Finite Group Databasebeta

← #11811913

Subgroups — #11811913

Order 1152.

1765 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3795C211yesmaximal, characteristic
2C2×C3×#875C211yesmaximal, characteristic, complemented
3C2×#63866C211yesmaximal, characteristic, complemented
4C2×#63856C211yesmaximal, characteristic, complemented
5#10592628C211yesmaximal, characteristic, complemented
6C3×C4×(C2×D3)⋊C4C211yesmaximal, characteristic, complemented
7#10591390C214yesmaximal, complemented
8#10591373C214yesmaximal, complemented
9#10591318C211yesmaximal, characteristic, complemented
10#82959—31nomaximal
11#73489—31nomaximal
12C22×#634C2211yescharacteristic
13C22×C3×D12C2211yescharacteristic
14C22×C3×C4×D3C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C22×#625C2212yes—
17C22×D3×Dic3C2212yes—
18C2×#3795—24no—
19C2×#3795—24no—
20C2×#3795—24no—
21C2×C32×C42C2211yescharacteristic
22C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
23C2×C3×(C2×D3)⋊C4C2212yes—
24C2×C3×(C2×D3)⋊C4C2214yes—
25C2×C3×C4×Dic3C2211yescharacteristic
26C3×#875—24no—
27C2×#3724C2212yes—
28C2×Dic3⋊Dic3C2214yes—
29C2×(C2×D3)⋊Dic3C2212yes—
30C2×(C2×D3)⋊Dic3C2214yes—
31#63866—24no—
32#63856—24no—
33#63658C2214yes—
34C3×C3⋊#95C2212yes—
35C22×#237—31no—
36C22×#235—31no—
37C2×C3×#124—31no—
38C2×#946—31no—
39C2×#944—31no—
40C2×#924—31no—
41C2×#875D311yescharacteristic, complemented
42C2×#869—31no—
43C3×C4×C22⋊C4—31no—
44#5221—31no—
45#4961—34no—
46#4959—34no—
47#4946—34no—
48#4939—31no—
49C4×(C2×D3)⋊C4D311yescharacteristic, complemented
50#4929—31no—
51#4929—31no—
52#4929—61no—
53#4668—34no—
54C23×C3×D3C2312yes—
55C23×C3×D3C2311yescharacteristic
56C22×C32×C4C2312yes—
57C22×C32×C4C2311yescharacteristic
58C22×C3⋊Dic3C2312yes—
59C22×C3⋊Dic3C2312yes—
60C2×#634—22no—
61C2×#634—22no—
62C2×#634D414yes—
63C22×C3×Dic3C2312yes—
64C22×C3×Dic3C2311yescharacteristic
65C22×C3×Dic3C2312yes—
66C2×C3×D12—22no—
67C2×C3×D12—22no—
68C2×C3×D12D414yes—
69C2×C3×C4×D3—22no—
70C2×C3×C4×D3—22no—
71C2×C3×C4×D3D414yes—
72C2×C3×Dic6—22no—
73C2×C3×Dic6—22no—
74C2×C3×Dic6—22no—
75C2×#625—24no—
76C2×#625—24no—
77C2×#625—24no—
78C2×D3×Dic3—24no—
79C2×D3×Dic3—24no—
80C2×D3×Dic3—24no—
81#3795—232no—
82C32×C42—22no—
83C4⋊(C3⋊Dic3)—22no—
84C3×(C2×D3)⋊C4—28no—
85C3×(C2×D3)⋊C4—24no—
86C3×C4×Dic3—22no—
87#3724—24no—
88Dic3⋊Dic3—28no—
89(C2×D3)⋊Dic3—24no—
90(C2×D3)⋊Dic3—28no—
91#2432—91noSylow
92C22×C3×Q8—31no—
93C22×C3×D4—31no—
94C23×C3×C4—31no—
95C22×#67—32no—
96C22×#67—32no—
97C23×Dic3—32no—
98C2×#237—64no—
99C2×#237—64no—
100C2×#237—64no—
101C2×#235—64no—
102C2×#235—64no—
103C2×#235—64no—
104C22×D12C2×D311yescharacteristic
105C22×C4×D3—32no—
106C22×C4×D3C2×D311yescharacteristic
107C22×Dic6—31no—
108C22×Dic6—31no—
109C22×Dic6—61no—
110C22×Dic6C2×D311yescharacteristic
111C3×#124—64no—
112C2×C3×C22⋊C4—34no—
113C2×C3×C22⋊C4—32no—
114C2×C3×C42—21no—
115C2×C3×C42—31no—
116C2×C3×C42C2×D311yescharacteristic
117C2×C3×C42C2×D311yescharacteristic
118C2×#217—34no—
119C2×#217—32no—
120#946—64no—
121#944—64no—
122C2×(C2×D3)⋊C4—34no—
123C2×(C2×D3)⋊C4—32no—
124C2×(C2×D3)⋊C4C2×D314yes—
125C2×(C2×D3)⋊C4C2×D312yes—
126C2×C4⋊Dic3—31no—
127C2×C4⋊Dic3—31no—
128C2×C4⋊Dic3—34no—
129C2×C4⋊Dic3—61no—
130#924—64no—
131C2×Dic3⋊C4—32no—
132C2×Dic3⋊C4—32no—
133C2×Dic3⋊C4—34no—
134C2×C4×Dic3C2×D311yescharacteristic
135#875—24no—
136#869—64no—
137C3×#95—32no—
138C3⋊#95—34no—
139C3⋊#95—34no—
140C3⋊#95—64no—
141C3⋊#95C2×D312yes—
142C23×C32C2411yescharacteristic
143C22×C3×D3—22no—
144C22×C3×D3—22no—
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C22×C3×D3—24no—
148C22×C3×D3—22no—
149C22×C3×D3—22no—
150C22×C3×D3—24no—
151C22×C3×D3C2×D412yes—
152C22×C3×D3C2×D414yes—
153C2×C32×C4—22no—
154C2×C32×C4—22no—
155C2×C32×C4—22no—
156C2×C32×C4C2×D412yes—
157C2×C32×C4C2×D412yes—
158C2×C32×C4C2×D412yes—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—22no—
161C2×C3⋊Dic3—22no—
162C2×C3⋊Dic3—24no—
163C2×C3⋊Dic3C2×D414yes—
164#634—28no—
165C2×C3×Dic3—22no—
166C2×C3×Dic3—21no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—22no—
169C2×C3×Dic3—22no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—22no—
172C2×C3×Dic3C2×D412yes—
173C2×C3×Dic3#4114yescomplemented
174C3×D12—28no—
175C3×C4×D3—28no—
176C3×Dic6—44no—
177#625—216no—
178D3×Dic3—216no—
179C22×#41—91no—
180C2×C4⋊Q8—91no—
181C2×#124—31no—
182C2×C22⋊Q8—91no—
183#394—94no—
184#391—91no—
185#390—91no—
186#385—94no—
187C4×C22⋊C4—31no—
188C24×C3—32no—
189C24×C3—31no—
190C23×D3C22×D311yescharacteristic
191C23×D3C22×D312yes—
192C2×C3×Q8—62no—
193C2×C3×Q8—62no—
194C2×C3×Q8—62no—
195C2×C3×D4—34no—
196C2×C3×D4—62no—
197C2×C3×D4—62no—
198C22×C3×C4—21no—
199C22×C3×C4—22no—
200C22×C3×C4—31no—
201C22×C3×C4—32no—
202C22×C3×C4—34no—
203C22×C3×C4—32no—
204C22×C3×C4—62no—
205C22×C3×C4—62no—
206C22×C3×C4C22×D312yes—
207C22×C3×C4C22×D312yes—
208C22×C3×C4C22×D311yescharacteristic
209C22×C3×C4C22×D311yescharacteristic
210C2×#67—64no—
211C2×#67—64no—
212C2×#67—64no—
213C2×#67—64no—
214C2×#67—64no—
215C2×#67—64no—
216C22×Dic3—32no—
217C22×Dic3—32no—
218C22×Dic3—32no—
219C22×Dic3—32no—
220C22×Dic3—62no—
221C22×Dic3—62no—
222C22×Dic3—64no—
223C22×Dic3—64no—
224C22×Dic3—64no—
225C22×Dic3C22×D312yes—
226C22×Dic3C22×D311yescharacteristic
227C22×Dic3C22×D312yes—
228#237—632no—
229#235—632no—
230C2×D12—22no—
231C2×D12—22no—
232C2×D12#6714yes—
233C2×C4×D3—22no—
234C2×C4×D3—22no—
235C2×C4×D3—64no—
236C2×C4×D3—64no—
237C2×C4×D3—64no—
238C2×C4×D3D1214yes—
239C2×Dic6—22no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6—34no—
243C2×Dic6—34no—
244C2×Dic6—64no—
245C2×Dic6—62no—
246C2×Dic6—62no—
247C2×Dic6—62no—
248C2×Dic6—62no—
249C2×Dic6—122no—
250C2×Dic6—122no—
251C3×C22⋊C4—64no—
252C3×C22⋊C4—68no—
253C3×C42—22no—
254C3×C42—22no—
255C3×C42—42no—
256C3×C42—62no—
257#217—64no—
258#217—68no—
259(C2×D3)⋊C4—28no—
260(C2×D3)⋊C4—24no—
261(C2×D3)⋊C4—68no—
262(C2×D3)⋊C4—64no—
263C4⋊Dic3—68no—
264C4⋊Dic3—62no—
265C4⋊Dic3—62no—
266C4⋊Dic3—122no—
267Dic3⋊C4—68no—
268Dic3⋊C4—64no—
269Dic3⋊C4—64no—
270C4×Dic3—22no—
271C22×C32C22×D411yescharacteristic
272C22×C32C22×D411yescharacteristic
273C22×C32C22×D411yescharacteristic
274C22×C32C2×#4112yes—
275C22×C322−1+412yes—
276C2×C3×D3—24no—
277C2×C3×D3—22no—
278C2×C3×D3—22no—
279C2×C3×D3—22no—
280C2×C3×D3—28no—
281C2×C3×D3—42no—
282C2×C3×D3—42no—
283C2×C3×D3—42no—
284C2×C3×D3—44no—
285C2×C3×D3—42no—
286C32×C4—42no—
287C32×C4#12014yes—
288C3⋊Dic3—24no—
289C3⋊Dic3—42no—
290C3×Dic3—22no—
291C3×Dic3—24no—
292C3×Dic3—42no—
293C2×#41—184no—
294C2×#41—184no—
295C2×#41—184no—
296C22×Q8—31no—
297C22×Q8—91no—
298C22×D4—31no—
299C22×D4—92no—
300C23×C4—31no—
301C23×C4—92no—
302C4⋊Q8—184no—
303#124—64no—
304C22⋊Q8—184no—
305C2×C4⋊C4—92no—
306C2×C4⋊C4—94no—
307C2×C4⋊C4—91no—
308C2×C22⋊C4—34no—
309C2×C22⋊C4—32no—
310C2×C22⋊C4—92no—
311C2×C22⋊C4—94no—
312C2×C42—31no—
313C2×C42D3211yescharacteristic
314#95—32no—
315#95—94no—
316C23×C3—21no—
317C23×C3—32no—
318C23×C3—34no—
319C23×C3—62no—
320C23×C3—64no—
321C23×C3—62no—
322C23×C3—61no—
323C23×C3—62no—
324C23×C3—62no—
325C23×C3—64no—
326C23×C3—61no—
327C23×C3C23×D311yescharacteristic
328C23×C3C23×D311yescharacteristic
329C22×D3—22no—
330C22×D3—21no—
331C22×D3—21no—
332C22×D3—22no—
333C22×D3—22no—
334C22×D3—24no—
335C22×D3—22no—
336C22×D3—24no—
337C22×D3C2×D1212yes—
338C22×D3C2×#6714yes—
339C3×Q8—124no—
340C3×D4—68no—
341C2×C3×C4—24no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—22no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—22no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—32no—
352C2×C3×C4—34no—
353C2×C3×C4—42no—
354C2×C3×C4—42no—
355C2×C3×C4—62no—
356C2×C3×C4—61no—
357C2×C3×C4—62no—
358C2×C3×C4—61no—
359C2×C3×C4—68no—
360C2×C3×C4—62no—
361C2×C3×C4—62no—
362C2×C3×C4—62no—
363C2×C3×C4C2×D1212yes—
364C2×C3×C4D3×D412yes—
365C2×C3×C4D3×D412yes—
366C2×C3×C4D3×D412yes—
367C2×C3×C4D3×D412yes—
368C2×C3×C4C2×#6712yes—
369#67—616no—
370#67—616no—
371C2×Dic3—22no—
372C2×Dic3—21no—
373C2×Dic3—21no—
374C2×Dic3—22no—
375C2×Dic3—22no—
376C2×Dic3—22no—
377C2×Dic3—22no—
378C2×Dic3—34no—
379C2×Dic3—34no—
380C2×Dic3—64no—
381C2×Dic3—616no—
382C2×Dic3—62no—
383C2×Dic3—62no—
384C2×Dic3—62no—
385C2×Dic3—62no—
386C2×Dic3—62no—
387C2×Dic3—64no—
388C2×Dic3—62no—
389C2×Dic3—64no—
390C2×Dic3—122no—
391C2×Dic3—122no—
392C2×Dic3—122no—
393C2×Dic3—124no—
394C2×Dic3C2×D1212yes—
395C2×Dic3#23514yescomplemented
396D12—28no—
397C4×D3—28no—
398C4×D3—616no—
399Dic6—44no—
400Dic6—68no—
401Dic6—68no—
402Dic6—128no—
403C2×C32C2×#12011yescharacteristic
404C2×C32#53211yescharacteristic
405C2×C32#53611yescharacteristic
406C2×C32#54712yes—
407C2×C32#54812yes—
408C3×D3—44no—
409C3×D3—42no—
410C24—31no—
411C24—32no—
412#41—1832no—
413C2×Q8—62no—
414C2×Q8—62no—
415C2×Q8—62no—
416C2×Q8—94no—
417C2×Q8—182no—
418C2×Q8—182no—
419C2×D4—34no—
420C2×D4—62no—
421C2×D4—62no—
422C2×D4—184no—
423C2×D4—184no—
424C2×D4—184no—
425C22×C4—31no—
426C22×C4—32no—
427C22×C4—34no—
428C22×C4—32no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—92no—
432C22×C4—92no—
433C22×C4—184no—
434C22×C4—184no—
435C22×C4—184no—
436C22×C4C2×D3212yes—
437C22×C4C2×D3211yescharacteristic
438C4⋊C4—184no—
439C4⋊C4—188no—
440C4⋊C4—182no—
441C22⋊C4—64no—
442C22⋊C4—68no—
443C22⋊C4—188no—
444C22⋊C4—184no—
445C42—22no—
446C42—62no—
447C22×C3—22no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—22no—
451C22×C3—21no—
452C22×C3—62no—
453C22×C3—62no—
454C22×C3—64no—
455C22×C3—62no—
456C22×C3—68no—
457C22×C3—122no—
458C22×C3—122no—
459C22×C3—122no—
460C22×C3—124no—
461C22×C3—122no—
462C22×C3C22×D1211yescharacteristic
463C22×C3C2×#23512yes—
464C22×C3C2×D3×D411yescharacteristic
465C22×C3C2×D3×D411yescharacteristic
466C22×C3C2×D3×D411yescharacteristic
467C22×C3C2×D3×D411yescharacteristic
468C22×C3D3×#4112yes—
469C22×C3#101012yes—
470C22×C3#101012yes—
471C22×C3C22×#6711yescharacteristic
472C2×D3—22no—
473C2×D3—22no—
474C2×D3—22no—
475C2×D3—24no—
476C2×D3—28no—
477C2×D3—42no—
478C2×D3—42no—
479C2×D3—44no—
480C2×D3—42no—
481C2×D3—42no—
482C3×C4—24no—
483C3×C4—44no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—64no—
487C3×C4—62no—
488C3×C4—122no—
489C3×C4#88214yes—
490C3×C4#93714yes—
491Dic3—22no—
492Dic3—24no—
493Dic3—42no—
494Dic3—64no—
495Dic3—64no—
496Dic3—124no—
497Dic3—122no—
498Dic3—122no—
499Dic3—242no—
500C32#243211yescharacteristic, complemented, Sylow
501C23—32no—
502C23—34no—
503C23—64no—
504C23—62no—
505C23—62no—
506C23—61no—
507C23—62no—
508C23—62no—
509C23—64no—
510C23—61no—
511C23C22×D3211yescharacteristic
512Q8—124no—
513Q8—188no—
514D4—68no—
515D4—1816no—
516C2×C4—22no—
517C2×C4—22no—
518C2×C4—22no—
519C2×C4—32no—
520C2×C4—34no—
521C2×C4—62no—
522C2×C4—62no—
523C2×C4—61no—
524C2×C4—61no—
525C2×C4—68no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—62no—
529C2×C4—94no—
530C2×C4—1816no—
531C2×C4—182no—
532C2×C4—182no—
533C2×C4—182no—
534C2×C4—184no—
535C2×C4D3×D1212yes—
536C2×C4#380212yes—
537C2×C4D3×#6712yes—
538C2×C3—22no—
539C2×C3—21no—
540C2×C3—22no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—122no—
544C2×C3—124no—
545C2×C3#547711yescharacteristic
546C2×C3C2×#88211yescharacteristic
547C2×C3#552611yescharacteristic
548C2×C3#555112yes—
549C2×C3#555212yes—
550C2×C3#566811yescharacteristic
551C2×C3#567511yescharacteristic
552C2×C3#567712yes—
553C2×C3C2×#93711yescharacteristic
554C2×C3#582412yes—
555D3—44no—
556D3—42no—
557C22—68no—
558C22—62no—
559C22—62no—
560C22—62no—
561C22—64no—
562C22—122no—
563C22—122no—
564C22—122no—
565C22—124no—
566C22—122no—
567C22#6429812yes—
568C22C2×D3×D1211yescharacteristic
569C22C2×#380211yescharacteristic
570C22D3×#23512yes—
571C22C2×D3×#6711yescharacteristic
572C4—42no—
573C4—64no—
574C4—62no—
575C4—122no—
576C4—184no—
577C4—362no—
578C4#6392214yes—
579C3—21no—
580C3#7348911yescharacteristic, complemented
581C3#8295911yescharacteristic, complemented
582C2—124no—
583C2—122no—
584C2#1059389411yescharacteristic
585C2#1059390512yes—
586C2#1059396911yescharacteristic
587C2#1059488712yes—
588C2C2×#6392211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C42C22
prime radicalC22×#3795C2
hypercenterC2×C42D32
hypocenterC32#2432
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C3×#875C2semidirect
3. C2×#63866C2semidirect
4. C2×#63856C2semidirect
5. #10592628C2semidirect
6. C3×C4×(C2×D3)⋊C4C2semidirect
7. #10591390C2semidirect
8. #10591373C2semidirect
9. #10591318C2semidirect
41. C2×#875D3semidirect
49. C4×(C2×D3)⋊C4D3semidirect
173. C2×C3×Dic3#41semidirect
395. C2×Dic3#235semidirect
500. C32#2432semidirect
580. C3#73489semidirect
581. C3#82959semidirect

Maximal subgroups

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

subgroupindex
C22×#37952
C2×C3×#8752
C2×#638662
C2×#638562
#105926282
C3×C4×(C2×D3)⋊C42
#105913902
#105913732
#105913182
#829593
#734893