Finite Group Databasebeta

← #11805650

Subgroups — #11805650

Order 1152.

1980 conjugacy classes of subgroups.

The subgroup lattice has 483 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64101C211yesmaximal, characteristic, complemented
2C2×C4⋊C4×C3⋊D3C211yesmaximal, characteristic, complemented
3C2×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic
4C32×C4×C4⋊C4C211yesmaximal, characteristic, complemented
5#10592686C211yesmaximal, characteristic, complemented
6#10592686C211yesmaximal, characteristic, complemented
7#10592639C212yesmaximal, complemented
8C4×#3752C212yesmaximal, complemented
9#10592635C211yesmaximal, characteristic, complemented
10#10592364C212yesmaximal
11#10592352C212yesmaximal, complemented
12#73771—34nomaximal
13C22×C4⋊(C3⋊D3)C2211yescharacteristic
14C22×C4×C3⋊D3C2212yes—
15C22×C4×C3⋊D3C2211yescharacteristic
16C22×C4×C3⋊D3C2211yescharacteristic
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×C32×C4⋊C4C2211yescharacteristic
19C2×C32×C42C2212yes—
20C2×C32×C42C2211yescharacteristic
21C2×#3752C2212yes—
22C2×#3752C2212yes—
23C2×#3752C2212yes—
24C2×#3752C2211yescharacteristic
25C2×#3752C2211yescharacteristic
26C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
27C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
29C2×#3750C2212yes—
30C2×C4×C3⋊Dic3C2212yes—
31#64101—24no—
32C4⋊C4×C3⋊D3—24no—
33C4×C4⋊(C3⋊D3)C418yescomplemented
34C32×#95C2212yes—
35#63658C2212yes—
36#63658C2212yes—
37#63658C2212yes—
38#63658C2212yes—
39C2×#892—34no—
40C2×D3×C4⋊C4—34no—
41C2×C4×D12—34no—
42C3×C4×C4⋊C4D314yescomplemented
43#4987—34no—
44#4987—34no—
45#4940—38no—
46C4×(C2×D3)⋊C4—38no—
47#4936—34no—
48#4665—38no—
49#4653—38no—
50C23×C3⋊D3C2311yescharacteristic
51C23×C3⋊D3C2311yescharacteristic
52C22×C32×C4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2311yescharacteristic
55C22×C32×C4C2312yes—
56C22×C32×C4C2311yescharacteristic
57C22×C3⋊Dic3C2312yes—
58C22×C3⋊Dic3C2312yes—
59C22×C3⋊Dic3C2311yescharacteristic
60C22×C3⋊Dic3C2311yescharacteristic
61C2×C4⋊(C3⋊D3)—22no—
62C2×C4⋊(C3⋊D3)—22no—
63C2×C4⋊(C3⋊D3)C2×C414yes—
64C2×C4×C3⋊D3—22no—
65C2×C4×C3⋊D3—22no—
66C2×C4×C3⋊D3—22no—
67C2×C4×C3⋊D3—22no—
68C2×C4×C3⋊D3—24no—
69C2×C4×C3⋊D3—24no—
70C2×C4×C3⋊D3—42no—
71C2×C4×C3⋊D3C2×C414yes—
72C2×C4×C3⋊D3C2×C414yescomplemented
73C32×C4⋊C4—22no—
74C32×C4⋊C4D414yescomplemented
75C32×C42—42no—
76C32×C42C2×C414yescomplemented
77#3752—24no—
78#3752—24no—
79#3752—42no—
80#3752C2×C414yes—
81#3752C2×C414yescomplemented
82C4⋊(C3⋊Dic3)—22no—
83C4⋊(C3⋊Dic3)—22no—
84C4⋊(C3⋊Dic3)C2×C414yescomplemented
85#3750—24no—
86C4×C3⋊Dic3—24no—
87#2352—91noSylow
88C22×D12—34no—
89C22×C4×D3—34no—
90C22×C4×D3—34no—
91C22×C4×D3—38no—
92C2×C3×C4⋊C4C2×D314yes—
93C2×C3×C4⋊C4C2×D314yes—
94C2×C3×C42C2×D318yes—
95C2×C3×C42C2×D314yes—
96C2×(C2×D3)⋊C4—34no—
97C2×(C2×D3)⋊C4—34no—
98C2×(C2×D3)⋊C4—38no—
99C2×(C2×D3)⋊C4—38no—
100C2×(C2×D3)⋊C4—38no—
101C2×C4⋊Dic3—34no—
102C2×C4⋊Dic3—34no—
103C2×C4⋊Dic3—34no—
104C2×Dic3⋊C4—38no—
105C2×C4×Dic3—38no—
106#892—616no—
107D3×C4⋊C4—616no—
108C4×D12—332no—
109C3×#95C2×D318yes—
110C3⋊#95—38no—
111C3⋊#95—38no—
112C3⋊#95—38no—
113C3⋊#95—38no—
114C23×C32C2411yescharacteristic
115C22×C3⋊D3—21no—
116C22×C3⋊D3—21no—
117C22×C3⋊D3—21no—
118C22×C3⋊D3—21no—
119C22×C3⋊D3—21no—
120C22×C3⋊D3—21no—
121C22×C3⋊D3—21no—
122C22×C3⋊D3—21no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3C22×C412yes—
126C22×C3⋊D3C22×C412yes—
127C22×C3⋊D3#4112yes—
128C22×C3⋊D3#4112yes—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—21no—
133C2×C32×C4—22no—
134C2×C32×C4—21no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4—22no—
138C2×C32×C4C22×C412yes—
139C2×C32×C4C22×C412yes—
140C2×C32×C4C22×C412yes—
141C2×C32×C4C2×D414yes—
142C2×C32×C4C2×D412yes—
143C2×C32×C4#4112yes—
144C2×C3⋊Dic3—21no—
145C2×C3⋊Dic3—21no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3—21no—
148C2×C3⋊Dic3—22no—
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151C2×C3⋊Dic3—22no—
152C2×C3⋊Dic3—22no—
153C2×C3⋊Dic3—22no—
154C2×C3⋊Dic3C22×C412yes—
155C2×C3⋊Dic3C22×C412yes—
156C4⋊(C3⋊D3)—28no—
157C4×C3⋊D3—28no—
158C4×C3⋊D3—28no—
159C4×C3⋊D3—48no—
160C2×C4×D4—91no—
161C2×#117—91no—
162C22×C4⋊C4—91no—
163#388—92no—
164#387—92no—
165#386—92no—
166#382—91no—
167#380—91no—
168#380—91no—
169C4×C4⋊C4C3⋊D311yescharacteristic, complemented
170C4×C22⋊C4—92no—
171C23×D3—34no—
172C23×D3—34no—
173C22×C3×C4C22×D318yes—
174C22×C3×C4C22×D314yes—
175C22×C3×C4C22×D314yes—
176C22×C3×C4C22×D318yes—
177C22×C3×C4C22×D314yes—
178C22×Dic3—34no—
179C22×Dic3—34no—
180C22×Dic3—38no—
181C22×Dic3—38no—
182C2×D12—316no—
183C2×D12—68no—
184C2×D12—68no—
185C2×C4×D3—316no—
186C2×C4×D3—316no—
187C2×C4×D3—68no—
188C2×C4×D3—68no—
189C2×C4×D3—68no—
190C2×C4×D3—68no—
191C2×C4×D3—616no—
192C2×C4×D3—616no—
193C2×C4×D3—128no—
194C3×C4⋊C4—28no—
195C3×C4⋊C4D12116yescomplemented
196C3×C42—48no—
197C3×C42C4×D3116yescomplemented
198(C2×D3)⋊C4—316no—
199(C2×D3)⋊C4—316no—
200(C2×D3)⋊C4—616no—
201(C2×D3)⋊C4—616no—
202(C2×D3)⋊C4—128no—
203C4⋊Dic3—316no—
204C4⋊Dic3—68no—
205C4⋊Dic3—68no—
206Dic3⋊C4—616no—
207C4×Dic3—616no—
208C22×C32C23×C411yescharacteristic
209C22×C32C22×D411yescharacteristic
210C22×C32C2×#4111yescharacteristic
211C22×C32C2×#4111yescharacteristic
212C22×C32C2×#4111yescharacteristic
213C22×C32H(5,𝔽2)11yescharacteristic
214C22×C322−1+411yescharacteristic
215C2×C3⋊D3—22no—
216C2×C3⋊D3—22no—
217C2×C3⋊D3—22no—
218C2×C3⋊D3—22no—
219C2×C3⋊D3—22no—
220C2×C3⋊D3—22no—
221C2×C3⋊D3—22no—
222C2×C3⋊D3—22no—
223C2×C3⋊D3—22no—
224C2×C3⋊D3—41no—
225C2×C3⋊D3—41no—
226C2×C3⋊D3—41no—
227C2×C3⋊D3—41no—
228C2×C3⋊D3#11714yescomplemented
229C32×C4—22no—
230C32×C4—22no—
231C32×C4—24no—
232C32×C4—42no—
233C32×C4C4×D414yes—
234C3⋊Dic3—22no—
235C3⋊Dic3—22no—
236C3⋊Dic3—42no—
237C3⋊Dic3—42no—
238C22×D4—91no—
239C23×C4—91no—
240C23×C4—91no—
241C23×C4—92no—
242C4×D4—98no—
243#117—184no—
244C2×C4⋊C4—91no—
245C2×C4⋊C4—91no—
246C2×C4⋊C4—92no—
247C2×C4⋊C4—91no—
248C2×C4⋊C4—184no—
249C2×C4⋊C4C2×C3⋊D311yescharacteristic
250C2×C4⋊C4C2×C3⋊D311yescharacteristic
251C2×C22⋊C4—91no—
252C2×C22⋊C4—91no—
253C2×C22⋊C4—92no—
254C2×C22⋊C4—92no—
255C2×C22⋊C4—92no—
256C2×C42—92no—
257C2×C42C2×C3⋊D312yes—
258C2×C42C2×C3⋊D311yescharacteristic
259#95—92no—
260#95—92no—
261#95—92no—
262#95—92no—
263#95C2×C3⋊D312yes—
264C23×C3C23×D314yes—
265C22×D3—38no—
266C22×D3—38no—
267C22×D3—38no—
268C22×D3—38no—
269C22×D3—64no—
270C22×D3—64no—
271C22×D3—64no—
272C22×D3—64no—
273C22×D3—64no—
274C22×D3—64no—
275C22×D3—64no—
276C22×D3—64no—
277C22×D3—64no—
278C22×D3—64no—
279C2×C3×C4—24no—
280C2×C3×C4—28no—
281C2×C3×C4—24no—
282C2×C3×C4—28no—
283C2×C3×C4—24no—
284C2×C3×C4—24no—
285C2×C3×C4—28no—
286C2×C3×C4—28no—
287C2×C3×C4—28no—
288C2×C3×C4C2×C4×D318yes—
289C2×C3×C4C2×C4×D318yes—
290C2×C3×C4C2×C4×D318yes—
291C2×C3×C4C2×D12116yes—
292C2×C3×C4C2×D1218yes—
293C2×C3×C4#23518yes—
294C2×Dic3—38no—
295C2×Dic3—38no—
296C2×Dic3—64no—
297C2×Dic3—64no—
298C2×Dic3—64no—
299C2×Dic3—64no—
300C2×Dic3—68no—
301C2×Dic3—68no—
302C2×Dic3—68no—
303C2×Dic3—68no—
304C2×Dic3—68no—
305C2×Dic3—68no—
306D12—632no—
307C4×D3—632no—
308C4×D3—632no—
309C4×D3—1232no—
310C2×C32C2×#11711yescharacteristic
311C2×C32C2×C4×D411yescharacteristic
312C2×C32#52011yescharacteristic
313C2×C32#54611yescharacteristic
314C2×C32#54811yescharacteristic
315C2×C32#55211yescharacteristic
316C2×C32#55311yescharacteristic
317C3⋊D3—24no—
318C3⋊D3—42no—
319C24—91no—
320C24—91no—
321C2×D4—94no—
322C2×D4—182no—
323C2×D4—182no—
324C22×C4—91no—
325C22×C4—91no—
326C22×C4—94no—
327C22×C4—94no—
328C22×C4—92no—
329C22×C4—92no—
330C22×C4—184no—
331C22×C4—182no—
332C22×C4—182no—
333C22×C4—182no—
334C22×C4—182no—
335C22×C4—184no—
336C22×C4—362no—
337C22×C4C22×C3⋊D312yes—
338C22×C4C22×C3⋊D311yescharacteristic
339C22×C4C22×C3⋊D311yescharacteristic
340C22×C4C22×C3⋊D312yes—
341C22×C4C22×C3⋊D311yescharacteristic
342C4⋊C4—22no—
343C4⋊C4—94no—
344C4⋊C4—184no—
345C4⋊C4—182no—
346C4⋊C4—182no—
347C4⋊C4C4⋊(C3⋊D3)14yescomplemented
348C22⋊C4—94no—
349C22⋊C4—94no—
350C22⋊C4—184no—
351C22⋊C4—184no—
352C22⋊C4—362no—
353C42—42no—
354C42—184no—
355C42C4×C3⋊D314yescomplemented
356C22×C3C22×C4×D314yes—
357C22×C3C22×D1214yes—
358C22×C3C2×#23514yes—
359C22×C3#100414yes—
360C22×C3#100714yes—
361C22×C3D3×#4114yes—
362C22×C3D3×#4114yes—
363C2×D3—316no—
364C2×D3—68no—
365C2×D3—68no—
366C2×D3—68no—
367C2×D3—68no—
368C2×D3—68no—
369C2×D3—68no—
370C2×D3—68no—
371C2×D3—68no—
372C2×D3—68no—
373C2×D3—124no—
374C2×D3—124no—
375C2×D3—124no—
376C2×D3—124no—
377C3×C4—28no—
378C3×C4—28no—
379C3×C4—216no—
380C3×C4—48no—
381C3×C4C4×D12116yes—
382Dic3—68no—
383Dic3—68no—
384Dic3—128no—
385Dic3—128no—
386C32#235211yescharacteristic, complemented, Sylow
387C23—92no—
388C23—92no—
389C23—92no—
390C23—92no—
391C23—181no—
392C23—181no—
393C23—181no—
394C23—181no—
395C23—181no—
396C23—181no—
397C23—181no—
398C23—181no—
399C23—181no—
400C23—181no—
401C23C23×C3⋊D311yescharacteristic
402D4—188no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—92no—
413C2×C4—92no—
414C2×C4—181no—
415C2×C4—181no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—188no—
419C2×C4—188no—
420C2×C4—182no—
421C2×C4—182no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4—182no—
425C2×C4—182no—
426C2×C4—368no—
427C2×C4C2×C4×C3⋊D312yes—
428C2×C4C2×C4×C3⋊D312yes—
429C2×C4C2×C4×C3⋊D312yes—
430C2×C4C2×C4⋊(C3⋊D3)14yes—
431C2×C4C2×C4⋊(C3⋊D3)12yes—
432C2×C4#382812yes—
433C2×C3C2×C4×D1214yes—
434C2×C3#550414yes—
435C2×C3D3×#11714yes—
436C2×C3#555414yes—
437C2×C3#555814yes—
438C2×C3#557614yes—
439C2×C3#558114yes—
440D3—616no—
441D3—128no—
442C22—94no—
443C22—182no—
444C22—182no—
445C22—182no—
446C22—182no—
447C22—182no—
448C22—182no—
449C22—182no—
450C22—182no—
451C22—182no—
452C22—361no—
453C22—361no—
454C22—361no—
455C22—361no—
456C22C22×C4×C3⋊D311yescharacteristic
457C22C22×C4⋊(C3⋊D3)11yescharacteristic
458C22C2×#382811yescharacteristic
459C22#6436111yescharacteristic
460C22#6436411yescharacteristic
461C22#41×C3⋊D311yescharacteristic
462C22#41×C3⋊D311yescharacteristic
463C4—22no—
464C4—22no—
465C4—24no—
466C4—42no—
467C4—182no—
468C4—182no—
469C4—362no—
470C4—362no—
471C4C4×C4⋊(C3⋊D3)14yes—
472C3#7377114yescomplemented
473C2—184no—
474C2—362no—
475C2C2×C4×C4⋊(C3⋊D3)11yescharacteristic
476C2#1059597511yescharacteristic
477C2C3⋊D3×#11711yescharacteristic
478C2#1059602511yescharacteristic
479C2#1059602911yescharacteristic
480C2#1059604711yescharacteristic
481C2#1059605211yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×C4×C4⋊C4C2
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC4×C4⋊C4C3⋊D3
hypocenterC32#2352
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#64101C2semidirect
2. C2×C4⋊C4×C3⋊D3C2semidirect
4. C32×C4×C4⋊C4C2semidirect
5. #10592686C2semidirect
6. #10592686C2semidirect
7. #10592639C2semidirect
8. C4×#3752C2semidirect
9. #10592635C2semidirect
11. #10592352C2semidirect
33. C4×C4⋊(C3⋊D3)C4semidirect
42. C3×C4×C4⋊C4D3semidirect
72. C2×C4×C3⋊D3C2×C4semidirect
74. C32×C4⋊C4D4semidirect
76. C32×C42C2×C4semidirect
81. #3752C2×C4semidirect
84. C4⋊(C3⋊Dic3)C2×C4semidirect
169. C4×C4⋊C4C3⋊D3semidirect
195. C3×C4⋊C4D12semidirect
197. C3×C42C4×D3semidirect
228. C2×C3⋊D3#117semidirect
347. C4⋊C4C4⋊(C3⋊D3)semidirect
355. C42C4×C3⋊D3semidirect
386. C32#2352semidirect
472. C3#73771semidirect

Maximal subgroups

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

subgroupindex
C2×#641012
C2×C4⋊C4×C3⋊D32
C2×C4×C4⋊(C3⋊D3)2
C32×C4×C4⋊C42
#105926862
#105926392
C4×#37522
#105926352
#105923642
#105923522
#737713