Finite Group Databasebeta

← #13178311

Subgroups — #13178311

Order 1728.

1364 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89040C212yesmaximal, complemented
2C3×#64106C212yesmaximal, complemented
3C3×#64096C211yesmaximal, characteristic, complemented
4C4⋊#6296C212yesmaximal, complemented
5C4×#6295C211yesmaximal, characteristic
6#11723692C214yesmaximal, complemented
7#11723685C212yesmaximal
8#11723678C211yesmaximal, characteristic, complemented
9#10594526—31nomaximal
10#10594383—32nomaximal
11C2×C3×C4×C3⋊D3C2212yes—
12C2×C3×#634C2212yes—
13C2×C3×#634C2211yescharacteristic
14C2×#6296C2214yes—
15C2×#6295C2211yescharacteristic
16C2×#6294C2214yes—
17#89040—28no—
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752C2214yes—
20C3×C4⋊(C3⋊Dic3)C2214yes—
21C3×#3750C2212yes—
22C3×C4×C3⋊Dic3C2211yescharacteristic
23C3×C4×C3⋊Dic3C2212yes—
24#88815C2212yes—
25#88815C2212yes—
26#88815C2211yescharacteristic
27#88814C2212yes—
28C2×#3797—32no—
29C2×D3×Dic6—32no—
30C2×D3×Dic6—32no—
31#64106D312yescomplemented
32#64096D311yescharacteristic, complemented
33C3×#897—32no—
34C3×#897—32no—
35C3×#897—62no—
36C3×#887—32no—
37C3×#887—61no—
38#63919—32no—
39#63918—34no—
40C4×#626—32no—
41#63902—32no—
42C4×#625—31no—
43#63899—34no—
44#63895—34no—
45#63893—34no—
46#63880—34no—
47#63873—31no—
48#63866—32no—
49C22×C3×C3⋊D3C2312yes—
50C2×C33×C4C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3×C3⋊Dic3C2311yescharacteristic
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2314yes—
55C2×C3×C3⋊Dic3C2312yes—
56C2×C3×C3⋊Dic3C2311yescharacteristic
57C3×C4×C3⋊D3—24no—
58C3×#634—24no—
59C3×#634D414yescomplemented
60#6296—28no—
61#6295Q814yescomplemented
62#6294—28no—
63#5632—92no—
64Q8×D12—91no—
65C2×C4×C3⋊D3C2×D312yes—
66C2×#634C2×D312yes—
67C2×#634C2×D311yescharacteristic
68C2×C3×C4×D3—32no—
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—62no—
71C2×C3×Dic6—32no—
72C2×C3×Dic6—32no—
73C2×C3×Dic6—32no—
74C2×C3×Dic6—61no—
75C2×C3×Dic6—62no—
76C2×#627—34no—
77C2×#627—34no—
78C2×#627—34no—
79C2×#626—32no—
80C2×#625—31no—
81C2×#624—34no—
82C2×D3×Dic3—34no—
83C2×D3×Dic3—34no—
84#3797—68no—
85D3×Dic6—68no—
86D3×Dic6—68no—
87C32×C4⋊C4—21no—
88C32×C4⋊C4—22no—
89C32×C4⋊C4—41no—
90C32×C4⋊C4C2×D312yes—
91C32×C4⋊C4C2×D311yescharacteristic
92#3752C2×D314yes—
93C4⋊(C3⋊Dic3)C2×D314yes—
94#3750C2×D312yes—
95C4×C3⋊Dic3C2×D311yescharacteristic
96C4×C3⋊Dic3C2×D312yes—
97C3×(C2×D3)⋊C4—34no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—64no—
100C3×C4⋊Dic3—34no—
101C3×C4⋊Dic3—34no—
102C3×C4⋊Dic3—64no—
103C3×Dic3⋊C4—34no—
104C3×Dic3⋊C4—62no—
105C3×C4×Dic3—32no—
106C3×C4×Dic3—34no—
107C3×C4×Dic3—62no—
108C3×C4×Dic3—61no—
109#3724—32no—
110#3724—32no—
111#3724—31no—
112Dic3⋊Dic3—34no—
113Dic3⋊Dic3—34no—
114Dic3⋊Dic3—32no—
115#3722—32no—
116(C2×D3)⋊Dic3—32no—
117(C2×D3)⋊Dic3—32no—
118C22×C33C2411yescharacteristic
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3C2×Q814yescomplemented
122C33×C4—22no—
123C33×C4C2×D412yes—
124C3×C3⋊Dic3—24no—
125C3×C3⋊Dic3—22no—
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3C2×D414yes—
128C3×C3⋊Dic3C2×Q812yescomplemented
129C2×D3×Q8—92no—
130C2×D3×Q8—92no—
131C22×Dic6—92no—
132C3×C22⋊Q8—92no—
133C3×C4×Q8—91no—
134#944—92no—
135C4×#67—92no—
136#897—32no—
137#897—32no—
138#897—62no—
139#897—94no—
140#896—94no—
141#888—94no—
142#887—32no—
143#887—61no—
144#67⋊C4—94no—
145#878—94no—
146C4×D12—92no—
147C4×D12—91no—
148#869—92no—
149#869—91no—
150C22×C3⋊D3C22×D312yes—
151C22×C3×D3—32no—
152C22×C3×D3—32no—
153C22×C3×D3—62no—
154C2×C32×C4—21no—
155C2×C32×C4—22no—
156C2×C32×C4—22no—
157C2×C32×C4—24no—
158C2×C32×C4—41no—
159C2×C32×C4—42no—
160C2×C32×C4C22×D314yes—
161C2×C32×C4C22×D312yes—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D312yes—
164C2×C3⋊Dic3C22×D311yescharacteristic
165C2×C3⋊Dic3C22×D312yes—
166C2×C3⋊Dic3C22×D314yes—
167C2×C3⋊Dic3C22×D312yes—
168C2×C3⋊Dic3C22×D311yescharacteristic
169C4×C3⋊D3—24no—
170#634—24no—
171#634D1214yescomplemented
172C2×C3×Dic3—32no—
173C2×C3×Dic3—34no—
174C2×C3×Dic3—32no—
175C2×C3×Dic3—32no—
176C2×C3×Dic3—32no—
177C2×C3×Dic3—34no—
178C2×C3×Dic3—34no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3—64no—
184C3×C4×D3—64no—
185C3×C4×D3—64no—
186C3×C4×D3—124no—
187C3×Dic6—38no—
188C3×Dic6—64no—
189C3×Dic6—64no—
190C3×Dic6—64no—
191C3×Dic6—124no—
192#627—68no—
193#627—68no—
194#627—68no—
195#626—38no—
196#625—34no—
197#624—68no—
198D3×Dic3—68no—
199D3×Dic3—68no—
200D4×Q8—271noSylow
201C2×C33C22×D411yescharacteristic
202C2×C33C22×Q811yescharacteristic
203C2×C332−1+411yescharacteristic
204C3×C3⋊D3—24no—
205C2×C3×Q8—91no—
206C2×C3×Q8—92no—
207C22×C3×C4—92no—
208C2×#67—92no—
209C22×Dic3—94no—
210D3×Q8—188no—
211D3×Q8—188no—
212C2×D12—91no—
213C2×C4×D3—32no—
214C2×C4×D3—32no—
215C2×C4×D3—62no—
216C2×C4×D3—94no—
217C2×C4×D3—94no—
218C2×Dic6—32no—
219C2×Dic6—32no—
220C2×Dic6—32no—
221C2×Dic6—61no—
222C2×Dic6—62no—
223C2×Dic6—94no—
224C2×Dic6—94no—
225C2×Dic6—94no—
226C2×Dic6—188no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—94no—
231C3×C4⋊C4—92no—
232C3×C4⋊C4D3212yes—
233C3×C4⋊C4D3211yescharacteristic
234C3×C22⋊C4—94no—
235C3×C42—92no—
236C3×C42—91no—
237#217—92no—
238(C2×D3)⋊C4—34no—
239(C2×D3)⋊C4—34no—
240(C2×D3)⋊C4—64no—
241(C2×D3)⋊C4—92no—
242(C2×D3)⋊C4—92no—
243C4⋊Dic3—34no—
244C4⋊Dic3—34no—
245C4⋊Dic3—64no—
246C4⋊Dic3—91no—
247C4⋊Dic3—92no—
248C4⋊Dic3—92no—
249Dic3⋊C4—34no—
250Dic3⋊C4—62no—
251Dic3⋊C4—94no—
252Dic3⋊C4—94no—
253Dic3⋊C4—92no—
254C4×Dic3—34no—
255C4×Dic3—32no—
256C4×Dic3—62no—
257C4×Dic3—61no—
258C22×C32—21no—
259C22×C32—22no—
260C22×C32—41no—
261C22×C32C23×D311yescharacteristic
262C22×C32C23×D312yes—
263C2×C3⋊D3—22no—
264C2×C3⋊D3—22no—
265C2×C3⋊D3D3×Q814yescomplemented
266C2×C3×D3—34no—
267C2×C3×D3—34no—
268C2×C3×D3—64no—
269C2×C3×D3—62no—
270C2×C3×D3—62no—
271C2×C3×D3—62no—
272C2×C3×D3—62no—
273C2×C3×D3—122no—
274C2×C3×D3—122no—
275C32×C4—24no—
276C32×C4—22no—
277C32×C4—24no—
278C32×C4—22no—
279C32×C4—42no—
280C32×C4—42no—
281C32×C4—44no—
282C32×C4—82no—
283C32×C4C2×D1212yes—
284C32×C4D3×D414yes—
285C3⋊Dic3—24no—
286C3⋊Dic3—22no—
287C3⋊Dic3—21no—
288C3⋊Dic3C2×D1214yes—
289C3⋊Dic3D3×Q812yescomplemented
290C3×Dic3—38no—
291C3×Dic3—34no—
292C3×Dic3—64no—
293C3×Dic3—62no—
294C3×Dic3—64no—
295C3×Dic3—62no—
296C3×Dic3—64no—
297C3×Dic3—62no—
298C3×Dic3—62no—
299C3×Dic3—124no—
300C3×Dic3—122no—
301C3×Dic3—121no—
302C22×Q8—272no—
303C4⋊Q8—272no—
304C4⋊Q8—271no—
305C22⋊Q8—92no—
306C22⋊Q8—274no—
307C4×Q8—91no—
308C4×D4—272no—
309C4×D4—271no—
310C33D4×Q811yescharacteristic, complemented, Sylow
311C23×C3—92no—
312C22×D3—32no—
313C22×D3—32no—
314C22×D3—62no—
315C3×Q8—94no—
316C3×Q8—184no—
317C2×C3×C4—21no—
318C2×C3×C4—22no—
319C2×C3×C4—22no—
320C2×C3×C4—24no—
321C2×C3×C4—41no—
322C2×C3×C4—42no—
323C2×C3×C4—91no—
324C2×C3×C4—92no—
325C2×C3×C4—94no—
326C2×C3×C4—92no—
327C2×C3×C4—91no—
328C2×C3×C4—184no—
329C2×C3×C4C2×D3214yes—
330C2×C3×C4C2×D3212yes—
331C2×C3×C4C2×D3212yes—
332C2×C3×C4C2×D3211yescharacteristic
333#67—98no—
334C2×Dic3—32no—
335C2×Dic3—34no—
336C2×Dic3—32no—
337C2×Dic3—34no—
338C2×Dic3—32no—
339C2×Dic3—32no—
340C2×Dic3—34no—
341C2×Dic3—61no—
342C2×Dic3—62no—
343C2×Dic3—61no—
344C2×Dic3—62no—
345C2×Dic3—64no—
346C2×Dic3—188no—
347D12—94no—
348C4×D3—64no—
349C4×D3—64no—
350C4×D3—124no—
351C4×D3—188no—
352C4×D3—188no—
353Dic6—38no—
354Dic6—64no—
355Dic6—64no—
356Dic6—64no—
357Dic6—124no—
358Dic6—188no—
359Dic6—188no—
360Dic6—188no—
361C2×C32—22no—
362C2×C32—21no—
363C2×C32—21no—
364C2×C32—22no—
365C2×C32—21no—
366C2×C32—22no—
367C2×C32—41no—
368C2×C32—41no—
369C2×C32—41no—
370C2×C32C22×D1211yescharacteristic
371C2×C32C2×D3×D412yes—
372C2×C32C2×D3×Q812yes—
373C2×C32C2×D3×Q811yescharacteristic
374C2×C32#101012yes—
375C2×C32#101011yescharacteristic
376C3⋊D3—24no—
377C3×D3—64no—
378C3×D3—64no—
379C3×D3—124no—
380C2×Q8—91no—
381C2×Q8—92no—
382C2×Q8—274no—
383C2×Q8—548no—
384C2×D4—271no—
385C22×C4—92no—
386C22×C4—274no—
387C4⋊C4—92no—
388C4⋊C4—94no—
389C4⋊C4—272no—
390C4⋊C4—271no—
391C4⋊C4—272no—
392C4⋊C4#111311yescharacteristic
393C22⋊C4—94no—
394C22⋊C4—272no—
395C42—92no—
396C42—91no—
397C22×C3—21no—
398C22×C3—22no—
399C22×C3—41no—
400C22×C3—94no—
401C22×C3—182no—
402C22×C3—182no—
403C22×C3C22×D3212yes—
404C22×C3C22×D3211yescharacteristic
405C2×D3—34no—
406C2×D3—34no—
407C2×D3—64no—
408C2×D3—62no—
409C2×D3—62no—
410C2×D3—62no—
411C2×D3—62no—
412C2×D3—122no—
413C2×D3—122no—
414C3×C4—24no—
415C3×C4—22no—
416C3×C4—22no—
417C3×C4—24no—
418C3×C4—42no—
419C3×C4—44no—
420C3×C4—42no—
421C3×C4—82no—
422C3×C4—94no—
423C3×C4—92no—
424C3×C4—182no—
425C3×C4—184no—
426C3×C4—181no—
427C3×C4D3×D1214yes—
428C3×C4#380212yes—
429Dic3—38no—
430Dic3—34no—
431Dic3—64no—
432Dic3—62no—
433Dic3—64no—
434Dic3—62no—
435Dic3—64no—
436Dic3—62no—
437Dic3—62no—
438Dic3—124no—
439Dic3—122no—
440Dic3—121no—
441C32—22no—
442C32—21no—
443C32—41no—
444C32Q8×D1211yescharacteristic, complemented
445C32#563212yescomplemented
446C23—92no—
447Q8—94no—
448Q8—184no—
449Q8—548no—
450D4—274no—
451C2×C4—91no—
452C2×C4—92no—
453C2×C4—91no—
454C2×C4—94no—
455C2×C4—92no—
456C2×C4—184no—
457C2×C4—548no—
458C2×C4C2×#111312yes—
459C2×C4C2×#111311yescharacteristic
460C2×C3—21no—
461C2×C3—22no—
462C2×C3—21no—
463C2×C3—22no—
464C2×C3—21no—
465C2×C3—22no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—184no—
470C2×C3C2×D3×D1212yes—
471C2×C3C2×#380211yescharacteristic
472C2×C3#6430912yes—
473C2×C3#6430911yescharacteristic
474C2×C3D32×Q812yes—
475C2×C3D32×Q811yescharacteristic
476D3—64no—
477D3—64no—
478D3—124no—
479C22—94no—
480C22—182no—
481C22—182no—
482C22C22×#111311yescharacteristic
483C4—22no—
484C4—94no—
485C4—92no—
486C4—182no—
487C4—184no—
488C4—181no—
489C4C4⋊#111312yes—
490C3—22no—
491C3—21no—
492C3—41no—
493C3#1059438312yescomplemented
494C3#1059452611yescharacteristic, complemented
495C2—184no—
496C2C2×C4⋊#111311yescharacteristic
497C2#1172462011yescharacteristic
498C2Q8×#111311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC2×#6295C22
hypercenterC4⋊C4#1113
hypocenterC33D4×Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#89040C2semidirect
2. C3×#64106C2semidirect
3. C3×#64096C2semidirect
4. C4⋊#6296C2semidirect
6. #11723692C2semidirect
8. #11723678C2semidirect
31. #64106D3semidirect
32. #64096D3semidirect
59. C3×#634D4semidirect
61. #6295Q8semidirect
121. C2×C3×C3⋊D3C2×Q8semidirect
128. C3×C3⋊Dic3C2×Q8semidirect
171. #634D12semidirect
265. C2×C3⋊D3D3×Q8semidirect
289. C3⋊Dic3D3×Q8semidirect
310. C33D4×Q8semidirect
444. C32Q8×D12semidirect
445. C32#5632semidirect
493. C3#10594383semidirect
494. C3#10594526semidirect

Maximal subgroups

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

subgroupindex
C2×#890402
C3×#641062
C3×#640962
C4⋊#62962
C4×#62952
#117236922
#117236852
#117236782
#105945263
#105943833