Finite Group Databasebeta

← #10690246

Subgroups — #10690246

Order 768.

660 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 660 conjugacy classes of subgroups — too many to draw. Only its 147 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×#2316C211yesmaximal, characteristic
2C2×#4981C211yesmaximal, characteristic
3C2×#4528C211yesmaximal, characteristic
4#69501C212yesmaximal
5#67624C212yesmaximal
6#9027—31nomaximal, Sylow
7C2×C3×#117—21no—
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×C4⋊Dic3C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C3×#389—24no—
12C3×#380—21no—
13C3×#380C2212yes—
14C2×C3⋊#95—21no—
15C2×C3⋊#95C2211yescharacteristic
16C2×#830C2212yes—
17#4981—22no—
18#4981C414yescomplemented
19#4529—24no—
20#4528—22no—
21C2×#398—31no—
22#2316D311yescharacteristic
23#1823—32no—
24C2×#324—31no—
25#1294—32no—
26C23×C3×C4D411yescharacteristic
27C23×C3×C4D411yescharacteristic
28C23×C3×C4C2311yescharacteristic
29C23×Dic3—21no—
30C23×Dic3—21no—
31C3×#117—42no—
32C3×#117—42no—
33C2×C3×C4⋊C4—22no—
34C2×C3×C4⋊C4—22no—
35C2×C3×C4⋊C4—22no—
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4C2×C412yes—
38C2×C3×C4⋊C4D412yes—
39C2×C3×C4⋊C4D414yes—
40C2×C3×C22⋊C4—22no—
41C2×C3×C42—22no—
42C2×C4⋊Dic3—22no—
43C2×C4⋊Dic3—22no—
44C2×C4⋊Dic3—21no—
45C2×C4⋊Dic3C2×C412yes—
46C2×Dic3⋊C4—24no—
47C2×Dic3⋊C4—42no—
48C2×Dic3⋊C4—42no—
49C22×C3⋊C8—22no—
50C3×#95—22no—
51C3×#95—22no—
52C3⋊#95—24no—
53C3⋊#95—21no—
54C3⋊#95C2×C412yes—
55#830—22no—
56#830—22no—
57C2×#117—21no—
58C22×C4⋊C4—31no—
59C22×C4⋊C4—61no—
60C22×C4⋊C4C2×D311yescharacteristic
61C2×C22⋊C8—32no—
62#398—34no—
63#398—62no—
64#389—24no—
65#380—21no—
66#380C2×D312yes—
67C2×#95—31no—
68C2×#95—61no—
69#340—64no—
70#324—62no—
71C24×C3C2×D411yescharacteristic
72C22×C3×C4—21no—
73C22×C3×C4—21no—
74C22×C3×C4—21no—
75C22×C3×C4—21no—
76C22×C3×C4—21no—
77C22×C3×C4—21no—
78C22×C3×C4—24no—
79C22×C3×C4—42no—
80C22×C3×C4—42no—
81C22×C3×C4C22⋊C412yes—
82C22×C3×C4C22⋊C412yes—
83C22×C3×C4D812yes—
84C22×C3×C4SD1612yes—
85C22×C3×C4C22×C411yescharacteristic
86C22×C3×C4C2×D411yescharacteristic
87C22×C3×C4C2×D412yes—
88C22×C3×C4#4112yes—
89C22×Dic3—24no—
90C22×Dic3—22no—
91C22×Dic3—22no—
92C22×Dic3—42no—
93C22×Dic3—41no—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C22×Dic3—41no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3—41no—
101C3×C4⋊C4—24no—
102C3×C4⋊C4—24no—
103C3×C4⋊C4—44no—
104C3×C4⋊C4—44no—
105C3×C4⋊C4—82no—
106C3×C22⋊C4—42no—
107C3×C22⋊C4—42no—
108C3×C42—44no—
109C4⋊Dic3—24no—
110C4⋊Dic3—42no—
111Dic3⋊C4—48no—
112C2×C3⋊C8—44no—
113C2×C3⋊C8—42no—
114C23×C4—61no—
115C23×C4—61no—
116C23×C4#6711yescharacteristic
117C23×C4#6711yescharacteristic
118C23×C4C22×D311yescharacteristic
119C22×C8—62no—
120#117—42no—
121#117—42no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—32no—
127C2×C4⋊C4—62no—
128C2×C4⋊C4—62no—
129C2×C4⋊C4—64no—
130C2×C4⋊C4—61no—
131C2×C4⋊C4—122no—
132C2×C4⋊C4—122no—
133C2×C4⋊C4C4×D312yes—
134C2×C4⋊C4D1212yes—
135C2×C4⋊C4#6714yes—
136C2×C22⋊C4—22no—
137C2×C42—22no—
138C22⋊C8—62no—
139C22⋊C8—62no—
140#95—22no—
141#95—22no—
142#95—32no—
143#95—64no—
144#95—61no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3C2×C22⋊C411yescharacteristic
148C23×C3#12011yescharacteristic
149C23×C3#12311yescharacteristic
150C23×C3C2×D811yescharacteristic
151C23×C3C2×SD1611yescharacteristic
152C23×C3AGL(1,ℤ/8)11yescharacteristic
153C23×C3#13711yescharacteristic
154C2×C3×C4—22no—
155C2×C3×C4—22no—
156C2×C3×C4—22no—
157C2×C3×C4—22no—
158C2×C3×C4—24no—
159C2×C3×C4—22no—
160C2×C3×C4—22no—
161C2×C3×C4—22no—
162C2×C3×C4—41no—
163C2×C3×C4—41no—
164C2×C3×C4—42no—
165C2×C3×C4—42no—
166C2×C3×C4—42no—
167C2×C3×C4—44no—
168C2×C3×C4—44no—
169C2×C3×C4—44no—
170C2×C3×C4—81no—
171C2×C3×C4#10214yes—
172C2×C3×C4C4×D412yes—
173C2×C3×C4#12112yes—
174C2×Dic3—24no—
175C2×Dic3—44no—
176C2×Dic3—44no—
177C2×Dic3—42no—
178C2×Dic3—42no—
179C2×Dic3—42no—
180C2×Dic3—42no—
181C2×Dic3—42no—
182C2×Dic3—81no—
183C2×Dic3—81no—
184C2×Dic3—81no—
185C2×Dic3—81no—
186C3⋊C8—82no—
187C24C2×#6711yescharacteristic
188C22×C4—21no—
189C22×C4—21no—
190C22×C4—21no—
191C22×C4—21no—
192C22×C4—21no—
193C22×C4—21no—
194C22×C4—24no—
195C22×C4—42no—
196C22×C4—42no—
197C22×C4—64no—
198C22×C4—62no—
199C22×C4—62no—
200C22×C4—122no—
201C22×C4—121no—
202C22×C4—121no—
203C22×C4—121no—
204C22×C4—121no—
205C22×C4—121no—
206C22×C4—121no—
207C22×C4—121no—
208C22×C4—121no—
209C22×C4(C2×D3)⋊C412yes—
210C22×C4(C2×D3)⋊C412yes—
211C22×C4#21312yes—
212C22×C4#21512yes—
213C22×C4C2×C4×D311yescharacteristic
214C22×C4C2×D1211yescharacteristic
215C22×C4#23512yes—
216C22×C4C2×#6712yes—
217C2×C8—124no—
218C2×C8—122no—
219C4⋊C4—24no—
220C4⋊C4—24no—
221C4⋊C4—44no—
222C4⋊C4—44no—
223C4⋊C4—64no—
224C4⋊C4—82no—
225C4⋊C4—128no—
226C4⋊C4—122no—
227C22⋊C4—42no—
228C22⋊C4—42no—
229C42—44no—
230C22×C3—22no—
231C22×C3—22no—
232C22×C3—22no—
233C22×C3—22no—
234C22×C3—21no—
235C22×C3—21no—
236C22×C3—21no—
237C22×C3—21no—
238C22×C3#38611yescharacteristic
239C22×C3C2×#10211yescharacteristic
240C22×C3#41711yescharacteristic
241C22×C3#44711yescharacteristic
242C22×C3#44811yescharacteristic
243C22×C3#45612yes—
244C22×C3#45812yes—
245C22×C3#48011yescharacteristic
246C22×C3#48111yescharacteristic
247C3×C4—24no—
248C3×C4—22no—
249C3×C4—22no—
250C3×C4—42no—
251C3×C4—84no—
252Dic3—44no—
253Dic3—82no—
254C23—22no—
255C23—22no—
256C23C2×(C2×D3)⋊C411yescharacteristic
257C23#92911yescharacteristic
258C23C2×#21311yescharacteristic
259C23#93211yescharacteristic
260C23C2×#21511yescharacteristic
261C23#94211yescharacteristic
262C23#95311yescharacteristic
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—24no—
270C2×C4—22no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—44no—
279C2×C4—64no—
280C2×C4—81no—
281C2×C4—124no—
282C2×C4—124no—
283C2×C4—122no—
284C2×C4—122no—
285C2×C4—122no—
286C2×C4—122no—
287C2×C4—122no—
288C2×C4—241no—
289C2×C4—241no—
290C2×C4—241no—
291C2×C4—241no—
292C2×C4#80914yes—
293C2×C4C4×#6712yes—
294C2×C4#93012yes—
295C8—242no—
296C2×C3—21no—
297C2×C3—21no—
298C2×C3—21no—
299C2×C3—21no—
300C2×C3#161012yes—
301C2×C3#161112yes—
302C2×C3#187411yescharacteristic
303C2×C3#188611yescharacteristic
304C2×C3#189011yescharacteristic
305C22—22no—
306C22—22no—
307C22—22no—
308C22—22no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22—21no—
313C22C2×#80911yescharacteristic
314C22#496611yescharacteristic
315C22#496711yescharacteristic
316C22#496811yescharacteristic
317C22#521311yescharacteristic
318C22#521711yescharacteristic
319C22#522711yescharacteristic
320C22#524312yes—
321C22#524612yes—
322C4—24no—
323C4—22no—
324C4—22no—
325C4—42no—
326C4—84no—
327C4—124no—
328C4—242no—
329C3#902711yescharacteristic, complemented, Sylow
330C2—21no—
331C2—21no—
332C2—21no—
333C2—21no—
334C2#7156011yescharacteristic
335C2#7165511yescharacteristic
336C2#7166011yescharacteristic
337C2#7187412yes—
338C2#7189812yes—
subgroupG/H
centerC23#953
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2316C2
socleC23×C3#120
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2316D3
hypocenterC3#9027
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2316

Complements

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

NH ≅ G/Ntype
18. #4981C4semidirect
329. C3#9027semidirect

Maximal subgroups

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

subgroupindex
C3×#23162
C2×#49812
C2×#45282
#695012
#676242
#90273