Finite Group Databasebeta

← #11878807

Subgroups — #11878807

Order 1152.

947 conjugacy classes of subgroups.

The subgroup lattice has 521 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5725C211yesmaximal, characteristic, complemented
2#10593970C211yesmaximal, characteristic, complemented
3#10592784C211yesmaximal, characteristic, complemented
4C3×#5079C211yesmaximal, characteristic, complemented
5#10591588C214yesmaximal, complemented
6#10591568C214yesmaximal, complemented
7#10590350C211yesmaximal, characteristic, complemented
8#10590277C211yesmaximal, characteristic, complemented
9#10590261C211yesmaximal, characteristic, complemented
10#83303—31nomaximal
11#81603—31nomaximal
12C2×C3×#239C2212yes—
13C2×#3799—21no—
14C32×C4⋊Q8C2211yescharacteristic
15C2×#3755C2214yes—
16#64083C2211yescharacteristic
17C3×#947C2212yes—
18C3×#895—21no—
19C3×#895—21no—
20C3×#893—21no—
21C3×#893—21no—
22C3×#874C2211yescharacteristic
23C3×C42×D3C2211yescharacteristic
24C3×#869C2211yescharacteristic
25#63909—21no—
26#63909C2212yes—
27#63894—21no—
28#63894C2212yes—
29#63864—21no—
30C2×#3714C2214yes—
31C4×C3⋊(C3⋊C8)C2211yescharacteristic
32C3×#835C2214yes—
33C3×C4×C3⋊C8C2211yescharacteristic
34Dic3⋊(C3⋊C8)C2212yes—
35#63562C2214yes—
36(C2×D3)⋊(C3⋊C8)C2212yes—
37C3×#532—31no—
38#5725—31no—
39#5725D311yescharacteristic, complemented
40#5477—31no—
41C3×#487—31no—
42#5238—34no—
43#5088—31no—
44#5085—31no—
45#5085—31no—
46#5085—61no—
47#5079D311yescharacteristic, complemented
48#5073—31no—
49#5070—31no—
50#5057—34no—
51#4925—31no—
52#4819—34no—
53#4810—34no—
54#4695—31no—
55#4685—31no—
56C2×C32×Q8C2312yes—
57C2×C4⋊(C3⋊D3)—21no—
58C2×C4⋊(C3⋊D3)—21no—
59C2×C4⋊(C3⋊D3)C2312yes—
60C3×#239—44no—
61C2×C3×D12—22no—
62C2×C3×D12—21no—
63C2×C3×D12—21no—
64C2×C3×D12C2312yes—
65C2×C3×C4×D3—22no—
66C2×C3×C4×D3D411yescharacteristic
67C2×C3×C4×D3D411yescharacteristic
68C2×C3×C4×D3C2311yescharacteristic
69C2×C3×Dic6—21no—
70C2×#626—22no—
71C2×#624—22no—
72#3799—44no—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C42C2311yescharacteristic
76#3755—44no—
77C2×C3⋊(C3⋊C8)C2312yes—
78C3×(C2×D3)⋊C4—22no—
79C3×(C2×D3)⋊C4—22no—
80C3×C4⋊Dic3—21no—
81C3×C4⋊Dic3C2312yes—
82C3×C4×Dic3D411yescharacteristic
83C3×C4×Dic3D411yescharacteristic
84C3×C4×Dic3C2311yescharacteristic
85C2×C3×C3⋊C8C2312yes—
86#3722—22no—
87#3722—22no—
88#3714—44no—
89#3207—91noSylow
90C2×C3×#41—32no—
91C2×#239—61no—
92C2×#239C2×D312yes—
93C2×#235—61no—
94C3×C4⋊Q8—21no—
95C3×C4⋊Q8—31no—
96C3×C4⋊Q8C2×D311yescharacteristic
97C3×C4⋊Q8C2×D311yescharacteristic
98C3×#127—31no—
99C3×#124—32no—
100C3×#121—61no—
101C3×#121—61no—
102C3×C4×D4—61no—
103C3×C4×D4—61no—
104C2×C3×C42—31no—
105#947—61no—
106#947C2×D312yes—
107C2×#215—34no—
108C2×#215—34no—
109C2×#215—64no—
110C2×#213—34no—
111#930—32no—
112#930—61no—
113C2×D24—34no—
114#895—21no—
115#895—21no—
116#895—32no—
117#895—61no—
118#893—21no—
119#893—21no—
120#893—32no—
121#893—61no—
122#875—61no—
123#874—31no—
124#874—31no—
125#874—61no—
126#874C2×D311yescharacteristic
127C4×D12—32no—
128C4×D12—61no—
129C42×D3C2×D311yescharacteristic
130#869C2×D311yescharacteristic
131C3×#103—34no—
132C3×C4×C8—31no—
133#835C2×D314yes—
134#832—34no—
135#830—32no—
136(C2×D3)⋊C8—32no—
137Dic3⋊C8—32no—
138#809—34no—
139C4⋊(C3⋊C8)—32no—
140C4×C3⋊C8—31no—
141C4×C3⋊C8—31no—
142C4×C3⋊C8—61no—
143C4×C3⋊C8C2×D311yescharacteristic
144C22×C3⋊D3—22no—
145C22×C3×D3—22no—
146C22×C3×D3C2×D411yescharacteristic
147C32×Q8—24no—
148C2×C32×C4—22no—
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2×D411yescharacteristic
151C2×C32×C4C2411yescharacteristic
152C4⋊(C3⋊D3)—24no—
153C4⋊(C3⋊D3)—42no—
154C4⋊(C3⋊D3)—42no—
155C2×C3×Dic3—22no—
156C2×C3×Dic3C2×D411yescharacteristic
157C2×C3×Dic3C2×D411yescharacteristic
158C2×C3×Dic3C2×D411yescharacteristic
159C3×D12—24no—
160C3×D12—44no—
161C3×D12—42no—
162C3×D12—42no—
163C3×C4×D3—22no—
164C3×C4×D3—22no—
165C3×C4×D3—22no—
166C3×C4×D3—44no—
167C3×Dic6—42no—
168#626—44no—
169#624—44no—
170C3⋊(C3⋊C8)—22no—
171C3×C3⋊C8—22no—
172#532—31no—
173#532—91no—
174#493—91no—
175#492—91no—
176#487—31no—
177#486—91no—
178#463—94no—
179#449—94no—
180#431—91no—
181C3×#41—124no—
182C2×C3×Q8—22no—
183C2×C3×Q8—61no—
184C2×C3×Q8C22×D312yes—
185C2×C3×Q8C22×D312yes—
186C2×C3×D4—32no—
187C2×C3×D4—62no—
188C2×C3×D4—61no—
189C2×C3×D4—61no—
190C22×C3×C4—31no—
191C22×C3×C4—31no—
192C22×C3×C4—31no—
193C22×C3×C4—62no—
194C2×#67—62no—
195#239—44no—
196#239—124no—
197#235—124no—
198C2×D12—22no—
199C2×D12—21no—
200C2×D12—21no—
201C2×D12—32no—
202C2×D12—32no—
203C2×D12—62no—
204C2×D12—62no—
205C2×D12—61no—
206C2×D12—61no—
207C2×D12—61no—
208C2×D12—61no—
209C2×D12—121no—
210C2×D12—121no—
211C2×D12C22×D312yes—
212C2×C4×D3—22no—
213C2×C4×D3—62no—
214C2×C4×D3—62no—
215C2×C4×D3#6711yescharacteristic
216C2×C4×D3#6711yescharacteristic
217C2×C4×D3C22×D311yescharacteristic
218C2×Dic6—21no—
219C2×C3×C8—32no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—32no—
225C3×C4⋊C4—41no—
226C3×C4⋊C4—41no—
227C3×C4⋊C4—61no—
228C3×C22⋊C4—62no—
229C3×C22⋊C4—62no—
230C3×C42—21no—
231C3×C42—31no—
232C3×C42—31no—
233C3×C42—31no—
234C3×C42C22×D311yescharacteristic
235C3×C42C22×D311yescharacteristic
236#215—124no—
237#215—124no—
238#215—128no—
239#213—124no—
240(C2×D3)⋊C4—22no—
241(C2×D3)⋊C4—22no—
242(C2×D3)⋊C4—62no—
243(C2×D3)⋊C4—62no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—62no—
246C4⋊Dic3—21no—
247C4⋊Dic3C22×D312yes—
248C4×Dic3#6711yescharacteristic
249C4×Dic3#6711yescharacteristic
250C4×Dic3C22×D311yescharacteristic
251C2×C3⋊C8—32no—
252C2×C3⋊C8—32no—
253C2×C3⋊C8—62no—
254C2×C3⋊C8C22×D312yes—
255D24—124no—
256C22×C32C22×D411yescharacteristic
257C2×C3⋊D3—42no—
258C2×C3⋊D3—42no—
259C2×C3⋊D3—42no—
260C2×C3×D3—21no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—42no—
264C2×C3×D3—42no—
265C2×C3×D3—42no—
266C32×C4—42no—
267C32×C4#13512yes—
268C32×C4AGL(1,ℤ/8)12yes—
269C32×C4H(5,𝔽2)12yes—
270C3×Dic3—21no—
271C3×Dic3—21no—
272C3×Dic3—21no—
273C3×Dic3—42no—
274C2×#41—32no—
275C2×#41—181no—
276C2×SD16—94no—
277C2×D8—94no—
278C4⋊Q8—31no—
279C4⋊Q8D3211yescharacteristic
280#127—31no—
281#127—91no—
282#124—32no—
283#124—181no—
284#121—61no—
285#121—61no—
286#121—92no—
287#121—181no—
288C4×D4—61no—
289C4×D4—61no—
290C4×D4—92no—
291C4×D4—181no—
292C2×C42—31no—
293C4⋊C8—92no—
294#103—34no—
295#102—94no—
296C22⋊C8—92no—
297C4×C8—31no—
298C4×C8—91no—
299C23×C3—31no—
300C23×C3—62no—
301C22×D3—22no—
302C22×D3—62no—
303C22×D3—62no—
304C22×D3—122no—
305C22×D3C2×#6711yescharacteristic
306C3×Q8—24no—
307C3×Q8—24no—
308C3×Q8—44no—
309C3×Q8—122no—
310C3×D4—64no—
311C3×D4—124no—
312C3×D4—122no—
313C3×D4—122no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—21no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—31no—
320C2×C3×C4—31no—
321C2×C3×C4—31no—
322C2×C3×C4—42no—
323C2×C3×C4—62no—
324C2×C3×C4—62no—
325C2×C3×C4—62no—
326C2×C3×C4—62no—
327C2×C3×C4—124no—
328C2×C3×C4D3×D411yescharacteristic
329C2×C3×C4D3×D411yescharacteristic
330C2×C3×C4C2×#6711yescharacteristic
331C2×C3×C4C2×#6711yescharacteristic
332C2×C3×C4C23×D311yescharacteristic
333C2×C3×C4C23×D311yescharacteristic
334#67—124no—
335C2×Dic3—22no—
336C2×Dic3C2×#6711yescharacteristic
337C2×Dic3C2×#6711yescharacteristic
338C2×Dic3C2×#6711yescharacteristic
339D12—24no—
340D12—44no—
341D12—42no—
342D12—42no—
343D12—64no—
344D12—64no—
345D12—124no—
346D12—124no—
347D12—122no—
348D12—122no—
349D12—122no—
350D12—122no—
351D12—242no—
352D12—242no—
353C4×D3—22no—
354C4×D3—22no—
355C4×D3—22no—
356C4×D3—44no—
357C4×D3—124no—
358C4×D3—124no—
359Dic6—42no—
360C3×C8—62no—
361C3⋊C8—22no—
362C3⋊C8—64no—
363C3⋊C8—62no—
364C3⋊C8—62no—
365C2×C32#53511yescharacteristic
366C2×C32C2×#13511yescharacteristic
367C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
368C3⋊D3—82no—
369C3×D3—41no—
370C3×D3—82no—
371#41—124no—
372#41—364no—
373C2×Q8—61no—
374C2×Q8C2×D3212yes—
375C2×D4—32no—
376C2×D4—62no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—92no—
380C2×D4—182no—
381C2×D4—181no—
382C2×D4—181no—
383C22×C4—31no—
384C22×C4—31no—
385C22×C4—31no—
386C22×C4—62no—
387C22×C4—182no—
388SD16—364no—
389D8—364no—
390C2×C8—32no—
391C2×C8—92no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—32no—
395C4⋊C4—61no—
396C22⋊C4—62no—
397C22⋊C4—62no—
398C22⋊C4—182no—
399C22⋊C4—182no—
400C42—31no—
401C42—31no—
402C42—31no—
403C42C2×D3211yescharacteristic
404C22×C3—21no—
405C22×C3—61no—
406C22×C3—61no—
407C22×C3—61no—
408C22×C3—122no—
409C22×C3—122no—
410C22×C3—122no—
411C22×C3C2×D3×D411yescharacteristic
412C22×C3C22×#6711yescharacteristic
413C2×D3—21no—
414C2×D3—21no—
415C2×D3—21no—
416C2×D3—42no—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—122no—
420C2×D3—122no—
421C2×D3—122no—
422C2×D3—122no—
423C2×D3—122no—
424C2×D3—122no—
425C2×D3—242no—
426C2×D3—242no—
427C2×D3—242no—
428C3×C4—22no—
429C3×C4—22no—
430C3×C4—22no—
431C3×C4—42no—
432C3×C4—42no—
433C3×C4—61no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—82no—
437C3×C4—122no—
438C3×C4#91412yes—
439C3×C4#91912yes—
440C3×C4#94912yes—
441C3×C4#95012yes—
442C3×C4#100412yes—
443C3×C4#100912yes—
444Dic3—21no—
445Dic3—21no—
446Dic3—21no—
447Dic3—42no—
448C32#320711yescharacteristic, complemented, Sylow
449C23—31no—
450C23—62no—
451C23—182no—
452Q8—24no—
453Q8—122no—
454D4—64no—
455D4—124no—
456D4—122no—
457D4—122no—
458D4—184no—
459D4—364no—
460D4—362no—
461D4—362no—
462C2×C4—22no—
463C2×C4—31no—
464C2×C4—31no—
465C2×C4—31no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—124no—
471C2×C4—364no—
472C2×C4D3×#6711yescharacteristic
473C2×C4D3×#6711yescharacteristic
474C2×C4C22×D3211yescharacteristic
475C8—62no—
476C8—182no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—121no—
481C2×C3—242no—
482C2×C3#567411yescharacteristic
483C2×C3C2×#91411yescharacteristic
484C2×C3C2×#91911yescharacteristic
485C2×C3#580511yescharacteristic
486C2×C3C2×#94911yescharacteristic
487C2×C3C2×#95011yescharacteristic
488D3—41no—
489D3—82no—
490D3—242no—
491D3—242no—
492D3—482no—
493C22—61no—
494C22—61no—
495C22—61no—
496C22—122no—
497C22—122no—
498C22—122no—
499C22—362no—
500C22—362no—
501C22—362no—
502C22C2×D3×#6711yescharacteristic
503C4—42no—
504C4—61no—
505C4—61no—
506C4—61no—
507C4—122no—
508C4#6393912yes—
509C4#6394912yes—
510C4#6431412yes—
511C3—21no—
512C3#8160311yescharacteristic, complemented
513C3#8330311yescharacteristic, complemented
514C2—121no—
515C2—242no—
516C2—722no—
517C2#1059523511yescharacteristic
518C2C2×#6393911yescharacteristic
519C2C2×#6394911yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
hypercenterC4⋊Q8D32
hypocenterC32#3207
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5725C2semidirect
2. #10593970C2semidirect
3. #10592784C2semidirect
4. C3×#5079C2semidirect
5. #10591588C2semidirect
6. #10591568C2semidirect
7. #10590350C2semidirect
8. #10590277C2semidirect
9. #10590261C2semidirect
39. #5725D3semidirect
47. #5079D3semidirect
448. C32#3207semidirect
512. C3#81603semidirect
513. C3#83303semidirect

Maximal subgroups

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

subgroupindex
C3×#57252
#105939702
#105927842
C3×#50792
#105915882
#105915682
#105903502
#105902772
#105902612
#833033
#816033