C22⋊C4—42no—
227C22×C3—22no—
228C22×C3—22no—
229C22×C3—22no—
230C22×C3—22no—
231C22×C3—21no—
232C22×C3—21no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—84no—
236C22×C3—84no—
237C22×C3—82no—
238C22×C3—82no—
239C22×C3—82no—
240C22×C3#39211yescharacteristic
241C22×C3#45012yes—
242C22×C3#45311yescharacteristic
243C22×C3#45411yescharacteristic
244C22×C3#45812yes—
245C22×C3#46012yes—
246C22×C3#49211yescharacteristic
247C22×C3#49611yescharacteristic
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—44no—
251C3×C4—82no—
252Dic3—82no—
253Dic3—82no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—44no—
258C23—42no—
259C23—42no—
260C23—44no—
261C23—42no—
262C23#93212yes—
263C23C2×#21412yes—
264C23#93811yescharacteristic
265C23#94011yescharacteristic
266C23#95311yescharacteristic
267D4—44no—
268D4—82no—
269D4—88no—
270D4—82no—
271D4—84no—
272D4—82no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—41no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—42no—
283C2×C4—82no—
284C2×C4—124no—
285C2×C4—122no—
286C2×C4—124no—
287C2×C4—122no—
288C2×C4—241no—
289C2×C4—242no—
290C2×C4—242no—
291C2×C4—241no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4#93712yes—
295C2×C4#93912yes—
296C8—124no—
297C2×C3—21no—
298C2×C3—22no—
299C2×C3—21no—
300C2×C3—162no—
301C2×C3#161312yes—
302C2×C3#161612yes—
303C2×C3#200611yescharacteristic
304C2×C3#201411yescharacteristic
305C2×C3#202211yescharacteristic
306C22—22no—
307C22—22no—
308C22—21no—
309C22—21no—
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—82no—
315C22—84no—
316C22—82no—
317C22—84no—
318C22—82no—
319C22#507411yescharacteristic
320C22#507512yes—
321C22#507611yescharacteristic
322C22#521812yes—
323C22#522311yescharacteristic
324C22#524111yescharacteristic
325C22#524312yes—
326C22#524511yescharacteristic
327C4—22no—
328C4—22no—
329C4—44no—
330C4—82no—
331C4—242no—
332C4—242no—
333C3#996811yescharacteristic, complemented, Sylow
334C2—21no—
335C2—22no—
336C2—21no—
337C2—162no—
338C2#7176811yescharacteristic
339C2#7184412yes—
340C2#7184911yescharacteristic
341C2#7187512yes—
342C2#7187911yescharacteristic
subgroupG/H
centerC23#953
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2656C2
socleC23×C3#120
hypercenter#2656D3
hypocenterC3#9968
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×#5034C2semidirect
3. C2×#4537C2semidirect
4. #69423C2semidirect
5. #67613C2semidirect
46. C2×C4⋊Dic3D4semidirect
53. #830D4semidirect
333. C3#9968semidirect

Maximal subgroups

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

subgroupindex
C3×#26562
C2×#50342
C2×#45372
#694232
#676132
#99683