Finite Group Databasebeta

← #11895864

Subgroups — #11895864

Order 1152.

798 conjugacy classes of subgroups.

The subgroup lattice has 533 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5542C211yesmaximal, characteristic, complemented
2Dic6×D12C211yesmaximal, characteristic, complemented
3#10592706C211yesmaximal, characteristic, complemented
4C3×C4×#213C211yesmaximal, characteristic
5#10591488C212yesmaximal, complemented
6#10591431C212yesmaximal
7#10590557C212yesmaximal, complemented
8#10590454C212yesmaximal, complemented
9#10590363C211yesmaximal, characteristic, complemented
10#10590287C211yesmaximal, characteristic, complemented
11#10590258C211yesmaximal, characteristic, complemented
12#83668—31nomaximal
13#79571—31nomaximal
14C2×D3×Dic6—21no—
15C32×C4×D4C2211yescharacteristic
16#64078C2211yescharacteristic
17C3×#936C2212yes—
18C2×C3×#213C2211yescharacteristic
19C3×#926—21no—
20C3×#924—21no—
21C3×#887C2212yes—
22C3×#879—22no—
23C3×C4×D12C2211yescharacteristic
24C3×#869C2211yescharacteristic
25C3×C4×Dic6C2211yescharacteristic
26#63918C2212yes—
27#63893—22no—
28Dic3×D12C2212yes—
29#63857—21no—
30C2×#3715C2212yes—
31#63659C2212yes—
32#63634C2212yes—
33C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
34C3×#832C2211yescharacteristic
35C3×#809C2211yescharacteristic
36C3×#807C2211yescharacteristic
37C3×C4×C3⋊C8C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63566C2212yes—
40#63566C2212yes—
41#63565C2212yes—
42#63554C2212yes—
43C3×#556—31no—
44Q8×D12—31no—
45#5542D311yescharacteristic, complemented
46D4×Dic6—31no—
47C3×C4×D8—31no—
48#5153—32no—
49#5083—31no—
50#5079—31no—
51#5057—32no—
52#5055—32no—
53#5026—32no—
54#5024—31no—
55#5021—32no—
56#5019—31no—
57C4×#213D311yescharacteristic
58#5007—31no—
59#5007—31no—
60#5007—61no—
61#4875—32no—
62#4765—32no—
63#4757—32no—
64#4681—31no—
65#4679—31no—
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—21no—
68C2×#634—21no—
69C2×C3×D12C2311yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×Dic6D411yescharacteristic
72C2×C3×Dic6C2312yes—
73C2×#627—22no—
74C2×D3×Dic3—22no—
75D3×Dic6—44no—
76C32×C4⋊C4C2311yescharacteristic
77C32×C22⋊C4—21no—
78C32×C42C2311yescharacteristic
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2312yes—
81C2×C3⋊(C3⋊C8)C2312yes—
82C3×#217—22no—
83C3×#217—22no—
84C3×#213—22no—
85C3×(C2×D3)⋊C4—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3D411yescharacteristic
89C3×C4⋊Dic3C2311yescharacteristic
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4D412yes—
92C3×C4×Dic3—22no—
93C3×C4×Dic3C2312yes—
94C2×C3×C3⋊C8C2311yescharacteristic
95C2×C3×C3⋊C8C2311yescharacteristic
96Dic3⋊Dic3—22no—
97(C2×D3)⋊Dic3—22no—
98#3715—24no—
99#3367—91noSylow
100C2×D3×Q8—61no—
101C22×Dic6—61no—
102C2×C3×D8—31no—
103C3×C4⋊Q8—31no—
104C3×#126—62no—
105C3×#124—32no—
106C3×C22⋊Q8—61no—
107C3×C4×Q8—31no—
108C3×C4×Q8—32no—
109C3×C4×D4—21no—
110C3×C4×D4—31no—
111C3×C4×D4C2×D311yescharacteristic
112C3×C4×D4C2×D311yescharacteristic
113C3×#117—61no—
114C2×#215—32no—
115#936C2×D312yes—
116D4×Dic3—32no—
117C2×#214—32no—
118C2×#213C2×D311yescharacteristic
119#926—21no—
120#924—21no—
121#924—61no—
122#897—62no—
123#897—61no—
124#888—32no—
125#887C2×D312yes—
126#879—22no—
127#878—62no—
128C4×D12—32no—
129C4×D12C2×D311yescharacteristic
130#869—31no—
131#869—31no—
132#869—32no—
133#869—61no—
134#869C2×D311yescharacteristic
135C4×Dic6C2×D311yescharacteristic
136C3×#107—31no—
137C3×#102—31no—
138C3×#102—31no—
139C3×C4×C8—31no—
140#835—32no—
141#835—32no—
142#835—32no—
143#832—32no—
144#832—32no—
145#832—62no—
146#832C2×D311yescharacteristic
147#816—32no—
148#816—32no—
149#815—32no—
150Dic3⋊C8—32no—
151#810—32no—
152#809C2×D311yescharacteristic
153#807—32no—
154#807—32no—
155#807—62no—
156#807C2×D311yescharacteristic
157C4⋊(C3⋊C8)—31no—
158C4⋊(C3⋊C8)—31no—
159C4⋊(C3⋊C8)—32no—
160C4⋊(C3⋊C8)—61no—
161#803—32no—
162C4×C3⋊C8C2×D311yescharacteristic
163C23×C32—21no—
164C22×C3×D3—21no—
165C32×D4—21no—
166C32×D4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—42no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2411yescharacteristic
172C2×C3⋊Dic3—22no—
173#634—42no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3C2×D412yes—
177C2×C3×Dic3C2×D412yes—
178C3×D12—21no—
179C3×D12#4112yes—
180C3×C4×D3—42no—
181C3×Dic6—22no—
182C3×Dic6—22no—
183C3×Dic6—22no—
184#627—44no—
185D3×Dic3—44no—
186C3⋊(C3⋊C8)—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189#556—31no—
190D4×Q8—91no—
191#489—92no—
192#487—91no—
193#477—92no—
194#474—91no—
195#463—92no—
196#461—91no—
197#446—91no—
198#440—92no—
199C4×D8—31no—
200C2×C3×Q8—31no—
201C2×C3×Q8—32no—
202C2×C3×D4—21no—
203C2×C3×D4—31no—
204C2×C3×D4C22×D311yescharacteristic
205C2×C3×D4C22×D311yescharacteristic
206C22×C3×C4—21no—
207C22×C3×C4—21no—
208C22×C3×C4—41no—
209C22×C3×C4—61no—
210C22×Dic3—62no—
211D3×Q8—124no—
212C2×D12C22×D311yescharacteristic
213C2×C4×D3—21no—
214C2×C4×D3—62no—
215C2×Dic6—62no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×Dic6—62no—
219C2×Dic6—121no—
220C2×Dic6—124no—
221C2×Dic6#6711yescharacteristic
222C2×Dic6C22×D312yes—
223C3×D8—62no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—32no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—62no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4C22×D311yescharacteristic
234C3×C4⋊C4C22×D311yescharacteristic
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—41no—
238C3×C22⋊C4—62no—
239C3×C22⋊C4—62no—
240C3×C22⋊C4—61no—
241C3×C42—21no—
242C3×C42—32no—
243C3×C42—62no—
244C3×C42C22×D311yescharacteristic
245C3×C42C22×D311yescharacteristic
246#217—22no—
247#217—22no—
248#217—62no—
249#215—64no—
250#214—64no—
251#213—22no—
252(C2×D3)⋊C4—21no—
253(C2×D3)⋊C4—62no—
254C4⋊Dic3—21no—
255C4⋊Dic3—21no—
256C4⋊Dic3—32no—
257C4⋊Dic3—32no—
258C4⋊Dic3—62no—
259C4⋊Dic3—62no—
260C4⋊Dic3—61no—
261C4⋊Dic3—61no—
262C4⋊Dic3—121no—
263C4⋊Dic3#6711yescharacteristic
264C4⋊Dic3C22×D311yescharacteristic
265Dic3⋊C4—22no—
266Dic3⋊C4—62no—
267Dic3⋊C4#6712yes—
268C4×Dic3—22no—
269C4×Dic3C22×D312yes—
270C2×C3⋊C8—32no—
271C2×C3⋊C8—32no—
272C2×C3⋊C8—62no—
273C2×C3⋊C8C22×D311yescharacteristic
274C2×C3⋊C8C22×D311yescharacteristic
275C22×C32—41no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32C22×D411yescharacteristic
279C2×C3×D3—22no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C32×C4—21no—
283C32×C4—41no—
284C32×C4AGL(1,ℤ/8)12yes—
285C32×C4C2×#4111yescharacteristic
286C32×C42−1+411yescharacteristic
287C3⋊Dic3—42no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—42no—
291C3×Dic3—41no—
292C22×Q8—181no—
293C2×SD16—92no—
294C2×D8—31no—
295C4⋊Q8—31no—
296C4⋊Q8—92no—
297C4⋊Q8—91no—
298#126—62no—
299#124—32no—
300C22⋊Q8—61no—
301C22⋊Q8—182no—
302C22⋊Q8—181no—
303C4×Q8—31no—
304C4×Q8—32no—
305C4×D4—31no—
306C4×D4—92no—
307C4×D4D3211yescharacteristic
308#117—61no—
309#107—31no—
310#107—92no—
311C4⋊C8—92no—
312C4⋊C8—91no—
313#103—92no—
314#103—92no—
315#103—92no—
316#102—31no—
317#102—31no—
318#102—92no—
319#97—92no—
320C4×C8—31no—
321C23×C3—21no—
322C23×C3—21no—
323C23×C3—41no—
324C23×C3—61no—
325C22×D3—21no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—62no—
329C3×D4—22no—
330C3×D4—21no—
331C3×D4—21no—
332C3×D4—21no—
333C3×D4—21no—
334C3×D4—32no—
335C3×D4—41no—
336C3×D4—61no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—32no—
343C2×C3×C4—32no—
344C2×C3×C4—41no—
345C2×C3×C4—42no—
346C2×C3×C4—42no—
347C2×C3×C4—62no—
348C2×C3×C4—61no—
349C2×C3×C4—82no—
350C2×C3×C4—122no—
351C2×C3×C4D3×D411yescharacteristic
352C2×C3×C4D3×D411yescharacteristic
353C2×C3×C4C2×#6711yescharacteristic
354C2×C3×C4C2×#6711yescharacteristic
355C2×C3×C4C23×D311yescharacteristic
356C2×C3×C4C23×D311yescharacteristic
357C2×Dic3—21no—
358C2×Dic3—22no—
359C2×Dic3—62no—
360C2×Dic3—62no—
361C2×Dic3—122no—
362C2×Dic3—124no—
363C2×Dic3C2×#6712yes—
364C2×Dic3C2×#6712yes—
365D12—21no—
366D12#23512yes—
367C4×D3—42no—
368C4×D3—124no—
369Dic6—22no—
370Dic6—22no—
371Dic6—22no—
372Dic6—124no—
373Dic6—124no—
374Dic6—122no—
375Dic6—122no—
376Dic6—242no—
377C3×C8—61no—
378C3×C8—61no—
379C3⋊C8—21no—
380C3⋊C8—21no—
381C3⋊C8—62no—
382C3⋊C8—62no—
383C3⋊C8—122no—
384C2×C32—81no—
385C2×C32#54811yescharacteristic
386C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
387C2×C32#57811yescharacteristic
388C3×D3—42no—
389C2×Q8—31no—
390C2×Q8—32no—
391C2×Q8—182no—
392C2×Q8—181no—
393C2×Q8—364no—
394C2×D4—31no—
395C2×D4C2×D3211yescharacteristic
396C22×C4—21no—
397C22×C4—61no—
398C22×C4—182no—
399SD16—184no—
400D8—62no—
401C2×C8—31no—
402C2×C8—31no—
403C2×C8—92no—
404C4⋊C4—31no—
405C4⋊C4—32no—
406C4⋊C4—31no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—61no—
410C4⋊C4—92no—
411C4⋊C4—182no—
412C4⋊C4—181no—
413C4⋊C4C2×D3211yescharacteristic
414C22⋊C4—21no—
415C22⋊C4—62no—
416C22⋊C4—62no—
417C22⋊C4—61no—
418C22⋊C4—182no—
419C42—32no—
420C42—62no—
421C42C2×D3211yescharacteristic
422C22×C3—21no—
423C22×C3—42no—
424C22×C3—41no—
425C22×C3—41no—
426C22×C3—41no—
427C22×C3—41no—
428C22×C3—41no—
429C22×C3—41no—
430C22×C3—62no—
431C22×C3—81no—
432C22×C3—81no—
433C22×C3—121no—
434C22×C3—121no—
435C22×C3C2×D3×D411yescharacteristic
436C22×C3C22×#6711yescharacteristic
437C2×D3—22no—
438C2×D3—41no—
439C2×D3—41no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—22no—
445C3×C4—41no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—81no—
451C3×C4—122no—
452C3×C4—121no—
453C3×C4#91112yes—
454C3×C4#94912yes—
455C3×C4C2×#23511yescharacteristic
456C3×C4D3×#4111yescharacteristic
457C3×C4#101011yescharacteristic
458C3×C4#101011yescharacteristic
459Dic3—22no—
460Dic3—22no—
461Dic3—41no—
462Dic3—42no—
463Dic3—122no—
464Dic3—122no—
465Dic3—242no—
466C32#336711yescharacteristic, complemented, Sylow
467C23—21no—
468C23—61no—
469Q8—62no—
470Q8—62no—
471Q8—62no—
472Q8—364no—
473Q8—362no—
474D4—21no—
475D4—21no—
476D4—32no—
477D4—61no—
478C2×C4—21no—
479C2×C4—32no—
480C2×C4—32no—
481C2×C4—42no—
482C2×C4—62no—
483C2×C4—61no—
484C2×C4—122no—
485C2×C4—182no—
486C2×C4—364no—
487C2×C4D3×#6711yescharacteristic
488C2×C4D3×#6711yescharacteristic
489C2×C4C22×D3211yescharacteristic
490C8—61no—
491C8—61no—
492C8—182no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—82no—
497C2×C3—81no—
498C2×C3—81no—
499C2×C3—122no—
500C2×C3#555211yescharacteristic
501C2×C3C2×#91111yescharacteristic
502C2×C3#577111yescharacteristic
503C2×C3C2×#94911yescharacteristic
504C2×C3#583111yescharacteristic
505C2×C3#583811yescharacteristic
506D3—42no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—62no—
511C22—121no—
512C22—121no—
513C22C2×D3×#6711yescharacteristic
514C4—21no—
515C4—41no—
516C4—62no—
517C4—62no—
518C4—121no—
519C4—122no—
520C4—362no—
521C4#6392512yes—
522C4#6429811yescharacteristic
523C4D3×#23511yescharacteristic
524C3—21no—
525C3#7957111yescharacteristic, complemented
526C3#8366811yescharacteristic, complemented
527C2—81no—
528C2—122no—
529C2#1059489911yescharacteristic
530C2C2×#6392511yescharacteristic
531C2#1059546711yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
prime radicalC2×C3×#213C22
hypercenterC4×D4D32
hypocenterC32#3367
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C3×#5542C2semidirect
2. Dic6×D12C2semidirect
3. #10592706C2semidirect
5. #10591488C2semidirect
7. #10590557C2semidirect
8. #10590454C2semidirect
9. #10590363C2semidirect
10. #10590287C2semidirect
11. #10590258C2semidirect
45. #5542D3semidirect
466. C32#3367semidirect
525. C3#79571semidirect
526. C3#83668semidirect

Maximal subgroups

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

subgroupindex
C3×#55422
Dic6×D122
#105927062
C3×C4×#2132
#105914882
#105914312
#105905572
#105904542
#105903632
#105902872
#105902582
#836683
#795713