Finite Group Databasebeta

← #10690958

Subgroups — #10690958

Order 768.

902 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 902 conjugacy classes of subgroups — too many to draw. Only its 143 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×#2848C211yesmaximal, characteristic
2C2×#5211C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4538C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9154—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×C22⋊Q8C412yescomplemented
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#217—21no—
11C3×#385—21no—
12C3×#385—22no—
13C3×#380C412yescomplemented
14C3×#380C2211yescharacteristic
15C3×#379—21no—
16#5211—24no—
17C2×C3⋊#95—21no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4538—24no—
22#2848D311yescharacteristic
23C2×#381—31no—
24#1779—31no—
25C2×#328—32no—
26#1294—32no—
27C25×C3D411yescharacteristic
28C22×C3×Q8C2×C411yescharacteristic
29C23×C3×C4—22no—
30C23×C3×C4—22no—
31C23×C3×C4—44no—
32C23×C3×C4C2×C412yes—
33C23×C3×C4D412yes—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C23×Dic3—22no—
37C3×C22⋊Q8—22no—
38C3×C22⋊Q8—22no—
39C3×C22⋊Q8—44no—
40C2×C3×C4⋊C4—22no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×#217—24no—
49C2×#217—44no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—21no—
53C3⋊#95—24no—
54C3⋊#95—22no—
55C3⋊#95D414yes—
56#830—22no—
57#830—22no—
58#830—44no—
59C24×C4C2×D311yescharacteristic
60C2×C22⋊Q8Dic312yescomplemented
61C2×C22⋊Q8C2×D311yescharacteristic
62C22×C22⋊C4—61no—
63C2×C22⋊C8—32no—
64C2×C22⋊C8—62no—
65#385—21no—
66#385—22no—
67#381—64no—
68#380Dic312yescomplemented
69#380C2×D311yescharacteristic
70#379—21no—
71C2×#95—32no—
72C2×#95—61no—
73#328—64no—
74C24×C3—24no—
75C24×C3—21no—
76C24×C3—21no—
77C24×C3—22no—
78C24×C3—22no—
79C24×C3—42no—
80C24×C3C22⋊C412yes—
81C24×C3C2×D411yescharacteristic
82C2×C3×Q8—22no—
83C2×C3×Q8—42no—
84C22×C3×C4—24no—
85C22×C3×C4—24no—
86C22×C3×C4—21no—
87C22×C3×C4—21no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—21no—
91C22×C3×C4—22no—
92C22×C3×C4—21no—
93C22×C3×C4—42no—
94C22×C3×C4—42no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—48no—
98C22×C3×C4—48no—
99C22×C3×C4C22⋊C412yes—
100C22×C3×C4C22×C411yescharacteristic
101C22×C3×C4C2×D412yes—
102C22×C3×C4C2×D411yescharacteristic
103C22×C3×C4#4112yes—
104C22×Dic3—24no—
105C22×Dic3—44no—
106C22×Dic3—44no—
107C22×Dic3—42no—
108C22×Dic3—42no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—42no—
111C3×C4⋊C4—44no—
112C3×C4⋊C4—42no—
113C3×C22⋊C4—44no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—41no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—42no—
118#217—48no—
119C2×C3⋊C8—44no—
120C2×C3⋊C8—42no—
121C25#6711yescharacteristic
122C22×Q8C2×Dic311yescharacteristic
123C23×C4—22no—
124C23×C4—22no—
125C23×C4—44no—
126C23×C4—62no—
127C23×C4C2×Dic312yes—
128C23×C4#6712yes—
129C23×C4#6711yescharacteristic
130C23×C4C22×D311yescharacteristic
131C22×C8—62no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—44no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4C2×Dic311yescharacteristic
138C2×C4⋊C4C2×Dic312yes—
139C2×C22⋊C4—22no—
140C2×C22⋊C4—22no—
141C2×C22⋊C4—21no—
142C2×C22⋊C4—21no—
143C2×C22⋊C4—64no—
144C2×C22⋊C4—124no—
145C22⋊C8—62no—
146C22⋊C8—62no—
147C22⋊C8—124no—
148#95—22no—
149#95—21no—
150#95—34no—
151#95—64no—
152#95—62no—
153C23×C3—22no—
154C23×C3—22no—
155C23×C3—24no—
156C23×C3—24no—
157C23×C3—22no—
158C23×C3—22no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—48no—
164C23×C3—48no—
165C23×C3—42no—
166C23×C3—42no—
167C23×C3—42no—
168C23×C3—42no—
169C23×C3#9914yes—
170C23×C3C2×C22⋊C411yescharacteristic
171C23×C3#12011yescharacteristic
172C23×C3#12311yescharacteristic
173C23×C3#13512yes—
174C23×C3#13712yes—
175C3×Q8—82no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—22no—
181C2×C3×C4—42no—
182C2×C3×C4—42no—
183C2×C3×C4—48no—
184C2×C3×C4—48no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—42no—
190C2×C3×C4—42no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4C4×D412yes—
195C2×C3×C4#12112yes—
196C2×Dic3—44no—
197C2×Dic3—44no—
198C2×Dic3—44no—
199C2×Dic3—84no—
200C2×Dic3—84no—
201C3⋊C8—82no—
202C24—24no—
203C24—21no—
204C24—21no—
205C24—22no—
206C24—22no—
207C24—42no—
208C24#21712yes—
209C24C2×#6711yescharacteristic
210C2×Q8—22no—
211C2×Q8—42no—
212C22×C4—24no—
213C22×C4—24no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—24no—
217C22×C4—24no—
218C22×C4—22no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—48no—
226C22×C4—48no—
227C22×C4—64no—
228C22×C4—124no—
229C22×C4—124no—
230C22×C4—122no—
231C22×C4—122no—
232C22×C4#21712yes—
233C22×C4D3×D412yes—
234C22×C4#23712yes—
235C22×C4C22×Dic311yescharacteristic
236C22×C4C2×#6711yescharacteristic
237C2×C8—124no—
238C2×C8—122no—
239C4⋊C4—22no—
240C4⋊C4—42no—
241C4⋊C4—44no—
242C4⋊C4—42no—
243C22⋊C4—44no—
244C22⋊C4—44no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—42no—
248C22⋊C4—128no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—24no—
254C22×C3—24no—
255C22×C3—24no—
256C22×C3—24no—
257C22×C3—24no—
258C22×C3—48no—
259C22×C3—48no—
260C22×C3—42no—
261C22×C3—42no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3#38611yescharacteristic
265C22×C3C2×#9912yes—
266C22×C3#41611yescharacteristic
267C22×C3#41911yescharacteristic
268C22×C3#42112yes—
269C22×C3#45212yes—
270C22×C3#48512yes—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—42no—
274C3×C4—81no—
275C3×C4—81no—
276C3×C4—82no—
277Dic3—84no—
278C23—22no—
279C23—22no—
280C23—24no—
281C23—24no—
282C23—22no—
283C23—22no—
284C23—22no—
285C23—22no—
286C23—22no—
287C23—22no—
288C23—48no—
289C23—48no—
290C23—42no—
291C23—42no—
292C23—42no—
293C23—42no—
294C23#83414yes—
295C23#93511yescharacteristic
296C23#93711yescharacteristic
297C23#94212yes—
298C23#95012yes—
299C23C2×#21711yescharacteristic
300Q8—82no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—42no—
307C2×C4—42no—
308C2×C4—48no—
309C2×C4—48no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—124no—
320C2×C4—124no—
321C2×C4—124no—
322C2×C4—244no—
323C2×C4—244no—
324C2×C4D4×Dic312yes—
325C2×C4#93912yes—
326C8—242no—
327C2×C3—22no—
328C2×C3—22no—
329C2×C3—44no—
330C2×C3#151112yes—
331C2×C3#151612yes—
332C2×C3#187311yescharacteristic
333C2×C3#189711yescharacteristic
334C2×C3#189811yescharacteristic
335C22—24no—
336C22—24no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—24no—
342C22—24no—
343C22—24no—
344C22—48no—
345C22—48no—
346C22—42no—
347C22—42no—
348C22—42no—
349C22—42no—
350C22#504512yes—
351C22#504712yes—
352C22C2×#83412yes—
353C22#522011yescharacteristic
354C22#522511yescharacteristic
355C22#523311yescharacteristic
356C22#523612yes—
357C4—22no—
358C4—22no—
359C4—42no—
360C4—82no—
361C4—81no—
362C4—81no—
363C4—244no—
364C3#915411yescharacteristic, complemented, Sylow
365C2—22no—
366C2—22no—
367C2—44no—
368C2#7167011yescharacteristic
369C2#7168912yes—
370C2#7169311yescharacteristic
371C2#7171012yes—
372C2#7171211yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2848C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenter#2848D3
hypocenterC3#9154
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ #2848

Complements

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

NH ≅ G/Ntype
4. C2×#4538C2semidirect
5. #67624C2semidirect
8. C2×C3×C22⋊Q8C4semidirect
13. C3×#380C4semidirect
60. C2×C22⋊Q8Dic3semidirect
68. #380Dic3semidirect
364. C3#9154semidirect

Maximal subgroups

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

subgroupindex
C3×#28482
C2×#52112
#721262
C2×#45382
#676242
#91543