Finite Group Databasebeta

← #10690918

Subgroups — #10690918

Order 768.

1010 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 151 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×C4×#217C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4C2×#4538C212yesmaximal, complemented
5#67624C212yesmaximal, complemented
6#9146—31nomaximal, Sylow
7C24×C3×C4C2211yescharacteristic
8C2×C3×C22⋊Q8C412yescomplemented
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×#217—21no—
11C22×C4×Dic3—21no—
12C3×#385—21no—
13C3×#385—22no—
14C3×#380C412yescomplemented
15C3×#380C2211yescharacteristic
16C3×#379—21no—
17C4×#217—24no—
18C2×C3⋊#95C2212yes—
19C2×#830—22no—
20C2×#830C2212yes—
21#4538—24no—
22#2848D311yescharacteristic
23C2×C4×C22⋊C4—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×C4D411yescharacteristic
34C23×C3×C4D412yes—
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—28no—
50C2×C4×Dic3—24no—
51C2×C4×Dic3—44no—
52C22×C3⋊C8—22no—
53C3×#95—22no—
54C3×#95—21no—
55C3⋊#95—22no—
56C3⋊#95D414yes—
57#830—22no—
58#830—22no—
59#830—44no—
60C24×C4C2×D311yescharacteristic
61C2×C22⋊Q8Dic312yescomplemented
62C2×C22⋊Q8C2×D311yescharacteristic
63C22×C22⋊C4—61no—
64C22×C42—61no—
65C2×C22⋊C8—32no—
66C2×C22⋊C8—62no—
67#385—21no—
68#385—22no—
69#380Dic312yescomplemented
70#380C2×D311yescharacteristic
71#379—21no—
72C4×C22⋊C4—64no—
73C2×#95—32no—
74#328—64no—
75C24×C3—24no—
76C24×C3—21no—
77C24×C3—21no—
78C24×C3—22no—
79C24×C3—22no—
80C24×C3—42no—
81C24×C3C22⋊C412yes—
82C24×C3C2×D411yescharacteristic
83C2×C3×Q8—22no—
84C2×C3×Q8—42no—
85C22×C3×C4—21no—
86C22×C3×C4—21no—
87C22×C3×C4—24no—
88C22×C3×C4—24no—
89C22×C3×C4—24no—
90C22×C3×C4—24no—
91C22×C3×C4—24no—
92C22×C3×C4—21no—
93C22×C3×C4—22no—
94C22×C3×C4—21no—
95C22×C3×C4—42no—
96C22×C3×C4—42no—
97C22×C3×C4—42no—
98C22×C3×C4—48no—
99C22×C3×C4—48no—
100C22×C3×C4C22⋊C412yes—
101C22×C3×C4C22×C411yescharacteristic
102C22×C3×C4C2×D412yes—
103C22×C3×C4C2×D411yescharacteristic
104C22×C3×C4#4112yes—
105C22×Dic3—28no—
106C22×Dic3—24no—
107C22×Dic3—44no—
108C22×Dic3—42no—
109C22×Dic3—42no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—44no—
113C3×C4⋊C4—42no—
114C3×C22⋊C4—44no—
115C3×C22⋊C4—44no—
116C3×C22⋊C4—41no—
117C3×C22⋊C4—41no—
118C3×C22⋊C4—42no—
119#217—216no—
120C4×Dic3—48no—
121C2×C3⋊C8—44no—
122C2×C3⋊C8—42no—
123C25#6711yescharacteristic
124C22×Q8C2×Dic311yescharacteristic
125C23×C4—22no—
126C23×C4—22no—
127C23×C4—44no—
128C23×C4—62no—
129C23×C4C2×Dic312yes—
130C23×C4#6711yescharacteristic
131C23×C4#6712yes—
132C23×C4C22×D311yescharacteristic
133C22×C8—62no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8—44no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4C2×Dic311yescharacteristic
140C2×C4⋊C4C2×Dic312yes—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—22no—
143C2×C22⋊C4—21no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—64no—
146C2×C22⋊C4—68no—
147C2×C42—64no—
148C2×C42—124no—
149C22⋊C8—62no—
150C22⋊C8—62no—
151C22⋊C8—124no—
152#95—22no—
153#95—21no—
154#95—34no—
155#95—62no—
156C23×C3—22no—
157C23×C3—22no—
158C23×C3—24no—
159C23×C3—24no—
160C23×C3—22no—
161C23×C3—22no—
162C23×C3—22no—
163C23×C3—22no—
164C23×C3—22no—
165C23×C3—22no—
166C23×C3—48no—
167C23×C3—48no—
168C23×C3—42no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3#10414yes—
173C23×C3C2×C22⋊C411yescharacteristic
174C23×C3#12011yescharacteristic
175C23×C3#12311yescharacteristic
176C23×C3#13512yes—
177C23×C3#13712yes—
178C3×Q8—82no—
179C2×C3×C4—24no—
180C2×C3×C4—24no—
181C2×C3×C4—22no—
182C2×C3×C4—22no—
183C2×C3×C4—22no—
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4—42no—
187C2×C3×C4—48no—
188C2×C3×C4—48no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4#9914yes—
197C2×C3×C4C4×D412yes—
198C2×C3×C4#12112yes—
199C2×Dic3—28no—
200C2×Dic3—44no—
201C2×Dic3—48no—
202C2×Dic3—48no—
203C2×Dic3—44no—
204C3⋊C8—82no—
205C24—24no—
206C24—21no—
207C24—21no—
208C24—22no—
209C24—22no—
210C24—42no—
211C24#21712yes—
212C24C2×#6711yescharacteristic
213C2×Q8—22no—
214C2×Q8—42no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—24no—
218C22×C4—24no—
219C22×C4—24no—
220C22×C4—24no—
221C22×C4—24no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—48no—
229C22×C4—48no—
230C22×C4—68no—
231C22×C4—64no—
232C22×C4—124no—
233C22×C4—122no—
234C22×C4—122no—
235C22×C4#21712yes—
236C22×C4D3×D412yes—
237C22×C4#23712yes—
238C22×C4C22×Dic311yescharacteristic
239C22×C4C2×#6711yescharacteristic
240C2×C8—124no—
241C2×C8—122no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—44no—
245C4⋊C4—42no—
246C22⋊C4—44no—
247C22⋊C4—44no—
248C22⋊C4—41no—
249C22⋊C4—41no—
250C22⋊C4—42no—
251C22⋊C4—616no—
252C42—128no—
253C22×C3—21no—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—21no—
257C22×C3—24no—
258C22×C3—24no—
259C22×C3—24no—
260C22×C3—24no—
261C22×C3—24no—
262C22×C3—48no—
263C22×C3—48no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3#38611yescharacteristic
269C22×C3C2×#9912yes—
270C22×C3#41611yescharacteristic
271C22×C3#41911yescharacteristic
272C22×C3C2×#10412yes—
273C22×C3#45212yes—
274C22×C3#48512yes—
275C3×C4—22no—
276C3×C4—24no—
277C3×C4—22no—
278C3×C4—81no—
279C3×C4—81no—
280C3×C4—82no—
281Dic3—48no—
282C23—22no—
283C23—22no—
284C23—24no—
285C23—24no—
286C23—22no—
287C23—22no—
288C23—22no—
289C23—22no—
290C23—22no—
291C23—22no—
292C23—48no—
293C23—48no—
294C23—42no—
295C23—42no—
296C23—42no—
297C23—42no—
298C23#83714yes—
299C23#93511yescharacteristic
300C23#93711yescharacteristic
301C23#94212yes—
302C23#95012yes—
303C23C2×#21711yescharacteristic
304Q8—82no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—48no—
314C2×C4—48no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—68no—
323C2×C4—124no—
324C2×C4—128no—
325C2×C4—128no—
326C2×C4—124no—
327C2×C4#83414yes—
328C2×C4D4×Dic312yes—
329C2×C4#93912yes—
330C8—242no—
331C2×C3—22no—
332C2×C3—22no—
333C2×C3—44no—
334C2×C3#150112yes—
335C2×C3#150812yes—
336C2×C3#187311yescharacteristic
337C2×C3#189611yescharacteristic
338C2×C3#189811yescharacteristic
339C22—24no—
340C22—24no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—24no—
346C22—24no—
347C22—24no—
348C22—48no—
349C22—48no—
350C22—42no—
351C22—42no—
352C22—42no—
353C22—42no—
354C22#504512yes—
355C22#504712yes—
356C22C2×#83412yes—
357C22#522011yescharacteristic
358C22#522511yescharacteristic
359C22#523311yescharacteristic
360C22C2×#83712yes—
361C4—22no—
362C4—24no—
363C4—22no—
364C4—82no—
365C4—81no—
366C4—81no—
367C4—128no—
368C3#914611yescharacteristic, complemented, Sylow
369C2—22no—
370C2—22no—
371C2—44no—
372C2#7167011yescharacteristic
373C2#7168812yes—
374C2#7169311yescharacteristic
375C2#7170712yes—
376C2#7171111yescharacteristic
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#9146
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
14. C3×#380C4semidirect
61. C2×C22⋊Q8Dic3semidirect
69. #380Dic3semidirect
368. C3#9146semidirect

Maximal subgroups

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

subgroupindex
C3×#28482
C2×C4×#2172
#721262
C2×#45382
#676242
#91463