Finite Group Databasebeta

← #11785015

Subgroups — #11785015

Order 1152.

1149 conjugacy classes of subgroups.

The subgroup lattice has 645 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#1010C211yesmaximal, characteristic
2#10592935C211yesmaximal, characteristic, complemented
3C3×#5236C211yesmaximal, characteristic, complemented
4#10591231C211yesmaximal, characteristic, complemented
5#10591100C211yesmaximal, characteristic, complemented
6#10591099C211yesmaximal, characteristic, complemented
7#10591076C211yesmaximal, characteristic, complemented
8#72144—31nomaximal
9#70415—31nomaximal
10C2×C32×#41C2211yescharacteristic
11C3×#1010—22no—
12C3×#1010—22no—
13C3×#1010—22no—
14C3×#1010—22no—
15C2×C3×D3×Q8C412yescomplemented
16C2×C3×#237—22no—
17C2×C3×#237C412yescomplemented
18C2×C3×#235—21no—
19C2×C3×#235C2211yescharacteristic
20C22×C3×Dic6—21no—
21C22×C3×Dic6C2211yescharacteristic
22#64135C2211yescharacteristic
23C2×#3748C2211yescharacteristic
24C3×#926C2211yescharacteristic
25C2×C3×#208C2211yescharacteristic
26#63664—22no—
27C3×#837—22no—
28#63580—22no—
29#63566—22no—
30#63565—22no—
31#63560—21no—
32#63558—21no—
33C2×C3×2−1+4—31no—
34C2×#1010D311yescharacteristic
35C3×#421—31no—
36#5240—31no—
37#5237—31no—
38#5236—31no—
39#5236—31no—
40#5236—61no—
41#5236D311yescharacteristic, complemented
42#5225—31no—
43#5225—31no—
44#5132—31no—
45#5120—31no—
46#5006—31no—
47#5004—31no—
48C32×#41—21no—
49C32×#41—21no—
50C32×#41D412yes—
51C32×#41D412yes—
52C2×C32×Q8C2×C411yescharacteristic
53C2×C32×D4—21no—
54C2×C32×D4C2×C411yescharacteristic
55C22×C32×C4—21no—
56C22×C32×C4C2311yescharacteristic
57C2×C3×#67—22no—
58C2×C3×#67D412yes—
59C22×C3×Dic3—21no—
60C22×C3×Dic3—22no—
61C3×D3×Q8—24no—
62C3×D3×Q8—24no—
63C3×#237—28no—
64C3×#237—28no—
65C3×#237—24no—
66C3×#237—24no—
67C3×#235—24no—
68C3×#235—24no—
69C3×#235—21no—
70C3×#235—21no—
71C3×#235—21no—
72C3×#235—21no—
73C2×C3×D12D411yescharacteristic
74C2×C3×C4×D3—22no—
75C2×C3×C4×D3C2×C412yes—
76C2×C3×Dic6—24no—
77C2×C3×Dic6—24no—
78C2×C3×Dic6—21no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—22no—
82C2×C3×Dic6C2×C412yes—
83C2×C3×Dic6D411yescharacteristic
84C2×C3×Dic6D412yes—
85C2×C3×Dic6D412yes—
86#3757—21no—
87C4⋊(C3⋊Dic3)—21no—
88C4×C3⋊Dic3—21no—
89#3748—21no—
90#3748—21no—
91C2×C3⋊(C3⋊C8)—21no—
92C3×#217—21no—
93C3×C4⋊Dic3—21no—
94C3×C4×Dic3—21no—
95C3×#208—21no—
96C3×#208—21no—
97C2×C3×C3⋊C8—21no—
98#1796—91noSylow
99C3×2−1+4—62no—
100C3×2−1+4—62no—
101C3×2−1+4—62no—
102C3×2−1+4—62no—
103C2×C3×#41—21no—
104C2×C3×#41—31no—
105C2×C3×#41—32no—
106C2×C3×#41—61no—
107C2×C3×#41—62no—
108C2×C3×#41C2×D311yescharacteristic
109C2×C3×#41C2×D311yescharacteristic
110C22×C3×Q8—32no—
111C22×C3×Q8—31no—
112C22×C3×Q8—61no—
113#1010—22no—
114#1010—22no—
115#1010—22no—
116#1010—22no—
117C2×D3×Q8Dic312yescomplemented
118C2×#237—22no—
119C2×#237Dic312yescomplemented
120C2×#235—21no—
121C2×#235C2×D311yescharacteristic
122C22×Dic6—21no—
123C22×Dic6C2×D311yescharacteristic
124C2×C3×M16—31no—
125C3×#117—31no—
126#926—31no—
127#926—31no—
128#926—61no—
129#926C2×D311yescharacteristic
130C2×#208—31no—
131C2×#208—31no—
132C2×#208—61no—
133C2×#208C2×D311yescharacteristic
134C3×#104—62no—
135#837—22no—
136#837—64no—
137#837—62no—
138#837—62no—
139#836—61no—
140#835—62no—
141#835—62no—
142#834—62no—
143#833—61no—
144#824—61no—
145#823—61no—
146#816—62no—
147#810—62no—
148#806—62no—
149C23×C32—21no—
150C23×C32C2×D411yescharacteristic
151C22×C3×D3C22⋊C412yes—
152C32×Q8—21no—
153C32×Q8C22⋊C412yes—
154C32×D4—22no—
155C32×D4—22no—
156C32×D4—21no—
157C32×D4C22⋊C412yes—
158C2×C32×C4—21no—
159C2×C32×C4—22no—
160C2×C32×C4—22no—
161C2×C32×C4C22×C411yescharacteristic
162C2×C32×C4C2×D411yescharacteristic
163C2×C32×C4C2×D411yescharacteristic
164C2×C32×C4C2×D411yescharacteristic
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C3⋊Dic3—41no—
168C3×#67—28no—
169C3×#67—22no—
170C3×#67—22no—
171C2×C3×Dic3—28no—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—41no—
176C2×C3×Dic3C22⋊C412yes—
177C3×D12—21no—
178C3×D12—21no—
179C3×C4×D3—28no—
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×Dic6—28no—
183C3×Dic6—22no—
184C3×Dic6—21no—
185C3×Dic6—22no—
186C3×Dic6—21no—
187C3×Dic6—22no—
188C3×Dic6C22⋊C414yes—
189C3⋊(C3⋊C8)—41no—
190C3×C3⋊C8—41no—
191C2×2−1+4—31no—
192#421—31no—
193#421—91no—
194#419—91no—
195#419—91no—
196#413—91no—
197#410—91no—
198C3×#41—21no—
199C3×#41—21no—
200C3×#41—21no—
201C3×#41—21no—
202C3×#41—22no—
203C3×#41—22no—
204C3×#41—41no—
205C3×#41—41no—
206C3×#41—64no—
207C3×#41—64no—
208C3×#41—64no—
209C3×#41—68no—
210C3×#41—68no—
211C3×#41—64no—
212C3×#41—61no—
213C3×#41—61no—
214C3×#41—61no—
215C3×#41—61no—
216C3×#41D1212yes—
217C3×#41#6712yes—
218C3×#41#6712yes—
219C3×#41#6712yes—
220C2×C3×Q8—21no—
221C2×C3×Q8—31no—
222C2×C3×Q8—32no—
223C2×C3×Q8—32no—
224C2×C3×Q8—32no—
225C2×C3×Q8—62no—
226C2×C3×Q8—64no—
227C2×C3×Q8—64no—
228C2×C3×Q8—64no—
229C2×C3×Q8—64no—
230C2×C3×Q8—61no—
231C2×C3×Q8—61no—
232C2×C3×Q8—61no—
233C2×C3×Q8C4×D311yescharacteristic
234C2×C3×Q8C2×Dic311yescharacteristic
235C2×C3×D4—21no—
236C2×C3×D4—21no—
237C2×C3×D4—21no—
238C2×C3×D4—32no—
239C2×C3×D4—31no—
240C2×C3×D4—41no—
241C2×C3×D4—62no—
242C2×C3×D4C4×D311yescharacteristic
243C2×C3×D4C2×Dic311yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4—21no—
247C22×C3×C4—32no—
248C22×C3×C4—41no—
249C22×C3×C4—62no—
250C22×C3×C4—62no—
251C22×C3×C4—61no—
252C22×C3×C4C22×D311yescharacteristic
253C22×C3×C4C22×D311yescharacteristic
254C2×#67—22no—
255C2×#67#6712yes—
256C22×Dic3—21no—
257C22×Dic3—22no—
258D3×Q8—24no—
259D3×Q8—24no—
260#237—28no—
261#237—28no—
262#237—24no—
263#237—24no—
264#235—24no—
265#235—24no—
266#235—21no—
267#235—21no—
268#235—21no—
269#235—21no—
270C2×D12#6711yescharacteristic
271C2×C4×D3—22no—
272C2×C4×D3C2×Dic312yes—
273C2×Dic6—24no—
274C2×Dic6—24no—
275C2×Dic6—21no—
276C2×Dic6—21no—
277C2×Dic6—21no—
278C2×Dic6—22no—
279C2×Dic6C2×Dic312yes—
280C2×Dic6#6711yescharacteristic
281C2×Dic6#6712yes—
282C2×Dic6#6712yes—
283C3×M16—61no—
284C3×M16—61no—
285C2×C3×C8—61no—
286C3×C4⋊C4—61no—
287C3×C22⋊C4—61no—
288C3×C42—61no—
289#217—21no—
290#217—61no—
291#217—61no—
292#217—121no—
293C4⋊Dic3—21no—
294C4⋊Dic3—61no—
295C4⋊Dic3—61no—
296C4⋊Dic3—121no—
297C4×Dic3—21no—
298C4×Dic3—62no—
299C4×Dic3—61no—
300C4×Dic3—61no—
301#208—21no—
302#208—21no—
303#208—62no—
304#208—61no—
305#208—61no—
306#208—61no—
307#208—61no—
308#208—121no—
309C2×C3⋊C8—21no—
310C2×C3⋊C8—61no—
311C2×C3⋊C8—61no—
312C2×C3⋊C8—121no—
313C22×C32—22no—
314C22×C32—41no—
315C22×C32—42no—
316C22×C32C2×C22⋊C411yescharacteristic
317C22×C32#12011yescharacteristic
318C22×C32#12011yescharacteristic
319C2×C3×D3—22no—
320C2×C3×D3—42no—
321C32×C4—22no—
322C32×C4C2×C22⋊C411yescharacteristic
323C32×C4C2×C22⋊C411yescharacteristic
324C32×C4#12011yescharacteristic
325C32×C4#12011yescharacteristic
326C3⋊Dic3—81no—
327C3×Dic3—22no—
328C3×Dic3—24no—
329C3×Dic3—81no—
3302−1+4—62no—
3312−1+4—62no—
3322−1+4—62no—
3332−1+4—62no—
334C2×#41—31no—
335C2×#41—32no—
336C2×#41—61no—
337C2×#41—62no—
338C2×#41D3211yescharacteristic
339C22×Q8—32no—
340C22×Q8—31no—
341C22×Q8—61no—
342C2×M16—31no—
343C2×M16—91no—
344#117—31no—
345#117—91no—
346#104—62no—
347#104—182no—
348#103—182no—
349#103—182no—
350#101—181no—
351#100—181no—
352#99—182no—
353C23×C3—21no—
354C23×C3—21no—
355C23×C3—21no—
356C23×C3—32no—
357C23×C3—41no—
358C23×C3D3×D411yescharacteristic
359C23×C3C2×#6711yescharacteristic
360C22×D3#21712yes—
361C3×Q8—21no—
362C3×Q8—21no—
363C3×Q8—22no—
364C3×Q8—34no—
365C3×Q8—41no—
366C3×Q8—68no—
367C3×Q8—61no—
368C3×Q8—62no—
369C3×Q8—62no—
370C3×Q8—61no—
371C3×Q8—62no—
372C3×Q8(C2×D3)⋊C412yes—
373C3×Q8#21712yes—
374C3×D4—22no—
375C3×D4—22no—
376C3×D4—22no—
377C3×D4—22no—
378C3×D4—21no—
379C3×D4—21no—
380C3×D4—22no—
381C3×D4—41no—
382C3×D4—42no—
383C3×D4—42no—
384C3×D4—68no—
385C3×D4—62no—
386C3×D4—62no—
387C3×D4—61no—
388C3×D4—61no—
389C3×D4(C2×D3)⋊C412yes—
390C3×D4#21712yes—
391C2×C3×C4—21no—
392C2×C3×C4—21no—
393C2×C3×C4—22no—
394C2×C3×C4—22no—
395C2×C3×C4—22no—
396C2×C3×C4—22no—
397C2×C3×C4—21no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—21no—
401C2×C3×C4—21no—
402C2×C3×C4—21no—
403C2×C3×C4—32no—
404C2×C3×C4—42no—
405C2×C3×C4—42no—
406C2×C3×C4—41no—
407C2×C3×C4—68no—
408C2×C3×C4—68no—
409C2×C3×C4—62no—
410C2×C3×C4—62no—
411C2×C3×C4—62no—
412C2×C3×C4—62no—
413C2×C3×C4—62no—
414C2×C3×C4—121no—
415C2×C3×C4C2×C4×D311yescharacteristic
416C2×C3×C4C2×D1211yescharacteristic
417C2×C3×C4D3×D411yescharacteristic
418C2×C3×C4D3×D411yescharacteristic
419C2×C3×C4D3×D411yescharacteristic
420C2×C3×C4C22×Dic311yescharacteristic
421C2×C3×C4C2×#6711yescharacteristic
422C2×C3×C4C2×#6711yescharacteristic
423C2×C3×C4C2×#6711yescharacteristic
424C2×C3×C4C2×#6711yescharacteristic
425C2×C3×C4C2×#6711yescharacteristic
426C2×C3×C4C2×#6711yescharacteristic
427#67—28no—
428#67—22no—
429#67—22no—
430C2×Dic3—28no—
431C2×Dic3—22no—
432C2×Dic3—22no—
433C2×Dic3—22no—
434C2×Dic3—41no—
435C2×Dic3—122no—
436C2×Dic3—121no—
437C2×Dic3—121no—
438C2×Dic3#21712yes—
439D12—21no—
440D12—21no—
441C4×D3—28no—
442C4×D3—22no—
443C4×D3—22no—
444Dic6—28no—
445Dic6—22no—
446Dic6—21no—
447Dic6—22no—
448Dic6—21no—
449Dic6—22no—
450Dic6#21714yes—
451C3×C8—121no—
452C3⋊C8—41no—
453C3⋊C8—122no—
454C3⋊C8—121no—
455C3⋊C8—121no—
456C2×C32—21no—
457C2×C32—21no—
458C2×C32—21no—
459C2×C32—42no—
460C2×C32#37911yescharacteristic
461C3×D3—42no—
462#41—21no—
463#41—21no—
464#41—64no—
465#41—64no—
466#41—64no—
467#41—68no—
468#41—68no—
469#41—64no—
470#41—61no—
471#41—61no—
472#41—61no—
473#41—61no—
474#41#62512yes—
475#41#62612yes—
476C2×Q8—31no—
477C2×Q8—32no—
478C2×Q8—32no—
479C2×Q8—32no—
480C2×Q8—62no—
481C2×Q8—64no—
482C2×Q8—64no—
483C2×Q8—64no—
484C2×Q8—64no—
485C2×Q8—61no—
486C2×Q8—61no—
487C2×Q8—61no—
488C2×Q8D3×Dic311yescharacteristic
489C2×D4—21no—
490C2×D4—32no—
491C2×D4—31no—
492C2×D4—62no—
493C2×D4D3×Dic311yescharacteristic
494C22×C4—21no—
495C22×C4—32no—
496C22×C4—62no—
497C22×C4—62no—
498C22×C4—61no—
499C22×C4C2×D3211yescharacteristic
500M16—61no—
501M16—61no—
502M16—181no—
503M16—181no—
504C2×C8—61no—
505C2×C8—181no—
506C4⋊C4—61no—
507C4⋊C4—181no—
508C22⋊C4—61no—
509C22⋊C4—181no—
510C42—61no—
511C42—181no—
512C22×C3—22no—
513C22×C3—22no—
514C22×C3—21no—
515C22×C3—21no—
516C22×C3—21no—
517C22×C3—42no—
518C22×C3—41no—
519C22×C3—41no—
520C22×C3—42no—
521C22×C3—42no—
522C22×C3—62no—
523C22×C3—81no—
524C22×C3—82no—
525C22×C3—122no—
526C22×C3D3×C22⋊C411yescharacteristic
527C22×C3#88211yescharacteristic
528C22×C3#93711yescharacteristic
529C22×C3C2×#21711yescharacteristic
530C22×C3#95311yescharacteristic
531C22×C3#95311yescharacteristic
532C2×D3—22no—
533C2×D3—42no—
534C3×C4—22no—
535C3×C4—22no—
536C3×C4—21no—
537C3×C4—21no—
538C3×C4—21no—
539C3×C4—21no—
540C3×C4—42no—
541C3×C4—62no—
542C3×C4—64no—
543C3×C4—241no—
544C3×C4D3×C22⋊C411yescharacteristic
545C3×C4#88211yescharacteristic
546C3×C4C2×(C2×D3)⋊C411yescharacteristic
547C3×C4#93711yescharacteristic
548C3×C4C2×#21711yescharacteristic
549C3×C4C2×#21711yescharacteristic
550C3×C4#95311yescharacteristic
551C3×C4#95311yescharacteristic
552Dic3—22no—
553Dic3—24no—
554Dic3—81no—
555Dic3—241no—
556Dic3—241no—
557Dic3—242no—
558C32#179611yescharacteristic, complemented, Sylow
559C23—21no—
560C23—32no—
561C23D3×#6711yescharacteristic
562Q8—21no—
563Q8—34no—
564Q8—68no—
565Q8—62no—
566Q8—61no—
567Q8—62no—
568Q8—61no—
569Q8—62no—
570Q8(C2×D3)⋊Dic312yes—
571D4—22no—
572D4—22no—
573D4—21no—
574D4—68no—
575D4—62no—
576D4—62no—
577D4—61no—
578D4—61no—
579D4(C2×D3)⋊Dic312yes—
580C2×C4—21no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—32no—
584C2×C4—68no—
585C2×C4—68no—
586C2×C4—62no—
587C2×C4—62no—
588C2×C4—62no—
589C2×C4—62no—
590C2×C4—62no—
591C2×C4—121no—
592C2×C4—361no—
593C2×C4C2×D3×Dic311yescharacteristic
594C2×C4C2×#62511yescharacteristic
595C2×C4C2×#62611yescharacteristic
596C2×C4D3×#6711yescharacteristic
597C2×C4D3×#6711yescharacteristic
598C2×C4D3×#6711yescharacteristic
599C8—121no—
600C8—361no—
601C2×C3—21no—
602C2×C3—21no—
603C2×C3—21no—
604C2×C3—21no—
605C2×C3—21no—
606C2×C3—21no—
607C2×C3—21no—
608C2×C3—41no—
609C2×C3—41no—
610C2×C3—41no—
611C2×C3—42no—
612C2×C3—42no—
613C2×C3—82no—
614C2×C3—122no—
615C2×C3#495511yescharacteristic
616C2×C3#524711yescharacteristic
617D3—42no—
618C22—22no—
619C22—41no—
620C22—42no—
621C22—62no—
622C22—122no—
623C22D3×#21711yescharacteristic
624C22#6397611yescharacteristic
625C22#6397711yescharacteristic
626C4—22no—
627C4—62no—
628C4—64no—
629C4—241no—
630C4—721no—
631C4C2×(C2×D3)⋊Dic311yescharacteristic
632C4D3×#21711yescharacteristic
633C4#6397611yescharacteristic
634C4#6397711yescharacteristic
635C3—21no—
636C3#7041511yescharacteristic, complemented
637C3#7214411yescharacteristic, complemented
638C2—21no—
639C2—21no—
640C2—21no—
641C2—42no—
642C2—122no—
643C2#1059170611yescharacteristic
subgroupG/H
centerC2#10591706
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC2×C32#379
prime radicalC2×C3×#1010C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1796
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #10592935C2semidirect
3. C3×#5236C2semidirect
4. #10591231C2semidirect
5. #10591100C2semidirect
6. #10591099C2semidirect
7. #10591076C2semidirect
15. C2×C3×D3×Q8C4semidirect
17. C2×C3×#237C4semidirect
41. #5236D3semidirect
117. C2×D3×Q8Dic3semidirect
119. C2×#237Dic3semidirect
558. C32#1796semidirect
636. C3#70415semidirect
637. C3#72144semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#10102
#105929352
C3×#52362
#105912312
#105911002
#105910992
#105910762
#721443
#704153