Finite Group Databasebeta

← #10690202

Subgroups — #10690202

Order 768.

928 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 928 conjugacy classes of subgroups — too many to draw. Only its 135 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
1C2×C3×#394C211yesmaximal, characteristic
2#82155C211yesmaximal, characteristic, complemented
3#72128C212yesmaximal
4C2×#4528C211yesmaximal, characteristic, complemented
5#67612C212yesmaximal, complemented
6#9018—31nomaximal, Sylow
7C23×C3×D4C2211yescharacteristic
8C22×C3×C4⋊C4C2211yescharacteristic
9C22×C3×C22⋊C4—21no—
10C22×C3×C22⋊C4—21no—
11C3×#394—22no—
12C3×#394C414yescomplemented
13C2×C3×#95C2211yescharacteristic
14C2×#832—24no—
15#5212C2212yes—
16#5211—21no—
17C4×#217—21no—
18C2×#830C2212yes—
19#4980—24no—
20#4528—22no—
21C2×#394D311yescharacteristic
22#2313—31no—
23#1786—32no—
24C2×#324—31no—
25#1294—32no—
26C25×C3—21no—
27C22×C3×D4—22no—
28C22×C3×D4—21no—
29C22×C3×D4—22no—
30C22×C3×D4—44no—
31C22×C3×D4C2×C412yes—
32C23×C3×C4—21no—
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C2×C3×C4⋊C4—22no—
37C2×C3×C4⋊C4—22no—
38C2×C3×C4⋊C4C2×C412yes—
39C2×C3×C4⋊C4D412yes—
40C2×C3×C22⋊C4—24no—
41C2×C3×C22⋊C4—24no—
42C2×C3×C22⋊C4—44no—
43C2×C3×C22⋊C4—44no—
44C2×#217—22no—
45C2×C4⋊Dic3D414yes—
46C2×C4×Dic3—22no—
47C22×C3⋊C8—22no—
48C3×#95—21no—
49C3×#95C2×C412yes—
50#832—48no—
51C3⋊#95—22no—
52C3⋊#95—24no—
53C3⋊#95—22no—
54#830—22no—
55#830—22no—
56C23×D4C2×D311yescharacteristic
57C22×C4⋊C4C2×D311yescharacteristic
58C22×C22⋊C4—21no—
59C22×C22⋊C4—21no—
60C2×#102—64no—
61C2×C22⋊C8—32no—
62#394—22no—
63#394Dic314yescomplemented
64#387—64no—
65#381—61no—
66#380—32no—
67C4×C22⋊C4—61no—
68C2×#95C2×D311yescharacteristic
69#324—62no—
70C24×C3—24no—
71C24×C3—22no—
72C24×C3—42no—
73C24×C3—42no—
74C24×C3—42no—
75C24×C3—41no—
76C24×C3—41no—
77C24×C3—42no—
78C24×C3—42no—
79C24×C3C2×D411yescharacteristic
80C2×C3×D4—28no—
81C2×C3×D4—24no—
82C2×C3×D4—44no—
83C2×C3×D4—44no—
84C2×C3×D4—48no—
85C2×C3×D4—44no—
86C2×C3×D4—42no—
87C22×C3×C4—21no—
88C22×C3×C4—21no—
89C22×C3×C4—21no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—42no—
99C22×C3×C4C22⋊C412yes—
100C22×C3×C4C22⋊C412yes—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4112yes—
105C22×Dic3—24no—
106C3×C4⋊C4—24no—
107C3×C4⋊C4—24no—
108C3×C22⋊C4—48no—
109C3×C22⋊C4—48no—
110#217—42no—
111#217—42no—
112C4⋊Dic3—44no—
113C4×Dic3—82no—
114C2×C3⋊C8—44no—
115C2×C3⋊C8—42no—
116C25—21no—
117C22×D4—22no—
118C22×D4—22no—
119C22×D4—21no—
120C22×D4—44no—
121C22×D4C2×Dic312yes—
122C23×C4—21no—
123C23×C4#6711yescharacteristic
124C23×C4#6711yescharacteristic
125C23×C4C22×D311yescharacteristic
126C22×C8—62no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—34no—
130C2×C4⋊C4C2×Dic312yes—
131C2×C4⋊C4#6712yes—
132C2×C22⋊C4—24no—
133C2×C22⋊C4—24no—
134C2×C22⋊C4—44no—
135C2×C22⋊C4—44no—
136C2×C22⋊C4—62no—
137C2×C42—62no—
138#102—128no—
139C22⋊C8—62no—
140C22⋊C8—62no—
141#95—21no—
142#95—62no—
143#95—64no—
144#95—62no—
145#95C2×Dic312yes—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—24no—
149C23×C3—42no—
150C23×C3—44no—
151C23×C3—48no—
152C23×C3—48no—
153C23×C3—42no—
154C23×C3—42no—
155C23×C3—42no—
156C23×C3—42no—
157C23×C3—42no—
158C23×C3—44no—
159C23×C3—84no—
160C23×C3—84no—
161C23×C3—82no—
162C23×C3—82no—
163C23×C3—82no—
164C23×C3—82no—
165C23×C3C2×C22⋊C411yescharacteristic
166C23×C3#12011yescharacteristic
167C23×C3#12311yescharacteristic
168C23×C3#13512yes—
169C23×C3AGL(1,ℤ/8)12yes—
170C3×D4—28no—
171C3×D4—48no—
172C3×D4—44no—
173C2×C3×C4—24no—
174C2×C3×C4—22no—
175C2×C3×C4—22no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—41no—
179C2×C3×C4—41no—
180C2×C3×C4—42no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—42no—
184C2×C3×C4—42no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—42no—
188C2×C3×C4—81no—
189C2×C3×C4—84no—
190C2×C3×C4C4×D412yes—
191C2×C3×C4#12112yes—
192C2×Dic3—44no—
193C2×Dic3—44no—
194C2×Dic3—44no—
195C3⋊C8—82no—
196C24—24no—
197C24—22no—
198C24—42no—
199C24—42no—
200C24—42no—
201C24—42no—
202C24—41no—
203C24—41no—
204C24—42no—
205C24C2×#6711yescharacteristic
206C2×D4—28no—
207C2×D4—24no—
208C2×D4—44no—
209C2×D4—44no—
210C2×D4—48no—
211C2×D4—44no—
212C2×D4—42no—
213C22×C4—22no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—22no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—64no—
226C22×C4#21712yes—
227C22×C4#21712yes—
228C22×C4D3×D412yes—
229C22×C4#23712yes—
230C22×C4C22×Dic311yescharacteristic
231C22×C4C2×#6711yescharacteristic
232C2×C8—124no—
233C2×C8—122no—
234C4⋊C4—24no—
235C4⋊C4—24no—
236C4⋊C4—124no—
237C22⋊C4—48no—
238C22⋊C4—48no—
239C22⋊C4—122no—
240C22⋊C4—122no—
241C42—242no—
242C22×C3—24no—
243C22×C3—24no—
244C22×C3—21no—
245C22×C3—21no—
246C22×C3—21no—
247C22×C3—21no—
248C22×C3—44no—
249C22×C3—44no—
250C22×C3—44no—
251C22×C3—82no—
252C22×C3—82no—
253C22×C3—88no—
254C22×C3—88no—
255C22×C3—82no—
256C22×C3—82no—
257C22×C3#38611yescharacteristic
258C22×C3#41611yescharacteristic
259C22×C3#41811yescharacteristic
260C22×C3#44912yes—
261C22×C3#45412yes—
262C22×C3B(4,𝔽2)12yes—
263C22×C3#48212yes—
264C3×C4—22no—
265C3×C4—22no—
266C3×C4—42no—
267C3×C4—42no—
268C3×C4—82no—
269Dic3—84no—
270C23—22no—
271C23—22no—
272C23—24no—
273C23—42no—
274C23—44no—
275C23—42no—
276C23—42no—
277C23—48no—
278C23—48no—
279C23—42no—
280C23—42no—
281C23—42no—
282C23—44no—
283C23—82no—
284C23—84no—
285C23—84no—
286C23—82no—
287C23—82no—
288C23—82no—
289C23#93212yes—
290C23#93511yescharacteristic
291C23#93711yescharacteristic
292C23#95012yes—
293C23C2×#21711yescharacteristic
294D4—28no—
295D4—48no—
296D4—44no—
297C2×C4—24no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—81no—
313C2×C4—84no—
314C2×C4—124no—
315C2×C4—124no—
316C2×C4—124no—
317C2×C4D4×Dic312yes—
318C2×C4#93912yes—
319C8—242no—
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—22no—
323C2×C3—84no—
324C2×C3#159714yes—
325C2×C3#187511yescharacteristic
326C2×C3#188811yescharacteristic
327C2×C3#188911yescharacteristic
328C22—21no—
329C22—21no—
330C22—24no—
331C22—24no—
332C22—21no—
333C22—21no—
334C22—44no—
335C22—44no—
336C22—44no—
337C22—82no—
338C22—82no—
339C22—88no—
340C22—82no—
341C22—88no—
342C22—82no—
343C22#503212yes—
344C22#503512yes—
345C22#503712yes—
346C22#506212yes—
347C22#521511yescharacteristic
348C22#522011yescharacteristic
349C22#523311yescharacteristic
350C4—22no—
351C4—22no—
352C4—42no—
353C4—42no—
354C4—82no—
355C4—244no—
356C3#901811yescharacteristic, complemented, Sylow
357C2—21no—
358C2—21no—
359C2—22no—
360C2—84no—
361C2#6970214yes—
362C2#7166911yescharacteristic
363C2#7167811yescharacteristic
364C2#7172411yescharacteristic
subgroupG/H
centerC23#937
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#394C2
socleC23×C3#120
prime radicalC23×C3×D4C22
cosocleC23×C3×C4C23
hypercenterC2×#394D3
hypocenterC3#9018
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#394

Complements

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

NH ≅ G/Ntype
2. #82155C2semidirect
4. C2×#4528C2semidirect
5. #67612C2semidirect
12. C3×#394C4semidirect
63. #394Dic3semidirect
356. C3#9018semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3942
#821552
#721282
C2×#45282
#676122
#90183