Finite Group Databasebeta

← #11868041

Subgroups — #11868041

Order 1152.

1037 conjugacy classes of subgroups.

The subgroup lattice has 603 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64110C211yesmaximal, characteristic, complemented
2C2×C3×#904C211yesmaximal, characteristic, complemented
3#10594795C212yesmaximal, complemented
4C2×C3×#819C211yesmaximal, characteristic, complemented
5#10591076C212yesmaximal, complemented
6#10590933C212yesmaximal, complemented
7#10590929C212yesmaximal, complemented
8#10590928C214yesmaximal, complemented
9#83928—31nomaximal
10#79256—31nomaximal
11C2×#3828C2211yescharacteristic
12C2×C3×#235C2212yes—
13C22×C32×C8C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15#64110—21no—
16#64110—21no—
17#64110C2212yes—
18#64110C2212yes—
19C2×#3743—21no—
20C2×C8×C3⋊D3—21no—
21C2×C3×#208C2212yes—
22C2×C3×Dic12C2211yescharacteristic, complemented
23C3×#904—21no—
24C3×#904—21no—
25C3×#904—21no—
26C3×#904—21no—
27C2×C3×D24C2211yescharacteristic, complemented
28C2×C3×#204C2212yescomplemented
29#63833C2214yes—
30#63831—22no—
31#63830C2214yes—
32#63829—24no—
33#63825—22no—
34C3×#819C2212yescomplemented
35C3×#819C2212yescomplemented
36#63560C2214yescomplemented
37#63558C2214yescomplemented
38C2×C3×#135—31no—
39#5822—32no—
40#5748—32no—
41C2×#904D311yescharacteristic, complemented
42C2×#901—31no—
43C2×#901—31no—
44C2×#901—61no—
45C2×C3×#108—31no—
46#5237—32no—
47#5192—32no—
48#5177—34no—
49#5160—32no—
50#5132—32no—
51C2×#819D311yescharacteristic, complemented
52#4898—34no—
53#4897—32no—
54#4896—32no—
55C22×C32×C4C2311yescharacteristic
56C2×#638—21no—
57#3828—21no—
58#3828—21no—
59#3828D412yes—
60#3828C2312yes—
61C2×C4⋊(C3⋊D3)D411yescharacteristic
62C2×C4×C3⋊D3—21no—
63C2×#634D411yescharacteristic
64C2×C3×#67—22no—
65C3×#235—22no—
66C3×#235—22no—
67C3×#235—22no—
68C3×#235—22no—
69C2×C3×D12C2312yes—
70C2×C3×C4×D3—22no—
71C2×C3×Dic6C2312yes—
72C2×C32×C8D411yescharacteristic
73C2×C32×C8D411yescharacteristic
74C2×C32×C8D411yescharacteristic
75C2×C32×C8D411yescharacteristic
76C2×C32×C8C2311yescharacteristic
77C2×C32×C8C2311yescharacteristic
78#3748—21no—
79#3748C2312yes—
80C2×C3⋊(C3⋊C8)—21no—
81#3743—22no—
82#3743—22no—
83C8×C3⋊D3—22no—
84C8×C3⋊D3—22no—
85C3×#208—22no—
86C3×#208C2314yes—
87C2×C3×C3⋊C8—22no—
88C3×Dic12—21no—
89C3×Dic12—21no—
90C3×D24—21no—
91C3×D24—21no—
92C3×#204—22no—
93C3×#204—22no—
94#3719—24no—
95#3717—24no—
96#3716—24no—
97#3714—24no—
98#3067—91noSylow
99C2×C3×#41—32no—
100C2×#235—31no—
101C2×#235—31no—
102C2×#235—61no—
103C2×#235C2×D312yes—
104C3×#135—61no—
105C3×#135—61no—
106C3×#135—61no—
107C3×#135—61no—
108C2×C3×Q16—31no—
109C2×C3×SD16—32no—
110C2×C3×D8—31no—
111C2×C3×M16—32no—
112C22×C3×C8—21no—
113C22×C3×C8C2×D311yescharacteristic
114C22×C3×C8C2×D311yescharacteristic
115#951—62no—
116#950—64no—
117#949—62no—
118#942—34no—
119#932—34no—
120C2×#208—31no—
121C2×#208—31no—
122C2×#208—61no—
123C2×#208C2×D312yes—
124#909—34no—
125#909—62no—
126#908—34no—
127#908—62no—
128C2×Dic12C2×D311yescharacteristic, complemented
129#904—21no—
130#904—21no—
131#904—21no—
132#904—21no—
133#904—64no—
134C2×D24C2×D311yescharacteristic, complemented
135C2×#204C2×D312yescomplemented
136#901—32no—
137#901—32no—
138#901—32no—
139#901—32no—
140#901—62no—
141#901—62no—
142#901—62no—
143#901—62no—
144#901—61no—
145#901—61no—
146#901—61no—
147#901—61no—
148C2×ΓL(1,𝔽25)—62no—
149C2×ΓL(1,𝔽25)—61no—
150C2×ΓL(1,𝔽25)—61no—
151C2×D3×C8—62no—
152C2×D3×C8—61no—
153C2×D3×C8—61no—
154C3×#108—32no—
155C3×#108—32no—
156#836—34no—
157#833—34no—
158#824—34no—
159#823—34no—
160#819C2×D312yescomplemented
161#819C2×D312yescomplemented
162C23×C32#4111yescharacteristic
163C22×C3⋊D3—21no—
164C22×C3×D3—22no—
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2×D411yescharacteristic
169C2×C32×C4#4111yescharacteristic
170C2×C32×C4C2411yescharacteristic
171#638—22no—
172#638—22no—
173C2×C3⋊Dic3—21no—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)C2×D412yes—
176C4×C3⋊D3—22no—
177C4×C3⋊D3—22no—
178#634—21no—
179#634C2×D412yes—
180C3×#67—42no—
181C3×#67—42no—
182C2×C3×Dic3—22no—
183C3×D12—22no—
184C3×D12—22no—
185C3×C4×D3—42no—
186C3×C4×D3—42no—
187C3×Dic6—22no—
188C3×Dic6—22no—
189C32×C8C2×D411yescharacteristic
190C32×C8C2×D411yescharacteristic
191C32×C8C2×D411yescharacteristic
192C32×C8C2×D411yescharacteristic
193C3⋊(C3⋊C8)—22no—
194C3×C3⋊C8—24no—
195#575—92no—
196C2×#135—31no—
197C2×#131—91no—
198#473—92no—
199#472—92no—
200#471—94no—
201C2×#108—31no—
202#413—92no—
203C3×#41—62no—
204C3×#41—62no—
205C3×#41—62no—
206C3×#41—62no—
207C2×C3×Q8—32no—
208C2×C3×D4—32no—
209C2×C3×D4—62no—
210C22×C3×C4—21no—
211C22×C3×C4—62no—
212C22×C3×C4C22×D311yescharacteristic
213C22×C3×C4C22×D311yescharacteristic
214C2×#67—22no—
215C2×#67—62no—
216C2×#67—61no—
217C2×#67—61no—
218#235—22no—
219#235—22no—
220#235—22no—
221#235—22no—
222#235—32no—
223#235—32no—
224#235—32no—
225#235—32no—
226#235—62no—
227#235—62no—
228#235—62no—
229#235—62no—
230#235—61no—
231#235—61no—
232#235—61no—
233#235—61no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—61no—
237C2×D12C22×D312yes—
238C2×C4×D3—22no—
239C2×C4×D3—62no—
240C2×C4×D3—61no—
241C2×C4×D3—61no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C2×Dic6C22×D312yes—
246C3×Q16—61no—
247C3×Q16—61no—
248C3×SD16—62no—
249C3×SD16—62no—
250C3×D8—61no—
251C3×D8—61no—
252C3×M16—34no—
253C3×M16—62no—
254C2×C3×C8—21no—
255C2×C3×C8—21no—
256C2×C3×C8—21no—
257C2×C3×C8—21no—
258C2×C3×C8—21no—
259C2×C3×C8—21no—
260C2×C3×C8—62no—
261C2×C3×C8D1211yescharacteristic
262C2×C3×C8D1211yescharacteristic
263C2×C3×C8D1211yescharacteristic
264C2×C3×C8D1211yescharacteristic
265C2×C3×C8#6711yescharacteristic
266C2×C3×C8#6711yescharacteristic
267C2×C3×C8#6711yescharacteristic
268C2×C3×C8#6711yescharacteristic
269C2×C3×C8C22×D311yescharacteristic
270C2×C3×C8C22×D311yescharacteristic
271C2×C3×C8C22×D311yescharacteristic
272C2×C3×C8C22×D311yescharacteristic
273#216—64no—
274#215—64no—
275#214—64no—
276#213—64no—
277#208—22no—
278#208—32no—
279#208—32no—
280#208—62no—
281#208—62no—
282#208—61no—
283#208—61no—
284#208C22×D314yes—
285C2×C3⋊C8—22no—
286C2×C3⋊C8—62no—
287C2×C3⋊C8—61no—
288C2×C3⋊C8—61no—
289Dic12—21no—
290Dic12—21no—
291Dic12—64no—
292D24—21no—
293D24—21no—
294D24—64no—
295#204—22no—
296#204—22no—
297#204—64no—
298#204—64no—
299ΓL(1,𝔽25)—64no—
300ΓL(1,𝔽25)—64no—
301ΓL(1,𝔽25)—62no—
302ΓL(1,𝔽25)—62no—
303ΓL(1,𝔽25)—62no—
304ΓL(1,𝔽25)—62no—
305D3×C8—64no—
306D3×C8—64no—
307D3×C8—62no—
308D3×C8—62no—
309D3×C8—62no—
310D3×C8—62no—
311C22×C32—41no—
312C22×C32#12111yescharacteristic
313C22×C32#12111yescharacteristic
314C22×C32C2×#4111yescharacteristic
315C2×C3⋊D3—22no—
316C2×C3⋊D3—42no—
317C2×C3×D3—42no—
318C2×C3×D3—82no—
319C32×C4#12111yescharacteristic
320C32×C4#12111yescharacteristic
321C32×C4C22×D411yescharacteristic
322C32×C4C22×D411yescharacteristic
323C3⋊Dic3—22no—
324C3×Dic3—42no—
325C2×#41—32no—
326C2×#41—91no—
327#137—94no—
328#137—182no—
329AGL(1,ℤ/8)—94no—
330AGL(1,ℤ/8)—182no—
331#135—61no—
332#135—61no—
333#135—61no—
334#135—61no—
335#135—184no—
336C2×Q16—31no—
337C2×SD16—32no—
338C2×D8—31no—
339#131—92no—
340#131—92no—
341#131—181no—
342#131—181no—
343C2×M16—32no—
344C2×M16—91no—
345C2×M16—181no—
346C22×C8—181no—
347C22×C8D3211yescharacteristic
348#108—32no—
349#108—32no—
350#101—94no—
351#100—94no—
352C23×C3—21no—
353C23×C3—62no—
354C23×C3#23711yescharacteristic
355C23×C3#23911yescharacteristic
356C22×D3—22no—
357C22×D3—62no—
358C22×D3—61no—
359C22×D3—61no—
360C3×Q8—62no—
361C3×Q8—62no—
362C3×D4—62no—
363C3×D4—62no—
364C3×D4—122no—
365C3×D4—122no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—62no—
373C2×C3×C4—122no—
374C2×C3×C4—122no—
375C2×C3×C4C2×D1211yescharacteristic
376C2×C3×C4C2×D1211yescharacteristic
377C2×C3×C4D3×D411yescharacteristic
378C2×C3×C4D3×D411yescharacteristic
379C2×C3×C4D3×D411yescharacteristic
380C2×C3×C4D3×D411yescharacteristic
381C2×C3×C4#23711yescharacteristic
382C2×C3×C4#23911yescharacteristic
383C2×C3×C4C2×#6711yescharacteristic
384C2×C3×C4C2×#6711yescharacteristic
385C2×C3×C4C23×D311yescharacteristic
386C2×C3×C4C23×D311yescharacteristic
387#67—42no—
388#67—42no—
389#67—64no—
390#67—64no—
391#67—62no—
392#67—62no—
393#67—62no—
394#67—62no—
395C2×Dic3—22no—
396C2×Dic3—62no—
397C2×Dic3—61no—
398C2×Dic3—61no—
399D12—22no—
400D12—22no—
401D12—32no—
402D12—32no—
403D12—62no—
404D12—62no—
405D12—61no—
406D12—61no—
407C4×D3—42no—
408C4×D3—42no—
409C4×D3—64no—
410C4×D3—64no—
411C4×D3—62no—
412C4×D3—62no—
413C4×D3—62no—
414C4×D3—62no—
415Dic6—22no—
416Dic6—22no—
417Dic6—32no—
418Dic6—32no—
419Dic6—62no—
420Dic6—62no—
421Dic6—61no—
422Dic6—61no—
423C3×C8—21no—
424C3×C8—21no—
425C3×C8—21no—
426C3×C8—21no—
427C3×C8—64no—
428C3×C8C2×D1211yescharacteristic
429C3×C8C2×D1211yescharacteristic
430C3×C8C2×D1211yescharacteristic
431C3×C8C2×D1211yescharacteristic
432C3×C8C2×#6711yescharacteristic
433C3×C8C2×#6711yescharacteristic
434C3×C8C2×#6711yescharacteristic
435C3×C8C2×#6711yescharacteristic
436C3⋊C8—24no—
437C3⋊C8—64no—
438C3⋊C8—62no—
439C3⋊C8—62no—
440C2×C32—21no—
441C2×C32—21no—
442C2×C32—21no—
443C2×C32C2×#12111yescharacteristic
444C3⋊D3—42no—
445C3×D3—82no—
446#41—62no—
447#41—62no—
448#41—62no—
449#41—62no—
450#41—92no—
451#41—92no—
452#41—181no—
453#41—181no—
454C2×Q8—32no—
455C2×Q8—91no—
456C2×D4—32no—
457C2×D4—62no—
458C2×D4—91no—
459C2×D4—181no—
460C22×C4—62no—
461C22×C4—181no—
462C22×C4C2×D3211yescharacteristic
463Q16—61no—
464Q16—61no—
465Q16—184no—
466SD16—62no—
467SD16—62no—
468SD16—184no—
469SD16—184no—
470D8—61no—
471D8—61no—
472D8—184no—
473M16—34no—
474M16—62no—
475M16—92no—
476M16—181no—
477M16—182no—
478M16—182no—
479C2×C8—62no—
480C2×C8—181no—
481C2×C8—182no—
482C2×C8—182no—
483C2×C8#62611yescharacteristic
484C2×C8#62611yescharacteristic
485C2×C8#62611yescharacteristic
486C2×C8#62611yescharacteristic
487C2×C8C2×D3211yescharacteristic
488C2×C8C2×D3211yescharacteristic
489C22×C3—21no—
490C22×C3—21no—
491C22×C3—21no—
492C22×C3—41no—
493C22×C3—41no—
494C22×C3—42no—
495C22×C3—122no—
496C22×C3—242no—
497C22×C3#89511yescharacteristic
498C22×C3#89511yescharacteristic
499C22×C3#93811yescharacteristic
500C22×C3#93811yescharacteristic
501C22×C3C2×#23711yescharacteristic
502C22×C3C2×#23911yescharacteristic
503C2×D3—42no—
504C2×D3—64no—
505C2×D3—62no—
506C2×D3—62no—
507C2×D3—82no—
508C2×D3—124no—
509C2×D3—122no—
510C2×D3—122no—
511C3×C4—21no—
512C3×C4—21no—
513C3×C4—21no—
514C3×C4—21no—
515C3×C4—122no—
516C3×C4#89511yescharacteristic
517C3×C4#89511yescharacteristic
518C3×C4#93811yescharacteristic
519C3×C4#93811yescharacteristic
520C3×C4C22×D1211yescharacteristic
521C3×C4C2×D3×D411yescharacteristic
522C3×C4C2×D3×D411yescharacteristic
523C3×C4C22×#6711yescharacteristic
524Dic3—42no—
525Dic3—64no—
526Dic3—62no—
527Dic3—62no—
528C32#306711yescharacteristic, complemented, Sylow
529C23—62no—
530C23—181no—
531C23#379611yescharacteristic
532Q8—62no—
533Q8—62no—
534Q8—92no—
535Q8—181no—
536D4—62no—
537D4—62no—
538D4—92no—
539D4—122no—
540D4—122no—
541D4—182no—
542D4—182no—
543D4—181no—
544C2×C4—62no—
545C2×C4—122no—
546C2×C4—122no—
547C2×C4—181no—
548C2×C4—182no—
549C2×C4—182no—
550C2×C4#379611yescharacteristic
551C2×C4#380211yescharacteristic
552C2×C4#380211yescharacteristic
553C2×C4C2×#62611yescharacteristic
554C2×C4C2×#62611yescharacteristic
555C2×C4C22×D3211yescharacteristic
556C8—64no—
557C8—182no—
558C8C2×#62611yescharacteristic
559C8C2×#62611yescharacteristic
560C8C2×#62611yescharacteristic
561C8C2×#62611yescharacteristic
562C2×C3—22no—
563C2×C3—21no—
564C2×C3—21no—
565C2×C3—21no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—21no—
570C2×C3—41no—
571C2×C3—41no—
572C2×C3—242no—
573C2×C3C2×#89511yescharacteristic
574C2×C3C2×#93811yescharacteristic
575D3—82no—
576D3—124no—
577D3—122no—
578D3—122no—
579C22—41no—
580C22—122no—
581C22—182no—
582C22—242no—
583C22—362no—
584C22#6391611yescharacteristic
585C22#6391611yescharacteristic
586C22C2×#379611yescharacteristic
587C4—122no—
588C4—182no—
589C4#6391611yescharacteristic
590C4#6391611yescharacteristic
591C4C2×#380211yescharacteristic
592C4C22×#62611yescharacteristic
593C3—21no—
594C3#7925611yescharacteristic, complemented
595C3#8392811yescharacteristic, complemented
596C2—21no—
597C2—21no—
598C2—21no—
599C2—242no—
600C2—362no—
601C2C2×#6391611yescharacteristic
subgroupG/H
centerC4#63916
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×C8C22
socleC2×C32C2×#121
hypercenterC22×C8D32
hypocenterC32#3067
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#64110C2semidirect
2. C2×C3×#904C2semidirect
3. #10594795C2semidirect
4. C2×C3×#819C2semidirect
5. #10591076C2semidirect
6. #10590933C2semidirect
7. #10590929C2semidirect
8. #10590928C2semidirect
22. C2×C3×Dic12C22semidirect
27. C2×C3×D24C22semidirect
28. C2×C3×#204C22semidirect
34. C3×#819C22semidirect
35. C3×#819C22semidirect
36. #63560C22semidirect
37. #63558C22semidirect
41. C2×#904D3semidirect
51. C2×#819D3semidirect
128. C2×Dic12C2×D3semidirect
134. C2×D24C2×D3semidirect
135. C2×#204C2×D3semidirect
160. #819C2×D3semidirect
161. #819C2×D3semidirect
528. C32#3067semidirect
594. C3#79256semidirect
595. C3#83928semidirect

Maximal subgroups

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

subgroupindex
C2×#641102
C2×C3×#9042
#105947952
C2×C3×#8192
#105910762
#105909332
#105909292
#105909282
#839283
#792563