Finite Group Databasebeta

← #11792740

Subgroups — #11792740

Order 1152.

873 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64117C211yesmaximal, characteristic
2C2×#63833C211yesmaximal, characteristic, complemented
3C22×#3717C211yesmaximal, characteristic, complemented
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C2×#63559C211yesmaximal, characteristic, complemented
6#10591073C211yesmaximal, characteristic, complemented
7C3×#4558C211yesmaximal, characteristic, complemented
8#70117—31nomaximal
9#68943—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×C3×Dic6C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14#64117—22no—
15#64117—22no—
16C2×#3745—21no—
17C2×#3744—21no—
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20C22×C3×C3⋊C8C2211yescharacteristic
21C2×#3719—21no—
22#63833—22no—
23#63833—22no—
24C2×#3717—22no—
25C2×#3717—21no—
26C2×#3717—22no—
27C2×#3717—22no—
28C3×#824—22no—
29#63559—22no—
30#63557—22no—
31C2×#942—31no—
32C22×#215—31no—
33C2×#909—31no—
34C2×#908—31no—
35C2×#908—31no—
36C2×#908—61no—
37C22×#204—31no—
38C2×C3×#101—31no—
39C2×#824D311yescharacteristic, complemented
40C2×#823—31no—
41C2×#823—31no—
42#5129—31no—
43#5110—31no—
44C3×#341—31no—
45#4558D311yescharacteristic, complemented
46C23×C3⋊D3—21no—
47C22×C32×C4C2311yescharacteristic
48C2×#638—21no—
49#3828—21no—
50#3828—21no—
51#3828—21no—
52#3828—21no—
53C2×C4⋊(C3⋊D3)—21no—
54C2×C4⋊(C3⋊D3)—21no—
55C2×C4⋊(C3⋊D3)—21no—
56C2×C4⋊(C3⋊D3)—21no—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)D412yes—
59C2×C4⋊(C3⋊D3)D411yescharacteristic
60C2×C4×C3⋊D3—21no—
61C2×#634D411yescharacteristic
62C22×C3×Dic3—21no—
63C2×C3×Dic6—21no—
64C2×C3×Dic6—21no—
65C2×C3×Dic6—21no—
66C2×C3×Dic6—21no—
67C2×C3×Dic6—21no—
68C2×C3×Dic6D412yes—
69C32×M16—21no—
70C32×M16D412yes—
71C2×C32×C8—21no—
72#3745—24no—
73#3744—24no—
74C3×#208—21no—
75C3×#208—21no—
76C3×#208—21no—
77C3×#208D412yes—
78C2×C3×C3⋊C8—21no—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8—21no—
82C2×C3×C3⋊C8D412yes—
83#3719—24no—
84#3717—28no—
85#3717—24no—
86#2019—91noSylow
87C22×C3×Q8—31no—
88C2×#235—31no—
89C2×#235—31no—
90C2×#235—61no—
91C22×D12—31no—
92C22×D12—31no—
93C22×D12—61no—
94C22×Dic6C2×D311yescharacteristic
95C2×C3×M16—21no—
96C2×C3×M16—31no—
97C2×C3×M16—31no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C2×D311yescharacteristic
100C22×C3×C8—31no—
101C2×#216—61no—
102#942—62no—
103#942—62no—
104C2×#215—62no—
105C2×#215—61no—
106C2×#215—62no—
107C2×#215—62no—
108C2×#208C2×D311yescharacteristic
109C2×#208C2×D311yescharacteristic
110C22×C3⋊C8C2×D311yescharacteristic
111#909—62no—
112#909—62no—
113#908—62no—
114#908—62no—
115#908—64no—
116#908—64no—
117#908—62no—
118#908—62no—
119C2×Dic12—61no—
120C2×D24—61no—
121C2×D24—61no—
122C2×D24—62no—
123C2×#204—61no—
124C2×#204—61no—
125C2×#204—62no—
126C2×#204—62no—
127C2×#204—62no—
128C2×#204—61no—
129C2×#204—62no—
130C3×#101—62no—
131#824—22no—
132#823—62no—
133#823—62no—
134(C2×D3)⋊C8—62no—
135(C2×D3)⋊C8—62no—
136C23×C32#4111yescharacteristic
137C22×C3⋊D3—21no—
138C22×C3⋊D3—22no—
139C22×C3⋊D3—41no—
140C22×C3⋊D3—41no—
141C22×C3⋊D3—82no—
142C2×C32×C4C2×D411yescharacteristic
143C2×C32×C4C2×D411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148#638—41no—
149#638—41no—
150C2×C3⋊Dic3—21no—
151C4⋊(C3⋊D3)—22no—
152C4⋊(C3⋊D3)—21no—
153C4⋊(C3⋊D3)—22no—
154C4⋊(C3⋊D3)—21no—
155C4⋊(C3⋊D3)—41no—
156C4⋊(C3⋊D3)—41no—
157C4×C3⋊D3—41no—
158C4×C3⋊D3—41no—
159#634—21no—
160#634—21no—
161C2×C3×Dic3—22no—
162C2×C3×Dic3—41no—
163C2×C3×Dic3—41no—
164C3×Dic6—22no—
165C3×Dic6—22no—
166C3×Dic6—41no—
167C3×Dic6—41no—
168C32×C8—22no—
169C3×C3⋊C8—22no—
170C3×C3⋊C8—22no—
171C3×C3⋊C8—41no—
172C2×#137—91no—
173C2×AGL(1,ℤ/8)—91no—
174C22×SD16—91no—
175C2×#101—31no—
176C2×#100—91no—
177#408—91no—
178#341—31no—
179C23×D3—61no—
180C23×D3—61no—
181C23×D3—121no—
182C2×C3×Q8—32no—
183C2×C3×Q8—61no—
184C2×C3×Q8—61no—
185C2×C3×Q8—61no—
186C2×C3×Q8—61no—
187C2×C3×Q8—61no—
188C22×C3×C4—21no—
189C22×C3×C4—61no—
190C22×C3×C4C22×D311yescharacteristic
191C22×C3×C4C22×D311yescharacteristic
192C2×#67—61no—
193C2×#67—61no—
194C2×#67—121no—
195C22×Dic3—21no—
196#235—62no—
197#235—62no—
198#235—61no—
199#235—61no—
200#235—61no—
201#235—61no—
202#235—61no—
203#235—61no—
204#235—61no—
205#235—61no—
206#235—121no—
207#235—121no—
208C2×D12—31no—
209C2×D12—32no—
210C2×D12—31no—
211C2×D12—32no—
212C2×D12—61no—
213C2×D12—62no—
214C2×D12—62no—
215C2×D12—62no—
216C2×D12—61no—
217C2×D12—61no—
218C2×D12—61no—
219C2×D12—61no—
220C2×D12—61no—
221C2×D12—61no—
222C2×D12—61no—
223C2×D12—61no—
224C2×D12—61no—
225C2×D12—61no—
226C2×D12—121no—
227C2×D12—121no—
228C2×D12—121no—
229C2×C4×D3—61no—
230C2×C4×D3—61no—
231C2×C4×D3—121no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6—31no—
238C2×Dic6—31no—
239C2×Dic6—61no—
240C2×Dic6D1212yes—
241C3×M16—22no—
242C3×M16—21no—
243C3×M16—21no—
244C3×M16—22no—
245C3×M16—32no—
246C3×M16—61no—
247C3×M16—61no—
248C3×M16—61no—
249C3×M16D1212yes—
250C3×M16#6712yes—
251C2×C3×C8—22no—
252C2×C3×C8—21no—
253C2×C3×C8—21no—
254C2×C3×C8—32no—
255C2×C3×C8—61no—
256C2×C3×C8—61no—
257C2×C3×C8—61no—
258C2×C3×C8—61no—
259#216—64no—
260#215—68no—
261#215—64no—
262#208—21no—
263#208—21no—
264#208—21no—
265#208D1212yes—
266C2×C3⋊C8—21no—
267C2×C3⋊C8—21no—
268C2×C3⋊C8—21no—
269C2×C3⋊C8—21no—
270C2×C3⋊C8D1212yes—
271Dic12—64no—
272D24—68no—
273D24—64no—
274D24—64no—
275#204—68no—
276#204—64no—
277#204—64no—
278#204—68no—
279#204—64no—
280C22×C32—22no—
281C22×C32#12111yescharacteristic
282C22×C32#12111yescharacteristic
283C22×C32#12111yescharacteristic
284C2×C3⋊D3—41no—
285C2×C3⋊D3—42no—
286C2×C3⋊D3—42no—
287C2×C3⋊D3—82no—
288C2×C3⋊D3—82no—
289C2×C3⋊D3—82no—
290C32×C4#12011yescharacteristic
291C32×C4#12011yescharacteristic
292C32×C4#12011yescharacteristic
293C32×C4#12711yescharacteristic
294C3⋊Dic3—41no—
295C3×Dic3—42no—
296C2×#41—91no—
297C22×Q8—31no—
298C22×D4—91no—
299#137—182no—
300#137—182no—
301AGL(1,ℤ/8)—182no—
302AGL(1,ℤ/8)—182no—
303C2×Q16—181no—
304C2×SD16—181no—
305C2×SD16—182no—
306C2×SD16—182no—
307C2×SD16—181no—
308C2×SD16—182no—
309C2×D8—181no—
310C2×M16—31no—
311C2×M16—31no—
312C2×M16D3211yescharacteristic
313C22×C8—31no—
314#101—62no—
315#100—182no—
316C22⋊C8—182no—
317C23×C3—21no—
318C23×C3#23511yescharacteristic
319C23×C3#23911yescharacteristic
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—62no—
323C22×D3—62no—
324C22×D3—121no—
325C22×D3—122no—
326C22×D3—122no—
327C22×D3—122no—
328C22×D3—121no—
329C22×D3—121no—
330C22×D3—121no—
331C22×D3—121no—
332C22×D3—244no—
333C22×D3—242no—
334C22×D3—242no—
335C3×Q8—62no—
336C3×Q8—62no—
337C3×Q8—121no—
338C3×Q8—121no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—62no—
346C2×C3×C4—121no—
347C2×C3×C4—121no—
348C2×C3×C4C2×D1211yescharacteristic
349C2×C3×C4C2×D1211yescharacteristic
350C2×C3×C4C2×D1211yescharacteristic
351C2×C3×C4C2×D1211yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4D3×D411yescharacteristic
354C2×C3×C4D3×D411yescharacteristic
355C2×C3×C4D3×D411yescharacteristic
356C2×C3×C4D3×D411yescharacteristic
357C2×C3×C4D3×D411yescharacteristic
358C2×C3×C4D3×D411yescharacteristic
359C2×C3×C4C2×#6711yescharacteristic
360#67—122no—
361#67—122no—
362#67—121no—
363#67—121no—
364#67—121no—
365#67—121no—
366C2×Dic3—22no—
367C2×Dic3—41no—
368C2×Dic3—41no—
369C2×Dic3—61no—
370C2×Dic3—61no—
371C2×Dic3—121no—
372D12—64no—
373D12—62no—
374D12—62no—
375D12—61no—
376D12—62no—
377D12—61no—
378D12—62no—
379D12—61no—
380D12—62no—
381D12—61no—
382D12—122no—
383D12—121no—
384D12—122no—
385D12—122no—
386D12—121no—
387D12—121no—
388D12—121no—
389D12—121no—
390C4×D3—122no—
391C4×D3—122no—
392C4×D3—121no—
393C4×D3—121no—
394C4×D3—121no—
395C4×D3—121no—
396Dic6—22no—
397Dic6—22no—
398Dic6—41no—
399Dic6—41no—
400Dic6—62no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—121no—
406C3×C8—24no—
407C3×C8—22no—
408C3×C8—22no—
409C3×C8—62no—
410C3×C8—62no—
411C3×C8—121no—
412C3⋊C8—22no—
413C3⋊C8—22no—
414C3⋊C8—41no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32#39211yescharacteristic
418C2×C32#47112yes—
419C3⋊D3—81no—
420C3⋊D3—82no—
421C24—181no—
422#41—181no—
423#41—181no—
424#41—181no—
425#41—181no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—91no—
433C2×D4—91no—
434C2×D4—92no—
435C2×D4—181no—
436C2×D4—181no—
437C2×D4—181no—
438C2×D4—181no—
439C2×D4—181no—
440C2×D4—181no—
441C22×C4—61no—
442C22×C4—181no—
443C22×C4C2×D3211yescharacteristic
444Q16—184no—
445SD16—184no—
446SD16—188no—
447SD16—184no—
448D8—184no—
449M16—21no—
450M16—32no—
451M16—61no—
452M16—61no—
453M16—61no—
454M16#62612yes—
455C2×C8—21no—
456C2×C8—32no—
457C2×C8—61no—
458C2×C8—61no—
459C2×C8—61no—
460C2×C8—61no—
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—42no—
467C22×C3#88411yescharacteristic
468C22×C3#88411yescharacteristic
469C22×C3#89511yescharacteristic
470C22×C3#89511yescharacteristic
471C22×C3#89511yescharacteristic
472C22×C3#93011yescharacteristic
473C2×D3—122no—
474C2×D3—124no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—122no—
478C2×D3—121no—
479C2×D3—122no—
480C2×D3—122no—
481C2×D3—242no—
482C2×D3—242no—
483C2×D3—242no—
484C2×D3—244no—
485C2×D3—244no—
486C2×D3—244no—
487C2×D3—242no—
488C2×D3—242no—
489C2×D3—242no—
490C2×D3—242no—
491C3×C4—21no—
492C3×C4—21no—
493C3×C4—21no—
494C3×C4—21no—
495C3×C4—122no—
496C3×C4#87411yescharacteristic
497C3×C4#88211yescharacteristic
498C3×C4#88211yescharacteristic
499C3×C4#88211yescharacteristic
500C3×C4#88211yescharacteristic
501C3×C4#88211yescharacteristic
502C3×C4#93711yescharacteristic
503C3×C4#94011yescharacteristic
504Dic3—42no—
505Dic3—122no—
506Dic3—121no—
507Dic3—121no—
508C32#201911yescharacteristic, complemented, Sylow
509C23—181no—
510C23—182no—
511C23—361no—
512C23—361no—
513C23—722no—
514C23#379911yescharacteristic
515Q8—62no—
516Q8—62no—
517Q8—121no—
518Q8—121no—
519Q8—181no—
520Q8—181no—
521D4—182no—
522D4—181no—
523D4—181no—
524D4—182no—
525D4—361no—
526D4—361no—
527D4—361no—
528D4—361no—
529C2×C4—62no—
530C2×C4—121no—
531C2×C4—121no—
532C2×C4—181no—
533C2×C4—361no—
534C2×C4—361no—
535C2×C4D3×D1211yescharacteristic
536C2×C4D3×D1211yescharacteristic
537C2×C4D3×D1211yescharacteristic
538C2×C4C2×#62611yescharacteristic
539C2×C4#381111yescharacteristic
540C2×C4#381111yescharacteristic
541C8—22no—
542C8—62no—
543C8—62no—
544C8—121no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—22no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3#467111yescharacteristic
554C2×C3#475412yes—
555C2×C3#489812yes—
556C2×C3#496011yescharacteristic
557D3—244no—
558D3—242no—
559D3—241no—
560D3—242no—
561D3—241no—
562D3—242no—
563C22—22no—
564C22—362no—
565C22—361no—
566C22—362no—
567C22—722no—
568C22—722no—
569C22—722no—
570C22#6390911yescharacteristic
571C22#6391011yescharacteristic
572C22#6391011yescharacteristic
573C4—122no—
574C4—361no—
575C4#6391111yescharacteristic
576C4#6392311yescharacteristic
577C4#6392311yescharacteristic
578C4#6398111yescharacteristic
579C3—21no—
580C3#6894311yescharacteristic, complemented
581C3#7011711yescharacteristic, complemented
582C2—21no—
583C2—21no—
584C2—721no—
585C2—722no—
586C2#1059099012yes—
587C2#1059139511yescharacteristic
subgroupG/H
centerC22#63910
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×#64117C2
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
2. C2×#63833C2semidirect
3. C22×#3717C2semidirect
4. C2×C3×#824C2semidirect
5. C2×#63559C2semidirect
6. #10591073C2semidirect
7. C3×#4558C2semidirect
39. C2×#824D3semidirect
45. #4558D3semidirect
508. C32#2019semidirect
580. C3#68943semidirect
581. C3#70117semidirect

Maximal subgroups

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

subgroupindex
C2×#641172
C2×#638332
C22×#37172
C2×C3×#8242
C2×#635592
#105910732
C3×#45582
#701173
#689433