Finite Group Databasebeta

← #11895882

Subgroups — #11895882

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
1Dic6×D12C211yesmaximal, characteristic, complemented
2#10593865C211yesmaximal, characteristic, complemented
3#10592545C211yesmaximal, characteristic, complemented
4C3×#4841C211yesmaximal, characteristic, complemented
5C3×#4831C211yesmaximal, characteristic, complemented
6#10590893C212yesmaximal, complemented
7#10590804C212yesmaximal, complemented
8#10590766C212yesmaximal, complemented
9#10590754C212yesmaximal
10C4×#3714C211yesmaximal, characteristic
11#10590287C211yesmaximal, characteristic, complemented
12#80416—31nomaximal
13#79970—31nomaximal
14C2×D3×Dic6—21no—
15C4×C4⋊(C3⋊D3)C2211yescharacteristic
16#64078C2211yescharacteristic
17C2×C3×#204C2212yes—
18C3×C4×D12C2211yescharacteristic
19C3×#869C2211yescharacteristic
20C3×C4×Dic6C2211yescharacteristic
21#63918C2212yes—
22#63893—22no—
23Dic3×D12C2212yes—
24#63871—21no—
25#63864C2212yes—
26#63857—21no—
27#63849—21no—
28#63848—22no—
29Dic3×Dic6C2212yes—
30C2×#3714C2211yescharacteristic
31C32×C4⋊C8C2211yescharacteristic
32#63648C2212yes—
33#63645C2212yes—
34C3×#816C2212yes—
35C3×#815C2212yes—
36C3×Dic3⋊C8C2212yes—
37C3×C4×C3⋊C8C2211yescharacteristic
38#63575C2211yescharacteristic
39#63566C2212yes—
40#63566C2212yes—
41#63564C2211yescharacteristic
42#63563C2211yescharacteristic
43#5732—31no—
44#5584—31no—
45Q8×D12—31no—
46D4×Dic6—31no—
47C3×#461—31no—
48C3×#446—31no—
49#5165—32no—
50#5164—32no—
51#5079—31no—
52C4×#213—31no—
53#4850—32no—
54#4846—31no—
55#4846—31no—
56#4846—61no—
57#4844—32no—
58#4841D311yescharacteristic, complemented
59#4831D311yescharacteristic, complemented
60#4794—32no—
61#4793—32no—
62#4714—32no—
63#4710—32no—
64C4×D24—31no—
65#4681—31no—
66C2×C4⋊(C3⋊D3)C2311yescharacteristic
67C2×C4×C3⋊D3—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—
76C2×C32×C8C2312yes—
77C32×C42C2311yescharacteristic
78#3752—21no—
79C4⋊(C3⋊Dic3)—21no—
80C4⋊(C3⋊Dic3)C2311yescharacteristic
81C4⋊(C3⋊Dic3)C2312yes—
82C3×(C2×D3)⋊C4—21no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3D411yescharacteristic
86C3×C4⋊Dic3C2311yescharacteristic
87C3×Dic3⋊C4D412yes—
88C3×C4×Dic3C2312yes—
89C2×C3×C3⋊C8C2311yescharacteristic
90C2×C3×C3⋊C8C2311yescharacteristic
91C3×#204—24no—
92Dic3⋊Dic3—22no—
93Dic3⋊Dic3—22no—
94#3722—22no—
95#3722—22no—
96(C2×D3)⋊Dic3—22no—
97Dic32—22no—
98#3714—22no—
99#3367—91noSylow
100C2×D3×Q8—61no—
101C22×Dic6—61no—
102C2×C3×SD16—32no—
103C3×C4⋊Q8—31no—
104C3×C4×Q8—31no—
105C3×C4×D4—31no—
106#947—32no—
107#946—61no—
108Q8×Dic3—32no—
109D4×Dic3—32no—
110C2×#213—31no—
111#924—61no—
112C2×D24—31no—
113C2×#204C2×D312yes—
114#898—62no—
115#898—62no—
116#897—62no—
117#897—61no—
118#897—61no—
119#892—61no—
120#892—61no—
121#888—32no—
122#878—62no—
123#875—32no—
124C4×D12—32no—
125C4×D12—31no—
126C4×D12—31no—
127C4×D12—61no—
128C4×D12C2×D311yescharacteristic
129#869—31no—
130#869—31no—
131#869—32no—
132#869—61no—
133#869C2×D311yescharacteristic
134C4×Dic6—32no—
135C4×Dic6C2×D311yescharacteristic
136C3×C4⋊C8—21no—
137C3×C4⋊C8—32no—
138C3×C4⋊C8C2×D311yescharacteristic
139C3×C4⋊C8C2×D311yescharacteristic
140C3×#103—32no—
141C3×#97—32no—
142C3×C4×C8—31no—
143#835—32no—
144#835—32no—
145#832—31no—
146#821—31no—
147#821—32no—
148#821—32no—
149#821—31no—
150#821—62no—
151#818—32no—
152#818—32no—
153#818—31no—
154#818—62no—
155#816—32no—
156#816—32no—
157#816C2×D312yes—
158#815C2×D312yes—
159Dic3⋊C8C2×D312yes—
160#809—31no—
161#807—31no—
162C4×C3⋊C8C2×D311yescharacteristic
163C22×C3⋊D3—21no—
164C22×C3×D3—21no—
165C2×C32×C4C2×D411yescharacteristic
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2411yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—22no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)—21no—
172C4×C3⋊D3—42no—
173#634—42no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—21no—
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—
186C32×C8—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8—21no—
189#556—91no—
190D4×Q8—91no—
191#489—92no—
192#487—91no—
193#477—92no—
194#474—91no—
195#463—92no—
196#461—31no—
197#446—31no—
198#440—92no—
199C4×D8—91no—
200C2×C3×Q8—31no—
201C2×C3×Q8—32no—
202C2×C3×D4—31no—
203C22×C3×C4—61no—
204C22×Dic3—62no—
205D3×Q8—124no—
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—61no—
209C2×D12C22×D311yescharacteristic
210C2×C4×D3—21no—
211C2×C4×D3—62no—
212C2×C4×D3—61no—
213C2×C4×D3—61no—
214C2×C4×D3—121no—
215C2×Dic6—62no—
216C2×Dic6—61no—
217C2×Dic6—61no—
218C2×Dic6—62no—
219C2×Dic6—121no—
220C2×Dic6—124no—
221C2×Dic6D1211yescharacteristic
222C2×Dic6C22×D312yes—
223C3×SD16—64no—
224C2×C3×C8—22no—
225C2×C3×C8—31no—
226C2×C3×C8—31no—
227C2×C3×C8C22×D312yes—
228C2×C3×C8C22×D312yes—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—61no—
233C3×C4⋊C4—61no—
234C3×C22⋊C4—61no—
235C3×C42—21no—
236C3×C42—32no—
237C3×C42C22×D311yescharacteristic
238C3×C42C22×D311yescharacteristic
239#217—62no—
240#213—62no—
241(C2×D3)⋊C4—21no—
242(C2×D3)⋊C4—62no—
243(C2×D3)⋊C4—62no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—61no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—62no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—121no—
250C4⋊Dic3—21no—
251C4⋊Dic3—21no—
252C4⋊Dic3—31no—
253C4⋊Dic3—32no—
254C4⋊Dic3—31no—
255C4⋊Dic3—32no—
256C4⋊Dic3—61no—
257C4⋊Dic3—62no—
258C4⋊Dic3—62no—
259C4⋊Dic3—61no—
260C4⋊Dic3—61no—
261C4⋊Dic3—62no—
262C4⋊Dic3—121no—
263C4⋊Dic3D1211yescharacteristic
264C4⋊Dic3C22×D311yescharacteristic
265Dic3⋊C4—62no—
266Dic3⋊C4—62no—
267Dic3⋊C4D1212yes—
268C4×Dic3—62no—
269C4×Dic3—62no—
270C4×Dic3C22×D312yes—
271C2×C3⋊C8C22×D311yescharacteristic
272C2×C3⋊C8C22×D311yescharacteristic
273D24—62no—
274#204—24no—
275C22×C32C22×D411yescharacteristic
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C2×C3×D3—22no—
280C2×C3×D3—41no—
281C2×C3×D3—41no—
282C32×C4—21no—
283C32×C4AGL(1,ℤ/8)12yes—
284C32×C4C2×#4111yescharacteristic
285C32×C42−1+411yescharacteristic
286C3⋊Dic3—42no—
287C3⋊Dic3—41no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3—42no—
291C3×Dic3—41no—
292C22×Q8—181no—
293C2×SD16—32no—
294C2×D8—91no—
295C4⋊Q8—31no—
296C4⋊Q8—92no—
297C4⋊Q8—91no—
298#126—182no—
299#124—92no—
300C22⋊Q8—182no—
301C22⋊Q8—181no—
302C22⋊Q8—181no—
303C4×Q8—31no—
304C4×Q8—92no—
305C4×D4—31no—
306C4×D4—92no—
307C4×D4—91no—
308#117—181no—
309#107—91no—
310#107—92no—
311C4⋊C8—32no—
312C4⋊C8D3211yescharacteristic
313#103—32no—
314#103—92no—
315#103—92no—
316#102—91no—
317#102—91no—
318#102—92no—
319#97—32no—
320C4×C8—31no—
321C23×C3—61no—
322C22×D3—21no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—121no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—62no—
329C3×D4—32no—
330C3×D4—61no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—62no—
337C2×C3×C4—61no—
338C2×C3×C4—122no—
339C2×C3×C4C2×D1211yescharacteristic
340C2×C3×C4C2×D1211yescharacteristic
341C2×C3×C4D3×D411yescharacteristic
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4C23×D311yescharacteristic
344C2×C3×C4C23×D311yescharacteristic
345C2×Dic3—22no—
346C2×Dic3—21no—
347C2×Dic3—61no—
348C2×Dic3—62no—
349C2×Dic3—61no—
350C2×Dic3—62no—
351C2×Dic3—122no—
352C2×Dic3—121no—
353C2×Dic3—124no—
354C2×Dic3C2×D1212yes—
355C2×Dic3C2×D1212yes—
356D12—21no—
357D12—62no—
358D12—61no—
359D12—61no—
360D12—61no—
361D12—61no—
362D12—121no—
363D12#23712yes—
364C4×D3—42no—
365C4×D3—124no—
366C4×D3—122no—
367C4×D3—122no—
368C4×D3—242no—
369Dic6—22no—
370Dic6—22no—
371Dic6—22no—
372Dic6—124no—
373Dic6—124no—
374Dic6—122no—
375Dic6—122no—
376Dic6—242no—
377C3×C8—22no—
378C3×C8—22no—
379C3×C8—42no—
380C3×C8—61no—
381C3×C8—61no—
382C3⋊C8—21no—
383C3⋊C8—21no—
384C2×C32#54811yescharacteristic
385C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
386C2×C32#57811yescharacteristic
387C3⋊D3—81no—
388C3×D3—42no—
389C2×Q8—31no—
390C2×Q8—32no—
391C2×Q8—182no—
392C2×Q8—181no—
393C2×Q8—364no—
394C2×D4—31no—
395C2×D4—91no—
396C22×C4—61no—
397C22×C4—182no—
398C22×C4—181no—
399SD16—64no—
400D8—182no—
401C2×C8—31no—
402C2×C8—31no—
403C2×C8C2×D3212yes—
404C4⋊C4—31no—
405C4⋊C4—32no—
406C4⋊C4—31no—
407C4⋊C4—61no—
408C4⋊C4—61no—
409C4⋊C4—92no—
410C4⋊C4—91no—
411C4⋊C4—182no—
412C4⋊C4—182no—
413C4⋊C4—181no—
414C22⋊C4—61no—
415C22⋊C4—182no—
416C22⋊C4—182no—
417C22⋊C4—182no—
418C22⋊C4—181no—
419C42—32no—
420C42—182no—
421C42C2×D3211yescharacteristic
422C22×C3—21no—
423C22×C3—62no—
424C22×C3—121no—
425C22×C3—121no—
426C22×C3C22×D1211yescharacteristic
427C22×C3C2×D3×D411yescharacteristic
428C2×D3—22no—
429C2×D3—41no—
430C2×D3—41no—
431C2×D3—122no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—241no—
439C2×D3—241no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—41no—
446C3×C4—62no—
447C3×C4—62no—
448C3×C4—121no—
449C3×C4—122no—
450C3×C4#90812yes—
451C3×C4#91412yes—
452C3×C4C2×#23711yescharacteristic
453C3×C4#100711yescharacteristic
454C3×C4D3×#4111yescharacteristic
455C3×C4#101011yescharacteristic
456Dic3—22no—
457Dic3—22no—
458Dic3—41no—
459Dic3—42no—
460Dic3—122no—
461Dic3—121no—
462Dic3—122no—
463Dic3—121no—
464Dic3—242no—
465Dic3—241no—
466C32#336711yescharacteristic, complemented, Sylow
467C23—61no—
468C23—181no—
469Q8—62no—
470Q8—62no—
471Q8—62no—
472Q8—364no—
473Q8—362no—
474D4—32no—
475D4—61no—
476D4—181no—
477D4—181no—
478C2×C4—32no—
479C2×C4—32no—
480C2×C4—62no—
481C2×C4—61no—
482C2×C4—122no—
483C2×C4—181no—
484C2×C4—182no—
485C2×C4—364no—
486C2×C4—362no—
487C2×C4D3×D1211yescharacteristic
488C2×C4D3×D1211yescharacteristic
489C2×C4C22×D3211yescharacteristic
490C8—22no—
491C8—61no—
492C8—61no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—122no—
497C2×C3#557611yescharacteristic
498C2×C3#559811yescharacteristic
499C2×C3C2×#90811yescharacteristic
500C2×C3#575311yescharacteristic
501C2×C3C2×#91411yescharacteristic
502C2×C3#577911yescharacteristic
503D3—42no—
504D3—242no—
505D3—241no—
506D3—241no—
507C22—62no—
508C22—121no—
509C22—121no—
510C22—361no—
511C22—361no—
512C22—361no—
513C22C2×D3×D1211yescharacteristic
514C4—21no—
515C4—62no—
516C4—62no—
517C4—122no—
518C4—121no—
519C4—362no—
520C4—361no—
521C4#6379412yes—
522C4D3×#23711yescharacteristic
523C4#6431511yescharacteristic
524C3—21no—
525C3#7997011yescharacteristic, complemented
526C3#8041611yescharacteristic, complemented
527C2—122no—
528C2—721no—
529C2#1059438211yescharacteristic
530C2C2×#6379411yescharacteristic
531C2#1059472111yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊C8C22
socleC22×C32C22×D4
prime radicalC2×#3714C22
hypercenterC4⋊C8D32
hypocenterC32#3367
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
1. Dic6×D12C2semidirect
2. #10593865C2semidirect
3. #10592545C2semidirect
4. C3×#4841C2semidirect
5. C3×#4831C2semidirect
6. #10590893C2semidirect
7. #10590804C2semidirect
8. #10590766C2semidirect
11. #10590287C2semidirect
58. #4841D3semidirect
59. #4831D3semidirect
466. C32#3367semidirect
525. C3#79970semidirect
526. C3#80416semidirect

Maximal subgroups

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

subgroupindex
Dic6×D122
#105938652
#105925452
C3×#48412
C3×#48312
#105908932
#105908042
#105907662
#105907542
C4×#37142
#105902872
#804163
#799703