Finite Group Databasebeta

← #10692503

Subgroups — #10692503

Order 768.

570 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2564C211yesmaximal, characteristic
2#82301C211yesmaximal, characteristic
3#71690C212yesmaximal, complemented
4#71619C211yesmaximal, characteristic
5#67612C212yesmaximal, complemented
6#9383—31nomaximal, Sylow
7C2×C3×C22⋊Q8C2211yescharacteristic
8C2×C3×C4×D4C2211yescharacteristic
9C3×#400C411yescharacteristic, complemented
10C3×#399C411yescharacteristic, complemented
11C3×#395C411yescharacteristic, complemented
12C3×#393C411yescharacteristic, complemented
13C3×#387—21no—
14C3×#386—21no—
15C3×#385—21no—
16C3×#380—21no—
17C2×C3×#95C2211yescharacteristic
18C2×#834—21no—
19#5212C2212yes—
20C2×#830—21no—
21C2×#830C2212yes—
22D4×C3⋊C8—24no—
23#4925—21no—
24#4538—24no—
25#4526—21no—
26#2972—31no—
27#2564D311yescharacteristic
28#1799—31no—
29#1520—32no—
30#1294—32no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4D411yescharacteristic
33C23×C3×C4—21no—
34C23×C3×C4D411yescharacteristic
35C23×C3×C4C2311yescharacteristic
36C3×C22⋊Q8—22no—
37C3×C22⋊Q8—22no—
38C3×C4×D4—22no—
39C3×C4×D4—22no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4—21no—
43C2×C3×C4⋊C4C2×C411yescharacteristic
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C22⋊C4—21no—
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4C2×C411yescharacteristic
49C2×C3×C22⋊C4D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×#217—22no—
52C2×C4⋊Dic3—22no—
53C22×C3⋊C8—22no—
54C22×C3⋊C8—22no—
55C3×#95—21no—
56C3×#95—21no—
57C3×#95—21no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95C2×C411yescharacteristic
61C3×#95C2×C411yescharacteristic
62#834—42no—
63C3⋊#95—22no—
64#830—22no—
65#830—22no—
66#830D414yes—
67C4⋊(C3⋊C8)—22no—
68C4×C3⋊C8—22no—
69C2×C22⋊Q8C2×D311yescharacteristic
70C2×C4×D4C2×D311yescharacteristic
71C8×D4—64no—
72#431—61no—
73C2×#99—61no—
74C2×C22⋊C8—32no—
75C2×C22⋊C8—61no—
76#400Dic311yescharacteristic, complemented
77#399Dic311yescharacteristic, complemented
78#395Dic311yescharacteristic, complemented
79#393Dic311yescharacteristic, complemented
80#387—21no—
81#386—21no—
82#385—21no—
83#380—21no—
84#380—32no—
85C2×#95C2×D311yescharacteristic
86#342—61no—
87#328—64no—
88C24×C3C22⋊C411yescharacteristic
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—81no—
92C2×C3×D4—22no—
93C2×C3×D4—22no—
94C2×C3×D4SD1612yes—
95C2×C3×D4Q1612yes—
96C22×C3×C4—21no—
97C22×C3×C4—21no—
98C22×C3×C4—21no—
99C22×C3×C4—21no—
100C22×C3×C4—21no—
101C22×C3×C4—21no—
102C22×C3×C4—42no—
103C22×C3×C4—42no—
104C22×C3×C4—81no—
105C22×C3×C4—81no—
106C22×C3×C4—81no—
107C22×C3×C4C22⋊C411yescharacteristic
108C22×C3×C4C22⋊C411yescharacteristic
109C22×C3×C4C22⋊C411yescharacteristic
110C22×C3×C4C22×C411yescharacteristic
111C22×C3×C4C2×D412yes—
112C22×C3×C4C2×D411yescharacteristic
113C22×C3×C4#4112yes—
114C22×Dic3—42no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—41no—
119C3×C4⋊C4—41no—
120C3×C4⋊C4—81no—
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—42no—
125C3×C22⋊C4—81no—
126C3×C42—21no—
127C3×C42—21no—
128#217—42no—
129#217—42no—
130C4⋊Dic3—82no—
131C2×C3⋊C8—24no—
132C2×C3⋊C8—82no—
133C2×C3⋊C8—82no—
134C22×Q8C2×Dic311yescharacteristic
135C22×D4#6711yescharacteristic
136C23×C4—21no—
137C23×C4#6711yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—62no—
140C22×C8—62no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×D4—22no—
144C4×D4—22no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—62no—
149C2×C4⋊C4C2×Dic311yescharacteristic
150C2×C4⋊C4C2×Dic311yescharacteristic
151C2×C22⋊C4—21no—
152C2×C22⋊C4—21no—
153C2×C22⋊C4—21no—
154C2×C22⋊C4—62no—
155C2×C22⋊C4C2×Dic311yescharacteristic
156C2×C22⋊C4#6711yescharacteristic
157C2×C42#6711yescharacteristic
158C4⋊C8—62no—
159#99—122no—
160C22⋊C8—34no—
161C22⋊C8—62no—
162C22⋊C8—62no—
163C4×C8—62no—
164#95—21no—
165#95—21no—
166#95—21no—
167#95—21no—
168#95—21no—
169#95—62no—
170#95C2×Dic311yescharacteristic
171#95C2×Dic311yescharacteristic
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—41no—
175C23×C3—41no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3#10312yes—
179C23×C3C2×C22⋊C411yescharacteristic
180C23×C3#12112yes—
181C23×C3C2×SD1611yescharacteristic
182C23×C3C2×Q1611yescharacteristic
183C23×C3#13512yes—
184C3×Q8—82no—
185C3×D4—24no—
186C3×D4—24no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—22no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—42no—
193C2×C3×C4—42no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—81no—
211C2×C3×C4—84no—
212C2×C3×C4#10312yes—
213C2×C3×C4C4×D412yes—
214C2×C3×C4#12011yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×Dic3—82no—
217C2×Dic3—82no—
218C2×Dic3—82no—
219C3⋊C8—44no—
220C24#21711yescharacteristic
221C24C2×#6711yescharacteristic
222C2×Q8—22no—
223C2×Q8—81no—
224C2×D4—22no—
225C2×D4—22no—
226C2×D4#21512yes—
227C2×D4#21612yes—
228C22×C4—21no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—42no—
235C22×C4—42no—
236C22×C4—81no—
237C22×C4—81no—
238C22×C4—81no—
239C22×C4—122no—
240C22×C4#21711yescharacteristic
241C22×C4#21711yescharacteristic
242C22×C4#21711yescharacteristic
243C22×C4D3×D412yes—
244C22×C4#23712yes—
245C22×C4C22×Dic311yescharacteristic
246C22×C4C2×#6711yescharacteristic
247C2×C8—64no—
248C2×C8—242no—
249C2×C8—242no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—42no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—81no—
256C4⋊C4—242no—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C22⋊C4—42no—
260C22⋊C4—42no—
261C22⋊C4—81no—
262C22⋊C4—122no—
263C22⋊C4—122no—
264C42—21no—
265C42—21no—
266C22×C3—22no—
267C22×C3—21no—
268C22×C3—22no—
269C22×C3—21no—
270C22×C3—22no—
271C22×C3—42no—
272C22×C3—41no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—81no—
278C22×C3#38611yescharacteristic
279C22×C3C2×#10311yescharacteristic
280C22×C3#41611yescharacteristic
281C22×C3#46512yes—
282C22×C3#46712yes—
283C3×C4—22no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—22no—
287C3×C4—42no—
288C3×C4—41no—
289C3×C4—41no—
290C3×C4—81no—
291C3×C4—81no—
292C3×C4—81no—
293Dic3—162no—
294C23—21no—
295C23—21no—
296C23—41no—
297C23—41no—
298C23—41no—
299C23—41no—
300C23#83512yes—
301C23#93912yes—
302C23C2×#21511yescharacteristic
303C23C2×#21611yescharacteristic
304C23#95012yes—
305C23C2×#21711yescharacteristic
306Q8—82no—
307D4—24no—
308D4—24no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—22no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—81no—
333C2×C4—84no—
334C2×C4—242no—
335C2×C4—242no—
336C2×C4—242no—
337C2×C4#83512yes—
338C2×C4D4×Dic312yes—
339C2×C4#93511yescharacteristic
340C2×C4#93711yescharacteristic
341C8—124no—
342C2×C3—22no—
343C2×C3—21no—
344C2×C3—21no—
345C2×C3—22no—
346C2×C3—42no—
347C2×C3#188411yescharacteristic
348C2×C3#189211yescharacteristic
349C2×C3#191311yescharacteristic
350C22—21no—
351C22—22no—
352C22—21no—
353C22—22no—
354C22—22no—
355C22—42no—
356C22—41no—
357C22—41no—
358C22—42no—
359C22—41no—
360C22—41no—
361C22—81no—
362C22#504812yes—
363C22#505112yes—
364C22#522011yescharacteristic
365C22C2×#83511yescharacteristic
366C22#523311yescharacteristic
367C4—22no—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—42no—
372C4—41no—
373C4—41no—
374C4—81no—
375C4—81no—
376C4—81no—
377C4—482no—
378C3#938311yescharacteristic, complemented, Sylow
379C2—21no—
380C2—22no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#7169611yescharacteristic
385C2#7181711yescharacteristic
386C2#7184811yescharacteristic
subgroupG/H
centerC22#5220
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2564C2
socleC22×C3#386
prime radicalC22×C3×D4D4
cosocleC23×C3×C4C23
hypercenter#2564D3
hypocenterC3#9383
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2564

Complements

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

NH ≅ G/Ntype
3. #71690C2semidirect
5. #67612C2semidirect
9. C3×#400C4semidirect
10. C3×#399C4semidirect
11. C3×#395C4semidirect
12. C3×#393C4semidirect
76. #400Dic3semidirect
77. #399Dic3semidirect
78. #395Dic3semidirect
79. #393Dic3semidirect
378. C3#9383semidirect

Maximal subgroups

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

subgroupindex
C3×#25642
#823012
#716902
#716192
#676122
#93833