Finite Group Databasebeta

← C2×#67641 #10710154

Subgroups — C2×#67641 #10710154

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#390C211yesmaximal, characteristic
2C22×#834C212yesmaximal
3#67641C214yesmaximal, direct product factor
4C2×#1393—31nomaximal, Sylow
5C23×C3×D4C2211yescharacteristic
6C22×C3×C4⋊C4C411yescharacteristic, complemented
7C22×C3×C22⋊C4—22no—
8C22×C3×C42C411yescharacteristic, complemented
9C22×#217—22no—
10C3×#390—22no—
11C3×#390C412yescomplemented
12C3×#390C2212yes—
13C2×#834—22no—
14C2×#834—24no—
15C2×#834C418yescomplemented
16C2×#834C2214yes—
17C22×#99—32no—
18C2×#390D311yescharacteristic
19#1393—34no—
20C25×C3D412yes—
21C22×C3×D4—42no—
22C22×C3×D4—41no—
23C22×C3×D4—41no—
24C22×C3×D4C2×C414yes—
25C22×C3×D4C2×C411yescharacteristic
26C22×C3×D4D412yes—
27C22×C3×D4D412yes—
28C22×C3×D4Q812yes—
29C22×C3×D4C2311yescharacteristic
30C23×C3×C4—21no—
31C23×C3×C4—21no—
32C23×C3×C4C2×C411yescharacteristic
33C23×Dic3—42no—
34C2×C3×C4⋊C4—21no—
35C2×C3×C4⋊C4—41no—
36C2×C3×C4⋊C4—41no—
37C2×C3×C4⋊C4C2×C412yescomplemented
38C2×C3×C22⋊C4—24no—
39C2×C3×C22⋊C4—24no—
40C2×C3×C22⋊C4—48no—
41C2×C3×C42—21no—
42C2×C3×C42—42no—
43C2×C3×C42C2×C412yescomplemented
44C2×#217—28no—
45C2×#217—24no—
46C2×#217—42no—
47C2×#217—44no—
48#834—28no—
49#834—24no—
50#834C2×C418yescomplemented
51C23×D4C2×D311yescharacteristic
52C22×C4⋊C4Dic311yescharacteristic, complemented
53C22×C22⋊C4—22no—
54C22×C22⋊C4—62no—
55C22×C42Dic311yescharacteristic, complemented
56C2×#99—38no—
57C2×#99—34no—
58C2×#99—64no—
59C2×#99—62no—
60#390—22no—
61#390Dic312yescomplemented
62#390C2×D312yes—
63C24×C3—22no—
64C24×C3—24no—
65C24×C3—22no—
66C24×C3—44no—
67C24×C3—44no—
68C24×C3C22⋊C414yes—
69C24×C3C22⋊C412yes—
70C24×C3C22⋊C411yescharacteristic
71C24×C3C4⋊C414yes—
72C24×C3C2×D412yes—
73C2×C3×D4—44no—
74C2×C3×D4—44no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—42no—
78C2×C3×D4—42no—
79C2×C3×D4—42no—
80C2×C3×D4—42no—
81C2×C3×D4—42no—
82C2×C3×D4—42no—
83C2×C3×D4—41no—
84C2×C3×D4—41no—
85C2×C3×D4C4212yes—
86C2×C3×D4C22⋊C414yes—
87C2×C3×D4C4⋊C414yes—
88C2×C3×D4C4⋊C412yes—
89C2×C3×D4C22×C412yes—
90C2×C3×D4C2×D411yescharacteristic
91C2×C3×D4C2×Q811yescharacteristic
92C22×C3×C4—22no—
93C22×C3×C4—21no—
94C22×C3×C4—21no—
95C22×C3×C4—24no—
96C22×C3×C4—22no—
97C22×C3×C4—22no—
98C22×C3×C4—42no—
99C22×C3×C4—42no—
100C22×C3×C4—44no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4C4212yes—
104C22×C3×C4C22⋊C411yescharacteristic
105C22×C3×C4C4⋊C412yes—
106C22×C3×C4C22×C411yescharacteristic
107C22×Dic3—48no—
108C22×Dic3—44no—
109C22×Dic3—84no—
110C22×Dic3—84no—
111C3×C4⋊C4—42no—
112C3×C4⋊C4—42no—
113C3×C22⋊C4—48no—
114C3×C22⋊C4—48no—
115C3×C42—42no—
116C3×C42—42no—
117#217—28no—
118#217—48no—
119#217—44no—
120C25#6712yes—
121C22×D4—42no—
122C22×D4—41no—
123C22×D4—41no—
124C22×D4Dic612yes—
125C22×D4C4×D314yes—
126C22×D4D1212yes—
127C22×D4C2×Dic311yescharacteristic
128C22×D4#6712yes—
129C22×D4C22×D311yescharacteristic
130C23×C4—21no—
131C23×C4—21no—
132C23×C4—122no—
133C23×C4C2×Dic311yescharacteristic
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4C2×Dic312yescomplemented
138C2×C22⋊C4—24no—
139C2×C22⋊C4—24no—
140C2×C22⋊C4—48no—
141C2×C22⋊C4—68no—
142C2×C22⋊C4—64no—
143C2×C22⋊C4—124no—
144C2×C22⋊C4—122no—
145C2×C42—21no—
146C2×C42—42no—
147C2×C42C2×Dic312yescomplemented
148#99—38no—
149#99—68no—
150#99—64no—
151C23×C3—22no—
152C23×C3—22no—
153C23×C3—24no—
154C23×C3—24no—
155C23×C3—22no—
156C23×C3—22no—
157C23×C3—24no—
158C23×C3—24no—
159C23×C3—22no—
160C23×C3—22no—
161C23×C3—42no—
162C23×C3—42no—
163C23×C3—48no—
164C23×C3—42no—
165C23×C3—48no—
166C23×C3—42no—
167C23×C3—44no—
168C23×C3—44no—
169C23×C3—82no—
170C23×C3—82no—
171C23×C3—88no—
172C23×C3#9512yes—
173C23×C3#9514yes—
174C23×C3#9911yescharacteristic
175C23×C3#9912yes—
176C23×C3#9911yescharacteristic
177C23×C3C2×C22⋊C412yes—
178C23×C3C2×C22⋊C411yescharacteristic
179C23×C3C2×C4⋊C412yes—
180C3×D4—42no—
181C3×D4—44no—
182C3×D4—44no—
183C3×D4—42no—
184C3×D4—42no—
185C3×D4—41no—
186C3×D4—41no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—22no—
190C2×C3×C4—22no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—24no—
196C2×C3×C4—42no—
197C2×C3×C4—48no—
198C2×C3×C4—42no—
199C2×C3×C4—44no—
200C2×C3×C4—42no—
201C2×C3×C4—82no—
202C2×C3×C4—82no—
203C2×C3×C4#9512yes—
204C2×C3×C4C2×C4211yescharacteristic
205C2×C3×C4C2×C4⋊C411yescharacteristic
206C2×Dic3—48no—
207C2×Dic3—84no—
208C2×Dic3—84no—
209C2×Dic3—816no—
210C24—22no—
211C24—22no—
212C24—24no—
213C24—44no—
214C24—44no—
215C24Dic3⋊C414yes—
216C24(C2×D3)⋊C414yes—
217C24#21712yes—
218C24#21711yescharacteristic
219C24C2×#6712yes—
220C2×D4—42no—
221C2×D4—42no—
222C2×D4—44no—
223C2×D4—44no—
224C2×D4—42no—
225C2×D4—42no—
226C2×D4—41no—
227C2×D4—41no—
228C2×D4—42no—
229C2×D4—42no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4C4×Dic312yes—
233C2×D4Dic3⋊C414yes—
234C2×D4C4⋊Dic312yes—
235C2×D4(C2×D3)⋊C414yes—
236C2×D4C2×Dic611yescharacteristic
237C2×D4C2×C4×D312yes—
238C2×D4C2×D1211yescharacteristic
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—24no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—42no—
246C22×C4—42no—
247C22×C4—44no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—124no—
251C22×C4—128no—
252C22×C4—244no—
253C22×C4—244no—
254C22×C4C4×Dic312yes—
255C22×C4C4⋊Dic312yes—
256C22×C4#21711yescharacteristic
257C22×C4C22×Dic311yescharacteristic
258C4⋊C4—42no—
259C4⋊C4—42no—
260C22⋊C4—48no—
261C22⋊C4—48no—
262C22⋊C4—68no—
263C22⋊C4—124no—
264C22⋊C4—128no—
265C42—42no—
266C42—42no—
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—21no—
271C22×C3—24no—
272C22×C3—24no—
273C22×C3—22no—
274C22×C3—24no—
275C22×C3—22no—
276C22×C3—44no—
277C22×C3—42no—
278C22×C3—42no—
279C22×C3—42no—
280C22×C3—42no—
281C22×C3—42no—
282C22×C3—42no—
283C22×C3—48no—
284C22×C3—48no—
285C22×C3—82no—
286C22×C3—82no—
287C22×C3—84no—
288C22×C3—84no—
289C22×C3#34212yes—
290C22×C3#34212yes—
291C22×C3#35312yes—
292C22×C3#35412yes—
293C22×C3C2×#9511yescharacteristic
294C22×C3C2×#9911yescharacteristic
295C22×C3C2×#9911yescharacteristic
296C3×C4—22no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—44no—
300C3×C4—82no—
301Dic3—88no—
302C23—22no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—24no—
307C23—24no—
308C23—22no—
309C23—22no—
310C23—24no—
311C23—24no—
312C23—42no—
313C23—42no—
314C23—48no—
315C23—48no—
316C23—42no—
317C23—42no—
318C23—44no—
319C23—44no—
320C23—82no—
321C23—88no—
322C23—82no—
323C23C3⋊#9512yes—
324C23C3⋊#9514yes—
325C23#83411yescharacteristic
326C23#83412yes—
327C23#83411yescharacteristic
328C23C2×Dic3⋊C412yes—
329C23C2×(C2×D3)⋊C412yes—
330C23C2×#21711yescharacteristic
331D4—42no—
332D4—44no—
333D4—42no—
334D4—44no—
335D4—41no—
336D4—42no—
337D4—41no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—22no—
343C2×C4—22no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—42no—
348C2×C4—48no—
349C2×C4—42no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—82no—
353C2×C4—82no—
354C2×C4—128no—
355C2×C4—244no—
356C2×C4—2416no—
357C2×C4—244no—
358C2×C4C3⋊#9512yes—
359C2×C4C2×C4×Dic311yescharacteristic
360C2×C4C2×C4⋊Dic311yescharacteristic
361C2×C3—21no—
362C2×C3—21no—
363C2×C3—22no—
364C2×C3—42no—
365C2×C3—44no—
366C2×C3—42no—
367C2×C3#139314yescomplemented
368C2×C3C2×#34211yescharacteristic
369C2×C3C2×#35311yescharacteristic
370C2×C3C2×#35411yescharacteristic
371C22—21no—
372C22—21no—
373C22—24no—
374C22—24no—
375C22—21no—
376C22—21no—
377C22—22no—
378C22—24no—
379C22—22no—
380C22—44no—
381C22—42no—
382C22—42no—
383C22—42no—
384C22—42no—
385C22—48no—
386C22—48no—
387C22—42no—
388C22—42no—
389C22—82no—
390C22—82no—
391C22—84no—
392C22—84no—
393C22#452612yes—
394C22#452612yes—
395C22#454112yes—
396C22#454512yes—
397C22C2×C3⋊#9511yescharacteristic
398C22C2×#83411yescharacteristic
399C22C2×#83411yescharacteristic
400C4—22no—
401C4—21no—
402C4—21no—
403C4—44no—
404C4—82no—
405C4—248no—
406C3C2×#139311yescharacteristic, complemented, Sylow
407C2—21no—
408C2—21no—
409C2—22no—
410C2—42no—
411C2—44no—
412C2—42no—
413C2#6764114yesdirect product factor
414C2C2×#452611yescharacteristic
415C2C2×#454111yescharacteristic
416C2C2×#454511yescharacteristic
subgroupG/H
centerC23#834
commutatorC2×C3×C4C2×C42
FrattiniC22×D4C22×D3
FittingC2×C3×#390C2
socleC23×C3#99
prime radicalC23×C3×D4C22
cosocleC22×C3×D4C23
hypercenterC2×#390D3
hypocenterC3C2×#1393
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24 ◁ C23×D4 ◁ C2×#390

Complements

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

NH ≅ G/Ntype
3. #67641C2direct (N × H)
6. C22×C3×C4⋊C4C4semidirect
8. C22×C3×C42C4semidirect
11. C3×#390C4semidirect
15. C2×#834C4semidirect
37. C2×C3×C4⋊C4C2×C4semidirect
43. C2×C3×C42C2×C4semidirect
50. #834C2×C4semidirect
52. C22×C4⋊C4Dic3semidirect
55. C22×C42Dic3semidirect
61. #390Dic3semidirect
137. C2×C4⋊C4C2×Dic3semidirect
147. C2×C42C2×Dic3semidirect
367. C2×C3#1393semidirect
406. C3C2×#1393semidirect
413. C2#67641direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#3902
C22×#8342
#676412
C2×#13933