Finite Group Databasebeta

← #11808790

Subgroups — #11808790

Order 1152.

1255 conjugacy classes of subgroups.

The subgroup lattice has 598 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#887C211yesmaximal, characteristic, complemented
2C2×#63894C211yesmaximal, characteristic
3#10592690C211yesmaximal, characteristic
4C3×#4986C211yesmaximal, characteristic, complemented
5#10591321C211yesmaximal, characteristic
6#10591310C212yesmaximal, complemented
7#10591239C211yesmaximal, characteristic, complemented
8#10591233C212yesmaximal, complemented
9#10591233C212yesmaximal, complemented
10#10591225C211yesmaximal, characteristic, complemented
11#10591204C212yesmaximal, complemented
12#82492—31nomaximal
13#78131—31nomaximal
14C22×C4⋊(C3⋊D3)C2211yescharacteristic
15C22×C3×Dic6C2211yescharacteristic
16C22×#624C2212yes—
17C2×C32×C4⋊C4C2211yescharacteristic
18C2×#3752C2212yes—
19C2×#3752C2212yes—
20C2×C3×C4⋊Dic3C2211yescharacteristic
21C2×C3×Dic3⋊C4C2211yescharacteristic
22C2×C3×Dic3⋊C4C2211yescharacteristic
23C2×C3×C4×Dic3C2211yescharacteristic
24C2×C3×C4×Dic3C2211yescharacteristic
25C2×C3×C4×Dic3C2211yescharacteristic
26C3×#887—24no—
27C2×#3722C2212yes—
28C2×#3722C2212yes—
29C2×#3722C2212yes—
30C2×Dic32C2212yes—
31#63894—24no—
32#63658C2211yescharacteristic
33C3×C3⋊#95C2212yes—
34C3×C3⋊#95C2212yes—
35C3×C3⋊#95C2211yescharacteristic
36#63579C2212yes—
37#63579C2212yes—
38#63579C2212yes—
39C2×C3×C4×Q8—31no—
40C2×#893—31no—
41C2×#887D311yescharacteristic, complemented
42C2×C4×D12—31no—
43C3×#400—31no—
44#5231—32no—
45#4992—32no—
46#4991—31no—
47#4991—31no—
48#4991—31no—
49#4991—61no—
50#4986—31no—
51#4986D311yescharacteristic, complemented
52#4985—31no—
53C4×(C2×D3)⋊C4—32no—
54C4×(C2×D3)⋊C4—32no—
55#4932—31no—
56#4675—32no—
57#4673—32no—
58#4672—31no—
59#4666—32no—
60#4666—32no—
61#4648—31no—
62C23×C3⋊D3C2312yes—
63C22×C32×C4C2311yescharacteristic
64C22×C32×C4C2311yescharacteristic
65C22×C32×C4C2311yescharacteristic
66C22×C3⋊Dic3C2312yes—
67C2×C4⋊(C3⋊D3)—24no—
68C2×C4⋊(C3⋊D3)—22no—
69C22×C3×Dic3C2311yescharacteristic
70C22×C3×Dic3C2312yes—
71C22×C3×Dic3C2312yes—
72C22×C3×Dic3C2311yescharacteristic
73C22×C3×Dic3C2311yescharacteristic
74C22×C3×Dic3C2311yescharacteristic
75C2×C3×Dic6—22no—
76C2×C3×Dic6—22no—
77C2×C3×Dic6—22no—
78C2×#624—24no—
79C2×#624—24no—
80C2×#624—24no—
81C32×C4⋊C4—22no—
82#3752—24no—
83#3752—24no—
84C3×C4⋊Dic3—22no—
85C3×Dic3⋊C4—22no—
86C3×Dic3⋊C4—22no—
87C3×C4×Dic3—22no—
88C3×C4×Dic3—22no—
89C3×C4×Dic3D414yes—
90#3722—24no—
91#3722—24no—
92#3722—24no—
93Dic32—24no—
94#2398—91noSylow
95C22×C3×Q8—31no—
96C22×D12—31no—
97C22×D12—31no—
98C22×D12—61no—
99C22×C4×D3—32no—
100C22×C4×D3—32no—
101C22×Dic6C2×D311yescharacteristic
102C3×C4×Q8—64no—
103C2×C3×C4⋊C4—21no—
104C2×C3×C4⋊C4—31no—
105C2×C3×C4⋊C4—31no—
106C2×C3×C4⋊C4—31no—
107C2×C3×C4⋊C4C2×D311yescharacteristic
108C2×C3×C4⋊C4C2×D311yescharacteristic
109C2×C3×C42—31no—
110C2×C3×C42—31no—
111C2×C3×C42—31no—
112C2×(C2×D3)⋊C4—32no—
113C2×(C2×D3)⋊C4—32no—
114C2×(C2×D3)⋊C4—32no—
115C2×(C2×D3)⋊C4—32no—
116C2×(C2×D3)⋊C4—32no—
117C2×(C2×D3)⋊C4—32no—
118C2×(C2×D3)⋊C4—32no—
119C2×(C2×D3)⋊C4—32no—
120C2×(C2×D3)⋊C4—32no—
121C2×(C2×D3)⋊C4—32no—
122C2×(C2×D3)⋊C4—62no—
123C2×(C2×D3)⋊C4—62no—
124C2×C4⋊Dic3C2×D311yescharacteristic
125C2×Dic3⋊C4C2×D311yescharacteristic
126C2×Dic3⋊C4C2×D311yescharacteristic
127C2×C4×Dic3—32no—
128C2×C4×Dic3—32no—
129C2×C4×Dic3C2×D311yescharacteristic
130C2×C4×Dic3C2×D311yescharacteristic
131C2×C4×Dic3C2×D311yescharacteristic
132#893—64no—
133#887—24no—
134C4×D12—64no—
135C3×#95—31no—
136C3×#95—32no—
137C3×#95—32no—
138C3⋊#95—31no—
139C3⋊#95—31no—
140C3⋊#95—32no—
141C3⋊#95—32no—
142C3⋊#95—32no—
143C3⋊#95—32no—
144C3⋊#95—32no—
145C3⋊#95—32no—
146C3⋊#95—61no—
147C3⋊#95C2×D311yescharacteristic
148C3⋊#95C2×D312yes—
149C3⋊#95C2×D312yes—
150C23×C32C2411yescharacteristic
151C22×C3⋊D3—22no—
152C22×C3⋊D3—22no—
153C22×C3⋊D3—22no—
154C22×C3⋊D3—22no—
155C22×C3⋊D3—22no—
156C22×C3⋊D3—22no—
157C22×C3⋊D3#4114yes—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—21no—
162C2×C32×C4—22no—
163C2×C32×C4C2×D412yes—
164C2×C32×C4#4112yes—
165C2×C3⋊Dic3—22no—
166C2×C3⋊Dic3—22no—
167C2×C3⋊Dic3—22no—
168C4⋊(C3⋊D3)—28no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3—22no—
179C2×C3×Dic3—22no—
180C2×C3×Dic3—22no—
181C2×C3×Dic3C2×D412yes—
182C2×C3×Dic3C2×D412yes—
183C2×C3×Dic3#4112yes—
184C2×C3×Dic3#4114yes—
185C3×Dic6—28no—
186#624—48no—
187C2×C4×Q8—31no—
188C2×C4×D4—91no—
189#400—31no—
190#400—91no—
191#397—92no—
192#394—91no—
193#394—91no—
194#393—92no—
195C4×C22⋊C4—92no—
196C4×C22⋊C4—92no—
197#376—91no—
198C23×D3—32no—
199C23×D3—32no—
200C23×D3—62no—
201C2×C3×Q8—62no—
202C2×C3×Q8—62no—
203C2×C3×Q8—62no—
204C22×C3×C4—21no—
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—31no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—32no—
211C22×C3×C4—31no—
212C22×C3×C4—32no—
213C22×C3×C4C22×D311yescharacteristic
214C22×C3×C4C22×D311yescharacteristic
215C22×C3×C4C22×D311yescharacteristic
216C22×C3×C4C22×D311yescharacteristic
217C22×C3×C4C22×D311yescharacteristic
218C22×C3×C4C22×D311yescharacteristic
219C22×Dic3—32no—
220C22×Dic3—32no—
221C22×Dic3—62no—
222C22×Dic3C22×D312yes—
223C22×Dic3C22×D311yescharacteristic
224C22×Dic3C22×D312yes—
225C22×Dic3C22×D311yescharacteristic
226C22×Dic3C22×D311yescharacteristic
227C22×Dic3C22×D311yescharacteristic
228C2×D12—64no—
229C2×D12—64no—
230C2×D12—62no—
231C2×D12—62no—
232C2×D12—122no—
233C2×D12—124no—
234C2×C4×D3—64no—
235C2×C4×D3—64no—
236C2×C4×D3—64no—
237C2×C4×D3—64no—
238C2×C4×D3—64no—
239C2×C4×D3—64no—
240C2×Dic6—22no—
241C2×Dic6—22no—
242C2×Dic6—22no—
243C3×C4⋊C4—22no—
244C3×C4⋊C4—22no—
245C3×C4⋊C4—42no—
246C3×C4⋊C4—62no—
247C3×C4⋊C4—62no—
248C3×C4⋊C4—62no—
249C3×C42—34no—
250C3×C42—62no—
251C3×C42—62no—
252(C2×D3)⋊C4—64no—
253(C2×D3)⋊C4—64no—
254(C2×D3)⋊C4—64no—
255(C2×D3)⋊C4—64no—
256(C2×D3)⋊C4—64no—
257(C2×D3)⋊C4—64no—
258(C2×D3)⋊C4—64no—
259(C2×D3)⋊C4—64no—
260(C2×D3)⋊C4—64no—
261(C2×D3)⋊C4—64no—
262(C2×D3)⋊C4—124no—
263(C2×D3)⋊C4—124no—
264C4⋊Dic3—22no—
265Dic3⋊C4—22no—
266Dic3⋊C4—22no—
267C4×Dic3—22no—
268C4×Dic3—22no—
269C4×Dic3—64no—
270C4×Dic3—64no—
271C4×Dic3#6714yes—
272C22×C32C22×D411yescharacteristic
273C22×C32C2×#4111yescharacteristic
274C22×C32C2×#4111yescharacteristic
275C22×C32C2×#4112yes—
276C22×C32C2×#4112yes—
277C2×C3⋊D3—24no—
278C2×C3⋊D3—24no—
279C2×C3⋊D3—24no—
280C2×C3⋊D3—42no—
281C2×C3⋊D3—42no—
282C2×C3⋊D3—42no—
283C2×C3⋊D3—42no—
284C32×C4—22no—
285C32×C4—22no—
286C32×C4—41no—
287C3⋊Dic3—42no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—41no—
291C3×Dic3—42no—
292C3×Dic3—42no—
293C3×Dic3#12414yes—
294C22×Q8—31no—
295C22×D4—91no—
296C23×C4—92no—
297C4×Q8—64no—
298C4×D4—184no—
299C2×C4⋊C4—31no—
300C2×C4⋊C4—31no—
301C2×C4⋊C4—31no—
302C2×C4⋊C4D3211yescharacteristic
303C2×C22⋊C4—92no—
304C2×C22⋊C4—92no—
305C2×C22⋊C4—92no—
306C2×C22⋊C4—92no—
307C2×C22⋊C4—92no—
308C2×C42—31no—
309C2×C42—31no—
310C2×C42—31no—
311C2×C42—92no—
312#95—31no—
313#95—32no—
314#95—32no—
315#95—92no—
316#95—92no—
317#95—92no—
318#95—91no—
319C23×C3—21no—
320C23×C3C23×D311yescharacteristic
321C23×C3C23×D311yescharacteristic
322C22×D3—34no—
323C22×D3—34no—
324C22×D3—64no—
325C22×D3—62no—
326C22×D3—62no—
327C22×D3—62no—
328C22×D3—62no—
329C22×D3—62no—
330C22×D3—62no—
331C22×D3—62no—
332C22×D3—62no—
333C22×D3—62no—
334C22×D3—62no—
335C22×D3—62no—
336C22×D3—62no—
337C22×D3—122no—
338C22×D3—122no—
339C22×D3—122no—
340C22×D3—122no—
341C22×D3—122no—
342C22×D3—122no—
343C3×Q8—68no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—22no—
347C2×C3×C4—22no—
348C2×C3×C4—21no—
349C2×C3×C4—21no—
350C2×C3×C4—21no—
351C2×C3×C4—22no—
352C2×C3×C4—22no—
353C2×C3×C4—21no—
354C2×C3×C4—22no—
355C2×C3×C4—22no—
356C2×C3×C4—32no—
357C2×C3×C4—34no—
358C2×C3×C4—32no—
359C2×C3×C4—32no—
360C2×C3×C4—41no—
361C2×C3×C4—42no—
362C2×C3×C4—41no—
363C2×C3×C4—42no—
364C2×C3×C4—41no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—62no—
370C2×C3×C4—61no—
371C2×C3×C4—62no—
372C2×C3×C4—62no—
373C2×C3×C4—62no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—62no—
377C2×C3×C4#23512yes—
378C2×C3×C4D3×D412yes—
379C2×C3×C4#23912yes—
380C2×C3×C4C2×#6712yes—
381C2×Dic3—21no—
382C2×Dic3—21no—
383C2×Dic3—21no—
384C2×Dic3—21no—
385C2×Dic3—22no—
386C2×Dic3—22no—
387C2×Dic3—22no—
388C2×Dic3—22no—
389C2×Dic3—22no—
390C2×Dic3—22no—
391C2×Dic3—22no—
392C2×Dic3—21no—
393C2×Dic3—62no—
394C2×Dic3—62no—
395C2×Dic3—62no—
396C2×Dic3—62no—
397C2×Dic3—62no—
398C2×Dic3—62no—
399C2×Dic3—122no—
400C2×Dic3—122no—
401C2×Dic3—122no—
402C2×Dic3#23512yes—
403C2×Dic3#23914yes—
404C2×Dic3C2×#6712yes—
405C2×Dic3C2×#6712yes—
406D12—68no—
407D12—68no—
408D12—128no—
409C4×D3—128no—
410C4×D3—128no—
411Dic6—28no—
412C2×C32#52511yescharacteristic
413C2×C32#52511yescharacteristic
414C2×C32C2×#12411yescharacteristic
415C2×C32#52912yes—
416C2×C32#52912yes—
417C3⋊D3—44no—
418C24—92no—
419C2×Q8—62no—
420C2×Q8—62no—
421C2×Q8—62no—
422C2×D4—184no—
423C2×D4—182no—
424C22×C4—31no—
425C22×C4—31no—
426C22×C4—31no—
427C22×C4—32no—
428C22×C4—32no—
429C22×C4—31no—
430C22×C4—92no—
431C22×C4—184no—
432C22×C4—184no—
433C22×C4—184no—
434C22×C4C2×D3211yescharacteristic
435C22×C4C2×D3211yescharacteristic
436C22×C4C2×D3211yescharacteristic
437C4⋊C4—22no—
438C4⋊C4—62no—
439C4⋊C4—62no—
440C4⋊C4—62no—
441C22⋊C4—184no—
442C22⋊C4—184no—
443C22⋊C4—184no—
444C22⋊C4—184no—
445C22⋊C4—184no—
446C42—34no—
447C42—62no—
448C42—62no—
449C42—184no—
450C22×C3—21no—
451C22×C3—21no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—21no—
455C22×C3C2×#23511yescharacteristic
456C22×C3C2×#23511yescharacteristic
457C22×C3C2×D3×D411yescharacteristic
458C22×C3C2×#23911yescharacteristic
459C22×C3C2×#23912yes—
460C22×C3D3×#4111yescharacteristic
461C22×C3D3×#4112yes—
462C22×C3D3×#4112yes—
463C22×C3D3×#4112yes—
464C22×C3C22×#6711yescharacteristic
465C2×D3—64no—
466C2×D3—64no—
467C2×D3—64no—
468C2×D3—64no—
469C2×D3—64no—
470C2×D3—64no—
471C2×D3—124no—
472C2×D3—124no—
473C2×D3—124no—
474C2×D3—122no—
475C2×D3—122no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—122no—
482C2×D3—242no—
483C2×D3—242no—
484C2×D3—242no—
485C2×D3—242no—
486C3×C4—22no—
487C3×C4—22no—
488C3×C4—22no—
489C3×C4—22no—
490C3×C4—34no—
491C3×C4—42no—
492C3×C4—42no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—62no—
496C3×C4—62no—
497C3×C4—81no—
498C3×C4—121no—
499C3×C4—122no—
500C3×C4—122no—
501Dic3—22no—
502Dic3—22no—
503Dic3—42no—
504Dic3—42no—
505Dic3—41no—
506Dic3—122no—
507Dic3—122no—
508Dic3—242no—
509Dic3#94714yes—
510C32#239811yescharacteristic, complemented, Sylow
511C23—94no—
512C23—182no—
513C23—182no—
514C23—182no—
515C23—182no—
516C23—182no—
517C23—182no—
518C23C22×D3211yescharacteristic
519Q8—68no—
520D4—188no—
521C2×C4—21no—
522C2×C4—22no—
523C2×C4—21no—
524C2×C4—22no—
525C2×C4—21no—
526C2×C4—34no—
527C2×C4—32no—
528C2×C4—32no—
529C2×C4—32no—
530C2×C4—61no—
531C2×C4—61no—
532C2×C4—61no—
533C2×C4—61no—
534C2×C4—62no—
535C2×C4—62no—
536C2×C4—62no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—62no—
540C2×C4—61no—
541C2×C4—62no—
542C2×C4—182no—
543C2×C4—182no—
544C2×C4—182no—
545C2×C4—368no—
546C2×C4#379912yes—
547C2×C4D3×#6712yes—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—22no—
551C2×C3—22no—
552C2×C3—21no—
553C2×C3#558012yes—
554C2×C3#558012yes—
555C2×C3#562711yescharacteristic
556C2×C3#565611yescharacteristic
557C2×C3D3×#12411yescharacteristic
558C2×C3#569112yes—
559C2×C3#570612yes—
560C2×C3#579111yescharacteristic
561C2×C3C2×#94711yescharacteristic
562C2×C3#582811yescharacteristic
563D3—124no—
564D3—124no—
565D3—244no—
566C22—184no—
567C22—184no—
568C22—184no—
569C22—362no—
570C22—362no—
571C22—362no—
572C22—362no—
573C22C2×#379911yescharacteristic
574C22D3×#23511yescharacteristic
575C22#6431212yes—
576C22D3×#23912yes—
577C22C2×D3×#6711yescharacteristic
578C4—22no—
579C4—22no—
580C4—34no—
581C4—41no—
582C4—62no—
583C4—62no—
584C4—121no—
585C4—122no—
586C4—122no—
587C4—362no—
588C3—21no—
589C3#7813111yescharacteristic, complemented
590C3#8249211yescharacteristic, complemented
591C2—364no—
592C2#1059442312yes—
593C2#1059446412yes—
594C2#1059500511yescharacteristic
595C2#1059515911yescharacteristic
596C2D3×#94711yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutator, socle, cosocleC23×C32C24
FittingC2×C32×C4⋊C4C22
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2398
LCSG ▷ C23×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#887C2semidirect
4. C3×#4986C2semidirect
6. #10591310C2semidirect
7. #10591239C2semidirect
8. #10591233C2semidirect
9. #10591233C2semidirect
10. #10591225C2semidirect
11. #10591204C2semidirect
41. C2×#887D3semidirect
51. #4986D3semidirect
510. C32#2398semidirect
589. C3#78131semidirect
590. C3#82492semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#8872
C2×#638942
#105926902
C3×#49862
#105913212
#105913102
#105912392
#105912332
#105912252
#105912042
#824923
#781313