Finite Group Databasebeta

← C3×#71605 #11785241

Subgroups — C3×#71605 #11785241

Order 1152.

925 conjugacy classes of subgroups.

The subgroup lattice has 453 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×C4×D4C211yesmaximal, characteristic
2C2×C3×#832C212yesmaximal
3C3×#4936C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
5C3×#4529C212yesmaximal, complemented
6C3×#1804—31nomaximal
7#71605C311yesmaximal, characteristic, direct product factor
8C22×C32×D4C2211yescharacteristic
9C23×C32×C4—21no—
10C32×C4×D4—22no—
11C32×C4×D4C414yescomplemented
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×C32×C22⋊C4—21no—
14C2×C32×C42C2211yescharacteristic
15C2×C3×C4⋊Dic3C2212yes—
16C22×C3×C3⋊C8C2212yes—
17C3×#832—24no—
18C3×C3⋊#95—21no—
19C3×C3⋊#95—21no—
20C3×C4⋊(C3⋊C8)—22no—
21C2×C3×C4×D4—21no—
22C2×C3×C4×D4D311yescharacteristic
23C2×C3×C4×D4C2×C311yescharacteristic
24C2×C3×C4⋊C8—31no—
25C2×C3×#102—32no—
26C3×#382—31no—
27C2×#832C2×C312yes—
28#4936C2×C311yescharacteristic, complemented
29C2×C4⋊(C3⋊C8)C2×C311yescharacteristic, complemented
30C3×#340—32no—
31#4529C2×C312yescomplemented
32C24×C32—21no—
33C2×C32×D4—21no—
34C2×C32×D4—21no—
35C2×C32×D4—21no—
36C2×C32×D4C2×C412yes—
37C2×C32×D4D412yes—
38C2×C32×D4Q812yes—
39C22×C32×C4—21no—
40C22×C32×C4—24no—
41C22×C32×C4—44no—
42C22×C32×C4D411yescharacteristic
43C22×C32×C4D411yescharacteristic
44C22×C32×C4C2311yescharacteristic
45C22×C3×Dic3—22no—
46C32×C4⋊C4—21no—
47C32×C4⋊C4C2×C412yes—
48C32×C22⋊C4—24no—
49C32×C42C2×C412yes—
50C32×C42D412yes—
51C3×C4⋊Dic3—22no—
52C3×C4⋊Dic3—22no—
53C2×C3×C3⋊C8—22no—
54C2×C3×C3⋊C8—22no—
55C2×C3×C3⋊C8—22no—
56#1804—31noSylow
57C22×C3×D4—21no—
58C22×C3×D4C2×D311yescharacteristic
59C22×C3×D4C22×C311yescharacteristic
60C23×C3×C4—22no—
61C23×C3×C4—21no—
62C23×C3×C4—21no—
63C22×C3×C8—32no—
64C3×C4×D4—24no—
65C3×C4×D4—22no—
66C3×C4×D4—22no—
67C3×C4×D4—24no—
68C3×C4×D4Dic314yescomplemented
69C3×C4×D4C3×C414yescomplemented
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—32no—
72C2×C3×C4⋊C4C2×D311yescharacteristic
73C2×C3×C4⋊C4C22×C311yescharacteristic
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4—21no—
76C2×C3×C22⋊C4—21no—
77C2×C3×C42—21no—
78C2×C3×C42C2×D311yescharacteristic
79C2×C3×C42C22×C311yescharacteristic
80C2×C4⋊Dic3C22×C312yes—
81C22×C3⋊C8C22×C312yes—
82C3×C4⋊C8—62no—
83C3×#102—64no—
84C3×#95—61no—
85C3×#95—61no—
86#832—24no—
87C3⋊#95—21no—
88C3⋊#95—21no—
89C4⋊(C3⋊C8)—22no—
90C23×C32—22no—
91C23×C32—22no—
92C23×C32—22no—
93C23×C32—42no—
94C23×C32—42no—
95C23×C32C2×D411yescharacteristic
96C32×D4—22no—
97C32×D4—22no—
98C32×D4—22no—
99C32×D4C4⋊C414yes—
100C2×C32×C4—22no—
101C2×C32×C4—22no—
102C2×C32×C4—22no—
103C2×C32×C4—48no—
104C2×C32×C4—42no—
105C2×C32×C4C22⋊C412yes—
106C2×C32×C4C22⋊C412yes—
107C2×C32×C4C22×C411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×Q811yescharacteristic
111C2×C32×C4#4111yescharacteristic
112C2×C32×C4#4111yescharacteristic
113C2×C3×Dic3—42no—
114C2×C3×Dic3—42no—
115C2×C3×Dic3—42no—
116C3×C3⋊C8—42no—
117C2×C4×D4C3×D311yescharacteristic
118C2×C4⋊C8—31no—
119C2×#102—32no—
120#382—31no—
121#340—32no—
122C24×C3—22no—
123C24×C3—21no—
124C24×C3—21no—
125C2×C3×D4—22no—
126C2×C3×D4—22no—
127C2×C3×D4—22no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—21no—
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4Dic612yes—
138C2×C3×D4D1212yes—
139C2×C3×D4C2×Dic312yes—
140C2×C3×D4C2×C3×C412yes—
141C2×C3×D4C3×D412yes—
142C2×C3×D4C3×Q812yes—
143C22×C3×C4—22no—
144C22×C3×C4—28no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—24no—
148C22×C3×C4—24no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—21no—
152C22×C3×C4—48no—
153C22×C3×C4—44no—
154C22×C3×C4—44no—
155C22×C3×C4—62no—
156C22×C3×C4#6711yescharacteristic
157C22×C3×C4#6711yescharacteristic
158C22×C3×C4C3×D411yescharacteristic
159C22×C3×C4C3×D411yescharacteristic
160C22×C3×C4C22×D311yescharacteristic
161C22×C3×C4C23×C311yescharacteristic
162C22×Dic3—22no—
163C2×C3×C8—62no—
164C2×C3×C8—62no—
165C2×C3×C8—62no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—62no—
171C3×C4⋊C4—62no—
172C3×C4⋊C4C2×Dic312yes—
173C3×C4⋊C4C2×C3×C412yes—
174C3×C22⋊C4—28no—
175C3×C22⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C42—22no—
178C3×C42—22no—
179C3×C42C2×Dic312yes—
180C3×C42#6712yes—
181C3×C42C2×C3×C412yes—
182C3×C42C3×D412yes—
183C4⋊Dic3—22no—
184C4⋊Dic3—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C22×C32—24no—
189C22×C32—42no—
190C22×C32—42no—
191C22×C32—44no—
192C22×C32—44no—
193C22×C32C2×C22⋊C411yescharacteristic
194C22×C32#12111yescharacteristic
195C22×C32C22⋊Q811yescharacteristic
196C22×C32#13512yes—
197C22×C32AGL(1,ℤ/8)12yes—
198C32×C4—22no—
199C32×C4—22no—
200C32×C4—42no—
201C32×C4C2×C4⋊C411yescharacteristic
202C32×C4#11711yescharacteristic
203C32×C4#12111yescharacteristic
204C32×C4C22⋊Q811yescharacteristic
205C3×Dic3—82no—
206C22×D4C2×C3×D311yescharacteristic
207C23×C4—21no—
208C22×C8—32no—
209C4×D4—22no—
210C4×D4C3×Dic314yescomplemented
211C2×C4⋊C4—32no—
212C2×C4⋊C4C2×C3×D311yescharacteristic
213C2×C22⋊C4—21no—
214C2×C42C2×C3×D311yescharacteristic
215C4⋊C8—62no—
216#102—64no—
217#95—61no—
218#95—61no—
219C23×C3—24no—
220C23×C3—24no—
221C23×C3—24no—
222C23×C3—22no—
223C23×C3—22no—
224C23×C3—22no—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3—22no—
228C23×C3—21no—
229C23×C3—44no—
230C23×C3—44no—
231C23×C3—42no—
232C23×C3—42no—
233C23×C3—42no—
234C23×C3—42no—
235C23×C3C2×#6711yescharacteristic
236C23×C3C2×C3×D411yescharacteristic
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—24no—
240C3×D4—22no—
241C3×D4—22no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—22no—
246C3×D4—24no—
247C3×D4C4⋊Dic314yes—
248C3×D4C3×C4⋊C414yes—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—24no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—42no—
266C2×C3×C4—416no—
267C2×C3×C4—48no—
268C2×C3×C4—48no—
269C2×C3×C4—44no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—122no—
273C2×C3×C4—122no—
274C2×C3×C4—122no—
275C2×C3×C4#21712yes—
276C2×C3×C4#21712yes—
277C2×C3×C4C3×C22⋊C412yes—
278C2×C3×C4C3×C22⋊C412yes—
279C2×C3×C4C2×Dic611yescharacteristic
280C2×C3×C4C2×D1211yescharacteristic
281C2×C3×C4#23511yescharacteristic
282C2×C3×C4#23511yescharacteristic
283C2×C3×C4C22×Dic311yescharacteristic
284C2×C3×C4C2×#6711yescharacteristic
285C2×C3×C4C22×C3×C411yescharacteristic
286C2×C3×C4C2×C3×D411yescharacteristic
287C2×C3×C4C2×C3×D411yescharacteristic
288C2×C3×C4C2×C3×Q811yescharacteristic
289C2×C3×C4C3×#4111yescharacteristic
290C2×C3×C4C3×#4111yescharacteristic
291C2×Dic3—42no—
292C2×Dic3—42no—
293C2×Dic3—42no—
294C3×C8—122no—
295C3⋊C8—42no—
296C2×C32—44no—
297C2×C32#38011yescharacteristic
298C2×C32#41611yescharacteristic
299C2×C32#41811yescharacteristic
300C2×C32#46312yes—
301C2×C32#47812yes—
302C24—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4C3×Dic612yes—
307C2×D4C3×D1212yes—
308C2×D4C2×C3×Dic312yes—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—44no—
312C22×C4—62no—
313C22×C4C3×#6711yescharacteristic
314C22×C4C3×#6711yescharacteristic
315C22×C4C22×C3×D311yescharacteristic
316C2×C8—62no—
317C2×C8—62no—
318C2×C8—62no—
319C4⋊C4—21no—
320C4⋊C4—62no—
321C4⋊C4—62no—
322C4⋊C4C2×C3×Dic312yes—
323C22⋊C4—24no—
324C42C2×C3×Dic312yes—
325C42C3×#6712yes—
326C22×C3—28no—
327C22×C3—24no—
328C22×C3—24no—
329C22×C3—21no—
330C22×C3—21no—
331C22×C3—22no—
332C22×C3—22no—
333C22×C3—21no—
334C22×C3—44no—
335C22×C3—44no—
336C22×C3—48no—
337C22×C3—48no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—44no—
341C22×C3—44no—
342C22×C3—42no—
343C22×C3—42no—
344C22×C3—44no—
345C22×C3—44no—
346C22×C3#92411yescharacteristic
347C22×C3#93011yescharacteristic
348C22×C3#93212yes—
349C22×C3#95012yes—
350C22×C3C2×#21711yescharacteristic
351C22×C3C2×C3×C22⋊C411yescharacteristic
352C22×C3C3×#12111yescharacteristic
353C22×C3C3×C22⋊Q811yescharacteristic
354C22×C3C3×#13512yes—
355C22×C3C3×AGL(1,ℤ/8)12yes—
356C3×C4—24no—
357C3×C4—22no—
358C3×C4—22no—
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—42no—
366C3×C4—44no—
367C3×C4—42no—
368C3×C4—42no—
369C3×C4—242no—
370C3×C4#92411yescharacteristic
371C3×C4C2×C4⋊Dic311yescharacteristic
372C3×C4#92611yescharacteristic
373C3×C4#93011yescharacteristic
374C3×C4C2×C3×C4⋊C411yescharacteristic
375C3×C4C3×#11711yescharacteristic
376C3×C4C3×#12111yescharacteristic
377C3×C4C3×C22⋊Q811yescharacteristic
378Dic3—82no—
379C32#180411yescharacteristic, complemented, Sylow
380C23—22no—
381C23—22no—
382C23—22no—
383C23—42no—
384C23—42no—
385C23C2×C3×#6711yescharacteristic
386D4—22no—
387D4—22no—
388D4—22no—
389D4C3×C4⋊Dic314yes—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—48no—
394C2×C4—42no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4C3×#21712yes—
399C2×C4C3×#21712yes—
400C2×C4C2×C3×Dic611yescharacteristic
401C2×C4C2×C3×D1211yescharacteristic
402C2×C4C3×#23511yescharacteristic
403C2×C4C3×#23511yescharacteristic
404C2×C4C22×C3×Dic311yescharacteristic
405C2×C4C2×C3×#6711yescharacteristic
406C8—122no—
407C2×C3—21no—
408C2×C3—21no—
409C2×C3—21no—
410C2×C3—22no—
411C2×C3—22no—
412C2×C3—48no—
413C2×C3—44no—
414C2×C3—44no—
415C2×C3#500912yes—
416C2×C3#501612yes—
417C2×C3#521211yescharacteristic
418C2×C3#521511yescharacteristic
419C2×C3#523311yescharacteristic
420C2×C3C3×#38011yescharacteristic
421C2×C3C3×#41611yescharacteristic
422C2×C3C3×#41811yescharacteristic
423C2×C3C3×#46312yes—
424C2×C3C3×#47812yes—
425C22—24no—
426C22—42no—
427C22—42no—
428C22—44no—
429C22—44no—
430C22C3×#92411yescharacteristic
431C22C3×#93011yescharacteristic
432C22C3×#93212yes—
433C22C3×#95012yes—
434C22C2×C3×#21711yescharacteristic
435C4—22no—
436C4—22no—
437C4—42no—
438C4—242no—
439C4C3×#92411yescharacteristic
440C4C2×C3×C4⋊Dic311yescharacteristic
441C4C3×#92611yescharacteristic
442C4C3×#93011yescharacteristic
443C3—21no—
444C3#7160511yescharacteristic, direct product factor
445C3C3×#180411yescharacteristic, complemented
446C2—44no—
447C2C3×#500912yes—
448C2C3×#501612yes—
449C2C3×#521211yescharacteristic
450C2C3×#521511yescharacteristic
451C2C3×#523311yescharacteristic
subgroupG/H
centerC23×C3C2×#67
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×C4×D4C2
socleC23×C32C2×D4
prime radicalC22×C32×D4C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×C4×D4D3
hypocenterC3C3×#1804
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23×C3 ◁ C2×C3×C42 ◁ C2×C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×#4936C2semidirect
4. C2×C3×C4⋊(C3⋊C8)C2semidirect
5. C3×#4529C2semidirect
7. #71605C3direct (N × H)
11. C32×C4×D4C4semidirect
28. #4936C2×C3semidirect
29. C2×C4⋊(C3⋊C8)C2×C3semidirect
31. #4529C2×C3semidirect
68. C3×C4×D4Dic3semidirect
69. C3×C4×D4C3×C4semidirect
210. C4×D4C3×Dic3semidirect
379. C32#1804semidirect
444. C3#71605direct (N × H)
445. C3C3×#1804semidirect

Maximal subgroups

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

subgroupindex
C2×C32×C4×D42
C2×C3×#8322
C3×#49362
C2×C3×C4⋊(C3⋊C8)2
C3×#45292
C3×#18043
#716053