Finite Group Databasebeta

← #11792744

Subgroups — #11792744

Order 1152.

797 conjugacy classes of subgroups.

The subgroup lattice has 561 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63831C211yesmaximal, characteristic, complemented
2C22×#3716C211yesmaximal, characteristic, complemented
3C2×#63823C211yesmaximal, characteristic
4C2×#63651C211yesmaximal, characteristic, complemented
5C2×C3×#823C211yesmaximal, characteristic, complemented
6C3×#5129C211yesmaximal, characteristic, complemented
7#10589166C211yesmaximal, characteristic, complemented
8#70091—31nomaximal
9#70007—31nomaximal
10C22×#634C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C22×C3×D12C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3719—21no—
18#63831—22no—
19#63831—22no—
20C2×#3716—22no—
21C2×#3716—22no—
22C2×#3716—21no—
23C2×#3716—22no—
24C2×#3715—21no—
25#63823—22no—
26#63823—22no—
27C2×#3713—21no—
28#63651—22no—
29C3×#823—22no—
30C3×(C2×D3)⋊C8—22no—
31C2×#951—31no—
32C2×#949—31no—
33C22×#214—31no—
34C2×#932—31no—
35C2×#909—31no—
36C22×#204—31no—
37C2×C3×#100—31no—
38C3×#408—31no—
39C2×#824—31no—
40C2×#824—31no—
41C2×#824—61no—
42C2×#823D311yescharacteristic, complemented
43#5129D311yescharacteristic, complemented
44#4558—31no—
45#4552—31no—
46C23×C3×D3—21no—
47C22×C32×C4C2311yescharacteristic
48C22×C3⋊Dic3—21no—
49C2×#634—21no—
50C2×#634—21no—
51C2×#634—21no—
52C2×#634—21no—
53C2×#634—21no—
54C2×#634D412yes—
55C2×C3×#67—21no—
56C3×#235—21no—
57C3×#235—21no—
58C3×#235—21no—
59C3×#235—21no—
60C2×C3×D12—21no—
61C2×C3×D12—21no—
62C2×C3×D12—21no—
63C2×C3×D12—21no—
64C2×C3×D12—21no—
65C2×C3×D12D411yescharacteristic
66C2×C3×D12D412yes—
67C2×C3×C4×D3—21no—
68C2×C3×Dic6D411yescharacteristic
69C32×M16—21no—
70C32×M16—21no—
71C2×C32×C8—21no—
72#3748—21no—
73#3748D412yes—
74C2×C3⋊(C3⋊C8)—21no—
75C3×#208—21no—
76C3×#208D412yes—
77C2×C3×C3⋊C8—21no—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8D412yes—
81#3719—24no—
82#3716—28no—
83#3716—24no—
84#3715—24no—
85#3713—24no—
86#2019—91noSylow
87C2×C3×#41—31no—
88C22×C3×D4—31no—
89C2×#235C2×D311yescharacteristic
90C22×D12C2×D311yescharacteristic
91C22×Dic6—31no—
92C22×Dic6—31no—
93C22×Dic6—61no—
94C2×C3×M16—21no—
95C2×C3×M16—31no—
96C2×C3×M16C2×D311yescharacteristic
97C2×C3×M16C2×D311yescharacteristic
98C22×C3×C8—31no—
99#951—62no—
100#951—62no—
101#949—62no—
102#949—62no—
103C2×#216—61no—
104C2×#215—61no—
105C2×#214—62no—
106C2×#214—62no—
107C2×#214—61no—
108C2×#214—61no—
109C2×#214—62no—
110#932—62no—
111#932—62no—
112C2×#213—61no—
113C2×#213—61no—
114C2×#208—31no—
115C2×#208—31no—
116C2×#208—61no—
117C2×#208C2×D311yescharacteristic
118C22×C3⋊C8C2×D311yescharacteristic
119#909—62no—
120#909—62no—
121C2×Dic12—61no—
122C2×#204—62no—
123C2×#204—62no—
124C2×#204—62no—
125C2×#204—61no—
126C3×#100—62no—
127C3×C22⋊C8—62no—
128#824—62no—
129#824—62no—
130#824—64no—
131#823—22no—
132(C2×D3)⋊C8—22no—
133C23×C32#4111yescharacteristic
134C22×C3×D3—21no—
135C22×C3×D3—22no—
136C22×C3×D3—41no—
137C22×C3×D3—41no—
138C22×C3×D3—82no—
139C2×C32×C4C2×D411yescharacteristic
140C2×C32×C4C2×D411yescharacteristic
141C2×C32×C4C2×D411yescharacteristic
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C3⋊Dic3—22no—
146C2×C3⋊Dic3—41no—
147C2×C3⋊Dic3—41no—
148#634—22no—
149#634—22no—
150#634—41no—
151#634—41no—
152C3×#67—41no—
153C3×#67—41no—
154C2×C3×Dic3—21no—
155C3×D12—21no—
156C3×D12—22no—
157C3×D12—22no—
158C3×D12—21no—
159C3×D12—41no—
160C3×D12—41no—
161C3×C4×D3—41no—
162C3×C4×D3—41no—
163C3×Dic6—21no—
164C3×Dic6—21no—
165C32×C8—41no—
166C3⋊(C3⋊C8)—22no—
167C3×C3⋊C8—22no—
168C3×C3⋊C8—22no—
169C2×#137—91no—
170C2×AGL(1,ℤ/8)—91no—
171C22×SD16—91no—
172C2×#101—91no—
173C2×#100—31no—
174#408—31no—
175#341—91no—
176C24×C3—61no—
177C23×D3—21no—
178C3×#41—61no—
179C3×#41—61no—
180C3×#41—61no—
181C3×#41—61no—
182C2×C3×Q8—31no—
183C2×C3×D4—31no—
184C2×C3×D4—32no—
185C2×C3×D4—61no—
186C2×C3×D4—61no—
187C2×C3×D4—61no—
188C2×C3×D4—61no—
189C2×C3×D4—61no—
190C2×C3×D4—61no—
191C22×C3×C4—21no—
192C22×C3×C4—61no—
193C22×C3×C4C22×D311yescharacteristic
194C22×C3×C4C22×D311yescharacteristic
195C2×#67—21no—
196C22×Dic3—62no—
197C22×Dic3—61no—
198C22×Dic3—61no—
199#235—21no—
200#235—21no—
201#235—21no—
202#235—21no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—21no—
206C2×D12—21no—
207C2×D12—21no—
208C2×D12D1211yescharacteristic
209C2×D12D1212yes—
210C2×C4×D3—21no—
211C2×Dic6—32no—
212C2×Dic6—32no—
213C2×Dic6—62no—
214C2×Dic6—61no—
215C2×Dic6—61no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×Dic6—61no—
219C2×Dic6—61no—
220C2×Dic6—61no—
221C2×Dic6—61no—
222C2×Dic6—62no—
223C2×Dic6—61no—
224C2×Dic6—61no—
225C2×Dic6—121no—
226C2×Dic6—121no—
227C2×Dic6—121no—
228C2×Dic6—121no—
229C2×Dic6D1211yescharacteristic
230C3×M16—22no—
231C3×M16—21no—
232C3×M16—21no—
233C3×M16—21no—
234C3×M16—21no—
235C3×M16—32no—
236C3×M16—41no—
237C3×M16—61no—
238C2×C3×C8—21no—
239C2×C3×C8—21no—
240C2×C3×C8—32no—
241C2×C3×C8—41no—
242C2×C3×C8—61no—
243C2×C3×C8—61no—
244C2×C3×C8—61no—
245#216—64no—
246#215—64no—
247#214—68no—
248#214—64no—
249#214—64no—
250#213—64no—
251#213—64no—
252#208—21no—
253#208—32no—
254#208—32no—
255#208—62no—
256#208—61no—
257#208—61no—
258#208—121no—
259#208#6712yes—
260C2×C3⋊C8—21no—
261C2×C3⋊C8—21no—
262C2×C3⋊C8—21no—
263C2×C3⋊C8—61no—
264C2×C3⋊C8—61no—
265C2×C3⋊C8—121no—
266C2×C3⋊C8#6712yes—
267Dic12—64no—
268#204—68no—
269#204—64no—
270C22×C32—22no—
271C22×C32#12111yescharacteristic
272C22×C32#12111yescharacteristic
273C22×C32#12111yescharacteristic
274C2×C3×D3—41no—
275C2×C3×D3—42no—
276C2×C3×D3—42no—
277C2×C3×D3—82no—
278C2×C3×D3—82no—
279C2×C3×D3—82no—
280C32×C4#12011yescharacteristic
281C32×C4#12011yescharacteristic
282C32×C4#12011yescharacteristic
283C32×C4#12711yescharacteristic
284C3⋊Dic3—42no—
285C3×Dic3—41no—
286C2×#41—31no—
287C22×Q8—91no—
288C22×D4—31no—
289#137—182no—
290#137—182no—
291AGL(1,ℤ/8)—182no—
292AGL(1,ℤ/8)—182no—
293C2×Q16—181no—
294C2×SD16—182no—
295C2×SD16—182no—
296C2×SD16—181no—
297C2×SD16—182no—
298C2×SD16—181no—
299C2×D8—181no—
300C2×M16—31no—
301C2×M16—91no—
302C2×M16D3211yescharacteristic
303C22×C8—31no—
304#101—182no—
305#100—62no—
306C22⋊C8—62no—
307C23×C3—21no—
308C23×C3—61no—
309C23×C3—62no—
310C23×C3—121no—
311C23×C3—121no—
312C23×C3—242no—
313C23×C3#23511yescharacteristic
314C23×C3#23511yescharacteristic
315C22×D3—21no—
316C22×D3—22no—
317C22×D3—41no—
318C22×D3—41no—
319C22×D3—82no—
320C3×Q8—61no—
321C3×Q8—61no—
322C3×D4—61no—
323C3×D4—62no—
324C3×D4—62no—
325C3×D4—61no—
326C3×D4—121no—
327C3×D4—121no—
328C3×D4—121no—
329C3×D4—121no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—61no—
337C2×C3×C4—121no—
338C2×C3×C4—121no—
339C2×C3×C4C2×D1211yescharacteristic
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4D3×D411yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C2×#6711yescharacteristic
351#67—41no—
352#67—41no—
353C2×Dic3—21no—
354C2×Dic3—64no—
355C2×Dic3—62no—
356C2×Dic3—62no—
357C2×Dic3—122no—
358C2×Dic3—122no—
359C2×Dic3—121no—
360C2×Dic3—121no—
361C2×Dic3—121no—
362C2×Dic3—121no—
363D12—22no—
364D12—21no—
365D12—21no—
366D12—22no—
367D12—41no—
368D12—41no—
369C4×D3—41no—
370C4×D3—41no—
371Dic6—21no—
372Dic6—21no—
373Dic6—62no—
374Dic6—62no—
375Dic6—62no—
376Dic6—62no—
377Dic6—122no—
378Dic6—122no—
379Dic6—122no—
380Dic6—122no—
381Dic6—121no—
382Dic6—121no—
383Dic6—121no—
384Dic6—121no—
385C3×C8—42no—
386C3×C8—41no—
387C3×C8—41no—
388C3×C8—62no—
389C3×C8—62no—
390C3⋊C8—22no—
391C3⋊C8—22no—
392C3⋊C8—62no—
393C3⋊C8—62no—
394C3⋊C8—122no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32#39211yescharacteristic
398C2×C32#47112yes—
399C3×D3—81no—
400C3×D3—82no—
401C24—61no—
402#41—61no—
403#41—61no—
404#41—61no—
405#41—61no—
406C2×Q8—31no—
407C2×Q8—92no—
408C2×Q8—181no—
409C2×Q8—181no—
410C2×Q8—181no—
411C2×Q8—181no—
412C2×Q8—181no—
413C2×D4—31no—
414C2×D4—32no—
415C2×D4—61no—
416C2×D4—61no—
417C2×D4—61no—
418C2×D4—61no—
419C2×D4—61no—
420C2×D4—61no—
421C22×C4—61no—
422C22×C4—181no—
423C22×C4C2×D3211yescharacteristic
424Q16—184no—
425SD16—188no—
426SD16—184no—
427SD16—184no—
428D8—184no—
429M16—21no—
430M16—21no—
431M16—32no—
432M16—61no—
433M16—92no—
434M16—181no—
435C2×C8—21no—
436C2×C8—32no—
437C2×C8—61no—
438C2×C8—61no—
439C2×C8—61no—
440C2×C8—181no—
441C22×C3—22no—
442C22×C3—22no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—42no—
447C22×C3—121no—
448C22×C3—122no—
449C22×C3—122no—
450C22×C3—242no—
451C22×C3—242no—
452C22×C3—242no—
453C22×C3#88411yescharacteristic
454C22×C3#88411yescharacteristic
455C22×C3#88411yescharacteristic
456C22×C3#88411yescharacteristic
457C22×C3#93011yescharacteristic
458C22×C3#93011yescharacteristic
459C2×D3—42no—
460C2×D3—41no—
461C2×D3—42no—
462C2×D3—82no—
463C2×D3—82no—
464C2×D3—82no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—121no—
470C3×C4#88211yescharacteristic
471C3×C4#88211yescharacteristic
472C3×C4#88211yescharacteristic
473C3×C4#88211yescharacteristic
474C3×C4#93711yescharacteristic
475C3×C4#93711yescharacteristic
476C3×C4#94011yescharacteristic
477C3×C4#94011yescharacteristic
478Dic3—41no—
479Dic3—124no—
480Dic3—122no—
481Dic3—122no—
482C32#201911yescharacteristic, complemented, Sylow
483C23—61no—
484C23—62no—
485C23—121no—
486C23—121no—
487C23—242no—
488C23#379811yescharacteristic
489Q8—61no—
490Q8—61no—
491Q8—182no—
492Q8—182no—
493Q8—361no—
494Q8—361no—
495D4—62no—
496D4—61no—
497D4—61no—
498D4—62no—
499D4—121no—
500D4—121no—
501D4—121no—
502D4—121no—
503C2×C4—61no—
504C2×C4—121no—
505C2×C4—121no—
506C2×C4—182no—
507C2×C4—361no—
508C2×C4—361no—
509C2×C4D3×D1211yescharacteristic
510C2×C4D3×D1211yescharacteristic
511C2×C4#380211yescharacteristic
512C2×C4D3×#6711yescharacteristic
513C2×C4D3×#6711yescharacteristic
514C2×C4#381111yescharacteristic
515C8—41no—
516C8—62no—
517C8—62no—
518C8—182no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—22no—
524C2×C3—21no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—242no—
528C2×C3—241no—
529C2×C3#474812yes—
530C2×C3#496011yescharacteristic
531C2×C3#496011yescharacteristic
532C2×C3#514112yes—
533D3—81no—
534D3—82no—
535C22—22no—
536C22—122no—
537C22—121no—
538C22—122no—
539C22—242no—
540C22—242no—
541C22—242no—
542C22#6390611yescharacteristic
543C22#6390611yescharacteristic
544C22#6391211yescharacteristic
545C4—121no—
546C4—362no—
547C4#6392211yescharacteristic
548C4#6392211yescharacteristic
549C4#6392311yescharacteristic
550C4#6397911yescharacteristic
551C3—21no—
552C3#7000711yescharacteristic, complemented
553C3#7009111yescharacteristic, complemented
554C2—21no—
555C2—21no—
556C2—241no—
557C2—242no—
558C2#1059099612yes—
559C2#1059139311yescharacteristic
subgroupG/H
centerC22#63906
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#63823C2
hypercenterC2×M16D32
hypocenterC32#2019
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C2×#63831C2semidirect
2. C22×#3716C2semidirect
4. C2×#63651C2semidirect
5. C2×C3×#823C2semidirect
6. C3×#5129C2semidirect
7. #10589166C2semidirect
42. C2×#823D3semidirect
43. #5129D3semidirect
482. C32#2019semidirect
552. C3#70007semidirect
553. C3#70091semidirect

Maximal subgroups

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

subgroupindex
C2×#638312
C22×#37162
C2×#638232
C2×#636512
C2×C3×#8232
C3×#51292
#105891662
#700913
#700073