Finite Group Databasebeta

← #13156488

Subgroups — #13156488

Order 1728.

564 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
1#11723424C211yesmaximal, characteristic
2C3×#63856C211yesmaximal, characteristic
3#11722080C211yesmaximal, characteristic, complemented
4#11722052C211yesmaximal, characteristic, complemented
5#11722038C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591559—31nomaximal
9#10591127—31nomaximal
10#10590711—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18(C2×D3)⋊(C3⋊Dic3)—21no—
19Dic3×C3⋊Dic3—21no—
20C3×#3724—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64135—31no—
23C3×#946—31no—
24C3×#924—31no—
25#63856D311yescharacteristic
26#63841—31no—
27#63841—31no—
28#63841—61no—
29#63662—31no—
30#63644—31no—
31C3×#835—31no—
32C3×#830—31no—
33C3×(C2×D3)⋊C8—31no—
34C3×#816—31no—
35#63576—31no—
36#63575—31no—
37#63575—31no—
38#63573—31no—
39#63573—31no—
40#63573—61no—
41#63566—31no—
42#63566—31no—
43#63566—61no—
44#63566D311yescharacteristic, complemented
45(C2×D3)⋊(C3⋊C8)D311yescharacteristic
46C22×C32×D3D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C32×Dic3—21no—
52C2×C32×Dic3D411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C32×C3⋊C8—21no—
59#5045—91no—
60#4815—91no—
61#4723—91no—
62C2×C32×Q8—31no—
63C22×C32×C4—31no—
64C2×C3×C4×D3—21no—
65C2×C3×C4×D3C2×D311yescharacteristic
66C2×C3×C4×D3C2×D311yescharacteristic
67C2×C3×Dic6—21no—
68C2×C3×Dic6C2×D311yescharacteristic
69C2×C3×Dic6C2×D311yescharacteristic
70C2×C32×C8—31no—
71#3757—61no—
72C4⋊(C3⋊Dic3)—31no—
73C4⋊(C3⋊Dic3)—31no—
74C4⋊(C3⋊Dic3)—31no—
75C4⋊(C3⋊Dic3)—61no—
76C4⋊(C3⋊Dic3)—61no—
77C4⋊(C3⋊Dic3)—61no—
78C4⋊(C3⋊Dic3)—121no—
79C4⋊(C3⋊Dic3)C2×D311yescharacteristic
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C4×C3⋊Dic3—61no—
82C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
83C3×#217—61no—
84C3×(C2×D3)⋊C4—61no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×Dic3⋊C4—61no—
92C3×Dic3⋊C4—61no—
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3724—21no—
100(C2×D3)⋊Dic3—21no—
101(C2×D3)⋊Dic3—62no—
102(C2×D3)⋊Dic3—61no—
103(C2×D3)⋊Dic3—61no—
104Dic32—62no—
105Dic32—61no—
106Dic32—61no—
107C22×C33C2×D411yescharacteristic
108C2×C32×D3—21no—
109C2×C32×D3—21no—
110C2×C32×D3—21no—
111C33×C4#4111yescharacteristic
112C33×C4#4111yescharacteristic
113C3⋊(C3⋊Dic3)—41no—
114C3×C3⋊Dic3—41no—
115C3×C3⋊Dic3—41no—
116C32×Dic3—21no—
117C32×Dic3—41no—
118C3×C22⋊Q8—91no—
119#946—31no—
120#926—91no—
121#926—91no—
122#926—181no—
123#924—31no—
124#892—91no—
125C3×#103—91no—
126C3×C22⋊C8—91no—
127#835—31no—
128#835—91no—
129#835—91no—
130#835—181no—
131#830—31no—
132(C2×D3)⋊C8—31no—
133#818—91no—
134#817—91no—
135#817—91no—
136#817—181no—
137#816—31no—
138#816—91no—
139#808—91no—
140#807—91no—
141#807—91no—
142C23×C32—31no—
143C22×C3×D3—21no—
144C22×C3×D3D1211yescharacteristic
145C22×C3×D3#6711yescharacteristic
146C32×Q8—61no—
147C32×Q8—61no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—21no—
151C2×C32×C4—31no—
152C2×C32×C4—41no—
153C2×C32×C4—61no—
154C2×C32×C4—61no—
155C2×C32×C4—61no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C3⋊Dic3—21no—
160C2×C3⋊Dic3—21no—
161C2×C3⋊Dic3—61no—
162C2×C3⋊Dic3—62no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—121no—
167C2×C3⋊Dic3—122no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—62no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3D1211yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C3×C4×D3—22no—
181C3×C4×D3—22no—
182C3×C4×D3—21no—
183C3×C4×D3—21no—
184C3×C4×D3—21no—
185C3×C4×D3—21no—
186C3×Dic6—22no—
187C3×Dic6—21no—
188C3×Dic6—21no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—41no—
192C32×C8—61no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200#485—271noSylow
201C2×C33#12311yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33#13711yescharacteristic
204C32×D3—41no—
205C2×C3×Q8—31no—
206C2×C3×Q8—31no—
207C2×C3×Q8—61no—
208C22×C3×C4—31no—
209C22×C3×C4—31no—
210C22×C3×C4—61no—
211C2×C4×D3D3211yescharacteristic
212C2×Dic6D3211yescharacteristic
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—91no—
218C3×C4⋊C4—91no—
219C3×C4⋊C4—181no—
220C3×C22⋊C4—181no—
221#217—61no—
222#217—181no—
223#217—181no—
224#217—182no—
225(C2×D3)⋊C4—61no—
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—361no—
240Dic3⋊C4—61no—
241Dic3⋊C4—61no—
242C4×Dic3—181no—
243C4×Dic3—181no—
244C4×Dic3—182no—
245C4×Dic3—181no—
246C2×C3⋊C8—31no—
247C2×C3⋊C8—31no—
248C2×C3⋊C8—61no—
249C2×C3⋊C8D3211yescharacteristic
250C22×C32—21no—
251C22×C32—21no—
252C22×C32—21no—
253C22×C32—41no—
254C22×C32—61no—
255C22×C32—61no—
256C22×C32—61no—
257C22×C32C2×D1211yescharacteristic
258C22×C32D3×D411yescharacteristic
259C22×C32C2×#6711yescharacteristic
260C2×C3×D3—22no—
261C2×C3×D3—21no—
262C2×C3×D3—21no—
263C2×C3×D3—21no—
264C2×C3×D3—21no—
265C2×C3×D3—21no—
266C2×C3×D3—21no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—41no—
276C32×C4—41no—
277C32×C4—61no—
278C32×C4—121no—
279C32×C4#23511yescharacteristic
280C32×C4#23711yescharacteristic
281C32×C4#23711yescharacteristic
282C32×C4#23711yescharacteristic
283C32×C4#23711yescharacteristic
284C32×C4#23711yescharacteristic
285C3⋊Dic3—41no—
286C3⋊Dic3—41no—
287C3⋊Dic3—121no—
288C3⋊Dic3—122no—
289C3⋊Dic3—121no—
290C3⋊Dic3—121no—
291C3⋊Dic3—241no—
292C3⋊Dic3—241no—
293C3⋊Dic3—242no—
294C3×Dic3—22no—
295C3×Dic3—21no—
296C3×Dic3—21no—
297C3×Dic3—42no—
298C3×Dic3—41no—
299C3×Dic3—41no—
300C3×Dic3—122no—
301C3×Dic3—121no—
302C3×Dic3—121no—
303C3×Dic3—121no—
304C3×Dic3—121no—
305C3×Dic3—241no—
306C22⋊Q8—91no—
307#117—271no—
308#107—271no—
309#106—271no—
310#103—91no—
311#103—271no—
312C22⋊C8—91no—
313C33#48511yescharacteristic, complemented, Sylow
314C23×C3—31no—
315C23×C3—31no—
316C23×C3—61no—
317C22×D3#62611yescharacteristic
318C3×Q8—62no—
319C3×Q8—61no—
320C3×Q8—61no—
321C3×Q8—61no—
322C3×Q8—61no—
323C3×Q8—121no—
324C2×C3×C4—21no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—31no—
328C2×C3×C4—31no—
329C2×C3×C4—41no—
330C2×C3×C4—61no—
331C2×C3×C4—62no—
332C2×C3×C4—62no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—61no—
336C2×C3×C4—61no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—121no—
340C2×C3×C4—181no—
341C2×C3×C4—181no—
342C2×C3×C4C2×D3211yescharacteristic
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×Dic3—21no—
346C2×Dic3—62no—
347C2×Dic3—61no—
348C2×Dic3—61no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—121no—
352C2×Dic3—182no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359C2×Dic3#62611yescharacteristic
360C4×D3—21no—
361C4×D3—21no—
362Dic6—21no—
363Dic6—21no—
364C3×C8—62no—
365C3×C8—61no—
366C3×C8—61no—
367C3×C8—181no—
368C3⋊C8—21no—
369C3⋊C8—61no—
370C3⋊C8—61no—
371C3⋊C8—121no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—21no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—41no—
382C2×C32—41no—
383C2×C32—41no—
384C2×C32—121no—
385C2×C32#88311yescharacteristic
386C2×C32#88611yescharacteristic
387C2×C32#90411yescharacteristic
388C2×C32#90911yescharacteristic
389C2×C32#91611yescharacteristic
390C2×C32#91811yescharacteristic
391C2×C32#93511yescharacteristic
392C2×C32#94211yescharacteristic
393C2×C32#95011yescharacteristic
394C3×D3—42no—
395C3×D3—41no—
396C3×D3—41no—
397C2×Q8—31no—
398C22×C4—31no—
399C2×C8—31no—
400C2×C8—91no—
401C4⋊C4—91no—
402C4⋊C4—91no—
403C4⋊C4—181no—
404C4⋊C4—271no—
405C22⋊C4—181no—
406C22⋊C4—541no—
407C42—541no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—41no—
412C22×C3—62no—
413C22×C3—61no—
414C22×C3—61no—
415C22×C3—61no—
416C22×C3—61no—
417C22×C3—61no—
418C22×C3—61no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3D3×D1211yescharacteristic
422C22×C3C2×#62611yescharacteristic
423C22×C3D3×#6711yescharacteristic
424C2×D3—21no—
425C2×D3—21no—
426C2×D3—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—62no—
436C3×C4—61no—
437C3×C4—61no—
438C3×C4—122no—
439C3×C4—121no—
440C3×C4—121no—
441C3×C4—361no—
442C3×C4—361no—
443C3×C4#379511yescharacteristic
444C3×C4#380411yescharacteristic
445C3×C4#380511yescharacteristic
446C3×C4#380511yescharacteristic
447C3×C4#380511yescharacteristic
448C3×C4#380511yescharacteristic
449Dic3—21no—
450Dic3—41no—
451Dic3—122no—
452Dic3—121no—
453Dic3—121no—
454Dic3—121no—
455Dic3—121no—
456Dic3—241no—
457Dic3—362no—
458Dic3—361no—
459Dic3—361no—
460Dic3—361no—
461Dic3—721no—
462Dic3—721no—
463Dic3—722no—
464C32—21no—
465C32—21no—
466C32—21no—
467C32—41no—
468C32#472311yescharacteristic, complemented
469C32#481511yescharacteristic, complemented
470C32#504511yescharacteristic, complemented
471C23—31no—
472Q8—61no—
473Q8—61no—
474C2×C4—31no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—181no—
479C2×C4—181no—
480C2×C4—541no—
481C2×C4D3311yescharacteristic
482C8—61no—
483C8—181no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3—122no—
497C2×C3—121no—
498C2×C3—121no—
499C2×C3#6380111yescharacteristic
500C2×C3#6380611yescharacteristic
501C2×C3#6382911yescharacteristic
502C2×C3#6383311yescharacteristic
503C2×C3#6389111yescharacteristic
504C2×C3#6394711yescharacteristic
505C2×C3#6394811yescharacteristic
506C2×C3#6396211yescharacteristic
507C2×C3#6396911yescharacteristic
508D3—41no—
509C22—61no—
510C22—61no—
511C22—61no—
512C22D3×#62611yescharacteristic
513C4—61no—
514C4—121no—
515C4—361no—
516C4—361no—
517C4—1081no—
518C4#8896111yescharacteristic
519C4#8896311yescharacteristic
520C3—21no—
521C3—21no—
522C3—21no—
523C3—41no—
524C3#1059071111yescharacteristic, complemented
525C3#1059112711yescharacteristic, complemented
526C3#1059155911yescharacteristic, complemented
527C2—121no—
528C2#1172285511yescharacteristic
529C2#1172288511yescharacteristic
530C2#1172298911yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypocenterC33#485
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. #11722080C2semidirect
4. #11722052C2semidirect
5. #11722038C2semidirect
6. C3×#63566C2semidirect
44. #63566D3semidirect
313. C33#485semidirect
468. C32#4723semidirect
469. C32#4815semidirect
470. C32#5045semidirect
524. C3#10590711semidirect
525. C3#10591127semidirect
526. C3#10591559semidirect

Maximal subgroups

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

subgroupindex
#117234242
C3×#638562
#117220802
#117220522
#117220382
C3×#635662
C3×(C2×D3)⋊(C3⋊C8)2
#105915593
#105911273
#105907113