Finite Group Databasebeta

← #11788145

Subgroups — #11788145

Order 1152.

1037 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64139C211yesmaximal, characteristic
2C22×C3×#208C211yesmaximal, characteristic, complemented
3C3×#5212C211yesmaximal, characteristic, complemented
4C2×#63564C212yesmaximal, complemented
5#10589164C212yesmaximal, complemented
6#71990—31nomaximal
7#69517—31nomaximal
8C23×C32×C4C2211yescharacteristic
9C22×#638—21no—
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11#64139—22no—
12#64139C414yescomplemented
13C2×#3752—21no—
14C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×#217—21no—
16C2×C3×C4⋊Dic3C2212yes—
17C2×C3×#208—22no—
18C2×C3×#208—24no—
19C22×C3×C3⋊C8C2212yes—
20C3×C3⋊#95—21no—
21#63564—24no—
22C22×C3×M16—31no—
23C2×#930—31no—
24C2×#930—31no—
25C2×#930—61no—
26C22×#208D311yescharacteristic, complemented
27C3×#380—31no—
28#5212D311yescharacteristic, complemented
29C2×#821—32no—
30C2×#809—32no—
31#4551—32no—
32#4529—32no—
33C24×C32D411yescharacteristic
34C23×C3⋊D3—21no—
35C22×C32×C4—22no—
36C22×C32×C4—22no—
37C22×C32×C4C2×C412yes—
38C22×C32×C4D411yescharacteristic
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×#638—24no—
42C2×#638—44no—
43C22×C3⋊Dic3—21no—
44C2×C4⋊(C3⋊D3)—22no—
45C2×C4⋊(C3⋊D3)—22no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)C2×C412yes—
48C22×C3×Dic3—22no—
49#3752—24no—
50C4⋊(C3⋊Dic3)—21no—
51C4⋊(C3⋊Dic3)C2×C412yes—
52C3×#217—42no—
53C3×C4⋊Dic3—22no—
54C3×C4⋊Dic3—22no—
55C3×#208—28no—
56C2×C3×C3⋊C8—22no—
57C2×C3×C3⋊C8—22no—
58C2×C3×C3⋊C8D414yescomplemented
59#1895—91noSylow
60C23×C3×C4—21no—
61C23×C3×C4C2×D311yescharacteristic
62C23×C3×C4C2×D311yescharacteristic
63C22×#67—62no—
64C22×#67—61no—
65C22×#67—61no—
66C22×D12—31no—
67C22×D12—31no—
68C22×D12—61no—
69C2×C3×M16—64no—
70C2×C3×M16—62no—
71C22×C3×C8—32no—
72C2×C3×C4⋊C4—32no—
73C2×C3×C22⋊C4—61no—
74C2×#217—21no—
75#930—34no—
76#930—34no—
77#930—64no—
78#930—64no—
79#930—62no—
80#930—62no—
81C2×(C2×D3)⋊C4—62no—
82C2×(C2×D3)⋊C4—61no—
83C2×(C2×D3)⋊C4—61no—
84C2×C4⋊Dic3—31no—
85C2×C4⋊Dic3—31no—
86C2×C4⋊Dic3—61no—
87C2×C4⋊Dic3C2×D312yes—
88C2×#208—22no—
89C2×#208—24no—
90C22×C3⋊C8C2×D312yes—
91C3×#95—61no—
92C3⋊#95—21no—
93#821—64no—
94#809—64no—
95C23×C32—22no—
96C23×C32—22no—
97C23×C32—21no—
98C23×C32—21no—
99C23×C32C22⋊C412yes—
100C23×C32C2×D411yescharacteristic
101C22×C3⋊D3—22no—
102C22×C3⋊D3—42no—
103C22×C3⋊D3—41no—
104C22×C3⋊D3—42no—
105C22×C3⋊D3—41no—
106C2×C32×C4—22no—
107C2×C32×C4—24no—
108C2×C32×C4—22no—
109C2×C32×C4—22no—
110C2×C32×C4C22⋊C412yes—
111C2×C32×C4C22×C411yescharacteristic
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D412yes—
114C2×C32×C4#4112yes—
115#638—48no—
116C2×C3⋊Dic3—22no—
117C2×C3⋊Dic3—42no—
118C4⋊(C3⋊D3)—24no—
119C4⋊(C3⋊D3)—41no—
120C4⋊(C3⋊D3)—41no—
121C2×C3×Dic3—42no—
122C2×C3×Dic3—42no—
123C2×C3×Dic3—42no—
124C3×C3⋊C8—24no—
125C22×M16—31no—
126C2×#121—91no—
127C2×#102—92no—
128#380—31no—
129#340—92no—
130C24×C3—21no—
131C24×C3D1211yescharacteristic
132C24×C3#6711yescharacteristic
133C23×D3—62no—
134C23×D3—61no—
135C23×D3—61no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—22no—
143C22×C3×C4—21no—
144C22×C3×C4—42no—
145C22×C3×C4—42no—
146C22×C3×C4—62no—
147C22×C3×C4C4×D312yes—
148C22×C3×C4C4×D312yes—
149C22×C3×C4D1211yescharacteristic
150C22×C3×C4D1212yes—
151C22×C3×C4#6711yescharacteristic
152C22×C3×C4#6712yes—
153C22×C3×C4C22×D311yescharacteristic
154C22×C3×C4C22×D311yescharacteristic
155C2×#67—68no—
156C2×#67—64no—
157C2×#67—64no—
158C2×#67—128no—
159C2×#67—124no—
160C2×#67—124no—
161C22×Dic3—22no—
162C22×Dic3—62no—
163C22×Dic3—61no—
164C22×Dic3—61no—
165C2×D12—32no—
166C2×D12—32no—
167C2×D12—62no—
168C2×D12—62no—
169C2×D12—62no—
170C2×D12—62no—
171C2×D12—62no—
172C2×D12—62no—
173C2×D12—61no—
174C2×D12—61no—
175C2×D12—122no—
176C2×D12—122no—
177C3×M16—68no—
178C2×C3×C8—34no—
179C2×C3×C8—62no—
180C2×C3×C8—62no—
181C3×C4⋊C4—62no—
182C3×C4⋊C4—62no—
183C3×C22⋊C4—122no—
184#217—42no—
185(C2×D3)⋊C4—68no—
186(C2×D3)⋊C4—64no—
187(C2×D3)⋊C4—64no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—32no—
191C4⋊Dic3—32no—
192C4⋊Dic3—62no—
193C4⋊Dic3—62no—
194C4⋊Dic3—61no—
195C4⋊Dic3—61no—
196#208—28no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199C2×C3⋊C8#6714yescomplemented
200C22×C32—22no—
201C22×C32—22no—
202C22×C32—22no—
203C22×C32—42no—
204C22×C32—42no—
205C22×C32C2×C22⋊C411yescharacteristic
206C22×C32#12112yes—
207C22×C32AGL(1,ℤ/8)11yescharacteristic
208C22×C32AGL(1,ℤ/8)12yes—
209C22×C32#13711yescharacteristic
210C2×C3⋊D3—42no—
211C2×C3⋊D3—42no—
212C2×C3⋊D3—42no—
213C2×C3⋊D3—82no—
214C2×C3⋊D3—82no—
215C32×C4—22no—
216C32×C4C4×D412yes—
217C32×C4#12011yescharacteristic
218C32×C4#12311yescharacteristic
219C3⋊Dic3—42no—
220C3×Dic3—82no—
221C22×D4—91no—
222C22×D4—181no—
223C23×C4D3211yescharacteristic
224C2×M16—64no—
225C2×M16—62no—
226C22×C8—32no—
227#121—94no—
228#121—182no—
229C2×C4⋊C4—32no—
230C2×C4⋊C4—91no—
231C2×C22⋊C4—61no—
232C2×C22⋊C4—181no—
233#102—184no—
234#95—61no—
235C23×C3—24no—
236C23×C3—21no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—21no—
241C23×C3—22no—
242C23×C3—21no—
243C23×C3—22no—
244C23×C3—22no—
245C23×C3—21no—
246C23×C3—42no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3(C2×D3)⋊C412yes—
250C23×C3(C2×D3)⋊C412yes—
251C23×C3C2×D1211yescharacteristic
252C23×C3C2×#6711yescharacteristic
253C22×D3—64no—
254C22×D3—62no—
255C22×D3—62no—
256C22×D3—124no—
257C22×D3—124no—
258C22×D3—122no—
259C22×D3—122no—
260C22×D3—122no—
261C22×D3—121no—
262C22×D3—121no—
263C22×D3—122no—
264C22×D3—121no—
265C22×D3—122no—
266C22×D3—121no—
267C22×D3—122no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—24no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—24no—
274C2×C3×C4—22no—
275C2×C3×C4—22no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—22no—
281C2×C3×C4—44no—
282C2×C3×C4—42no—
283C2×C3×C4—42no—
284C2×C3×C4—42no—
285C2×C3×C4—122no—
286C2×C3×C4—122no—
287C2×C3×C4—122no—
288C2×C3×C4(C2×D3)⋊C412yes—
289C2×C3×C4(C2×D3)⋊C412yes—
290C2×C3×C4C2×C4×D311yescharacteristic
291C2×C3×C4C2×C4×D311yescharacteristic
292C2×C3×C4C2×D1211yescharacteristic
293C2×C3×C4#23512yes—
294C2×C3×C4D3×D412yes—
295C2×C3×C4#23712yes—
296C2×C3×C4C2×#6711yescharacteristic
297C2×C3×C4C2×#6712yes—
298#67—1216no—
299#67—128no—
300#67—128no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—64no—
305C2×Dic3—62no—
306C2×Dic3—62no—
307C2×Dic3—124no—
308C2×Dic3—122no—
309C2×Dic3—122no—
310D12—64no—
311D12—64no—
312D12—124no—
313D12—122no—
314D12—122no—
315D12—121no—
316D12—121no—
317D12—121no—
318D12—121no—
319C3×C8—64no—
320C3⋊C8—24no—
321C2×C32—42no—
322C2×C32#38611yescharacteristic
323C2×C32#41711yescharacteristic
324C2×C32#41811yescharacteristic
325C2×C32#46812yes—
326C2×C32#46912yes—
327C3⋊D3—82no—
328C24—181no—
329C24#62611yescharacteristic
330C2×D4—92no—
331C2×D4—184no—
332C2×D4—182no—
333C2×D4—181no—
334C2×D4—182no—
335C2×D4—364no—
336C22×C4—22no—
337C22×C4—22no—
338C22×C4—62no—
339C22×C4—181no—
340C22×C4#62412yes—
341C22×C4#62611yescharacteristic
342C22×C4#62612yes—
343C22×C4C2×D3211yescharacteristic
344M16—68no—
345C2×C8—34no—
346C2×C8—62no—
347C2×C8—62no—
348C4⋊C4—62no—
349C4⋊C4—62no—
350C4⋊C4—92no—
351C4⋊C4—181no—
352C22⋊C4—122no—
353C22⋊C4—184no—
354C22×C3—24no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—22no—
361C22×C3—22no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—42no—
367C22×C3—42no—
368C22×C3—44no—
369C22×C3—44no—
370C22×C3—42no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3#90812yes—
375C22×C3#90811yescharacteristic
376C22×C3#90911yescharacteristic
377C22×C3C2×(C2×D3)⋊C411yescharacteristic
378C22×C3C2×(C2×D3)⋊C411yescharacteristic
379C22×C3#93012yes—
380C22×C3#93211yescharacteristic
381C22×C3#93912yes—
382C22×C3#94211yescharacteristic
383C22×C3#94912yes—
384C2×D3—124no—
385C2×D3—124no—
386C2×D3—124no—
387C2×D3—122no—
388C2×D3—122no—
389C2×D3—122no—
390C2×D3—122no—
391C2×D3—122no—
392C2×D3—122no—
393C2×D3—244no—
394C2×D3—244no—
395C2×D3—242no—
396C2×D3—242no—
397C2×D3—242no—
398C2×D3—242no—
399C3×C4—22no—
400C3×C4—22no—
401C3×C4—22no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—42no—
405C3×C4—242no—
406C3×C4#67⋊C412yes—
407C3×C4#88211yescharacteristic
408C3×C4#88611yescharacteristic
409C3×C4C4×#6712yes—
410C3×C4#92911yescharacteristic
411C3×C4#95311yescharacteristic
412Dic3—82no—
413Dic3—124no—
414Dic3—122no—
415Dic3—122no—
416C32#189511yescharacteristic, complemented, Sylow
417C23—22no—
418C23—21no—
419C23—21no—
420C23—22no—
421C23—182no—
422C23—362no—
423C23—361no—
424C23—361no—
425C23—362no—
426C23#372212yes—
427C23C2×#62611yescharacteristic
428D4—184no—
429D4—368no—
430D4—361no—
431D4—361no—
432C2×C4—22no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—22no—
436C2×C4—122no—
437C2×C4—122no—
438C2×C4—122no—
439C2×C4—182no—
440C2×C4—362no—
441C2×C4#372212yes—
442C2×C4#380412yes—
443C2×C4C2×#62411yescharacteristic
444C2×C4C2×#62611yescharacteristic
445C2×C4D3×#6712yes—
446C8—64no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—22no—
450C2×C3—22no—
451C2×C3—21no—
452C2×C3—44no—
453C2×C3—42no—
454C2×C3—42no—
455C2×C3#495811yescharacteristic
456C2×C3#496611yescharacteristic
457C2×C3#500111yescharacteristic
458C2×C3#503912yes—
459C2×C3#504912yes—
460C2×C3#513111yescharacteristic
461C2×C3#513311yescharacteristic
462C2×C3#513412yes—
463C2×C3#513512yes—
464C2×C3#521311yescharacteristic
465D3—244no—
466D3—242no—
467D3—242no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—42no—
472C22—42no—
473C22—362no—
474C22—362no—
475C22—362no—
476C22—722no—
477C22—722no—
478C22#6382512yes—
479C22#6383011yescharacteristic
480C22#6383311yescharacteristic
481C22C2×#372211yescharacteristic
482C22#6397812yes—
483C4—22no—
484C4—242no—
485C4—362no—
486C4#6395212yes—
487C4#6396611yescharacteristic
488C4#6397711yescharacteristic
489C3—21no—
490C3#6951711yescharacteristic, complemented
491C3#7199011yescharacteristic, complemented
492C2—42no—
493C2—722no—
494C2#1059109611yescharacteristic
495C2#1059110611yescharacteristic
496C2#1059116512yes—
497C2#1059117812yes—
498C2#1059171111yescharacteristic
subgroupG/H
centerC23C2×#626
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC23×C32×C4C22
socleC23×C32C2×D4
prime radicalC2×#64139C2
cosocleC22×C32×C4C23
hypercenterC23×C4D32
hypocenterC32#1895
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
2. C22×C3×#208C2semidirect
3. C3×#5212C2semidirect
4. C2×#63564C2semidirect
5. #10589164C2semidirect
12. #64139C4semidirect
26. C22×#208D3semidirect
28. #5212D3semidirect
58. C2×C3×C3⋊C8D4semidirect
199. C2×C3⋊C8#67semidirect
416. C32#1895semidirect
490. C3#69517semidirect
491. C3#71990semidirect

Maximal subgroups

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

subgroupindex
C2×#641392
C22×C3×#2082
C3×#52122
C2×#635642
#105891642
#719903
#695173