Finite Group Databasebeta

← #10694314

Subgroups — #10694314

Order 768.

856 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2856C211yesmaximal, characteristic
2C2×#4537C214yesmaximal, complemented
3C4×C3⋊#95C211yesmaximal, characteristic, complemented
4#71530C211yesmaximal, characteristic, complemented
5#9712—31nomaximal, Sylow
6C2×C3×#127C411yescharacteristic, complemented
7C2×C3×#121—21no—
8C2×C3×#121—21no—
9C2×C3×#121C2212yes—
10C22×C3×C42C2211yescharacteristic
11C22×C4×Dic3—21no—
12C3×#390—21no—
13C3×#390—21no—
14C3×#390C412yescomplemented
15C3×#384C411yescharacteristic, complemented
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830C2212yes—
19C2×C4⋊(C3⋊C8)—21no—
20#4537—28no—
21#4524—21no—
22#2856D311yescharacteristic
23#1797—31no—
24C2×#327—34no—
25C4×#95—31no—
26C22×C3×D4—22no—
27C22×C3×D4—21no—
28C22×C3×D4—21no—
29C22×C3×D4C2×C412yes—
30C23×C3×C4D411yescharacteristic
31C23×C3×C4D411yescharacteristic
32C23×C3×C4C2311yescharacteristic
33C23×Dic3—22no—
34C3×#127—42no—
35C3×#121—24no—
36C3×#121—24no—
37C3×#121—44no—
38C3×#121—42no—
39C3×#121—42no—
40C2×C3×C4⋊C4—21no—
41C2×C3×C4⋊C4—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C22⋊C4—22no—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C42—42no—
47C2×C3×C42C2×C411yescharacteristic
48C2×C3×C42C2×C411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42D411yescharacteristic
51C2×C4×Dic3—24no—
52C2×C4×Dic3—42no—
53C2×C4×Dic3—42no—
54C22×C3⋊C8—22no—
55C3⋊#95—24no—
56C3⋊#95—22no—
57C3⋊#95D414yescomplemented
58#830—24no—
59C4⋊(C3⋊C8)—42no—
60C2×#127Dic311yescharacteristic, complemented
61C2×#121—21no—
62C2×#121—21no—
63C2×#121C2×D312yes—
64C22×C42—61no—
65C22×C42C2×D311yescharacteristic
66C2×C4⋊C8—61no—
67C2×C22⋊C8—32no—
68#390—21no—
69#390—21no—
70#390Dic312yescomplemented
71#384Dic311yescharacteristic, complemented
72C2×#95—32no—
73C2×#95—61no—
74#336—61no—
75#327—68no—
76C24×C3—22no—
77C24×C3C2×D411yescharacteristic
78C2×C3×D4—24no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—44no—
84C2×C3×D4—42no—
85C2×C3×D4—42no—
86C2×C3×D4—44no—
87C2×C3×D4—42no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4C22⋊C411yescharacteristic
100C22×C3×C4C22⋊C411yescharacteristic
101C22×C3×C4C22⋊C411yescharacteristic
102C22×C3×C4C22⋊C411yescharacteristic
103C22×C3×C4D812yes—
104C22×C3×C4SD1612yes—
105C22×C3×C4C22×C411yescharacteristic
106C22×C3×C4C2×D412yes—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4112yes—
109C22×Dic3—24no—
110C22×Dic3—24no—
111C22×Dic3—24no—
112C22×Dic3—42no—
113C22×Dic3—42no—
114C22×Dic3—42no—
115C22×Dic3—42no—
116C3×C4⋊C4—24no—
117C3×C4⋊C4—42no—
118C3×C4⋊C4—42no—
119C3×C22⋊C4—44no—
120C3×C22⋊C4—44no—
121C3×C22⋊C4—42no—
122C3×C22⋊C4—42no—
123C3×C42—22no—
124C3×C42—22no—
125C3×C42—41no—
126C3×C42—41no—
127C4×Dic3—48no—
128C2×C3⋊C8—42no—
129C2×C3⋊C8—42no—
130C2×C3⋊C8—42no—
131C22×D4—22no—
132C22×D4—21no—
133C22×D4—21no—
134C22×D4C2×Dic312yes—
135C23×C4—62no—
136C23×C4#6711yescharacteristic
137C23×C4#6711yescharacteristic
138C23×C4C22×D311yescharacteristic
139C22×C8—62no—
140#127—42no—
141#121—24no—
142#121—24no—
143#121—44no—
144#121—42no—
145#121—42no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4C2×Dic312yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—22no—
152C2×C42—42no—
153C2×C42—64no—
154C2×C42—122no—
155C2×C42—122no—
156C2×C42C2×Dic311yescharacteristic
157C2×C42C2×Dic311yescharacteristic
158C2×C42#6711yescharacteristic
159C2×C42#6711yescharacteristic
160C4⋊C8—122no—
161C22⋊C8—64no—
162#95—34no—
163#95—64no—
164#95—62no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—42no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3—42no—
174C23×C3—42no—
175C23×C3—42no—
176C23×C3C2×C22⋊C411yescharacteristic
177C23×C3#12112yes—
178C23×C3C2×D811yescharacteristic
179C23×C3C2×SD1611yescharacteristic
180C23×C3AGL(1,ℤ/8)12yes—
181C3×D4—84no—
182C3×D4—84no—
183C3×D4—84no—
184C3×D4—84no—
185C3×D4—84no—
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—24no—
195C2×C3×C4—24no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—42no—
200C2×C3×C4—42no—
201C2×C3×C4—44no—
202C2×C3×C4—42no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4#9912yes—
206C2×C3×C4#10212yes—
207C2×C3×C4#10212yes—
208C2×C3×C4#10412yes—
209C2×C3×C4#10412yes—
210C2×C3×C4C4×D412yes—
211C2×C3×C4#12411yescharacteristic
212C2×C3×C4#12711yescharacteristic
213C2×Dic3—28no—
214C2×Dic3—44no—
215C2×Dic3—44no—
216C2×Dic3—44no—
217C2×Dic3—44no—
218C2×Dic3—44no—
219C2×Dic3—44no—
220C3⋊C8—82no—
221C24—22no—
222C24C2×#6711yescharacteristic
223C2×D4—24no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—42no—
227C2×D4—42no—
228C2×D4—42no—
229C2×D4—44no—
230C2×D4—42no—
231C2×D4—44no—
232C2×D4—42no—
233C2×D4—42no—
234C2×D4—42no—
235C2×D4—42no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—22no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—64no—
245C22×C4—64no—
246C22×C4—64no—
247C22×C4—122no—
248C22×C4—122no—
249C22×C4—122no—
250C22×C4—122no—
251C22×C4#21312yes—
252C22×C4#21412yes—
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—122no—
262C2×C8—122no—
263C2×C8—122no—
264C4⋊C4—24no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C22⋊C4—44no—
268C22⋊C4—44no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C42—22no—
272C42—22no—
273C42—41no—
274C42—41no—
275C42—128no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—21no—
279C22×C3—21no—
280C22×C3—21no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—22no—
284C22×C3—82no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—82no—
288C22×C3—82no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3#39011yescharacteristic
292C22×C3C2×#9911yescharacteristic
293C22×C3#41011yescharacteristic
294C22×C3C2×#10211yescharacteristic
295C22×C3#41811yescharacteristic
296C22×C3C2×#10411yescharacteristic
297C22×C3C2×#10411yescharacteristic
298C22×C3#45912yes—
299C22×C3#46012yes—
300C3×C4—21no—
301C3×C4—22no—
302C3×C4—21no—
303C3×C4—22no—
304C3×C4—22no—
305C3×C4—41no—
306C3×C4—82no—
307C3×C4#32714yescomplemented
308Dic3—48no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—42no—
314C23—42no—
315C23—42no—
316C23—42no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23C2×#21311yescharacteristic
321C23#93212yes—
322C23C2×#21411yescharacteristic
323C23#93812yes—
324C23C2×#21711yescharacteristic
325D4—84no—
326D4—84no—
327D4—84no—
328D4—84no—
329D4—84no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—68no—
350C2×C4—124no—
351C2×C4—124no—
352C2×C4—124no—
353C2×C4—124no—
354C2×C4—124no—
355C2×C4—124no—
356C2×C4#83212yes—
357C2×C4#83212yes—
358C2×C4#83412yes—
359C2×C4#83712yes—
360C2×C4#83712yes—
361C2×C4D4×Dic312yes—
362C2×C4#93611yescharacteristic
363C2×C4#94011yescharacteristic
364C8—242no—
365C2×C3—22no—
366C2×C3—22no—
367C2×C3—162no—
368C2×C3C2×#32711yescharacteristic
369C2×C3#150211yescharacteristic
370C2×C3#150511yescharacteristic
371C2×C3#150711yescharacteristic
372C2×C3#196211yescharacteristic
373C2×C3#196411yescharacteristic
374C2×C3#198211yescharacteristic
375C22—22no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22—22no—
380C22—21no—
381C22—21no—
382C22—22no—
383C22—82no—
384C22—82no—
385C22—82no—
386C22—82no—
387C22—82no—
388C22—82no—
389C22—82no—
390C22#507212yes—
391C22#507512yes—
392C22C2×#83211yescharacteristic
393C22#521511yescharacteristic
394C22C2×#83411yescharacteristic
395C22#522111yescharacteristic
396C22C2×#83711yescharacteristic
397C22C2×#83711yescharacteristic
398C22#524011yescharacteristic
399C4—21no—
400C4—22no—
401C4—21no—
402C4—22no—
403C4—22no—
404C4—41no—
405C4—82no—
406C4—128no—
407C4#453714yescomplemented
408C3#971211yescharacteristic, complemented, Sylow
409C2—22no—
410C2—22no—
411C2—162no—
412C2C2×#453711yescharacteristic
413C2#7167311yescharacteristic
414C2#7168611yescharacteristic
415C2#7176111yescharacteristic
416C2#7181111yescharacteristic
417C2#7181211yescharacteristic
418C2#7181511yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
Fitting, prime radicalC3×#2856C2
socleC23×C3C2×C22⋊C4
cosocleC23×C3×C4C23
hypercenter#2856D3
hypocenterC3#9712
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2856

Complements

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

NH ≅ G/Ntype
2. C2×#4537C2semidirect
3. C4×C3⋊#95C2semidirect
4. #71530C2semidirect
6. C2×C3×#127C4semidirect
14. C3×#390C4semidirect
15. C3×#384C4semidirect
57. C3⋊#95D4semidirect
60. C2×#127Dic3semidirect
70. #390Dic3semidirect
71. #384Dic3semidirect
307. C3×C4#327semidirect
407. C4#4537semidirect
408. C3#9712semidirect

Maximal subgroups

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

subgroupindex
C3×#28562
C2×#45372
C4×C3⋊#952
#715302
#97123