Finite Group Databasebeta

← #11806885

Subgroups — #11806885

Order 1152.

2496 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64149C211yesmaximal, characteristic
2C2×C4×#638C212yesmaximal, complemented
3C2×#64090C212yesmaximal, complemented
4C32×#386C211yesmaximal, characteristic, complemented
5#10592919C211yesmaximal, characteristic, complemented
6#10592656C212yesmaximal, complemented
7#10592656C212yesmaximal, complemented
8#10592364C212yesmaximal, complemented
9#10592348C212yesmaximal, complemented
10#74679—34nomaximal
11C22×C32×D4C2211yescharacteristic
12C23×C32×C4C2211yescharacteristic
13C22×#638C2212yes—
14C22×#638C2212yes—
15C23×C3⋊Dic3C2211yescharacteristic
16C22×C4⋊(C3⋊D3)C2211yescharacteristic
17C22×C4×C3⋊D3C2212yes—
18C2×C32×C22⋊C4C2212yes—
19C2×C32×C22⋊C4C2211yescharacteristic
20C2×#3757C2212yes—
21C2×#3757C2211yescharacteristic
22#64149C418yescomplemented
23C4×#638—28no—
24C2×#3752C2212yes—
25C2×#3752C2212yes—
26C2×#3752C2212yes—
27C2×#3750C2212yes—
28C2×#3750C2212yes—
29C2×C4×C3⋊Dic3C2212yes—
30C2×C4×C3⋊Dic3C2212yes—
31C2×C4×C3⋊Dic3C2211yescharacteristic
32#64090—28no—
33C32×#95C2212yes—
34#63658C2212yes—
35C2×#940—34no—
36C2×C4×#67—38no—
37C2×#67⋊C4—38no—
38C3×#386D314yescomplemented
39#5220—34no—
40#4957—38no—
41#4957—38no—
42#4665—38no—
43#4649—38no—
44C24×C32C2311yescharacteristic
45C24×C32C2311yescharacteristic
46C23×C3⋊D3C2312yes—
47C2×C32×D4—41no—
48C2×C32×D4—41no—
49C2×C32×D4C2×C414yescomplemented
50C22×C32×C4—24no—
51C22×C32×C4—41no—
52C22×C32×C4C2312yes—
53C22×C32×C4C2311yescharacteristic
54C22×C32×C4C2312yes—
55C2×#638—24no—
56C2×#638—24no—
57C2×#638—24no—
58C2×#638—24no—
59C2×#638C2×C418yescomplemented
60C2×#638C2×C418yescomplemented
61C22×C3⋊Dic3—24no—
62C22×C3⋊Dic3—41no—
63C22×C3⋊Dic3C2312yes—
64C22×C3⋊Dic3C2312yes—
65C22×C3⋊Dic3C2312yes—
66C2×C4⋊(C3⋊D3)—41no—
67C2×C4⋊(C3⋊D3)—41no—
68C2×C4⋊(C3⋊D3)C2×C414yescomplemented
69C2×C4×C3⋊D3—24no—
70C2×C4×C3⋊D3—24no—
71C2×C4×C3⋊D3—42no—
72C32×C22⋊C4—24no—
73C32×C22⋊C4—41no—
74#3757—24no—
75#3757—41no—
76#3752—24no—
77#3752—24no—
78#3752—42no—
79#3750—24no—
80#3750D418yescomplemented
81C4×C3⋊Dic3—24no—
82C4×C3⋊Dic3—24no—
83C4×C3⋊Dic3C2×C414yescomplemented
84#2366—91noSylow
85C22×C3×D4C2×D314yes—
86C23×C3×C4C2×D314yes—
87C22×#67—38no—
88C22×#67—38no—
89C23×Dic3—34no—
90C22×D12—34no—
91C22×C4×D3—38no—
92C2×C3×C22⋊C4C2×D318yes—
93C2×C3×C22⋊C4C2×D314yes—
94C2×#217—38no—
95C2×#217—34no—
96#940—332no—
97C4×#67—632no—
98C2×(C2×D3)⋊C4—38no—
99C2×(C2×D3)⋊C4—38no—
100C2×(C2×D3)⋊C4—38no—
101C2×Dic3⋊C4—38no—
102C2×Dic3⋊C4—38no—
103C2×C4×Dic3—34no—
104C2×C4×Dic3—38no—
105C2×C4×Dic3—38no—
106#67⋊C4—632no—
107C3×#95C2×D318yes—
108C3⋊#95—38no—
109C23×C32—22no—
110C23×C32—22no—
111C23×C32—22no—
112C23×C32—22no—
113C23×C32—21no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32C22×C412yes—
118C23×C32C22×C412yes—
119C23×C32C2411yescharacteristic
120C22×C3⋊D3—22no—
121C22×C3⋊D3—22no—
122C22×C3⋊D3—22no—
123C22×C3⋊D3—22no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—22no—
126C22×C3⋊D3C22×C414yes—
127C32×D4—44no—
128C2×C32×C4—22no—
129C2×C32×C4—22no—
130C2×C32×C4—21no—
131C2×C32×C4—21no—
132C2×C32×C4—22no—
133C2×C32×C4—22no—
134C2×C32×C4—22no—
135C2×C32×C4—44no—
136C2×C32×C4C22×C412yes—
137C2×C32×C4C2×D414yes—
138#638—216no—
139#638—216no—
140C2×C3⋊Dic3—22no—
141C2×C3⋊Dic3—22no—
142C2×C3⋊Dic3—22no—
143C2×C3⋊Dic3—22no—
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—44no—
146C2×C3⋊Dic3C22×C414yes—
147C2×C3⋊Dic3C2×D414yes—
148C2×C3⋊Dic3C2×D414yes—
149C2×C3⋊Dic3#4114yes—
150C4⋊(C3⋊D3)—44no—
151C4×C3⋊D3—48no—
152C2×#127—91no—
153C2×C4×D4—92no—
154C2×C4×D4—92no—
155#390—92no—
156#390—92no—
157#386—91no—
158#386—92no—
159#386C3⋊D311yescharacteristic, complemented
160C4×C4⋊C4—92no—
161C24×C3C22×D314yes—
162C24×C3C22×D314yes—
163C23×D3—38no—
164C2×C3×D4—44no—
165C2×C3×D4—44no—
166C2×C3×D4C4×D3116yescomplemented
167C22×C3×C4—216no—
168C22×C3×C4—44no—
169C22×C3×C4C22×D318yes—
170C22×C3×C4C22×D314yes—
171C22×C3×C4C22×D318yes—
172C2×#67—332no—
173C2×#67—332no—
174C2×#67—616no—
175C2×#67—616no—
176C2×#67—616no—
177C2×#67—616no—
178C22×Dic3—38no—
179C22×Dic3—38no—
180C22×Dic3—38no—
181C22×Dic3—616no—
182C22×Dic3—124no—
183C2×D12—316no—
184C2×D12—124no—
185C2×D12—124no—
186C2×C4×D3—616no—
187C2×C4×D3—616no—
188C2×C4×D3—128no—
189C3×C22⋊C4—216no—
190C3×C22⋊C4—44no—
191#217—616no—
192#217—124no—
193(C2×D3)⋊C4—616no—
194(C2×D3)⋊C4—616no—
195(C2×D3)⋊C4—128no—
196Dic3⋊C4—332no—
197Dic3⋊C4—616no—
198C4×Dic3—316no—
199C4×Dic3—616no—
200C4×Dic3—616no—
201C22×C32—24no—
202C22×C32—24no—
203C22×C32—22no—
204C22×C32—22no—
205C22×C32—41no—
206C22×C32—41no—
207C22×C32—42no—
208C22×C32—42no—
209C22×C32—41no—
210C22×C32—41no—
211C22×C32C23×C411yescharacteristic
212C22×C32C22×D412yes—
213C22×C32C2×#4112yes—
214C22×C32H(5,𝔽2)11yescharacteristic
215C22×C32H(5,𝔽2)11yescharacteristic
216C2×C3⋊D3—24no—
217C2×C3⋊D3—24no—
218C2×C3⋊D3—24no—
219C2×C3⋊D3—42no—
220C2×C3⋊D3—42no—
221C2×C3⋊D3—42no—
222C2×C3⋊D3—42no—
223C32×C4—22no—
224C32×C4—24no—
225C32×C4—42no—
226C3⋊Dic3—24no—
227C3⋊Dic3—42no—
228C3⋊Dic3C4×D418yescomplemented
229C22×D4—92no—
230C22×D4—92no—
231C22×D4—91no—
232C22×D4C2×C3⋊D311yescharacteristic
233C23×C4—91no—
234C23×C4—92no—
235C23×C4C2×C3⋊D311yescharacteristic
236#127—98no—
237C4×D4—188no—
238C4×D4—188no—
239C2×C4⋊C4—92no—
240C2×C4⋊C4—92no—
241C2×C22⋊C4—92no—
242C2×C22⋊C4—92no—
243C2×C22⋊C4—92no—
244C2×C22⋊C4—91no—
245C2×C22⋊C4—92no—
246C2×C22⋊C4C2×C3⋊D312yes—
247C2×C22⋊C4C2×C3⋊D311yescharacteristic
248C2×C42—91no—
249C2×C42—92no—
250C2×C42—92no—
251#95—92no—
252#95C2×C3⋊D312yes—
253C23×C3—28no—
254C23×C3—28no—
255C23×C3—24no—
256C23×C3—24no—
257C23×C3—24no—
258C23×C3—24no—
259C23×C3—28no—
260C23×C3—28no—
261C23×C3C2×C4×D318yes—
262C23×C3C2×C4×D318yes—
263C23×C3C23×D314yes—
264C22×D3—316no—
265C22×D3—68no—
266C22×D3—68no—
267C22×D3—68no—
268C22×D3—68no—
269C22×D3—68no—
270C22×D3—68no—
271C3×D4—416no—
272C2×C3×C4—28no—
273C2×C3×C4—28no—
274C2×C3×C4—24no—
275C2×C3×C4—24no—
276C2×C3×C4—28no—
277C2×C3×C4—28no—
278C2×C3×C4—28no—
279C2×C3×C4—416no—
280C2×C3×C4C2×C4×D318yes—
281C2×C3×C4D3×D4116yes—
282#67—664no—
283#67—664no—
284C2×Dic3—316no—
285C2×Dic3—316no—
286C2×Dic3—316no—
287C2×Dic3—316no—
288C2×Dic3—68no—
289C2×Dic3—68no—
290C2×Dic3—68no—
291C2×Dic3—68no—
292C2×Dic3—68no—
293C2×Dic3—1216no—
294D12—1216no—
295C4×D3—1232no—
296C2×C32—42no—
297C2×C32—42no—
298C2×C32C2×C4×D412yes—
299C2×C32#51811yescharacteristic
300C2×C32D4211yescharacteristic
301C2×C32#55012yes—
302C2×C32#55911yescharacteristic
303C3⋊D3—44no—
304C24—92no—
305C24C22×C3⋊D311yescharacteristic
306C24C22×C3⋊D311yescharacteristic
307C2×D4—41no—
308C2×D4—41no—
309C2×D4—98no—
310C2×D4—98no—
311C2×D4—94no—
312C2×D4—184no—
313C2×D4—184no—
314C2×D4—184no—
315C2×D4—184no—
316C2×D4—361no—
317C2×D4—361no—
318C2×D4C4×C3⋊D314yescomplemented
319C22×C4—24no—
320C22×C4—41no—
321C22×C4—92no—
322C22×C4—92no—
323C22×C4—92no—
324C22×C4—184no—
325C22×C4—184no—
326C22×C4—184no—
327C22×C4—362no—
328C22×C4—361no—
329C22×C4C22×C3⋊D312yes—
330C22×C4C22×C3⋊D311yescharacteristic
331C22×C4C22×C3⋊D312yes—
332C4⋊C4—98no—
333C4⋊C4—184no—
334C22⋊C4—24no—
335C22⋊C4—41no—
336C22⋊C4—184no—
337C22⋊C4—184no—
338C22⋊C4—184no—
339C22⋊C4—362no—
340C22⋊C4—361no—
341C42—94no—
342C42—184no—
343C42—184no—
344C22×C3—28no—
345C22×C3—28no—
346C22×C3—216no—
347C22×C3—216no—
348C22×C3—44no—
349C22×C3—44no—
350C22×C3—44no—
351C22×C3—44no—
352C22×C3—48no—
353C22×C3—48no—
354C22×C3C22×C4×D314yes—
355C22×C3C2×D3×D418yes—
356C22×C3#100414yes—
357C22×C3#100414yes—
358C22×C3D3×#4118yes—
359C2×D3—616no—
360C2×D3—616no—
361C2×D3—616no—
362C2×D3—128no—
363C2×D3—128no—
364C2×D3—128no—
365C2×D3—128no—
366C3×C4—28no—
367C3×C4—216no—
368C3×C4—48no—
369Dic3—332no—
370Dic3—616no—
371Dic3—128no—
372C32#236611yescharacteristic, complemented, Sylow
373C23—22no—
374C23—22no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—22no—
380C23—22no—
381C23—94no—
382C23—182no—
383C23—182no—
384C23—182no—
385C23—182no—
386C23—182no—
387C23—182no—
388C23C2×C4×C3⋊D312yes—
389C23C2×C4×C3⋊D312yes—
390C23C23×C3⋊D311yescharacteristic
391D4—44no—
392D4—1816no—
393D4—1816no—
394D4—364no—
395C2×C4—22no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—44no—
403C2×C4—94no—
404C2×C4—94no—
405C2×C4—94no—
406C2×C4—94no—
407C2×C4—182no—
408C2×C4—182no—
409C2×C4—182no—
410C2×C4—182no—
411C2×C4—182no—
412C2×C4—368no—
413C2×C4—364no—
414C2×C4C2×C4×C3⋊D312yes—
415C2×C4D4×C3⋊D314yes—
416C2×C3—48no—
417C2×C3—48no—
418C2×C3#548614yes—
419C2×C3C4×D3×D418yes—
420C2×C3#559014yes—
421C2×C3#559918yes—
422C2×C3#566414yes—
423D3—1216no—
424C22—22no—
425C22—22no—
426C22—24no—
427C22—24no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—42no—
433C22—42no—
434C22—184no—
435C22—184no—
436C22—184no—
437C22—362no—
438C22—362no—
439C22—362no—
440C22—362no—
441C22C22×C4×C3⋊D311yescharacteristic
442C22C2×D4×C3⋊D312yes—
443C22#6436111yescharacteristic
444C22#6436111yescharacteristic
445C22#41×C3⋊D312yes—
446C4—22no—
447C4—24no—
448C4—42no—
449C4—98no—
450C4—184no—
451C4—362no—
452C3#7467914yescomplemented
453C2—42no—
454C2—42no—
455C2—364no—
456C2#1059595711yescharacteristic
457C2C4×D4×C3⋊D312yes—
458C2#1059606111yescharacteristic
459C2#1059607012yes—
460C2#1059613511yescharacteristic
subgroupG/H
center, FrattiniC23C23×C3⋊D3
commutatorC22×C32C23×C4
FittingC32×#386C2
socle, cosocleC23×C32C24
prime radicalC2×#64149C2
hypercenter#386C3⋊D3
hypocenterC32#2366
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×#638C2semidirect
3. C2×#64090C2semidirect
4. C32×#386C2semidirect
5. #10592919C2semidirect
6. #10592656C2semidirect
7. #10592656C2semidirect
8. #10592364C2semidirect
9. #10592348C2semidirect
22. #64149C4semidirect
38. C3×#386D3semidirect
49. C2×C32×D4C2×C4semidirect
59. C2×#638C2×C4semidirect
60. C2×#638C2×C4semidirect
68. C2×C4⋊(C3⋊D3)C2×C4semidirect
80. #3750D4semidirect
83. C4×C3⋊Dic3C2×C4semidirect
159. #386C3⋊D3semidirect
166. C2×C3×D4C4×D3semidirect
228. C3⋊Dic3C4×D4semidirect
318. C2×D4C4×C3⋊D3semidirect
372. C32#2366semidirect
452. C3#74679semidirect

Maximal subgroups

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

subgroupindex
C2×#641492
C2×C4×#6382
C2×#640902
C32×#3862
#105929192
#105926562
#105923642
#105923482
#746793