Finite Group Databasebeta

← #13178309

Subgroups — #13178309

Order 1728.

1436 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89041C212yesmaximal, complemented
2C3×#64106C212yesmaximal, complemented
3C3×#64102C211yesmaximal, characteristic, complemented
4#11723695C212yesmaximal, complemented
5C4×#6295C211yesmaximal, characteristic, complemented
6#11723690C214yesmaximal, complemented
7#11723689C212yesmaximal, complemented
8#11723678C211yesmaximal, characteristic, complemented
9#10594530—31nomaximal
10#10594381—32nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×C3×C4×C3⋊D3C2212yes—
14C2×C3×#634C2212yes—
15C2×#6295C2214yes—
16C2×#6295C2211yescharacteristic
17C2×#6294C2214yes—
18#89041—28no—
19C33×C4⋊C4C2211yescharacteristic
20C3×#3752C2212yes—
21C3×#3752C2214yes—
22C3×C4⋊(C3⋊Dic3)C2214yes—
23C3×C4×C3⋊Dic3C2211yescharacteristic
24#88815C2212yes—
25#88815C2212yes—
26#88815C2211yescharacteristic
27#88814C2212yes—
28C2×#3799—32no—
29C2×#3796—32no—
30C2×#3795—32no—
31#64106D312yescomplemented
32#64102D311yescharacteristic, complemented
33C3×#897—32no—
34C3×#897—32no—
35C3×#897—62no—
36C3×#893—32no—
37C3×#893—61no—
38#63912—32no—
39#63906—34no—
40C4×#626—32no—
41C4×#625—31no—
42#63900—32no—
43#63891—34no—
44#63890—34no—
45D3×C4⋊Dic3—34no—
46#63873—31no—
47#63866—32no—
48#63854—34no—
49C22×C3×C3⋊D3C2312yes—
50C22×C3×C3⋊D3C2312yes—
51C2×C33×C4C2312yes—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3C2311yescharacteristic
54C2×C3×C3⋊Dic3C2312yes—
55C2×C3×C3⋊Dic3C2314yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4⋊(C3⋊D3)D414yescomplemented
58C3×C4×C3⋊D3—24no—
59C3×C4×C3⋊D3—24no—
60C3×#634—24no—
61#6295—28no—
62#6295—22no—
63#6294—28no—
64#5631—92no—
65#5555—91no—
66C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
67C2×C4×C3⋊D3C2×D312yes—
68C2×C4×C3⋊D3C2×D312yes—
69C2×#634C2×D312yes—
70C2×C3×D12—32no—
71C2×C3×D12—61no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—32no—
74C2×C3×C4×D3—32no—
75C2×C3×C4×D3—62no—
76C2×C3×C4×D3—62no—
77C2×C3×Dic6—32no—
78C2×C3×Dic6—32no—
79C2×C3×Dic6—62no—
80C2×#626—34no—
81C2×#626—34no—
82C2×#626—32no—
83C2×#625—34no—
84C2×#625—31no—
85C2×#624—34no—
86C2×D3×Dic3—34no—
87C2×D3×Dic3—34no—
88#3799—68no—
89#3796—68no—
90#3795—68no—
91C32×C4⋊C4—21no—
92C32×C4⋊C4—22no—
93C32×C4⋊C4—41no—
94C32×C4⋊C4C2×D312yes—
95C32×C4⋊C4C2×D311yescharacteristic
96#3752C2×D312yes—
97#3752C2×D314yes—
98C4⋊(C3⋊Dic3)C2×D314yes—
99C4×C3⋊Dic3C2×D311yescharacteristic
100C3×(C2×D3)⋊C4—34no—
101C3×(C2×D3)⋊C4—34no—
102C3×(C2×D3)⋊C4—34no—
103C3×(C2×D3)⋊C4—62no—
104C3×(C2×D3)⋊C4—64no—
105C3×C4⋊Dic3—34no—
106C3×C4⋊Dic3—34no—
107C3×C4⋊Dic3—64no—
108C3×C4×Dic3—32no—
109C3×C4×Dic3—61no—
110#3724—32no—
111#3724—32no—
112#3724—31no—
113Dic3⋊Dic3—34no—
114Dic3⋊Dic3—34no—
115Dic3⋊Dic3—32no—
116#3722—32no—
117(C2×D3)⋊Dic3—32no—
118(C2×D3)⋊Dic3—32no—
119C22×C33C2411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—22no—
123C2×C3×C3⋊D3—22no—
124C2×C3×C3⋊D3—22no—
125C2×C3×C3⋊D3C2×D414yes—
126C33×C4—22no—
127C33×C4C2×D412yes—
128C3×C3⋊Dic3—22no—
129C3×C3⋊Dic3—24no—
130C3×C3⋊Dic3—21no—
131C3×C3⋊Dic3#4112yescomplemented
132C2×#239—92no—
133C2×#237—92no—
134C2×#235—92no—
135C3×C22⋊Q8—92no—
136C3×C4×D4—91no—
137#944—92no—
138#930—92no—
139C4×#67—92no—
140C2×C4⋊Dic3—92no—
141#897—32no—
142#897—32no—
143#897—62no—
144#894—94no—
145#893—32no—
146#893—61no—
147D3×C4⋊C4—94no—
148#886—94no—
149#884—94no—
150#883—94no—
151C4×D12—91no—
152#869—91no—
153C22×C3⋊D3C22×D312yes—
154C22×C3⋊D3C22×D312yes—
155C22×C3×D3—34no—
156C22×C3×D3—32no—
157C22×C3×D3—32no—
158C22×C3×D3—62no—
159C22×C3×D3—62no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—24no—
163C2×C32×C4—22no—
164C2×C32×C4—41no—
165C2×C32×C4—42no—
166C2×C32×C4C22×D312yes—
167C2×C32×C4C22×D314yes—
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D312yes—
170C2×C3⋊Dic3C22×D311yescharacteristic
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D314yes—
173C2×C3⋊Dic3C22×D311yescharacteristic
174C4⋊(C3⋊D3)D1214yescomplemented
175C4×C3⋊D3—24no—
176C4×C3⋊D3—24no—
177#634—24no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—34no—
183C2×C3×Dic3—34no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—62no—
187C2×C3×Dic3—64no—
188C3×D12—38no—
189C3×D12—64no—
190C3×C4×D3—68no—
191C3×C4×D3—64no—
192C3×C4×D3—64no—
193C3×C4×D3—124no—
194C3×C4×D3—124no—
195C3×Dic6—64no—
196C3×Dic6—64no—
197C3×Dic6—124no—
198#626—64no—
199#626—68no—
200#626—68no—
201#625—68no—
202#625—62no—
203#624—68no—
204D3×Dic3—68no—
205D3×Dic3—68no—
206#547—271noSylow
207C2×C33C22×D411yescharacteristic
208C2×C33C2×#4111yescharacteristic
209C2×C332−1+411yescharacteristic
210C3×C3⋊D3—24no—
211C3×C3⋊D3—42no—
212C2×C3×Q8—92no—
213C2×C3×D4—91no—
214C22×C3×C4—92no—
215C22×C3×C4—92no—
216C2×#67—94no—
217C2×#67—92no—
218C2×#67—94no—
219C22×Dic3—94no—
220#239—188no—
221#237—188no—
222#235—188no—
223C2×D12—32no—
224C2×D12—61no—
225C2×D12—91no—
226C2×D12—94no—
227C2×C4×D3—32no—
228C2×C4×D3—34no—
229C2×C4×D3—32no—
230C2×C4×D3—62no—
231C2×C4×D3—62no—
232C2×C4×D3—94no—
233C2×C4×D3—94no—
234C2×Dic6—32no—
235C2×Dic6—32no—
236C2×Dic6—62no—
237C3×C4⋊C4—21no—
238C3×C4⋊C4—22no—
239C3×C4⋊C4—41no—
240C3×C4⋊C4—94no—
241C3×C4⋊C4D3212yes—
242C3×C4⋊C4D3211yescharacteristic
243C3×C22⋊C4—94no—
244C3×C22⋊C4—92no—
245C3×C42—91no—
246#217—92no—
247(C2×D3)⋊C4—34no—
248(C2×D3)⋊C4—34no—
249(C2×D3)⋊C4—34no—
250(C2×D3)⋊C4—62no—
251(C2×D3)⋊C4—64no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—92no—
254C4⋊Dic3—34no—
255C4⋊Dic3—34no—
256C4⋊Dic3—64no—
257C4⋊Dic3—92no—
258C4⋊Dic3—92no—
259C4⋊Dic3—91no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—92no—
263C4×Dic3—32no—
264C4×Dic3—61no—
265C22×C32—22no—
266C22×C32—21no—
267C22×C32—41no—
268C22×C32C23×D311yescharacteristic
269C22×C32C23×D312yes—
270C2×C3⋊D3—22no—
271C2×C3⋊D3—22no—
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3⋊D3C2×D1214yes—
276C2×C3×D3—38no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—62no—
286C2×C3×D3—62no—
287C2×C3×D3—122no—
288C2×C3×D3—122no—
289C2×C3×D3—122no—
290C2×C3×D3—122no—
291C32×C4—22no—
292C32×C4—24no—
293C32×C4—24no—
294C32×C4—22no—
295C32×C4—42no—
296C32×C4—42no—
297C32×C4—44no—
298C32×C4—82no—
299C32×C4C2×D1212yes—
300C32×C4D3×D414yes—
301C3⋊Dic3—22no—
302C3⋊Dic3—24no—
303C3⋊Dic3—21no—
304C3⋊Dic3#23712yescomplemented
305C3×Dic3—34no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—64no—
309C3×Dic3—62no—
310C3×Dic3—64no—
311C3×Dic3—62no—
312C3×Dic3—124no—
313C3×Dic3—122no—
314C3×Dic3—121no—
315C2×#41—272no—
316C4⋊Q8—271no—
317#123—274no—
318C22⋊Q8—92no—
319#121—272no—
320C4×D4—91no—
321C4×D4—271no—
322C2×C4⋊C4—272no—
323C33#54711yescharacteristic, complemented, Sylow
324C23×C3—92no—
325C23×C3—92no—
326C22×D3—34no—
327C22×D3—32no—
328C22×D3—32no—
329C22×D3—62no—
330C22×D3—62no—
331C3×Q8—184no—
332C3×D4—94no—
333C2×C3×C4—21no—
334C2×C3×C4—22no—
335C2×C3×C4—24no—
336C2×C3×C4—22no—
337C2×C3×C4—41no—
338C2×C3×C4—42no—
339C2×C3×C4—91no—
340C2×C3×C4—94no—
341C2×C3×C4—92no—
342C2×C3×C4—91no—
343C2×C3×C4—184no—
344C2×C3×C4—184no—
345C2×C3×C4C2×D3212yes—
346C2×C3×C4C2×D3214yes—
347C2×C3×C4C2×D3212yes—
348C2×C3×C4C2×D3211yescharacteristic
349#67—184no—
350#67—188no—
351#67—188no—
352C2×Dic3—32no—
353C2×Dic3—32no—
354C2×Dic3—34no—
355C2×Dic3—32no—
356C2×Dic3—32no—
357C2×Dic3—34no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—62no—
361C2×Dic3—64no—
362C2×Dic3—188no—
363D12—38no—
364D12—64no—
365D12—182no—
366D12—188no—
367C4×D3—68no—
368C4×D3—64no—
369C4×D3—64no—
370C4×D3—124no—
371C4×D3—124no—
372C4×D3—188no—
373C4×D3—188no—
374Dic6—64no—
375Dic6—64no—
376Dic6—124no—
377C2×C32—22no—
378C2×C32—21no—
379C2×C32—22no—
380C2×C32—21no—
381C2×C32—22no—
382C2×C32—21no—
383C2×C32—41no—
384C2×C32—41no—
385C2×C32—41no—
386C2×C32C22×D1211yescharacteristic
387C2×C32C2×D3×D412yes—
388C2×C32C2×#23711yescharacteristic
389C2×C32#100712yes—
390C2×C32D3×#4112yes—
391C2×C32#101011yescharacteristic
392C3⋊D3—24no—
393C3⋊D3—42no—
394C3×D3—68no—
395C3×D3—124no—
396C3×D3—124no—
397C3×D3—122no—
398C3×D3—122no—
399#41—548no—
400C2×Q8—92no—
401C2×D4—91no—
402C2×D4—271no—
403C2×D4—274no—
404C22×C4—92no—
405C22×C4—92no—
406C22×C4—274no—
407C4⋊C4—94no—
408C4⋊C4—271no—
409C4⋊C4—272no—
410C4⋊C4—272no—
411C4⋊C4#111311yescharacteristic
412C22⋊C4—92no—
413C22⋊C4—94no—
414C22⋊C4—272no—
415C42—91no—
416C22×C3—22no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3—94no—
420C22×C3—182no—
421C22×C3—182no—
422C22×C3—182no—
423C22×C3—182no—
424C22×C3—182no—
425C22×C3C22×D3212yes—
426C22×C3C22×D3211yescharacteristic
427C2×D3—38no—
428C2×D3—64no—
429C2×D3—64no—
430C2×D3—64no—
431C2×D3—64no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—62no—
435C2×D3—62no—
436C2×D3—62no—
437C2×D3—62no—
438C2×D3—122no—
439C2×D3—122no—
440C2×D3—122no—
441C2×D3—122no—
442C3×C4—24no—
443C3×C4—22no—
444C3×C4—24no—
445C3×C4—22no—
446C3×C4—42no—
447C3×C4—44no—
448C3×C4—42no—
449C3×C4—82no—
450C3×C4—92no—
451C3×C4—182no—
452C3×C4—184no—
453C3×C4—181no—
454C3×C4D3×D1214yes—
455C3×C4#380212yes—
456Dic3—34no—
457Dic3—62no—
458Dic3—64no—
459Dic3—62no—
460Dic3—64no—
461Dic3—62no—
462Dic3—62no—
463Dic3—124no—
464Dic3—122no—
465Dic3—121no—
466C32—22no—
467C32—21no—
468C32—41no—
469C32#555511yescharacteristic, complemented
470C32#563112yescomplemented
471C23—92no—
472C23—92no—
473Q8—184no—
474D4—94no—
475D4—542no—
476D4—548no—
477C2×C4—91no—
478C2×C4—91no—
479C2×C4—94no—
480C2×C4—92no—
481C2×C4—184no—
482C2×C4—184no—
483C2×C4—548no—
484C2×C4C2×#111312yes—
485C2×C4C2×#111311yescharacteristic
486C2×C3—22no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—22no—
490C2×C3—21no—
491C2×C3—22no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—184no—
496C2×C3—362no—
497C2×C3C2×D3×D1212yes—
498C2×C3C2×#380211yescharacteristic
499C2×C3D3×#23712yes—
500C2×C3#6431211yescharacteristic
501C2×C3#6431512yes—
502C2×C3#6431611yescharacteristic
503D3—68no—
504D3—124no—
505D3—124no—
506D3—122no—
507D3—122no—
508C22—94no—
509C22—182no—
510C22—182no—
511C22—182no—
512C22—182no—
513C22—182no—
514C22C22×#111311yescharacteristic
515C4—22no—
516C4—92no—
517C4—182no—
518C4—184no—
519C4—181no—
520C4C4⋊#111312yes—
521C3—21no—
522C3—22no—
523C3—41no—
524C3#1059438112yescomplemented
525C3#1059453011yescharacteristic, complemented
526C2—184no—
527C2—362no—
528C2C2×C4⋊#111311yescharacteristic
529C2#1172462211yescharacteristic
530C2#1172462411yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
hypercenterC4⋊C4#1113
hypocenterC33#547
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89041C2semidirect
2. C3×#64106C2semidirect
3. C3×#64102C2semidirect
4. #11723695C2semidirect
5. C4×#6295C2semidirect
6. #11723690C2semidirect
7. #11723689C2semidirect
8. #11723678C2semidirect
31. #64106D3semidirect
32. #64102D3semidirect
57. C3×C4⋊(C3⋊D3)D4semidirect
131. C3×C3⋊Dic3#41semidirect
174. C4⋊(C3⋊D3)D12semidirect
304. C3⋊Dic3#237semidirect
323. C33#547semidirect
469. C32#5555semidirect
470. C32#5631semidirect
524. C3#10594381semidirect
525. C3#10594530semidirect

Maximal subgroups

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

subgroupindex
C2×#890412
C3×#641062
C3×#641022
#117236952
C4×#62952
#117236902
#117236892
#117236782
#105945303
#105943813