Finite Group Databasebeta

← #13155488

Subgroups — #13155488

Order 1728.

662 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723667C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic
3C2×C3×#3712C211yesmaximal, characteristic
4#11722055C211yesmaximal, characteristic, complemented
5#11722052C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63573C211yesmaximal, characteristic, complemented
8#10590683—31nomaximal
9#10590681—31nomaximal
10#10590583—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88813—21no—
21#88807—21no—
22C3×#3712C414yescomplemented
23C4⋊C4×C3⋊D3—31no—
24C2×C3×ΓL(1,𝔽25)—31no—
25C2×C3×ΓL(1,𝔽25)—31no—
26D3×C4⋊Dic3—31no—
27D3×C4⋊Dic3—31no—
28D3×C4⋊Dic3—61no—
29#63849—31no—
30#63841—31no—
31#63841—31no—
32C2×#3712D311yescharacteristic
33#63645—31no—
34#63644—31no—
35C3×#818—31no—
36C3×#817—31no—
37C3×#808—31no—
38C3×#807—31no—
39#63575—31no—
40#63575—31no—
41#63575—61no—
42#63575D311yescharacteristic, complemented
43#63573—31no—
44#63573—31no—
45#63573—61no—
46#63573D311yescharacteristic, complemented
47C22×C3×C3⋊D3D411yescharacteristic
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D411yescharacteristic
53C3×C4×C3⋊D3C2×C412yes—
54C3×C4×C3⋊D3Q812yes—
55C2×C32×Dic3—21no—
56Dic3×C3⋊D3—42no—
57C32×C3⋊C8C2×C412yescomplemented
58C32×C3⋊C8C2×C412yescomplemented
59#5121—91no—
60#4881—91no—
61#4861—91no—
62C2×C4×C3⋊D3—61no—
63C2×C4×C3⋊D3C2×D311yescharacteristic
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—31no—
66C2×C3×C4×D3—61no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—61no—
69C2×D3×Dic3—62no—
70C2×C32×C8—31no—
71C2×C32×C8—31no—
72C32×C4⋊C4—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—31no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—61no—
79C4⋊(C3⋊Dic3)—121no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82#3750—61no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C2×C3×C3⋊C8—21no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C3×ΓL(1,𝔽25)—34no—
99C3×ΓL(1,𝔽25)—34no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—62no—
103#3722—61no—
104(C2×D3)⋊Dic3—61no—
105(C2×D3)⋊Dic3—61no—
106Dic32—61no—
107Dic32—61no—
108Dic32—61no—
109#3712Dic314yescomplemented
110C22×C33C2×D411yescharacteristic
111C2×C3×C3⋊D3—21no—
112C2×C3×C3⋊D3—21no—
113C2×C3×C3⋊D3C4⋊C412yes—
114C33×C4C22×C411yescharacteristic
115C33×C4C2×Q811yescharacteristic
116C3⋊(C3⋊Dic3)—41no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C3×C3⋊Dic3C4⋊C412yes—
120C32×Dic3—41no—
121C2×C3×M16—91no—
122#926—91no—
123C2×C4⋊Dic3—91no—
124C2×ΓL(1,𝔽25)—31no—
125C2×ΓL(1,𝔽25)—31no—
126#892—91no—
127#892—91no—
128D3×C4⋊C4—91no—
129D3×C4⋊C4—91no—
130D3×C4⋊C4—181no—
131C3×#107—91no—
132C3×#106—91no—
133#818—31no—
134#818—91no—
135#818—91no—
136#818—181no—
137#817—31no—
138#817—91no—
139#817—91no—
140#817—181no—
141#808—31no—
142#808—91no—
143#807—31no—
144#807—91no—
145C22×C3⋊D3D1211yescharacteristic
146C22×C3×D3—31no—
147C22×C3×D3—31no—
148C22×C3×D3—61no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—21no—
152C2×C32×C4—41no—
153C2×C32×C4—61no—
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C3⋊Dic3—21no—
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3—61no—
160C2×C3⋊Dic3—62no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—121no—
164C2×C3⋊Dic3—121no—
165C2×C3⋊Dic3—122no—
166C2×C3⋊Dic3D1211yescharacteristic
167C4×C3⋊D3—122no—
168C4×C3⋊D3Dic612yes—
169C4×C3⋊D3C2×Dic312yes—
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—32no—
183C3×C4×D3—32no—
184C3×C4×D3—32no—
185C3×C4×D3—32no—
186C3×C4×D3—62no—
187C3×C4×D3—62no—
188D3×Dic3—124no—
189D3×Dic3—122no—
190D3×Dic3—122no—
191C32×C8—32no—
192C32×C8—32no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8C4×D312yescomplemented
196C3×C3⋊C8C4×D312yescomplemented
197C3×C3⋊C8C2×Dic312yescomplemented
198C3×C3⋊C8C2×Dic312yescomplemented
199#428—271noSylow
200C2×C33C2×C4⋊C411yescharacteristic
201C2×C33AGL(1,ℤ/8)11yescharacteristic
202C2×C33#13711yescharacteristic
203C3×C3⋊D3—22no—
204C22×C3×C4—91no—
205C22×Dic3—181no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—61no—
209C2×C4×D3—181no—
210C2×C4×D3—181no—
211C2×C4×D3—182no—
212C3×M16—94no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—91no—
223C3×C4⋊C4—91no—
224#217—181no—
225(C2×D3)⋊C4—181no—
226(C2×D3)⋊C4—181no—
227C4⋊Dic3—31no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—31no—
231C4⋊Dic3—61no—
232C4⋊Dic3—61no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—181no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—181no—
240C4⋊Dic3—361no—
241C4⋊Dic3D3211yescharacteristic
242Dic3⋊C4—181no—
243Dic3⋊C4—181no—
244Dic3⋊C4—182no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—181no—
248C2×C3⋊C8D3211yescharacteristic
249C2×C3⋊C8D3211yescharacteristic
250ΓL(1,𝔽25)—34no—
251ΓL(1,𝔽25)—34no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32C2×D1211yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32D3×D411yescharacteristic
259C2×C3⋊D3—21no—
260C2×C3⋊D3—21no—
261C2×C3⋊D3C4⋊Dic312yes—
262C2×C3×D3—32no—
263C2×C3×D3—32no—
264C2×C3×D3—62no—
265C2×C3×D3—62no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—121no—
280C32×C4C2×Dic611yescharacteristic
281C32×C4C2×C4×D311yescharacteristic
282C32×C4C2×C4×D311yescharacteristic
283C32×C4D3×Q811yescharacteristic
284C32×C4D3×Q811yescharacteristic
285C32×C4C22×Dic311yescharacteristic
286C3⋊Dic3—41no—
287C3⋊Dic3—41no—
288C3⋊Dic3—121no—
289C3⋊Dic3—122no—
290C3⋊Dic3—121no—
291C3⋊Dic3—121no—
292C3⋊Dic3—241no—
293C3⋊Dic3—241no—
294C3⋊Dic3—242no—
295C3⋊Dic3C4⋊Dic312yes—
296C3×Dic3—32no—
297C3×Dic3—32no—
298C3×Dic3—42no—
299C3×Dic3—41no—
300C3×Dic3—41no—
301C3×Dic3—62no—
302C3×Dic3—121no—
303C3×Dic3—121no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C3×Dic3—241no—
307C3×Dic3—241no—
308C2×M16—91no—
309#117—271no—
310C2×C4⋊C4—271no—
311#107—91no—
312#107—271no—
313#106—91no—
314#106—271no—
315C33#42811yescharacteristic, complemented, Sylow
316C23×C3—91no—
317C22×D3—31no—
318C22×D3—31no—
319C22×D3—61no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—62no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—91no—
328C2×C3×C4—92no—
329C2×C3×C4—92no—
330C2×C3×C4—181no—
331C2×C3×C4—181no—
332C2×C3×C4C2×D3211yescharacteristic
333C2×C3×C4C2×D3211yescharacteristic
334C2×C3×C4C2×D3211yescharacteristic
335C2×Dic3—21no—
336C2×Dic3—31no—
337C2×Dic3—31no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—121no—
344C2×Dic3—121no—
345C2×Dic3—182no—
346C2×Dic3—181no—
347C2×Dic3—181no—
348C2×Dic3—181no—
349C2×Dic3—361no—
350C2×Dic3—361no—
351C2×Dic3—362no—
352C2×Dic3—362no—
353C4×D3—32no—
354C4×D3—32no—
355C4×D3—32no—
356C4×D3—32no—
357C4×D3—62no—
358C4×D3—62no—
359C4×D3—364no—
360C4×D3—362no—
361C4×D3—362no—
362C3×C8—32no—
363C3×C8—32no—
364C3×C8—32no—
365C3×C8—32no—
366C3×C8—62no—
367C3×C8—62no—
368C3⋊C8D3×Dic312yescomplemented
369C3⋊C8D3×Dic312yescomplemented
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—41no—
380C2×C32—41no—
381C2×C32—41no—
382C2×C32D3×C4⋊C411yescharacteristic
383C2×C32D3×C4⋊C411yescharacteristic
384C2×C32#90811yescharacteristic
385C2×C32#90911yescharacteristic
386C2×C32#91111yescharacteristic
387C2×C32#91411yescharacteristic
388C2×C32#91511yescharacteristic
389C2×C32#91811yescharacteristic
390C2×C32C2×C4⋊Dic311yescharacteristic
391C3⋊D3—22no—
392C3×D3—64no—
393C3×D3—62no—
394C3×D3—62no—
395C22×C4—91no—
396C22×C4—541no—
397M16—94no—
398C2×C8—31no—
399C2×C8—31no—
400C4⋊C4—31no—
401C4⋊C4—91no—
402C4⋊C4—91no—
403C4⋊C4—271no—
404C4⋊C4—541no—
405C22⋊C4—541no—
406C42—541no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—41no—
411C22×C3—92no—
412C22×C3—181no—
413C22×C3—181no—
414C22×C3D3×D1211yescharacteristic
415C22×C3D3×D1211yescharacteristic
416C22×C3#381111yescharacteristic
417C2×D3—32no—
418C2×D3—32no—
419C2×D3—62no—
420C2×D3—62no—
421C2×D3—62no—
422C2×D3—61no—
423C2×D3—61no—
424C2×D3—61no—
425C2×D3—61no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—92no—
435C3×C4—122no—
436C3×C4—121no—
437C3×C4—121no—
438C3×C4—361no—
439C3×C4—361no—
440C3×C4D3×Dic611yescharacteristic
441C3×C4D3×Dic611yescharacteristic
442C3×C4#379711yescharacteristic
443C3×C4C4×D3211yescharacteristic
444C3×C4C2×D3×Dic311yescharacteristic
445C3×C4C2×D3×Dic311yescharacteristic
446Dic3—32no—
447Dic3—32no—
448Dic3—41no—
449Dic3—62no—
450Dic3—121no—
451Dic3—121no—
452Dic3—121no—
453Dic3—121no—
454Dic3—241no—
455Dic3—241no—
456Dic3—362no—
457Dic3—361no—
458Dic3—361no—
459Dic3—361no—
460Dic3—721no—
461Dic3—721no—
462Dic3—722no—
463C32—21no—
464C32—21no—
465C32—21no—
466C32—41no—
467C32#486111yescharacteristic, complemented
468C32#488111yescharacteristic, complemented
469C32#512111yescharacteristic, complemented
470C23—91no—
471C2×C4—61no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—541no—
478C2×C4—1082no—
479C2×C4D3311yescharacteristic
480C8—32no—
481C8—32no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—41no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—182no—
495C2×C3#6379411yescharacteristic
496C2×C3#6379511yescharacteristic
497C2×C3#6380011yescharacteristic
498C2×C3#6380111yescharacteristic
499C2×C3#6388311yescharacteristic
500C2×C3D3×C4⋊Dic311yescharacteristic
501C2×C3D3×C4⋊Dic311yescharacteristic
502C2×C3#6393711yescharacteristic
503C2×C3#6395111yescharacteristic
504D3—64no—
505D3—62no—
506D3—62no—
507C22—92no—
508C22—181no—
509C22—181no—
510C22#8895511yescharacteristic
511C4—92no—
512C4—121no—
513C4—361no—
514C4—361no—
515C4—1081no—
516C4D32×Dic311yescharacteristic
517C4#8895711yescharacteristic
518C3—21no—
519C3—21no—
520C3—21no—
521C3—41no—
522C3#1059058311yescharacteristic, complemented
523C3#1059068111yescharacteristic, complemented
524C3#1059068311yescharacteristic, complemented
525C2—182no—
526C2#1172287911yescharacteristic
527C2#1172289611yescharacteristic
528C2#1172290811yescharacteristic
subgroupG/H
centerC22#88955
commutatorC33×C4C22×C4
Frattini, hypercenterC2×C4D33
Fitting, cosocleC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypocenterC33#428
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722055C2semidirect
5. #11722052C2semidirect
6. C3×#63575C2semidirect
7. C3×#63573C2semidirect
22. C3×#3712C4semidirect
42. #63575D3semidirect
46. #63573D3semidirect
57. C32×C3⋊C8C2×C4semidirect
58. C32×C3⋊C8C2×C4semidirect
109. #3712Dic3semidirect
195. C3×C3⋊C8C4×D3semidirect
196. C3×C3⋊C8C4×D3semidirect
197. C3×C3⋊C8C2×Dic3semidirect
198. C3×C3⋊C8C2×Dic3semidirect
315. C33#428semidirect
368. C3⋊C8D3×Dic3semidirect
369. C3⋊C8D3×Dic3semidirect
467. C32#4861semidirect
468. C32#4881semidirect
469. C32#5121semidirect
522. C3#10590583semidirect
523. C3#10590681semidirect
524. C3#10590683semidirect

Maximal subgroups

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

subgroupindex
#117236672
C3⋊D3×C4⋊Dic32
C2×C3×#37122
#117220552
#117220522
C3×#635752
C3×#635732
#105906833
#105906813
#105905833