Finite Group Databasebeta

← #10690768

Subgroups — #10690768

Order 768.

612 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2855C211yesmaximal, characteristic
2C2×#4926C211yesmaximal, characteristic
3C2×#4538C212yesmaximal
4#71534C211yesmaximal, characteristic
5#67624C212yesmaximal
6#9116—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C22×C3×C42C2211yescharacteristic
9C3×#391C411yescharacteristic, complemented
10C3×#391C411yescharacteristic, complemented
11C3×#389C411yescharacteristic, complemented
12C3×#389C411yescharacteristic, complemented
13C3×#387—21no—
14C3×#387—21no—
15C3×#385—21no—
16C3×#381—21no—
17C3×#380C2211yescharacteristic
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2212yes—
21C2×#830C2212yes—
22#4926—24no—
23C2×C4⋊(C3⋊C8)—21no—
24C2×#803—21no—
25#4538—24no—
26C2×#432—31no—
27#2855D311yescharacteristic
28C2×#328—32no—
29#1434—31no—
30#1294—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C23×C3×C4D411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4C2311yescharacteristic
35C23×Dic3—22no—
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4C2×C411yescharacteristic
42C2×C3×C4⋊C4C2×C411yescharacteristic
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C22⋊C4—21no—
45C2×C3×C22⋊C4—21no—
46C2×C3×C42—22no—
47C2×C3×C42—22no—
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42C2×C411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C3×C42D411yescharacteristic
52C22×C3⋊C8—22no—
53C3×#95—21no—
54C3×#95—21no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3⋊#95—24no—
61C3⋊#95—22no—
62C3⋊#95—22no—
63C3⋊#95—42no—
64#830—22no—
65#830D414yes—
66C4⋊(C3⋊C8)—24no—
67#803—24no—
68C2×C22⋊Q8C2×D311yescharacteristic
69C22×C42C2×D311yescharacteristic
70#432—64no—
71C2×C4⋊C8—61no—
72C2×C22⋊C8—32no—
73C2×#97—61no—
74#391Dic311yescharacteristic, complemented
75#391Dic311yescharacteristic, complemented
76#389Dic311yescharacteristic, complemented
77#389Dic311yescharacteristic, complemented
78#387—21no—
79#387—21no—
80#385—21no—
81#381—21no—
82#380C2×D311yescharacteristic
83C2×#95—32no—
84C2×#95—61no—
85C2×#95—61no—
86#328—64no—
87C24×C3C2×D411yescharacteristic
88C2×C3×Q8—22no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—22no—
95C22×C3×C4—21no—
96C22×C3×C4—22no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C22⋊C411yescharacteristic
106C22×C3×C4C22⋊C411yescharacteristic
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22×C411yescharacteristic
109C22×C3×C4C2×D412yes—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4#4112yes—
112C22×Dic3—24no—
113C22×Dic3—42no—
114C22×Dic3—42no—
115C22×Dic3—42no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—41no—
121C3×C4⋊C4—41no—
122C3×C4⋊C4—42no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—41no—
125C3×C4⋊C4—41no—
126C3×C22⋊C4—42no—
127C3×C22⋊C4—41no—
128C3×C22⋊C4—41no—
129C3×C42—22no—
130C3×C42—22no—
131C3×C42—21no—
132C3×C42—21no—
133C3×C42—21no—
134C3×C42—21no—
135C2×C3⋊C8—24no—
136C2×C3⋊C8—42no—
137C2×C3⋊C8—42no—
138C22×Q8C2×Dic311yescharacteristic
139C23×C4—62no—
140C23×C4#6711yescharacteristic
141C23×C4#6711yescharacteristic
142C23×C4C22×D311yescharacteristic
143C22×C8—62no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4C2×Dic311yescharacteristic
150C2×C4⋊C4C2×Dic311yescharacteristic
151C2×C4⋊C4C2×Dic311yescharacteristic
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C42—22no—
155C2×C42—22no—
156C2×C42C2×Dic311yescharacteristic
157C2×C42C2×Dic311yescharacteristic
158C2×C42#6711yescharacteristic
159C2×C42#6711yescharacteristic
160C4⋊C8—64no—
161C22⋊C8—34no—
162C22⋊C8—62no—
163#97—64no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—21no—
170#95—21no—
171#95—64no—
172#95—62no—
173#95—62no—
174#95—122no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12112yes—
181C23×C3AGL(1,ℤ/8)11yescharacteristic
182C23×C3#13712yes—
183C23×C3#13711yescharacteristic
184C3×Q8—82no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—42no—
201C2×C3×C4—42no—
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4#10412yes—
218C2×C3×C4C4×D412yes—
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12311yescharacteristic
221C2×Dic3—44no—
222C2×Dic3—44no—
223C2×Dic3—44no—
224C2×Dic3—82no—
225C2×Dic3—82no—
226C2×Dic3—82no—
227C2×Dic3—82no—
228C3⋊C8—44no—
229C24C2×#6711yescharacteristic
230C2×Q8—22no—
231C2×Q8—41no—
232C2×Q8—41no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—21no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—21no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4—64no—
247C22×C4—122no—
248C22×C4—122no—
249C22×C4—122no—
250C22×C4—122no—
251C22×C4—122no—
252C22×C4—122no—
253C22×C4#21711yescharacteristic
254C22×C4#21711yescharacteristic
255C22×C4#21711yescharacteristic
256C22×C4#21711yescharacteristic
257C22×C4D3×D412yes—
258C22×C4#23712yes—
259C22×C4C22×Dic311yescharacteristic
260C22×C4C2×#6711yescharacteristic
261C2×C8—64no—
262C2×C8—122no—
263C2×C8—122no—
264C4⋊C4—22no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C22⋊C4—42no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C42—22no—
275C42—22no—
276C42—21no—
277C42—21no—
278C42—21no—
279C42—21no—
280C22×C3—22no—
281C22×C3—22no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—21no—
286C22×C3—21no—
287C22×C3—22no—
288C22×C3#38611yescharacteristic
289C22×C3#41011yescharacteristic
290C22×C3#41011yescharacteristic
291C22×C3#41711yescharacteristic
292C22×C3#41911yescharacteristic
293C22×C3C2×#10411yescharacteristic
294C22×C3#42111yescharacteristic
295C22×C3#46812yes—
296C22×C3#47012yes—
297C3×C4—22no—
298C3×C4—21no—
299C3×C4—22no—
300C3×C4—21no—
301C3×C4—41no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4—81no—
307C3×C4—81no—
308Dic3—84no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23#93912yes—
314C23#94212yes—
315C23#94911yescharacteristic
316C23#95111yescharacteristic
317C23C2×#21711yescharacteristic
318Q8—82no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—44no—
337C2×C4—42no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—124no—
352C2×C4—124no—
353C2×C4—124no—
354C2×C4—242no—
355C2×C4—242no—
356C2×C4—242no—
357C2×C4—242no—
358C2×C4#83712yes—
359C2×C4D4×Dic312yes—
360C2×C4#93511yescharacteristic
361C2×C4#93711yescharacteristic
362C8—124no—
363C2×C3—22no—
364C2×C3—22no—
365C2×C3#150611yescharacteristic
366C2×C3#150911yescharacteristic
367C2×C3#151211yescharacteristic
368C2×C3#151511yescharacteristic
369C2×C3#187111yescharacteristic
370C2×C3#189411yescharacteristic
371C2×C3#189911yescharacteristic
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22#504912yes—
381C22#505212yes—
382C22#522011yescharacteristic
383C22#522511yescharacteristic
384C22#523211yescharacteristic
385C22C2×#83711yescharacteristic
386C22#523611yescharacteristic
387C22#524011yescharacteristic
388C22#524011yescharacteristic
389C4—21no—
390C4—22no—
391C4—21no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—81no—
397C4—81no—
398C4—81no—
399C4—81no—
400C4—244no—
401C3#911611yescharacteristic, complemented, Sylow
402C2—22no—
403C2—22no—
404C2#7169711yescharacteristic
405C2#7170511yescharacteristic
406C2#7171311yescharacteristic
407C2#7175511yescharacteristic
408C2#7175611yescharacteristic
409C2#7183611yescharacteristic
410C2#7183711yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2855C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#2855D3
hypocenterC3#9116
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
9. C3×#391C4semidirect
10. C3×#391C4semidirect
11. C3×#389C4semidirect
12. C3×#389C4semidirect
74. #391Dic3semidirect
75. #391Dic3semidirect
76. #389Dic3semidirect
77. #389Dic3semidirect
401. C3#9116semidirect

Maximal subgroups

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

subgroupindex
C3×#28552
C2×#49262
C2×#45382
#715342
#676242
#91163