Finite Group Databasebeta

← #11866356

Subgroups — #11866356

Order 1152.

1109 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#64151C211yesmaximal, characteristic, complemented
2C3×#5812C211yesmaximal, characteristic, complemented
3C22×C3×#215C211yesmaximal, characteristic, complemented
4#10595042C211yesmaximal, characteristic
5#10591617C214yesmaximal, complemented
6#10591604C214yesmaximal, complemented
7#10591116C211yesmaximal, characteristic, complemented
8#10591105C211yesmaximal, characteristic, complemented
9#10591085C211yesmaximal, characteristic, complemented
10#83849—31nomaximal
11#82551—31nomaximal
12C22×C32×Q8C2211yescharacteristic
13C2×#3828C2211yescharacteristic
14C22×C3×D12C2211yescharacteristic
15C2×#3756C2212yescomplemented
16#64151—24no—
17C2×#3755C2212yescomplemented
18C2×#3748C2211yescharacteristic
19C3×Q8×Dic3—22no—
20C3×Q8×Dic3C2214yescomplemented
21C2×C3×#215—24no—
22C2×C3×#215C2214yes—
23C3×#926—22no—
24C3×#926C2211yescharacteristic
25C3×#926C2211yescharacteristic
26C22×C3×C3⋊C8C2211yescharacteristic
27#63977—22no—
28#63916—22no—
29#63911C2212yes—
30#63860C2212yes—
31Dic3⋊(C3⋊C8)C2214yescomplemented
32#63567C2212yescomplemented
33#63564C2212yescomplemented
34#63562C2214yes—
35C22×C3×SD16—31no—
36C3×#519—31no—
37#5812D311yescharacteristic, complemented
38C2×#942—31no—
39C2×#942—31no—
40C2×#942—61no—
41C22×#215D311yescharacteristic, complemented
42#5805—31no—
43#5525—31no—
44#5215—31no—
45#5166—34no—
46#5163—34no—
47#5113—31no—
48#5102—31no—
49#5028—34no—
50#5027—34no—
51#5002—31no—
52#5001—31no—
53#4999—31no—
54C23×C3×D3—21no—
55C2×C32×Q8—21no—
56C2×C32×Q8—22no—
57C2×C32×Q8D414yescomplemented
58C2×C32×Q8C2312yes—
59C22×C32×C4—21no—
60C22×C32×C4C2311yescharacteristic
61C2×#638—21no—
62#3828—22no—
63#3828—22no—
64C2×C4⋊(C3⋊D3)C2311yescharacteristic
65C2×C4×C3⋊D3—21no—
66C2×#634C2311yescharacteristic
67C2×C3×D12—22no—
68C2×C3×D12—22no—
69C2×C3×D12—21no—
70C2×C3×D12C2312yes—
71C2×#626—24no—
72#3756—24no—
73#3755—24no—
74#3748—22no—
75C2×C3⋊(C3⋊C8)C2312yes—
76C3×#217—21no—
77C3×#217D412yes—
78C3×#215—28no—
79C3×C4⋊Dic3—24no—
80C3×C4⋊Dic3D412yes—
81C3×C4⋊Dic3C2312yes—
82C3×C4×Dic3—21no—
83C3×C4×Dic3—24no—
84C3×C4×Dic3C2312yes—
85C2×C3×C3⋊C8—22no—
86C2×C3×C3⋊C8C2312yes—
87(C2×D3)⋊Dic3—24no—
88#3043—91noSylow
89C22×C3×Q8—21no—
90C22×C3×Q8C2×D311yescharacteristic
91C22×C3×Q8C2×D311yescharacteristic
92C22×C3×D4—31no—
93C2×#235—31no—
94C2×#235—31no—
95C2×#235—61no—
96C22×D12C2×D311yescharacteristic
97C2×C3×SD16—34no—
98C2×C3×SD16—64no—
99C22×C3×C8—31no—
100C3×C4×Q8—34no—
101C3×C4×Q8—62no—
102C3×#117—31no—
103C3×#117—31no—
104C3×#117—62no—
105#953—62no—
106Q8×Dic3—22no—
107Q8×Dic3C2×D314yescomplemented
108C2×#216—32no—
109C2×#216—32no—
110C2×#216—62no—
111#942—68no—
112#942—64no—
113#942—64no—
114C2×#215—24no—
115C2×#215—32no—
116C2×#215—32no—
117C2×#215—62no—
118C2×#215C2×D314yes—
119#940—32no—
120#938—62no—
121#936—32no—
122#926—22no—
123#926C2×D311yescharacteristic
124#926C2×D311yescharacteristic
125C2×#208—31no—
126C2×#208—31no—
127C2×#208—61no—
128C22×C3⋊C8C2×D311yescharacteristic
129#895—62no—
130#882—62no—
131#875—32no—
132#874—32no—
133#832—34no—
134#821—32no—
135#816—32no—
136Dic3⋊C8—34no—
137#810—32no—
138#809—34no—
139#809—32no—
140C4⋊(C3⋊C8)—34no—
141C23×C32C2×D411yescharacteristic
142C22×C3⋊D3—21no—
143C22×C3×D3—22no—
144C22×C3×D3—42no—
145C22×C3×D3—82no—
146C32×Q8—24no—
147C32×Q8—22no—
148C32×Q8C2×D414yes—
149C2×C32×C4—24no—
150C2×C32×C4—22no—
151C2×C32×C4C2×D412yes—
152C2×C32×C4C2×D411yescharacteristic
153C2×C32×C4#4112yes—
154C2×C32×C4C2411yescharacteristic
155#638—42no—
156C2×C3⋊Dic3—21no—
157C4⋊(C3⋊D3)—21no—
158C4⋊(C3⋊D3)—21no—
159C4×C3⋊D3—42no—
160#634—21no—
161#634—21no—
162C2×C3×Dic3—22no—
163C2×C3×Dic3C2×D414yes—
164C3×D12—22no—
165C3×D12—22no—
166C3×D12—42no—
167#626—84no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—22no—
170C2×#137—91no—
171C22×SD16—31no—
172#535—91no—
173#519—31no—
174#464—94no—
175#460—94no—
176#424—91no—
177#418—91no—
178#416—91no—
179C24×C3—61no—
180C23×D3—21no—
181C2×C3×Q8—22no—
182C2×C3×Q8—21no—
183C2×C3×Q8—22no—
184C2×C3×Q8—21no—
185C2×C3×Q8—24no—
186C2×C3×Q8—22no—
187C2×C3×Q8—42no—
188C2×C3×Q8—41no—
189C2×C3×Q8D1214yescomplemented
190C2×C3×Q8#6714yescomplemented
191C2×C3×Q8C22×D312yes—
192C2×C3×Q8C22×D312yes—
193C2×C3×D4—32no—
194C2×C3×D4—62no—
195C2×C3×D4—62no—
196C2×C3×D4—61no—
197C22×C3×C4—21no—
198C22×C3×C4—21no—
199C22×C3×C4—21no—
200C22×C3×C4—41no—
201C22×C3×C4C22×D311yescharacteristic
202C22×C3×C4C22×D311yescharacteristic
203C2×#67—61no—
204C2×#67—61no—
205C2×#67—64no—
206C2×#67—121no—
207#235—64no—
208#235—62no—
209#235—62no—
210#235—62no—
211#235—62no—
212#235—122no—
213C2×D12—22no—
214C2×D12—22no—
215C2×D12—21no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—61no—
219C2×D12—64no—
220C2×D12C22×D312yes—
221C2×C4×D3—61no—
222C2×C4×D3—61no—
223C2×C4×D3—121no—
224C2×Dic6—31no—
225C2×Dic6—31no—
226C2×Dic6—61no—
227C3×SD16—68no—
228C2×C3×C8—32no—
229C2×C3×C8—62no—
230C3×C4⋊C4—32no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—64no—
233C3×C22⋊C4—32no—
234C3×C22⋊C4—61no—
235C3×C42—32no—
236C3×C42—64no—
237C3×C42—61no—
238#217—21no—
239#217—64no—
240#217#6712yes—
241#216—68no—
242#216—64no—
243#216—64no—
244#215—28no—
245#215—68no—
246#215—64no—
247#215—64no—
248(C2×D3)⋊C4—64no—
249C4⋊Dic3—24no—
250C4⋊Dic3#6712yes—
251C4⋊Dic3C22×D312yes—
252C4×Dic3—21no—
253C4×Dic3—24no—
254C4×Dic3C22×D312yes—
255#208—64no—
256#208—62no—
257#208—62no—
258C2×C3⋊C8—22no—
259C2×C3⋊C8—32no—
260C2×C3⋊C8—32no—
261C2×C3⋊C8—62no—
262C2×C3⋊C8C22×D312yes—
263C22×C32—22no—
264C22×C32#12112yes—
265C22×C32C22×D411yescharacteristic
266C2×C3⋊D3—41no—
267C2×C3⋊D3—42no—
268C2×C3×D3—42no—
269C2×C3×D3—44no—
270C2×C3×D3—84no—
271C32×C4—24no—
272C32×C4#12112yes—
273C32×C4C22×D411yescharacteristic
274C32×C4C2×#4111yescharacteristic
275C3⋊Dic3—41no—
276C3×Dic3—24no—
277C3×Dic3—42no—
278C2×#41—91no—
279C22×Q8D3211yescharacteristic
280C22×D4—31no—
281#137—184no—
282C2×Q16—92no—
283C2×SD16—34no—
284C2×SD16—64no—
285C2×SD16—92no—
286C2×M16—91no—
287C22×C8—31no—
288#127—92no—
289#124—92no—
290#121—182no—
291#120—182no—
292C4×Q8—34no—
293C4×Q8—62no—
294#117—31no—
295#117—31no—
296#117—62no—
297C4⋊C8—94no—
298#103—92no—
299#102—92no—
300#102—94no—
301C23×C3—21no—
302C23×C3—62no—
303C23×C3—122no—
304C23×C3—242no—
305C23×C3D3×D411yescharacteristic
306C23×C3C2×#6711yescharacteristic
307C22×D3—22no—
308C22×D3—42no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—82no—
312C22×D3—121no—
313C3×Q8—24no—
314C3×Q8—24no—
315C3×Q8—22no—
316C3×Q8—22no—
317C3×Q8—24no—
318C3×Q8—42no—
319C3×Q8—44no—
320C3×Q8C2×D1214yes—
321C3×Q8C2×#6714yes—
322C3×D4—62no—
323C3×D4—62no—
324C3×D4—122no—
325C2×C3×C4—24no—
326C2×C3×C4—24no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—34no—
334C2×C3×C4—42no—
335C2×C3×C4—44no—
336C2×C3×C4—62no—
337C2×C3×C4C2×D1212yes—
338C2×C3×C4#23512yes—
339C2×C3×C4D3×D411yescharacteristic
340C2×C3×C4#23712yes—
341C2×C3×C4C2×#6712yes—
342C2×C3×C4C2×#6711yescharacteristic
343C2×C3×C4C23×D311yescharacteristic
344C2×C3×C4C23×D311yescharacteristic
345#67—124no—
346#67—122no—
347#67—122no—
348#67—244no—
349C2×Dic3—22no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—121no—
353C2×Dic3C2×#6714yes—
354D12—22no—
355D12—22no—
356D12—42no—
357D12—62no—
358D12—61no—
359D12—61no—
360D12—61no—
361D12—61no—
362D12—121no—
363D12—244no—
364C4×D3—124no—
365C4×D3—122no—
366C4×D3—122no—
367Dic6—62no—
368Dic6—61no—
369Dic6—61no—
370Dic6—61no—
371Dic6—61no—
372Dic6—121no—
373C3×C8—62no—
374C3⋊C8—22no—
375C3⋊C8—64no—
376C3⋊C8—62no—
377C3⋊C8—62no—
378C2×C32—22no—
379C2×C32C2×#12111yescharacteristic
380C2×C32#57712yes—
381C3⋊D3—81no—
382C3×D3—82no—
383C24—61no—
384#41—182no—
385#41—182no—
386C2×Q8—22no—
387C2×Q8—21no—
388C2×Q8—91no—
389C2×Q8#62614yescomplemented
390C2×Q8C2×D3212yes—
391C2×D4—32no—
392C2×D4—62no—
393C2×D4—62no—
394C2×D4—61no—
395C2×D4—91no—
396C2×D4—184no—
397C2×D4—181no—
398C22×C4—21no—
399C22×C4—181no—
400C22×C4C2×D3211yescharacteristic
401Q16—184no—
402SD16—68no—
403SD16—184no—
404M16—182no—
405C2×C8—32no—
406C2×C8—62no—
407C2×C8—92no—
408C4⋊C4—32no—
409C4⋊C4—32no—
410C4⋊C4—64no—
411C22⋊C4—32no—
412C22⋊C4—61no—
413C22⋊C4—184no—
414C42—32no—
415C42—64no—
416C42—61no—
417C22×C3—22no—
418C22×C3—22no—
419C22×C3—21no—
420C22×C3—22no—
421C22×C3—42no—
422C22×C3—122no—
423C22×C3—124no—
424C22×C3—244no—
425C22×C3#93012yes—
426C22×C3#93912yes—
427C22×C3C2×D3×D411yescharacteristic
428C22×C3C22×#6711yescharacteristic
429C2×D3—44no—
430C2×D3—42no—
431C2×D3—84no—
432C2×D3—122no—
433C2×D3—122no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—122no—
437C2×D3—242no—
438C3×C4—24no—
439C3×C4—24no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—22no—
443C3×C4—44no—
444C3×C4—64no—
445C3×C4—122no—
446C3×C4#93012yes—
447C3×C4#93912yes—
448C3×C4C22×D1211yescharacteristic
449C3×C4C2×#23511yescharacteristic
450C3×C4C2×#23711yescharacteristic
451C3×C4C22×#6711yescharacteristic
452Dic3—24no—
453Dic3—42no—
454Dic3—122no—
455Dic3—121no—
456Dic3—121no—
457C32#304311yescharacteristic, complemented, Sylow
458C23—62no—
459C23—122no—
460C23—181no—
461C23—242no—
462C23D3×#6711yescharacteristic
463Q8—22no—
464Q8—24no—
465Q8—181no—
466Q8—181no—
467Q8C2×#62614yes—
468D4—62no—
469D4—62no—
470D4—122no—
471D4—181no—
472D4—181no—
473D4—362no—
474D4—724no—
475C2×C4—22no—
476C2×C4—24no—
477C2×C4—34no—
478C2×C4—62no—
479C2×C4—181no—
480C2×C4—362no—
481C2×C4#380412yes—
482C2×C4C2×#62612yes—
483C2×C4D3×#6711yescharacteristic
484C2×C4C22×D3211yescharacteristic
485C8—62no—
486C8—182no—
487C2×C3—22no—
488C2×C3—22no—
489C2×C3—21no—
490C2×C3—22no—
491C2×C3—42no—
492C2×C3—242no—
493C2×C3#576212yes—
494C2×C3C2×#93011yescharacteristic
495C2×C3C2×#93911yescharacteristic
496C2×C3#583712yes—
497D3—82no—
498D3—242no—
499D3—241no—
500D3—241no—
501C22—22no—
502C22—122no—
503C22—124no—
504C22—244no—
505C22—361no—
506C22—362no—
507C22#6397812yes—
508C22C2×D3×#6711yescharacteristic
509C4—24no—
510C4—64no—
511C4—122no—
512C4—361no—
513C4#6397812yes—
514C4C2×#380411yescharacteristic
515C4C22×#62611yescharacteristic
516C3—21no—
517C3#8255111yescharacteristic, complemented
518C3#8384911yescharacteristic, complemented
519C2—22no—
520C2—242no—
521C2—721no—
522C2#1059531812yes—
523C2C2×#6397811yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC22×C32×Q8C22
socleC22×C32C22×D4
prime radical#10595042C2
hypercenterC22×Q8D32
hypocenterC32#3043
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#64151C2semidirect
2. C3×#5812C2semidirect
3. C22×C3×#215C2semidirect
5. #10591617C2semidirect
6. #10591604C2semidirect
7. #10591116C2semidirect
8. #10591105C2semidirect
9. #10591085C2semidirect
15. C2×#3756C22semidirect
17. C2×#3755C22semidirect
20. C3×Q8×Dic3C22semidirect
31. Dic3⋊(C3⋊C8)C22semidirect
32. #63567C22semidirect
33. #63564C22semidirect
37. #5812D3semidirect
41. C22×#215D3semidirect
57. C2×C32×Q8D4semidirect
107. Q8×Dic3C2×D3semidirect
189. C2×C3×Q8D12semidirect
190. C2×C3×Q8#67semidirect
389. C2×Q8#626semidirect
457. C32#3043semidirect
517. C3#82551semidirect
518. C3#83849semidirect

Maximal subgroups

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

subgroupindex
C2×#641512
C3×#58122
C22×C3×#2152
#105950422
#105916172
#105916042
#105911162
#105911052
#105910852
#838493
#825513