Finite Group Databasebeta

← #11802970

Subgroups — #11802970

Order 1152.

1317 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×C3×Dic3⋊C4C211yesmaximal, characteristic
2C2×#63955C211yesmaximal, characteristic
3C2×#63955C211yesmaximal, characteristic
4#10592909C211yesmaximal, characteristic
5C3×#5210C211yesmaximal, characteristic
6#10591718C211yesmaximal, characteristic
7#10591693C211yesmaximal, characteristic
8#10591246C212yesmaximal, complemented
9#10591245C212yesmaximal, complemented
10#10591244C212yesmaximal, complemented
11#10591234C212yesmaximal, complemented
12#82157—31nomaximal
13#74382—31nomaximal
14C23×C32×C4C2211yescharacteristic
15C23×C3×Dic3C2211yescharacteristic
16C23×C3×Dic3C2211yescharacteristic
17C2×#3757C2211yescharacteristic
18C2×#3757C2211yescharacteristic
19C2×#3750C2212yes—
20C2×C3×#217C2211yescharacteristic
21C2×C3×#217C2211yescharacteristic
22C2×C3×Dic3⋊C4—24no—
23C2×C3×Dic3⋊C4C2212yes—
24C2×C3×Dic3⋊C4C2212yes—
25C2×C3×Dic3⋊C4C2212yes—
26C2×#3724C2212yes—
27C2×Dic3⋊Dic3C2211yescharacteristic
28C2×Dic3⋊Dic3C2211yescharacteristic
29C2×Dic3⋊Dic3C2212yes—
30C2×Dic3⋊Dic3C2212yes—
31#63955—24no—
32#63955C418yescomplemented
33C2×Dic32C2211yescharacteristic
34C2×Dic32C2211yescharacteristic
35C2×Dic32C2212yes—
36#63658C2212yes—
37C3×C3⋊#95C2212yes—
38#63579C2212yes—
39#63579C2212yes—
40C22×C3×C4⋊C4—31no—
41C2×#926—31no—
42C2×#926—31no—
43C22×Dic3⋊C4D311yescharacteristic
44C2×#877—31no—
45C2×#877—31no—
46C3×#385—31no—
47#5210—31no—
48#5210—31no—
49#5210—31no—
50#5210—31no—
51#5210—61no—
52#5210D311yescharacteristic
53#4977—32no—
54#4976—32no—
55#4975—32no—
56#4975—32no—
57#4972—32no—
58#4972—32no—
59#4950—31no—
60#4946—31no—
61#4650—32no—
62#4647—32no—
63C24×C32C2311yescharacteristic
64C22×C32×C4—22no—
65C22×C32×C4—42no—
66C22×C32×C4C2312yes—
67C22×C3⋊Dic3C2311yescharacteristic
68C22×C3⋊Dic3C2311yescharacteristic
69C22×C3⋊Dic3C2312yes—
70C22×C3×Dic3—22no—
71C22×C3×Dic3—42no—
72C22×C3×Dic3C2×C414yes—
73C22×C3×Dic3D414yes—
74C22×C3×Dic3Q814yes—
75C22×C3×Dic3C2311yescharacteristic
76C22×C3×Dic3C2312yes—
77C22×C3×Dic3C2311yescharacteristic
78C22×C3×Dic3C2312yes—
79C22×C3×Dic3C2311yescharacteristic
80C22×C3×Dic3C2311yescharacteristic
81#3757—22no—
82#3757C2×C414yes—
83#3750—42no—
84C3×#217—22no—
85C3×#217C2×C414yes—
86C3×Dic3⋊C4—24no—
87C3×Dic3⋊C4—24no—
88C3×Dic3⋊C4—42no—
89#3724—42no—
90Dic3⋊Dic3—22no—
91Dic3⋊Dic3—22no—
92Dic3⋊Dic3—42no—
93Dic3⋊Dic3C2×C418yescomplemented
94Dic32—22no—
95Dic32—22no—
96Dic32C2×C418yescomplemented
97#2317—91noSylow
98C23×C3×C4—21no—
99C23×C3×C4—31no—
100C23×C3×C4—31no—
101C23×C3×C4C2×D311yescharacteristic
102C23×C3×C4C2×D311yescharacteristic
103C23×Dic3C2×D311yescharacteristic
104C23×Dic3C2×D311yescharacteristic
105C2×C3×C4⋊C4—32no—
106C2×C3×C4⋊C4—32no—
107C2×C3×C4⋊C4—32no—
108C2×C3×C4⋊C4—64no—
109C2×C3×C22⋊C4—31no—
110C2×C3×C22⋊C4—31no—
111C2×#217—31no—
112C2×#217—31no—
113C2×#217—31no—
114C2×#217—31no—
115C2×#217—61no—
116C2×#217—61no—
117C2×#217C2×D311yescharacteristic
118C2×#217C2×D311yescharacteristic
119#926—38no—
120#926—64no—
121C2×C4⋊Dic3—32no—
122C2×C4⋊Dic3—31no—
123C2×C4⋊Dic3—31no—
124C2×C4⋊Dic3—32no—
125C2×Dic3⋊C4—24no—
126C2×Dic3⋊C4—32no—
127C2×Dic3⋊C4—31no—
128C2×Dic3⋊C4—31no—
129C2×Dic3⋊C4—32no—
130C2×Dic3⋊C4—32no—
131C2×Dic3⋊C4—32no—
132C2×Dic3⋊C4—32no—
133C2×Dic3⋊C4—32no—
134C2×Dic3⋊C4—62no—
135C2×Dic3⋊C4C2×D312yes—
136C2×Dic3⋊C4C2×D312yes—
137C2×Dic3⋊C4C2×D312yes—
138C2×C4×Dic3—32no—
139C2×C4×Dic3—32no—
140C2×C4×Dic3—31no—
141C2×C4×Dic3—31no—
142C2×C4×Dic3—31no—
143C2×C4×Dic3—31no—
144#877—38no—
145#877—64no—
146C3×#95—32no—
147C3⋊#95—32no—
148C3⋊#95—32no—
149C3⋊#95—32no—
150C3⋊#95—32no—
151C3⋊#95—32no—
152C3⋊#95—32no—
153C3⋊#95—62no—
154C3⋊#95C2×D312yes—
155C23×C32—22no—
156C23×C32—22no—
157C23×C32C22×C412yes—
158C23×C32C2×D412yes—
159C23×C32C2×Q812yes—
160C23×C32C2411yescharacteristic
161C2×C32×C4—22no—
162C2×C32×C4—22no—
163C2×C32×C4—22no—
164C2×C32×C4—44no—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—21no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—22no—
170C2×C3⋊Dic3—22no—
171C2×C3⋊Dic3C22×C414yes—
172C2×C3×Dic3—22no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—21no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3—44no—
182C2×C3×Dic3C4⋊C4116yescomplemented
183C2×C3×Dic3C22×C414yes—
184C2×C3×Dic3C22×C412yes—
185C2×C3×Dic3C22×C412yes—
186C2×C3×Dic3C2×D412yes—
187C2×C3×Dic3C2×D412yes—
188C2×C3×Dic3C2×Q812yes—
189C2×C3×Dic3C2×Q812yes—
190C2×#117—91no—
191C2×#117—91no—
192C22×C4⋊C4—31no—
193#389—92no—
194#389—92no—
195#385—31no—
196#385—91no—
197#385—91no—
198#385—91no—
199#384—92no—
200#382—92no—
201C24×C3—21no—
202C24×C3C22×D311yescharacteristic
203C24×C3C22×D311yescharacteristic
204C22×C3×C4—22no—
205C22×C3×C4—22no—
206C22×C3×C4—22no—
207C22×C3×C4—34no—
208C22×C3×C4—34no—
209C22×C3×C4—34no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—32no—
213C22×C3×C4—31no—
214C22×C3×C4—32no—
215C22×C3×C4—31no—
216C22×C3×C4—42no—
217C22×C3×C4—42no—
218C22×C3×C4—42no—
219C22×C3×C4—44no—
220C22×C3×C4—62no—
221C22×C3×C4—122no—
222C22×C3×C4C22×D312yes—
223C22×C3×C4C22×D312yes—
224C22×Dic3—22no—
225C22×Dic3—32no—
226C22×Dic3—32no—
227C22×Dic3—31no—
228C22×Dic3—31no—
229C22×Dic3—31no—
230C22×Dic3—31no—
231C22×Dic3—42no—
232C22×Dic3—62no—
233C22×Dic3—61no—
234C22×Dic3—61no—
235C22×Dic3Dic614yes—
236C22×Dic3C4×D314yes—
237C22×Dic3#6714yes—
238C22×Dic3C22×D311yescharacteristic
239C22×Dic3C22×D312yes—
240C22×Dic3C22×D311yescharacteristic
241C22×Dic3C22×D312yes—
242C22×Dic3C22×D311yescharacteristic
243C22×Dic3C22×D311yescharacteristic
244C3×C4⋊C4—64no—
245C3×C4⋊C4—64no—
246C3×C4⋊C4—122no—
247C3×C22⋊C4—34no—
248C3×C22⋊C4—62no—
249#217—22no—
250#217—34no—
251#217—34no—
252#217—64no—
253#217—62no—
254#217—62no—
255#217—122no—
256#217C4×D314yes—
257C4⋊Dic3—38no—
258C4⋊Dic3—62no—
259C4⋊Dic3—62no—
260C4⋊Dic3—122no—
261Dic3⋊C4—24no—
262Dic3⋊C4—24no—
263Dic3⋊C4—38no—
264Dic3⋊C4—42no—
265Dic3⋊C4—62no—
266Dic3⋊C4—62no—
267Dic3⋊C4—122no—
268Dic3⋊C4—122no—
269Dic3⋊C4—122no—
270Dic3⋊C4—122no—
271Dic3⋊C4—122no—
272Dic3⋊C4—124no—
273C4×Dic3—38no—
274C4×Dic3—38no—
275C4×Dic3—62no—
276C4×Dic3—62no—
277C4×Dic3—62no—
278C4×Dic3—62no—
279C22×C32—24no—
280C22×C32—22no—
281C22×C32—22no—
282C22×C32—24no—
283C22×C32C2×C4⋊C414yes—
284C22×C32C23×C411yescharacteristic
285C22×C32C22×D411yescharacteristic
286C22×C32C22×Q811yescharacteristic
287C22×C32H(5,𝔽2)12yes—
288C22×C322−1+412yes—
289C32×C4—42no—
290C3⋊Dic3—24no—
291C3⋊Dic3—41no—
292C3⋊Dic3—41no—
293C3×Dic3—24no—
294C3×Dic3—41no—
295C3×Dic3—42no—
296C3×Dic3—41no—
297C3×Dic3C2×C4⋊C414yes—
298C3×Dic3C2×C4⋊C414yes—
299C23×C4—31no—
300C23×C4—31no—
301C23×C4D3211yescharacteristic
302#117—98no—
303#117—184no—
304C2×C4⋊C4—32no—
305C2×C4⋊C4—32no—
306C2×C4⋊C4—32no—
307C2×C4⋊C4—64no—
308C2×C4⋊C4—92no—
309C2×C4⋊C4—91no—
310C2×C4⋊C4—91no—
311C2×C4⋊C4—92no—
312C2×C4⋊C4—92no—
313C2×C4⋊C4—92no—
314C2×C22⋊C4—31no—
315C2×C22⋊C4—31no—
316C2×C22⋊C4—91no—
317C2×C22⋊C4—91no—
318C2×C42—92no—
319C2×C42—91no—
320C2×C42—91no—
321#95—32no—
322#95—92no—
323#95—92no—
324#95—92no—
325C23×C3—22no—
326C23×C3—22no—
327C23×C3—22no—
328C23×C3—22no—
329C23×C3—21no—
330C23×C3—22no—
331C23×C3—22no—
332C23×C3—22no—
333C23×C3—42no—
334C23×C3—42no—
335C23×C3C2×Dic612yes—
336C23×C3C2×C4×D312yes—
337C23×C3C2×C4×D312yes—
338C23×C3D3×D412yes—
339C23×C3D3×Q812yes—
340C23×C3C2×#6712yes—
341C23×C3C23×D311yescharacteristic
342C23×C3C23×D311yescharacteristic
343C2×C3×C4—24no—
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—316no—
351C2×C3×C4—32no—
352C2×C3×C4—32no—
353C2×C3×C4—32no—
354C2×C3×C4—32no—
355C2×C3×C4—32no—
356C2×C3×C4—32no—
357C2×C3×C4—34no—
358C2×C3×C4—42no—
359C2×C3×C4—42no—
360C2×C3×C4—44no—
361C2×C3×C4—44no—
362C2×C3×C4—48no—
363C2×C3×C4—62no—
364C2×C3×C4—62no—
365C2×C3×C4—62no—
366C2×C3×C4—62no—
367C2×C3×C4—62no—
368C2×C3×C4—62no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4—61no—
372C2×C3×C4—124no—
373C2×Dic3—22no—
374C2×Dic3—22no—
375C2×Dic3—21no—
376C2×Dic3—21no—
377C2×Dic3—22no—
378C2×Dic3—22no—
379C2×Dic3—22no—
380C2×Dic3—22no—
381C2×Dic3—22no—
382C2×Dic3—34no—
383C2×Dic3—34no—
384C2×Dic3—44no—
385C2×Dic3—64no—
386C2×Dic3—62no—
387C2×Dic3—62no—
388C2×Dic3—62no—
389C2×Dic3—62no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—62no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—61no—
397C2×Dic3—61no—
398C2×Dic3—122no—
399C2×Dic3—122no—
400C2×Dic3—122no—
401C2×Dic3—121no—
402C2×Dic3—121no—
403C2×Dic3—122no—
404C2×Dic3Dic3⋊C4116yescomplemented
405C2×Dic3C2×Dic612yes—
406C2×Dic3C2×Dic612yes—
407C2×Dic3C2×C4×D314yes—
408C2×Dic3C2×C4×D312yes—
409C2×Dic3C2×C4×D312yes—
410C2×Dic3C2×#6712yes—
411C2×Dic3C2×#6712yes—
412C2×C32—24no—
413C2×C32C22×C4⋊C411yescharacteristic
414C2×C32#52012yes—
415C2×C32#53511yescharacteristic
416C2×C32#53611yescharacteristic
417C2×C32#54412yes—
418C24C2×D3211yescharacteristic
419C22×C4—22no—
420C22×C4—31no—
421C22×C4—31no—
422C22×C4—34no—
423C22×C4—34no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—31no—
429C22×C4—42no—
430C22×C4—62no—
431C22×C4—92no—
432C22×C4—91no—
433C22×C4—91no—
434C22×C4—122no—
435C22×C4C2×D3212yes—
436C4⋊C4—64no—
437C4⋊C4—64no—
438C4⋊C4—98no—
439C4⋊C4—122no—
440C4⋊C4—182no—
441C4⋊C4—182no—
442C4⋊C4—362no—
443C4⋊C4—362no—
444C4⋊C4—362no—
445C22⋊C4—34no—
446C22⋊C4—62no—
447C22⋊C4—94no—
448C22⋊C4—182no—
449C42—98no—
450C42—182no—
451C42—182no—
452C22×C3—22no—
453C22×C3—24no—
454C22×C3—24no—
455C22×C3—24no—
456C22×C3—22no—
457C22×C3—22no—
458C22×C3—22no—
459C22×C3—24no—
460C22×C3—21no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—21no—
464C22×C3—22no—
465C22×C3—24no—
466C22×C3—42no—
467C22×C3—44no—
468C22×C3—42no—
469C22×C3—44no—
470C22×C3D3×C4⋊C414yes—
471C22×C3C2×Dic3⋊C414yes—
472C22×C3C22×Dic611yescharacteristic
473C22×C3C22×C4×D311yescharacteristic
474C22×C3C22×C4×D311yescharacteristic
475C22×C3C2×D3×D411yescharacteristic
476C22×C3#100412yes—
477C22×C3#100412yes—
478C22×C3C2×D3×Q811yescharacteristic
479C22×C3#100712yes—
480C22×C3#101012yes—
481C22×C3C22×#6711yescharacteristic
482C3×C4—34no—
483C3×C4—34no—
484C3×C4—44no—
485C3×C4—42no—
486C3×C4—42no—
487C3×C4—64no—
488C3×C4—121no—
489C3×C4—121no—
490C3×C4—122no—
491Dic3—24no—
492Dic3—41no—
493Dic3—42no—
494Dic3—41no—
495Dic3—64no—
496Dic3—64no—
497Dic3—124no—
498Dic3—121no—
499Dic3—121no—
500Dic3—121no—
501Dic3—121no—
502Dic3—241no—
503Dic3—241no—
504Dic3C2×Dic3⋊C414yes—
505Dic3C2×Dic3⋊C414yes—
506C32#231711yescharacteristic, complemented, Sylow
507C23—22no—
508C23—22no—
509C23D3×Dic612yes—
510C23C4×D3212yes—
511C23D3×#6712yes—
512C23C22×D3211yescharacteristic
513C2×C4—22no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—32no—
517C2×C4—32no—
518C2×C4—32no—
519C2×C4—32no—
520C2×C4—32no—
521C2×C4—32no—
522C2×C4—316no—
523C2×C4—34no—
524C2×C4—44no—
525C2×C4—62no—
526C2×C4—62no—
527C2×C4—62no—
528C2×C4—61no—
529C2×C4—61no—
530C2×C4—62no—
531C2×C4—62no—
532C2×C4—62no—
533C2×C4—62no—
534C2×C4—94no—
535C2×C4—124no—
536C2×C4—182no—
537C2×C4—182no—
538C2×C4—182no—
539C2×C4—181no—
540C2×C4—181no—
541C2×C4—182no—
542C2×C3—24no—
543C2×C3—24no—
544C2×C3—21no—
545C2×C3—22no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—22no—
549C2×C3—44no—
550C2×C3#550012yes—
551C2×C3C2×D3×C4⋊C411yescharacteristic
552C2×C3#550412yes—
553C2×C3#554612yes—
554C2×C3#559211yescharacteristic
555C2×C3#562312yes—
556C2×C3#564811yescharacteristic
557C2×C3C22×Dic3⋊C411yescharacteristic
558C2×C3#580511yescharacteristic
559C2×C3#581611yescharacteristic
560C22—24no—
561C22—22no—
562C22—22no—
563C22—24no—
564C22D3×Dic3⋊C414yes—
565C22C2×D3×Dic611yescharacteristic
566C22#6429712yes—
567C22C2×C4×D3211yescharacteristic
568C22C2×D3×#6711yescharacteristic
569C22#6433012yes—
570C4—34no—
571C4—34no—
572C4—42no—
573C4—64no—
574C4—121no—
575C4—122no—
576C4—121no—
577C4—184no—
578C4—361no—
579C4—361no—
580C3—21no—
581C3#7438211yescharacteristic, complemented
582C3#8215711yescharacteristic, complemented
583C2—24no—
584C2#1059481212yes—
585C2C2×D3×Dic3⋊C411yescharacteristic
586C2#1059487412yes—
587C2#1059489711yescharacteristic
588C2#1059551311yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC23×C32×C4C22
socle, cosocleC23×C32C24
prime radicalC24×C32C23
hypercenterC23×C4D32
hypocenterC32#2317
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
8. #10591246C2semidirect
9. #10591245C2semidirect
10. #10591244C2semidirect
11. #10591234C2semidirect
32. #63955C4semidirect
93. Dic3⋊Dic3C2×C4semidirect
96. Dic32C2×C4semidirect
182. C2×C3×Dic3C4⋊C4semidirect
404. C2×Dic3Dic3⋊C4semidirect
506. C32#2317semidirect
581. C3#74382semidirect
582. C3#82157semidirect

Maximal subgroups

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

subgroupindex
C22×C3×Dic3⋊C42
C2×#639552
#105929092
C3×#52102
#105917182
#105916932
#105912462
#105912452
#105912442
#105912342
#821573
#743823