Finite Group Databasebeta

← #13156458

Subgroups — #13156458

Order 1728.

612 conjugacy classes of subgroups.

The subgroup lattice has 542 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723674C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3#11722052C211yesmaximal, characteristic, complemented
4#11722040C211yesmaximal, characteristic, complemented
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63567C211yesmaximal, characteristic, complemented
7C3×#63557C211yesmaximal, characteristic
8#10590675—31nomaximal
9#10590674—31nomaximal
10#10590529—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×Dic3×C3⋊D3—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88815—21no—
20#88814—21no—
21#88807—21no—
22C4⋊C4×C3⋊D3—31no—
23D3×C4⋊Dic3—31no—
24D3×C4⋊Dic3—31no—
25D3×C4⋊Dic3—61no—
26#63871—31no—
27#63857—31no—
28#63856—31no—
29#63644—31no—
30#63643—31no—
31C3×(C2×D3)⋊C8—31no—
32C3×(C2×D3)⋊C8—31no—
33C3×#817—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#808—31no—
37#63573—31no—
38#63573—31no—
39#63573—61no—
40#63573D311yescharacteristic, complemented
41#63567D311yescharacteristic, complemented
42#63566—31no—
43#63566—31no—
44#63566—61no—
45#63557D311yescharacteristic
46C22×C3×C3⋊D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57Dic3×C3⋊D3—42no—
58C32×C3⋊C8—21no—
59C32×C3⋊C8—21no—
60#4862—91no—
61#4805—91no—
62#4721—91no—
63C2×C4×C3⋊D3—61no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×#634C2×D311yescharacteristic
66C2×C3×C4×D3—31no—
67C2×C3×C4×D3—31no—
68C2×C3×C4×D3—61no—
69C2×C3×Dic6—31no—
70C2×C3×Dic6—31no—
71C2×C3×Dic6—61no—
72C2×D3×Dic3—61no—
73C2×D3×Dic3—61no—
74C2×D3×Dic3—62no—
75C2×C32×C8—31no—
76C2×C32×C8—31no—
77C32×C4⋊C4—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—31no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—61no—
84C4⋊(C3⋊Dic3)—121no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic
86#3750—61no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—21no—
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—62no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105#3722—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—61no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3SD1612yes—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119#946—91no—
120C2×C4⋊Dic3—91no—
121#924—91no—
122#897—91no—
123D3×C4⋊C4—91no—
124D3×C4⋊C4—91no—
125D3×C4⋊C4—181no—
126C3×#106—91no—
127C3×#103—91no—
128C3×C22⋊C8—91no—
129#835—91no—
130(C2×D3)⋊C8—31no—
131(C2×D3)⋊C8—31no—
132#817—31no—
133#817—91no—
134#817—91no—
135#817—181no—
136#816—31no—
137#816—91no—
138#816—91no—
139#816—181no—
140#810—31no—
141#808—31no—
142#808—91no—
143C22×C3⋊D3D1211yescharacteristic
144C22×C3×D3—31no—
145C22×C3×D3—31no—
146C22×C3×D3—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4C22×D311yescharacteristic
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C3⋊Dic3—21no—
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—61no—
158C2×C3⋊Dic3—62no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—121no—
162C2×C3⋊Dic3—121no—
163C2×C3⋊Dic3—122no—
164C2×C3⋊Dic3D1211yescharacteristic
165C4×C3⋊D3—21no—
166C4×C3⋊D3—21no—
167C4×C3⋊D3—122no—
168#634—21no—
169#634—21no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—31no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—61no—
178C2×C3×Dic3—61no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—121no—
181C2×C3×Dic3—121no—
182C3×C4×D3—62no—
183C3×C4×D3—62no—
184C3×C4×D3—61no—
185C3×C4×D3—61no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×Dic6—62no—
189C3×Dic6—61no—
190C3×Dic6—61no—
191C3×Dic6—61no—
192C3×Dic6—61no—
193C3×Dic6—121no—
194D3×Dic3—122no—
195D3×Dic3—122no—
196D3×Dic3—124no—
197C32×C8—61no—
198C32×C8—61no—
199C3×C3⋊C8—22no—
200C3×C3⋊C8—21no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—21no—
204C3×C3⋊C8—41no—
205#483—271noSylow
206C2×C33#12311yescharacteristic
207C2×C33C2×SD1611yescharacteristic
208C2×C33#13711yescharacteristic
209C3×C3⋊D3—22no—
210C2×C3×Q8—91no—
211C22×C3×C4—91no—
212C22×Dic3—181no—
213C2×C4×D3—31no—
214C2×C4×D3—31no—
215C2×C4×D3—61no—
216C2×C4×D3—181no—
217C2×C4×D3—181no—
218C2×C4×D3—182no—
219C2×Dic6—31no—
220C2×Dic6—31no—
221C2×Dic6—61no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8—61no—
227C2×C3×C8—61no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—61no—
231C3×C4⋊C4—91no—
232#217—181no—
233(C2×D3)⋊C4—181no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C4⋊Dic3—91no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—181no—
242C4⋊Dic3—181no—
243C4⋊Dic3—181no—
244C4⋊Dic3—181no—
245C4⋊Dic3—181no—
246C4⋊Dic3—361no—
247C4⋊Dic3D3211yescharacteristic
248Dic3⋊C4—181no—
249Dic3⋊C4—181no—
250Dic3⋊C4—181no—
251Dic3⋊C4—181no—
252Dic3⋊C4—182no—
253C2×C3⋊C8D3211yescharacteristic
254C2×C3⋊C8D3211yescharacteristic
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32C2×D1211yescharacteristic
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C2×C3⋊D3—21no—
263C2×C3⋊D3—21no—
264C2×C3⋊D3#20412yes—
265C2×C3×D3—32no—
266C2×C3×D3—32no—
267C2×C3×D3—62no—
268C2×C3×D3—62no—
269C2×C3×D3—62no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—61no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—121no—
283C32×C4#23511yescharacteristic
284C32×C4#23511yescharacteristic
285C32×C4#23711yescharacteristic
286C32×C4#23711yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23911yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—41no—
291C3⋊Dic3—41no—
292C3⋊Dic3—121no—
293C3⋊Dic3—122no—
294C3⋊Dic3—121no—
295C3⋊Dic3—121no—
296C3⋊Dic3—241no—
297C3⋊Dic3—241no—
298C3⋊Dic3—242no—
299C3×Dic3—42no—
300C3×Dic3—41no—
301C3×Dic3—41no—
302C3×Dic3—62no—
303C3×Dic3—61no—
304C3×Dic3—61no—
305C3×Dic3—122no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—241no—
311C22⋊Q8—271no—
312C2×C4⋊C4—271no—
313#106—91no—
314#106—271no—
315#103—91no—
316#103—271no—
317C22⋊C8—91no—
318C33#48311yescharacteristic, complemented, Sylow
319C23×C3—91no—
320C22×D3—31no—
321C22×D3—31no—
322C22×D3—61no—
323C3×Q8—181no—
324C3×Q8—181no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—61no—
331C2×C3×C4—61no—
332C2×C3×C4—91no—
333C2×C3×C4—181no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×C3×C4C2×D3211yescharacteristic
340C2×Dic3—21no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3—61no—
348C2×Dic3—121no—
349C2×Dic3—121no—
350C2×Dic3—182no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—361no—
355C2×Dic3—361no—
356C2×Dic3—362no—
357C2×Dic3—362no—
358C4×D3—62no—
359C4×D3—62no—
360C4×D3—61no—
361C4×D3—61no—
362C4×D3—61no—
363C4×D3—61no—
364C4×D3—362no—
365C4×D3—362no—
366C4×D3—364no—
367Dic6—62no—
368Dic6—61no—
369Dic6—61no—
370Dic6—61no—
371Dic6—61no—
372Dic6—121no—
373C3×C8—62no—
374C3×C8—61no—
375C3×C8—61no—
376C3×C8—61no—
377C3×C8—61no—
378C3×C8—121no—
379C3⋊C8—21no—
380C3⋊C8—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32#88311yescharacteristic
394C2×C32#88611yescharacteristic
395C2×C32#89411yescharacteristic
396C2×C32C2×#20411yescharacteristic
397C2×C32#90911yescharacteristic
398C2×C32D3×SD1611yescharacteristic
399C2×C32D3×SD1611yescharacteristic
400C2×C32#91511yescharacteristic
401C2×C32#91811yescharacteristic
402C3⋊D3—22no—
403C3×D3—64no—
404C3×D3—62no—
405C3×D3—62no—
406C2×Q8—91no—
407C22×C4—91no—
408C22×C4—541no—
409C2×C8—31no—
410C2×C8—31no—
411C4⋊C4—31no—
412C4⋊C4—91no—
413C4⋊C4—271no—
414C4⋊C4—541no—
415C4⋊C4—541no—
416C22⋊C4—541no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—92no—
422C22×C3—181no—
423C22×C3—181no—
424C22×C3D3×D1211yescharacteristic
425C22×C3D3×D1211yescharacteristic
426C22×C3#381111yescharacteristic
427C2×D3—32no—
428C2×D3—32no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—122no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—181no—
448C3×C4—361no—
449C3×C4—361no—
450C3×C4#379511yescharacteristic
451C3×C4#379611yescharacteristic
452C3×C4#379611yescharacteristic
453C3×C4#379811yescharacteristic
454C3×C4#380411yescharacteristic
455C3×C4#380511yescharacteristic
456Dic3—41no—
457Dic3—62no—
458Dic3—61no—
459Dic3—61no—
460Dic3—122no—
461Dic3—121no—
462Dic3—121no—
463Dic3—121no—
464Dic3—121no—
465Dic3—241no—
466Dic3—362no—
467Dic3—361no—
468Dic3—361no—
469Dic3—361no—
470Dic3—721no—
471Dic3—721no—
472Dic3—722no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#472111yescharacteristic, complemented
478C32#480511yescharacteristic, complemented
479C32#486211yescharacteristic, complemented
480C23—91no—
481Q8—181no—
482Q8—181no—
483C2×C4—61no—
484C2×C4—91no—
485C2×C4—181no—
486C2×C4—181no—
487C2×C4—181no—
488C2×C4—181no—
489C2×C4—541no—
490C2×C4—1082no—
491C2×C4D3311yescharacteristic
492C8—61no—
493C8—61no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—182no—
507C2×C3D3×#20411yescharacteristic
508C2×C3D3×#20411yescharacteristic
509C2×C3#6380011yescharacteristic
510C2×C3#6380111yescharacteristic
511C2×C3#6385311yescharacteristic
512C2×C3#6389011yescharacteristic
513C2×C3#6389111yescharacteristic
514C2×C3#6393011yescharacteristic
515C2×C3#6395111yescharacteristic
516D3—64no—
517D3—62no—
518D3—62no—
519C22—92no—
520C22—181no—
521C22—181no—
522C22#8895511yescharacteristic
523C4—121no—
524C4—181no—
525C4—361no—
526C4—361no—
527C4—1081no—
528C4#8896011yescharacteristic
529C4#8896411yescharacteristic
530C3—21no—
531C3—21no—
532C3—21no—
533C3—41no—
534C3#1059052911yescharacteristic, complemented
535C3#1059067411yescharacteristic, complemented
536C3#1059067511yescharacteristic, complemented
537C2—182no—
538C2#1172284211yescharacteristic
539C2#1172289611yescharacteristic
540C2#1172299711yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#483
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
3. #11722052C2semidirect
4. #11722040C2semidirect
5. C3×#63573C2semidirect
6. C3×#63567C2semidirect
40. #63573D3semidirect
41. #63567D3semidirect
318. C33#483semidirect
477. C32#4721semidirect
478. C32#4805semidirect
479. C32#4862semidirect
534. C3#10590529semidirect
535. C3#10590674semidirect
536. C3#10590675semidirect

Maximal subgroups

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

subgroupindex
#117236742
C3⋊D3×C4⋊Dic32
#117220522
#117220402
C3×#635732
C3×#635672
C3×#635572
#105906753
#105906743
#105905293