Finite Group Databasebeta

← #11788210

Subgroups — #11788210

Order 1152.

771 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#872C211yesmaximal, characteristic, complemented
2#10595022C211yesmaximal, characteristic
3C2×C3×#805C211yesmaximal, characteristic, complemented
4C2×#63569C211yesmaximal, characteristic, complemented
5#10591069C211yesmaximal, characteristic, complemented
6#10589389C211yesmaximal, characteristic, complemented
7#10589162C211yesmaximal, characteristic, complemented
8#71591—31nomaximal
9#70383—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C22×C3×C4×D3C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13#64135C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×(C2×D3)⋊C4—21no—
16C2×C3×Dic3⋊C4—21no—
17C2×C3×C4×Dic3C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#872—22no—
20C3×#872—22no—
21#63976—21no—
22#63961—21no—
23#63908C412yescomplemented
24C4×#625—22no—
25#63856C412yescomplemented
26C3×#805—22no—
27#63569—22no—
28(C2×D3)⋊(C3⋊C8)—22no—
29C2×C3×#117—31no—
30#5828—31no—
31#5791—31no—
32C2×#872D311yescharacteristic, complemented
33C2×C3×#104—31no—
34C2×#837—31no—
35#5207—31no—
36#5126—31no—
37C2×#805—31no—
38C2×#805D311yescharacteristic, complemented
39#4555—31no—
40#4523—31no—
41#4523—31no—
42#4523—31no—
43#4523—61no—
44C23×C3×D3D411yescharacteristic
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C22×C3×Dic3—21no—
49C22×C3×Dic3D411yescharacteristic
50C3×#235—21no—
51C3×#235—21no—
52C3×#235—21no—
53C3×#235—21no—
54C2×C3×D12C2×C411yescharacteristic
55C2×C3×C4×D3—21no—
56C2×C3×C4×D3—21no—
57C2×C3×C4×D3—21no—
58C2×C3×C4×D3—21no—
59C2×C3×C4×D3—21no—
60C2×C3×C4×D3C2×C412yes—
61C2×C3×C4×D3D412yes—
62C2×C3×Dic6C2×C411yescharacteristic
63C2×#625—22no—
64C32×C42—21no—
65C32×C42D412yescomplemented
66#3757—21no—
67C4⋊(C3⋊Dic3)C2×C412yes—
68C4×C3⋊Dic3—21no—
69#3748—21no—
70#3748—21no—
71C2×C3⋊(C3⋊C8)—21no—
72C3×(C2×D3)⋊C4—24no—
73C3×Dic3⋊C4—24no—
74C3×C4×Dic3—21no—
75C3×C4×Dic3D412yescomplemented
76C3×#208—21no—
77C3×#208—21no—
78C2×C3×C3⋊C8—21no—
79#3724—22no—
80(C2×D3)⋊Dic3—22no—
81(C2×D3)⋊Dic3—22no—
82#1897—91noSylow
83C2×C3×#41—31no—
84C23×C3×C4—31no—
85C2×#235C2×D311yescharacteristic
86C22×C4×D3C2×D311yescharacteristic
87C2×C3×M16—31no—
88C3×#117—62no—
89C3×#117—62no—
90C2×C3×C4⋊C4—61no—
91C2×C3×C22⋊C4—61no—
92C2×C3×C42—21no—
93C2×C3×C42—31no—
94C2×C3×C42C2×D311yescharacteristic
95C2×C3×C42C2×D311yescharacteristic
96#953—61no—
97#946—32no—
98#938—32no—
99#937—61no—
100#935—61no—
101#930—32no—
102#929—61no—
103C4×#67—62no—
104C4×#67—62no—
105C2×(C2×D3)⋊C4—21no—
106#926—31no—
107#926—31no—
108#926—61no—
109#924—32no—
110C2×Dic3⋊C4—21no—
111C2×C4×Dic3C2×D311yescharacteristic
112C2×#208—31no—
113C2×#208—31no—
114C2×#208—61no—
115C2×#208C2×D311yescharacteristic
116#872—22no—
117#872—22no—
118C3×#104—62no—
119#837—62no—
120#830—62no—
121(C2×D3)⋊C8—62no—
122#805—22no—
123#805—62no—
124C23×C32C2×D411yescharacteristic
125C22×C3×D3—21no—
126C22×C3×D3—21no—
127C22×C3×D3—21no—
128C22×C3×D3—42no—
129C22×C3×D3C22⋊C412yes—
130C2×C32×C4—22no—
131C2×C32×C4—42no—
132C2×C32×C4C22×C411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4#4111yescharacteristic
137C2×C32×C4#4111yescharacteristic
138C2×C3⋊Dic3—22no—
139C3×#67—41no—
140C3×#67—41no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—42no—
146C2×C3×Dic3C22⋊C412yes—
147C3×D12—21no—
148C3×D12—21no—
149C3×C4×D3—22no—
150C3×C4×D3—22no—
151C3×C4×D3—22no—
152C3×C4×D3—22no—
153C3×C4×D3—41no—
154C3×C4×D3—41no—
155C3×Dic6—21no—
156C3×Dic6—21no—
157#625—44no—
158C3⋊(C3⋊C8)—41no—
159C3×C3⋊C8—41no—
160#525—91no—
161C2×#117—31no—
162C2×#104—31no—
163C2×#104—91no—
164#407—91no—
165#339—91no—
166#339—91no—
167C24×C3—31no—
168C23×D3#6711yescharacteristic
169C3×#41—61no—
170C3×#41—61no—
171C3×#41—61no—
172C3×#41—61no—
173C2×C3×Q8—31no—
174C2×C3×D4—31no—
175C2×C3×D4—61no—
176C22×C3×C4—21no—
177C22×C3×C4—21no—
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—32no—
181C22×C3×C4—32no—
182C22×C3×C4—41no—
183C22×C3×C4—61no—
184C22×C3×C4—61no—
185C22×C3×C4—61no—
186C22×C3×C4—61no—
187C22×C3×C4—61no—
188C22×C3×C4—61no—
189C22×C3×C4C22×D311yescharacteristic
190C22×C3×C4C22×D311yescharacteristic
191C2×#67—21no—
192C2×#67—62no—
193C2×#67—62no—
194C22×Dic3—21no—
195C22×Dic3#6711yescharacteristic
196#235—21no—
197#235—21no—
198#235—21no—
199#235—21no—
200C2×D12C4×D311yescharacteristic
201C2×C4×D3—21no—
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3C4×D312yes—
207C2×C4×D3D1212yes—
208C2×Dic6C4×D311yescharacteristic
209C3×M16—61no—
210C3×M16—61no—
211C2×C3×C8—61no—
212C3×C4⋊C4—64no—
213C3×C22⋊C4—64no—
214C3×C42—21no—
215C3×C42—21no—
216C3×C42—22no—
217C3×C42—32no—
218C3×C42—41no—
219C3×C42—61no—
220C3×C42#6712yescomplemented
221C3×C42#6712yescomplemented
222#217—62no—
223#217—61no—
224#217—61no—
225#217—62no—
226#217—121no—
227(C2×D3)⋊C4—24no—
228(C2×D3)⋊C4—62no—
229(C2×D3)⋊C4—62no—
230C4⋊Dic3—32no—
231C4⋊Dic3—32no—
232C4⋊Dic3—62no—
233Dic3⋊C4—24no—
234Dic3⋊C4—62no—
235Dic3⋊C4—62no—
236C4×Dic3—21no—
237C4×Dic3—61no—
238C4×Dic3—61no—
239C4×Dic3—121no—
240C4×Dic3#6712yescomplemented
241#208—21no—
242#208—21no—
243#208—62no—
244#208—61no—
245#208—61no—
246#208—61no—
247#208—61no—
248#208—121no—
249C2×C3⋊C8—21no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8—61no—
252C2×C3⋊C8—121no—
253C22×C32—22no—
254C22×C32C2×C22⋊C411yescharacteristic
255C22×C32#12011yescharacteristic
256C22×C32#12311yescharacteristic
257C2×C3×D3—22no—
258C2×C3×D3—22no—
259C2×C3×D3—22no—
260C2×C3×D3—42no—
261C2×C3×D3—42no—
262C2×C3×D3—41no—
263C2×C3×D3—42no—
264C32×C4—42no—
265C32×C4C4×D411yescharacteristic
266C32×C4C4×D411yescharacteristic
267C32×C4#12111yescharacteristic
268C32×C4#12111yescharacteristic
269C3⋊Dic3—42no—
270C3×Dic3—22no—
271C3×Dic3—42no—
272C3×Dic3—41no—
273C2×#41—31no—
274C23×C4—31no—
275C2×M16—31no—
276C2×M16—91no—
277#123—181no—
278C22⋊Q8—92no—
279#121—92no—
280#120—181no—
281C4×D4—182no—
282#117—62no—
283#117—62no—
284#117—91no—
285C2×C4⋊C4—61no—
286C2×C22⋊C4—61no—
287C2×C42—31no—
288C2×C42D3211yescharacteristic
289#104—62no—
290#104—182no—
291C22⋊C8—182no—
292C23×C3—21no—
293C23×C3—32no—
294C23×C3—61no—
295C23×C3—61no—
296C23×C3—61no—
297C23×C3—122no—
298C23×C3D3×D411yescharacteristic
299C23×C3C2×#6711yescharacteristic
300C22×D3—21no—
301C22×D3—21no—
302C22×D3—21no—
303C22×D3—42no—
304C22×D3(C2×D3)⋊C412yes—
305C3×Q8—61no—
306C3×Q8—61no—
307C3×D4—61no—
308C3×D4—61no—
309C3×D4—121no—
310C3×D4—121no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—32no—
320C2×C3×C4—42no—
321C2×C3×C4—42no—
322C2×C3×C4—42no—
323C2×C3×C4—44no—
324C2×C3×C4—61no—
325C2×C3×C4—61no—
326C2×C3×C4—62no—
327C2×C3×C4—62no—
328C2×C3×C4—62no—
329C2×C3×C4—62no—
330C2×C3×C4—61no—
331C2×C3×C4—62no—
332C2×C3×C4—122no—
333C2×C3×C4—121no—
334C2×C3×C4—121no—
335C2×C3×C4C2×C4×D311yescharacteristic
336C2×C3×C4C2×C4×D311yescharacteristic
337C2×C3×C4C2×D1211yescharacteristic
338C2×C3×C4#23511yescharacteristic
339C2×C3×C4#23511yescharacteristic
340C2×C3×C4#23511yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4#23711yescharacteristic
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C2×#6711yescharacteristic
347#67—41no—
348#67—41no—
349#67—124no—
350#67—124no—
351C2×Dic3—21no—
352C2×Dic3—21no—
353C2×Dic3—21no—
354C2×Dic3—22no—
355C2×Dic3—42no—
356C2×Dic3—62no—
357C2×Dic3—62no—
358C2×Dic3—122no—
359C2×Dic3(C2×D3)⋊C412yes—
360D12—21no—
361D12—21no—
362C4×D3—22no—
363C4×D3—22no—
364C4×D3—22no—
365C4×D3—22no—
366C4×D3—41no—
367C4×D3—41no—
368Dic6—21no—
369Dic6—21no—
370C3×C8—121no—
371C3⋊C8—41no—
372C3⋊C8—122no—
373C3⋊C8—121no—
374C3⋊C8—121no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32#38611yescharacteristic
378C2×C32#42112yes—
379C3×D3—42no—
380C3×D3—81no—
381C24—31no—
382#41—61no—
383#41—61no—
384#41—61no—
385#41—61no—
386C2×Q8—31no—
387C2×D4—31no—
388C2×D4—61no—
389C2×D4—182no—
390C22×C4—21no—
391C22×C4—31no—
392C22×C4—32no—
393C22×C4—32no—
394C22×C4—61no—
395C22×C4—61no—
396C22×C4—61no—
397C22×C4—61no—
398C22×C4—61no—
399C22×C4—61no—
400C22×C4C2×D3211yescharacteristic
401M16—61no—
402M16—61no—
403M16—181no—
404M16—181no—
405C2×C8—61no—
406C2×C8—181no—
407C4⋊C4—64no—
408C4⋊C4—92no—
409C4⋊C4—182no—
410C22⋊C4—64no—
411C22⋊C4—182no—
412C22⋊C4—182no—
413C22⋊C4—181no—
414C42—21no—
415C42—32no—
416C42—61no—
417C42—181no—
418C42#62512yescomplemented
419C22×C3—22no—
420C22×C3—22no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—42no—
425C22×C3—62no—
426C22×C3—62no—
427C22×C3—62no—
428C22×C3—122no—
429C22×C3—122no—
430C22×C3—121no—
431C22×C3—122no—
432C22×C3D3×C22⋊C411yescharacteristic
433C22×C3#88311yescharacteristic
434C22×C3C2×(C2×D3)⋊C411yescharacteristic
435C22×C3#92911yescharacteristic
436C22×C3#93711yescharacteristic
437C22×C3#95311yescharacteristic
438C2×D3—22no—
439C2×D3—22no—
440C2×D3—22no—
441C2×D3—42no—
442C2×D3—42no—
443C2×D3—41no—
444C2×D3—42no—
445C3×C4—21no—
446C3×C4—21no—
447C3×C4—21no—
448C3×C4—21no—
449C3×C4—42no—
450C3×C4—42no—
451C3×C4—44no—
452C3×C4—62no—
453C3×C4—122no—
454C3×C4—121no—
455C3×C4#67⋊C411yescharacteristic
456C3×C4#88411yescharacteristic
457C3×C4C4×#6711yescharacteristic
458C3×C4C4×#6711yescharacteristic
459C3×C4C4×#6711yescharacteristic
460C3×C4#93011yescharacteristic
461C3×C4#93011yescharacteristic
462C3×C4#93811yescharacteristic
463Dic3—22no—
464Dic3—42no—
465Dic3—41no—
466Dic3—122no—
467Dic3—122no—
468Dic3—242no—
469C32#189711yescharacteristic, complemented, Sylow
470C23—32no—
471C23—61no—
472C23—61no—
473C23—61no—
474C23—122no—
475C23D3×#6711yescharacteristic
476Q8—61no—
477Q8—61no—
478D4—61no—
479D4—61no—
480D4—121no—
481D4—121no—
482D4—364no—
483C2×C4—22no—
484C2×C4—32no—
485C2×C4—42no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—62no—
489C2×C4—62no—
490C2×C4—62no—
491C2×C4—62no—
492C2×C4—61no—
493C2×C4—62no—
494C2×C4—122no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—182no—
498C2×C4#379511yescharacteristic
499C2×C4#379811yescharacteristic
500C2×C4C4×D3211yescharacteristic
501C2×C4D3×D1211yescharacteristic
502C2×C4C2×#62511yescharacteristic
503C2×C4D3×#6711yescharacteristic
504C8—121no—
505C8—361no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—22no—
512C2×C3—41no—
513C2×C3—41no—
514C2×C3—122no—
515C2×C3—241no—
516C2×C3#482112yes—
517C2×C3#495611yescharacteristic
518C2×C3#500512yes—
519C2×C3#521311yescharacteristic
520D3—42no—
521D3—81no—
522C22—22no—
523C22—62no—
524C22—62no—
525C22—62no—
526C22—122no—
527C22—122no—
528C22—121no—
529C22—122no—
530C22#6390511yescharacteristic
531C22D3×(C2×D3)⋊C411yescharacteristic
532C22#6397611yescharacteristic
533C4—42no—
534C4—62no—
535C4—122no—
536C4—121no—
537C4—362no—
538C4#6387911yescharacteristic
539C4C4×#62511yescharacteristic
540C4#6390611yescharacteristic
541C4#6390811yescharacteristic
542C3—21no—
543C3#7038311yescharacteristic, complemented
544C3#7159111yescharacteristic, complemented
545C2—21no—
546C2—21no—
547C2—122no—
548C2—241no—
549C2#1059039212yes—
550C2#1059138111yescharacteristic
subgroupG/H
centerC2×C4D3×D12
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C42C22
socleC22×C32C2×C22⋊C4
prime radical#10595022C2
cosocleC22×C32×C4C23
hypercenterC2×C42D32
hypocenterC32#1897
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
1. C2×C3×#872C2semidirect
3. C2×C3×#805C2semidirect
4. C2×#63569C2semidirect
5. #10591069C2semidirect
6. #10589389C2semidirect
7. #10589162C2semidirect
23. #63908C4semidirect
25. #63856C4semidirect
32. C2×#872D3semidirect
38. C2×#805D3semidirect
65. C32×C42D4semidirect
75. C3×C4×Dic3D4semidirect
220. C3×C42#67semidirect
221. C3×C42#67semidirect
240. C4×Dic3#67semidirect
418. C42#625semidirect
469. C32#1897semidirect
543. C3#70383semidirect
544. C3#71591semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8722
#105950222
C2×C3×#8052
C2×#635692
#105910692
#105893892
#105891622
#715913
#703833