Finite Group Databasebeta

← #13161981

Subgroups — #13161981

Order 1728.

1092 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 1092 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊#1113C211yesmaximal, characteristic, complemented
2C3×#64147C211yesmaximal, characteristic, complemented
3C2×C3×#3753C212yesmaximal, complemented
4#11722075C212yesmaximal, complemented
5#11722072C211yesmaximal, characteristic, complemented
6#10591516—31nomaximal
7#10591515—32nomaximal
8C22×#1113—21no—
9C2×C33×D4C2211yescharacteristic, complemented
10C2×C3×#638—21no—
11C2×C3×C4⋊(C3⋊D3)C2212yes—
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×#6295—21no—
14C4⋊#1113—24no—
15C3×#3757—21no—
16C3×#3753—24no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic, complemented
18C2×C3×C3⋊(C3⋊C8)C2212yescomplemented
19C2×#3802—31no—
20C2×D3×D12—32no—
21#64147D311yescharacteristic, complemented
22C2×#3753D312yescomplemented
23C3×#938—32no—
24C3×#938—61no—
25C2×C3×#213—32no—
26C2×C3×#213—32no—
27C2×C3×#213—62no—
28#63564—32no—
29#63563—32no—
30#63562—32no—
31#63557—31no—
32(C2×D3)⋊(C3⋊C8)—32no—
33C23×C33—21no—
34C22×C3×C3⋊D3—22no—
35C22×C3×C3⋊D3D411yescharacteristic
36C2×#1113—24no—
37C2×#1113—42no—
38C2×#1113—42no—
39C33×D4—22no—
40C2×C33×C4C2311yescharacteristic
41C3×#638—42no—
42C2×C3×C3⋊Dic3—21no—
43C2×C3×C3⋊Dic3D411yescharacteristic
44C3×C4⋊(C3⋊D3)—22no—
45C3×C4⋊(C3⋊D3)D414yescomplemented
46C3×C4×C3⋊D3—22no—
47#6295—24no—
48C3×C3⋊(C3⋊C8)—22no—
49#5156—92no—
50#5036—91no—
51C22×D32—62no—
52C22×D32—61no—
53C2×C32×D4—22no—
54C2×C32×D4—21no—
55C2×C32×D4—41no—
56C2×C32×D4C2×D311yescharacteristic, complemented
57C2×C32×D4C2×D312yescomplemented
58C2×#638—21no—
59C2×C4⋊(C3⋊D3)C2×D312yes—
60C2×C4×C3⋊D3C2×D311yescharacteristic
61C2×C3×#67—62no—
62C2×C3×#67—62no—
63C2×C3×D12—32no—
64C2×C3×D12—32no—
65C2×C3×D12—62no—
66C2×C3×C4×D3—32no—
67C2×C3×C4×D3—61no—
68C2×#626—62no—
69C2×#625—61no—
70#3802—64no—
71D3×D12—64no—
72D3×D12—64no—
73#3757—21no—
74#3753—24no—
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
76C2×C3⋊(C3⋊C8)C2×D312yescomplemented
77C3×#217—62no—
78C3×#217—62no—
79C3×#213—64no—
80C3×#213—64no—
81C3×#213—68no—
82C3×C4⋊Dic3—32no—
83C3×C4⋊Dic3—61no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—32no—
86C2×C3×C3⋊C8—62no—
87C22×C33—41no—
88C22×C33—41no—
89C22×C33—41no—
90C22×C33C2×D411yescharacteristic
91C2×C3×C3⋊D3—21no—
92C2×C3×C3⋊D3—21no—
93C2×C3×C3⋊D3—24no—
94C2×C3×C3⋊D3—42no—
95C2×C3×C3⋊D3—42no—
96C2×C3×C3⋊D3D812yescomplemented
97#1113—48no—
98C33×C4C2×D412yes—
99C3×C3⋊Dic3—21no—
100C3×C3⋊Dic3—41no—
101C2×D3×D4—92no—
102C22×D12—91no—
103C2×C3×D8—92no—
104C3×#121—91no—
105#938—32no—
106#938—61no—
107C2×#213—32no—
108C2×#213—32no—
109C2×#213—62no—
110#832—92no—
111#830—91no—
112#821—92no—
113(C2×D3)⋊C8—92no—
114#809—92no—
115C23×C32—21no—
116C23×C32—22no—
117C23×C32—22no—
118C23×C32—42no—
119C23×C32—42no—
120C22×C3⋊D3—22no—
121C22×C3⋊D3#6711yescharacteristic
122C22×C3×D3—32no—
123C22×C3×D3—61no—
124C22×C3×D3—62no—
125C22×C3×D3—62no—
126C22×C3×D3—122no—
127C2×D32—68no—
128C2×D32—64no—
129C2×D32—124no—
130C2×D32—124no—
131C2×D32—122no—
132C2×D32—122no—
133C32×D4—22no—
134C32×D4—24no—
135C32×D4—22no—
136C32×D4—22no—
137C32×D4—42no—
138C32×D4—42no—
139C32×D4—44no—
140C2×C32×C4—22no—
141C2×C32×C4—21no—
142C2×C32×C4—41no—
143C2×C32×C4C22×D311yescharacteristic
144C2×C32×C4C22×D312yes—
145#638—42no—
146C2×C3⋊Dic3—21no—
147C2×C3⋊Dic3#6711yescharacteristic
148C4⋊(C3⋊D3)—22no—
149C4⋊(C3⋊D3)#6714yescomplemented
150C4×C3⋊D3—22no—
151C3×#67—124no—
152C3×#67—124no—
153C2×C3×Dic3—32no—
154C2×C3×Dic3—61no—
155C2×C3×Dic3—62no—
156C2×C3×Dic3—62no—
157C3×D12—34no—
158C3×D12—34no—
159C3×D12—64no—
160C3×D12—62no—
161C3×D12—62no—
162C3×D12—122no—
163C3×C4×D3—62no—
164C3×C4×D3—62no—
165C3×C4×D3—122no—
166#626—68no—
167#625—64no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—64no—
170C3×C3⋊C8—62no—
171C3×C3⋊C8—62no—
172#447—271noSylow
173C2×C33—81no—
174C2×C33#12011yescharacteristic
175C2×C33C2×D811yescharacteristic
176C2×C33AGL(1,ℤ/8)11yescharacteristic
177C3×C3⋊D3—22no—
178C3×C3⋊D3—44no—
179C23×D3—182no—
180C23×D3—181no—
181C2×C3×D4—22no—
182C2×C3×D4—21no—
183C2×C3×D4—41no—
184C2×C3×D4—92no—
185C2×C3×D4—181no—
186C2×C3×D4D3211yescharacteristic, complemented
187C2×C3×D4D3212yescomplemented
188C22×C3×C4—91no—
189C2×#67—62no—
190C2×#67—62no—
191C2×#67—182no—
192D3×D4—184no—
193D3×D4—184no—
194C2×D12—32no—
195C2×D12—32no—
196C2×D12—62no—
197C2×D12—184no—
198C2×D12—181no—
199C2×C4×D3—32no—
200C2×C4×D3—61no—
201C3×D8—184no—
202C2×C3×C8—92no—
203C3×C4⋊C4—91no—
204C3×C22⋊C4—181no—
205#217—62no—
206#217—62no—
207#213—64no—
208#213—64no—
209#213—68no—
210C4⋊Dic3—32no—
211C4⋊Dic3—61no—
212C2×C3⋊C8—32no—
213C2×C3⋊C8—32no—
214C2×C3⋊C8—62no—
215C22×C32—22no—
216C22×C32—21no—
217C22×C32—41no—
218C22×C32—44no—
219C22×C32—42no—
220C22×C32—44no—
221C22×C32—41no—
222C22×C32—42no—
223C22×C32—41no—
224C22×C32—41no—
225C22×C32—42no—
226C22×C32—42no—
227C22×C32—42no—
228C22×C32—42no—
229C22×C32—82no—
230C22×C32—82no—
231C22×C32—82no—
232C22×C32—82no—
233C22×C32D3×D412yes—
234C22×C32C2×#6711yescharacteristic
235C2×C3⋊D3—21no—
236C2×C3⋊D3—21no—
237C2×C3⋊D3—24no—
238C2×C3⋊D3—42no—
239C2×C3⋊D3—42no—
240C2×C3⋊D3#21312yescomplemented
241C2×C3×D3—34no—
242C2×C3×D3—62no—
243C2×C3×D3—62no—
244C2×C3×D3—62no—
245C2×C3×D3—64no—
246C2×C3×D3—64no—
247C2×C3×D3—121no—
248C2×C3×D3—121no—
249C2×C3×D3—124no—
250C2×C3×D3—122no—
251C2×C3×D3—122no—
252C2×C3×D3—122no—
253C2×C3×D3—122no—
254C2×C3×D3—242no—
255C2×C3×D3—242no—
256D32—1216no—
257D32—128no—
258C32×C4—22no—
259C32×C4—22no—
260C32×C4—22no—
261C32×C4—42no—
262C32×C4D3×D412yes—
263C32×C4D3×D412yes—
264C32×C4C2×#6712yes—
265C3⋊Dic3—21no—
266C3⋊Dic3—41no—
267C3×Dic3—62no—
268C3×Dic3—121no—
269C3×Dic3—122no—
270C3×Dic3—122no—
271C22×D4—271no—
272C2×D8—92no—
273#121—91no—
274#102—272no—
275C22⋊C8—271no—
276C33#44711yescharacteristic, complemented, Sylow
277C23×C3—22no—
278C23×C3—22no—
279C23×C3—21no—
280C23×C3—42no—
281C23×C3—42no—
282C23×C3—91no—
283C23×C3—182no—
284C22×D3—32no—
285C22×D3—61no—
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—122no—
289C22×D3—184no—
290C22×D3—188no—
291C22×D3—364no—
292C22×D3—364no—
293C22×D3—362no—
294C22×D3—362no—
295C3×D4—24no—
296C3×D4—22no—
297C3×D4—22no—
298C3×D4—22no—
299C3×D4—42no—
300C3×D4—42no—
301C3×D4—44no—
302C3×D4—94no—
303C3×D4—182no—
304C3×D4—362no—
305C2×C3×C4—22no—
306C2×C3×C4—21no—
307C2×C3×C4—41no—
308C2×C3×C4—91no—
309C2×C3×C4—182no—
310C2×C3×C4—181no—
311C2×C3×C4C2×D3211yescharacteristic
312C2×C3×C4C2×D3212yes—
313#67—124no—
314#67—124no—
315#67—188no—
316C2×Dic3—32no—
317C2×Dic3—61no—
318C2×Dic3—62no—
319C2×Dic3—62no—
320D12—34no—
321D12—34no—
322D12—64no—
323D12—62no—
324D12—62no—
325D12—122no—
326D12—184no—
327C4×D3—62no—
328C4×D3—62no—
329C4×D3—122no—
330C3×C8—182no—
331C3⋊C8—64no—
332C3⋊C8—62no—
333C3⋊C8—62no—
334C2×C32—22no—
335C2×C32—22no—
336C2×C32—21no—
337C2×C32—21no—
338C2×C32—21no—
339C2×C32—22no—
340C2×C32—41no—
341C2×C32—41no—
342C2×C32—41no—
343C2×C32—82no—
344C2×C32—84no—
345C2×C32—84no—
346C2×C32—82no—
347C2×C32—81no—
348C2×C32D3×D812yes—
349C2×C32#91112yes—
350C2×C32C2×#21311yescharacteristic
351C2×C32#93712yes—
352C2×C32#93711yescharacteristic
353C2×C32#94911yescharacteristic
354C3⋊D3—22no—
355C3⋊D3—44no—
356C3×D3—64no—
357C3×D3—122no—
358C3×D3—124no—
359C3×D3—124no—
360C3×D3—244no—
361C24—541no—
362C2×D4—92no—
363C2×D4—181no—
364C2×D4—541no—
365C2×D4—544no—
366C2×D4#111311yescharacteristic, complemented
367C22×C4—91no—
368D8—184no—
369C2×C8—92no—
370C4⋊C4—91no—
371C22⋊C4—181no—
372C22×C3—21no—
373C22×C3—22no—
374C22×C3—42no—
375C22×C3—44no—
376C22×C3—42no—
377C22×C3—42no—
378C22×C3—44no—
379C22×C3—42no—
380C22×C3—42no—
381C22×C3—41no—
382C22×C3—41no—
383C22×C3—42no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—82no—
387C22×C3—82no—
388C22×C3—82no—
389C22×C3—82no—
390C22×C3—92no—
391C22×C3—181no—
392C22×C3—181no—
393C22×C3—184no—
394C22×C3—362no—
395C22×C3—362no—
396C22×C3D3×#6712yes—
397C22×C3#381111yescharacteristic
398C2×D3—34no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—64no—
403C2×D3—64no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—124no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—122no—
410C2×D3—122no—
411C2×D3—242no—
412C2×D3—242no—
413C2×D3—3616no—
414C2×D3—368no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—181no—
420C3×C4—361no—
421C3×C4D3×#6712yes—
422C3×C4D3×#6712yes—
423C3×C4#381112yes—
424Dic3—62no—
425Dic3—121no—
426Dic3—122no—
427Dic3—122no—
428C32—22no—
429C32—21no—
430C32—41no—
431C32#503611yescharacteristic, complemented
432C32#515612yescomplemented
433C23—21no—
434C23—91no—
435C23—182no—
436C23—544no—
437C23—1082no—
438C23—1082no—
439D4—22no—
440D4—94no—
441D4—182no—
442D4—362no—
443D4—544no—
444C2×C4—91no—
445C2×C4—182no—
446C2×C4—181no—
447C2×C4C2×#111311yescharacteristic
448C8—182no—
449C2×C3—21no—
450C2×C3—22no—
451C2×C3—21no—
452C2×C3—22no—
453C2×C3—21no—
454C2×C3—22no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—41no—
458C2×C3—82no—
459C2×C3—84no—
460C2×C3—84no—
461C2×C3—82no—
462C2×C3—81no—
463C2×C3—182no—
464C2×C3—364no—
465C2×C3D3×#21312yes—
466C2×C3#6392512yes—
467C2×C3#6392611yescharacteristic
468C2×C3#6392711yescharacteristic
469C2×C3#6398312yes—
470C2×C3#6398311yescharacteristic
471D3—64no—
472D3—122no—
473D3—124no—
474D3—124no—
475D3—244no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—92no—
480C22—181no—
481C22—181no—
482C22—184no—
483C22—362no—
484C22—362no—
485C22—1088no—
486C22#8904911yescharacteristic
487C4—181no—
488C4—361no—
489C4#8904912yes—
490C3—21no—
491C3—22no—
492C3—41no—
493C3#1059151512yescomplemented
494C3#1059151611yescharacteristic, complemented
495C2—81no—
496C2—182no—
497C2—364no—
498C2#1172370211yescharacteristic
499C2#1172370311yescharacteristic
500C2#1172372911yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4#1113
hypocenterC33#447
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×C4⋊#1113C2semidirect
2. C3×#64147C2semidirect
3. C2×C3×#3753C2semidirect
4. #11722075C2semidirect
5. #11722072C2semidirect
9. C2×C33×D4C22semidirect
17. C3×C4⋊(C3⋊Dic3)C22semidirect
18. C2×C3×C3⋊(C3⋊C8)C22semidirect
21. #64147D3semidirect
22. C2×#3753D3semidirect
45. C3×C4⋊(C3⋊D3)D4semidirect
56. C2×C32×D4C2×D3semidirect
57. C2×C32×D4C2×D3semidirect
75. C4⋊(C3⋊Dic3)C2×D3semidirect
76. C2×C3⋊(C3⋊C8)C2×D3semidirect
96. C2×C3×C3⋊D3D8semidirect
149. C4⋊(C3⋊D3)#67semidirect
186. C2×C3×D4D32semidirect
187. C2×C3×D4D32semidirect
240. C2×C3⋊D3#213semidirect
276. C33#447semidirect
366. C2×D4#1113semidirect
431. C32#5036semidirect
432. C32#5156semidirect
493. C3#10591515semidirect
494. C3#10591516semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊#11132
C3×#641472
C2×C3×#37532
#117220752
#117220722
#105915163
#105915153