Finite Group Databasebeta

← #11788347

Subgroups — #11788347

Order 1152.

725 conjugacy classes of subgroups.

The subgroup lattice has 581 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×M16C211yesmaximal, characteristic
2C2×#63856C211yesmaximal, characteristic
3C2×C3×#824C211yesmaximal, characteristic, complemented
4C2×#63560C211yesmaximal, characteristic, complemented
5#10591069C211yesmaximal, characteristic
6#10589420C211yesmaximal, characteristic, complemented
7#10589168C211yesmaximal, characteristic, complemented
8#72003—31nomaximal
9#70115—31nomaximal
10C22×C3×C4×D3C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×C4⋊(C3⋊Dic3)C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×D3×M16—22no—
18C3×D3×M16—22no—
19C2×C3×ΓL(1,𝔽25)—21no—
20C2×C3×D3×C8—21no—
21C2×#3724C412yes—
22C2×(C2×D3)⋊Dic3C412yes—
23#63856—21no—
24#63856—21no—
25#63856—21no—
26#63856—21no—
27C3×#824—22no—
28#63560—22no—
29(C2×D3)⋊(C3⋊C8)—21no—
30(C2×D3)⋊(C3⋊C8)—21no—
31C22×C3×M16—31no—
32C2×#946—31no—
33C2×#924—31no—
34C2×D3×M16D311yescharacteristic
35C2×C3×#101—31no—
36C2×#836—31no—
37#5207—31no—
38C2×#824—31no—
39C2×#824D311yescharacteristic, complemented
40#5126—31no—
41#4554—31no—
42#4554—31no—
43#4554—31no—
44#4554—31no—
45#4554—61no—
46C23×C3×D3C2×C411yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C22×C3⋊Dic3C2×C412yes—
49C22×C3×Dic3C2×C412yes—
50C22×C3×Dic3C2×C411yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C3×C4×D3—21no—
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—21no—
55C2×C3×C4×D3D411yescharacteristic
56C2×C3×C4×D3D411yescharacteristic
57C2×C3×C4×D3D411yescharacteristic
58C2×C3×C4×D3D411yescharacteristic
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C32×M16—21no—
66C32×M16D412yes—
67C2×C32×C8—21no—
68C4⋊(C3⋊Dic3)—21no—
69C4⋊(C3⋊Dic3)—21no—
70#3748—21no—
71#3748—21no—
72C2×C3⋊(C3⋊C8)—21no—
73C3×#208—21no—
74C3×#208—21no—
75C3×#208—21no—
76C3×#208D412yes—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C3×ΓL(1,𝔽25)—24no—
80C3×D3×C8—24no—
81#3724—22no—
82#3724—22no—
83(C2×D3)⋊Dic3—22no—
84(C2×D3)⋊Dic3—22no—
85#1901—91noSylow
86C22×C3×Q8—31no—
87C23×C3×C4—31no—
88C22×C4×D3C2×D311yescharacteristic
89C22×Dic6C2×D311yescharacteristic
90C2×C3×M16—21no—
91C2×C3×M16—31no—
92C2×C3×M16—31no—
93C2×C3×M16—61no—
94C2×C3×M16—62no—
95C2×C3×M16—62no—
96C2×C3×M16C2×D311yescharacteristic
97C2×C3×M16C2×D311yescharacteristic
98C22×C3×C8—61no—
99C2×#217—32no—
100#946—61no—
101#946—61no—
102#946—61no—
103#946—61no—
104C2×(C2×D3)⋊C4—32no—
105C2×C4⋊Dic3—31no—
106C2×C4⋊Dic3—31no—
107C2×C4⋊Dic3—61no—
108#924—61no—
109#924—61no—
110#924—61no—
111#924—61no—
112C2×Dic3⋊C4—32no—
113C2×Dic3⋊C4—32no—
114C2×#208—31no—
115C2×#208—31no—
116C2×#208—61no—
117C2×#208C2×D311yescharacteristic
118C2×#208C2×D311yescharacteristic
119D3×M16—22no—
120D3×M16—22no—
121C2×ΓL(1,𝔽25)—21no—
122C2×D3×C8—21no—
123C3×#101—62no—
124#836—62no—
125#830—61no—
126#830—61no—
127#824—22no—
128#824—62no—
129(C2×D3)⋊C8—61no—
130(C2×D3)⋊C8—61no—
131C23×C32C22×C411yescharacteristic
132C22×C3×D3—21no—
133C22×C3×D3—21no—
134C22×C3×D3—22no—
135C22×C3×D3—41no—
136C22×C3×D3C22⋊C411yescharacteristic
137C22×C3×D3C22⋊C411yescharacteristic
138C2×C32×C4C2×D411yescharacteristic
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4#4111yescharacteristic
143C2×C32×C4#4111yescharacteristic
144C2×C3⋊Dic3—22no—
145C2×C3⋊Dic3—42no—
146C2×C3×Dic3—21no—
147C2×C3×Dic3—21no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3C22⋊C411yescharacteristic
152C2×C3×Dic3C22⋊C411yescharacteristic
153C3×C4×D3—21no—
154C3×C4×D3—21no—
155C3×C4×D3—21no—
156C3×C4×D3—21no—
157C3×C4×D3—21no—
158C3×C4×D3—21no—
159C3×C4×D3—21no—
160C3×C4×D3—21no—
161C3×Dic6—41no—
162C3×Dic6—41no—
163C3×Dic6—41no—
164C3×Dic6—41no—
165C32×C8—22no—
166C3⋊(C3⋊C8)—41no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—41no—
169C22×M16—31no—
170C2×C22⋊Q8—91no—
171C2×#101—31no—
172C2×#101—91no—
173#407—91no—
174#343—91no—
175#343—91no—
176C24×C3—31no—
177C23×D3C4×D311yescharacteristic
178C2×C3×Q8—61no—
179C2×C3×Q8—61no—
180C2×C3×Q8—61no—
181C2×C3×Q8—61no—
182C2×C3×Q8—61no—
183C2×C3×Q8—61no—
184C22×C3×C4—21no—
185C22×C3×C4—31no—
186C22×C3×C4—31no—
187C22×C3×C4—31no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—32no—
191C22×C3×C4—61no—
192C22×C3×C4—61no—
193C22×C3×C4—61no—
194C22×C3×C4—61no—
195C22×C3×C4C22×D311yescharacteristic
196C22×C3×C4C22×D311yescharacteristic
197C22×Dic3—32no—
198C22×Dic3—32no—
199C22×Dic3—62no—
200C22×Dic3C4×D312yes—
201C22×Dic3C4×D311yescharacteristic
202C2×C4×D3—21no—
203C2×C4×D3—21no—
204C2×C4×D3—21no—
205C2×C4×D3—21no—
206C2×C4×D3D1211yescharacteristic
207C2×C4×D3D1211yescharacteristic
208C2×C4×D3#6711yescharacteristic
209C2×C4×D3#6711yescharacteristic
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—21no—
213C2×Dic6—21no—
214C2×Dic6—21no—
215C2×Dic6—21no—
216C3×M16—21no—
217C3×M16—21no—
218C3×M16—22no—
219C3×M16—32no—
220C3×M16—41no—
221C3×M16—64no—
222C3×M16—61no—
223C3×M16—61no—
224C3×M16—61no—
225C3×M16#6712yes—
226C3×M16#6712yes—
227C2×C3×C8—21no—
228C2×C3×C8—21no—
229C2×C3×C8—41no—
230C2×C3×C8—64no—
231C2×C3×C8—61no—
232C2×C3×C8—61no—
233#217—62no—
234#217—62no—
235(C2×D3)⋊C4—62no—
236(C2×D3)⋊C4—62no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—121no—
242C4⋊Dic3—121no—
243Dic3⋊C4—62no—
244Dic3⋊C4—62no—
245Dic3⋊C4—62no—
246Dic3⋊C4—62no—
247#208—21no—
248#208—21no—
249#208—21no—
250#208—62no—
251#208—61no—
252#208—61no—
253#208—61no—
254#208—61no—
255#208—121no—
256#208#6712yes—
257C2×C3⋊C8—21no—
258C2×C3⋊C8—21no—
259C2×C3⋊C8—61no—
260C2×C3⋊C8—61no—
261C2×C3⋊C8—121no—
262ΓL(1,𝔽25)—24no—
263D3×C8—24no—
264C22×C32—21no—
265C22×C32—21no—
266C22×C32C2×C22⋊C411yescharacteristic
267C22×C32C4×D411yescharacteristic
268C22×C32C4×D411yescharacteristic
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3—42no—
275C2×C3×D3—42no—
276C2×C3×D3#10112yescomplemented
277C32×C4#12011yescharacteristic
278C32×C4#12111yescharacteristic
279C32×C4#12111yescharacteristic
280C32×C4#12311yescharacteristic
281C3⋊Dic3—42no—
282C3×Dic3—21no—
283C3×Dic3—21no—
284C3×Dic3—42no—
285C22×Q8—31no—
286C23×C4—31no—
287C2×M16—31no—
288C2×M16—31no—
289C2×M16—61no—
290C2×M16—62no—
291C2×M16—62no—
292C2×M16—91no—
293C2×M16D3211yescharacteristic
294C22×C8—61no—
295C22⋊Q8—181no—
296C22⋊Q8—181no—
297C22⋊Q8—181no—
298C22⋊Q8—181no—
299C2×C4⋊C4—92no—
300C2×C4⋊C4—91no—
301C2×C22⋊C4—92no—
302#101—62no—
303#101—182no—
304C22⋊C8—181no—
305C22⋊C8—181no—
306C23×C3—21no—
307C23×C3—31no—
308C23×C3—31no—
309C23×C3—62no—
310C23×C3—61no—
311C23×C3—61no—
312C23×C3—121no—
313C23×C3C2×C4×D311yescharacteristic
314C23×C3C2×C4×D311yescharacteristic
315C22×D3—21no—
316C22×D3—21no—
317C22×D3—22no—
318C22×D3—41no—
319C22×D3(C2×D3)⋊C411yescharacteristic
320C22×D3(C2×D3)⋊C411yescharacteristic
321C3×Q8—121no—
322C3×Q8—121no—
323C3×Q8—121no—
324C3×Q8—121no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—62no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4C2×D1211yescharacteristic
347C2×C3×C4#23511yescharacteristic
348C2×C3×C4#23511yescharacteristic
349C2×C3×C4#23511yescharacteristic
350C2×C3×C4D3×D411yescharacteristic
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4#23711yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C2×#6711yescharacteristic
356C2×C3×C4C2×#6711yescharacteristic
357C2×C3×C4C2×#6711yescharacteristic
358C2×Dic3—21no—
359C2×Dic3—21no—
360C2×Dic3—22no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—62no—
364C2×Dic3—62no—
365C2×Dic3—122no—
366C2×Dic3—122no—
367C2×Dic3—122no—
368C2×Dic3—242no—
369C2×Dic3(C2×D3)⋊C411yescharacteristic
370C2×Dic3(C2×D3)⋊C411yescharacteristic
371C4×D3—21no—
372C4×D3—21no—
373C4×D3—21no—
374C4×D3—21no—
375C4×D3—21no—
376C4×D3—21no—
377C4×D3—21no—
378C4×D3—21no—
379Dic6—41no—
380Dic6—41no—
381Dic6—41no—
382Dic6—41no—
383C3×C8—22no—
384C3×C8—22no—
385C3×C8—42no—
386C3×C8—62no—
387C3×C8—121no—
388C3⋊C8—22no—
389C3⋊C8—41no—
390C3⋊C8—122no—
391C3⋊C8—121no—
392C3⋊C8—121no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32#38611yescharacteristic
396C2×C32C2×#10111yescharacteristic
397C2×C32#41311yescharacteristic
398C3×D3—22no—
399C3×D3—41no—
400C24—31no—
401C2×Q8—61no—
402C2×Q8—61no—
403C2×Q8—61no—
404C2×Q8—61no—
405C2×Q8—61no—
406C2×Q8—61no—
407C22×C4—31no—
408C22×C4—31no—
409C22×C4—31no—
410C22×C4—31no—
411C22×C4—31no—
412C22×C4—32no—
413C22×C4—61no—
414C22×C4—61no—
415C22×C4—61no—
416C22×C4—61no—
417C22×C4—92no—
418C22×C4C2×D3211yescharacteristic
419M16—21no—
420M16—32no—
421M16—64no—
422M16—61no—
423M16—61no—
424M16—61no—
425M16—181no—
426M16—181no—
427M16#62512yes—
428C2×C8—21no—
429C2×C8—64no—
430C2×C8—61no—
431C2×C8—61no—
432C2×C8—181no—
433C4⋊C4—182no—
434C4⋊C4—182no—
435C4⋊C4—181no—
436C4⋊C4—181no—
437C22⋊C4—182no—
438C22⋊C4—182no—
439C22×C3—21no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—32no—
447C22×C3—41no—
448C22×C3—41no—
449C22×C3—61no—
450C22×C3—61no—
451C22×C3—61no—
452C22×C3—61no—
453C22×C3—61no—
454C22×C3—122no—
455C22×C3—122no—
456C22×C3D3×C22⋊C411yescharacteristic
457C22×C3#67⋊C411yescharacteristic
458C22×C3C2×(C2×D3)⋊C411yescharacteristic
459C22×C3C4×#6711yescharacteristic
460C22×C3C4×#6711yescharacteristic
461C22×C3C4×#6711yescharacteristic
462C2×D3—21no—
463C2×D3—21no—
464C2×D3—21no—
465C2×D3—21no—
466C2×D3—21no—
467C2×D3—42no—
468C2×D3—42no—
469C2×D3#82412yescomplemented
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—61no—
475C3×C4—61no—
476C3×C4—122no—
477C3×C4#88311yescharacteristic
478C3×C4#88411yescharacteristic
479C3×C4#92911yescharacteristic
480C3×C4#93011yescharacteristic
481C3×C4#93011yescharacteristic
482C3×C4#93711yescharacteristic
483C3×C4#93811yescharacteristic
484C3×C4#95311yescharacteristic
485Dic3—21no—
486Dic3—21no—
487Dic3—42no—
488Dic3—122no—
489Dic3—122no—
490Dic3—242no—
491C32#190111yescharacteristic, complemented, Sylow
492C23—31no—
493C23—31no—
494C23—62no—
495C23—61no—
496C23—61no—
497C23—121no—
498C23C4×D3211yescharacteristic
499Q8—121no—
500Q8—121no—
501Q8—121no—
502Q8—121no—
503C2×C4—31no—
504C2×C4—31no—
505C2×C4—61no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—61no—
509C2×C4—61no—
510C2×C4—61no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—61no—
515C2×C4—62no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—182no—
519C2×C4—362no—
520C2×C4#379511yescharacteristic
521C2×C4#379811yescharacteristic
522C2×C4D3×D1211yescharacteristic
523C2×C4C2×#62511yescharacteristic
524C2×C4D3×#6711yescharacteristic
525C2×C4D3×#6711yescharacteristic
526C8—22no—
527C8—62no—
528C8—121no—
529C8—361no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—62no—
540C2×C3—121no—
541C2×C3D3×#10111yescharacteristic
542C2×C3#475111yescharacteristic
543C2×C3#495611yescharacteristic
544C2×C3#513211yescharacteristic
545C2×C3C2×#82411yescharacteristic
546C2×C3#521311yescharacteristic
547D3—22no—
548D3—41no—
549C22—21no—
550C22—21no—
551C22—32no—
552C22—61no—
553C22—61no—
554C22—61no—
555C22—61no—
556C22—61no—
557C22—122no—
558C22—122no—
559C22#6387911yescharacteristic
560C22C4×#62511yescharacteristic
561C22D3×(C2×D3)⋊C411yescharacteristic
562C4—61no—
563C4—61no—
564C4—122no—
565C4—362no—
566C4#6390511yescharacteristic
567C4#6390611yescharacteristic
568C4#6390811yescharacteristic
569C4#6397611yescharacteristic
570C3—21no—
571C3#7011511yescharacteristic, complemented
572C3#7200311yescharacteristic, complemented
573C2—21no—
574C2—21no—
575C2—62no—
576C2—121no—
577C2D3×#82411yescharacteristic
578C2#1059100711yescharacteristic
579C2#1059138111yescharacteristic
subgroupG/H
centerC22D3×(C2×D3)⋊C4
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C2×C22⋊C4
prime radicalC23×C3×D3C2×C4
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1901
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C2×C3×#824C2semidirect
4. C2×#63560C2semidirect
6. #10589420C2semidirect
7. #10589168C2semidirect
39. C2×#824D3semidirect
276. C2×C3×D3#101semidirect
469. C2×D3#824semidirect
491. C32#1901semidirect
571. C3#70115semidirect
572. C3#72003semidirect

Maximal subgroups

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

subgroupindex
C2×C3×D3×M162
C2×#638562
C2×C3×#8242
C2×#635602
#105910692
#105894202
#105891682
#720033
#701153