Finite Group Databasebeta

← #13156005

Subgroups — #13156005

Order 1728.

684 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic
3C3×#63866C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic
5#11722040C211yesmaximal, characteristic, complemented
6#11722038C211yesmaximal, characteristic, complemented
7C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
8#10591564—31nomaximal
9#10590752—31nomaximal
10#10590379—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×Dic3×C3⋊D3—21no—
15C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20#88803—21no—
21C3×Dic3⋊Dic3—21no—
22C3×Dic3⋊Dic3—21no—
23C3×#3717—22no—
24C4×C4⋊(C3⋊D3)—31no—
25C3×#944—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28C3×#869—31no—
29Dic3×D12—31no—
30Dic3×D12—31no—
31Dic3×D12—61no—
32#63866D311yescharacteristic, complemented
33C2×#3717—31no—
34C2×#3717D311yescharacteristic
35C2×#3716—31no—
36C2×#3715—31no—
37#63662—31no—
38#63643—31no—
39C3×Dic3⋊C8—31no—
40C3×C4⋊(C3⋊C8)—31no—
41Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
42#63566—31no—
43#63566—31no—
44#63566—31no—
45#63566—31no—
46#63566—61no—
47#63566—61no—
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)D412yes—
54C3×#634—21no—
55C3×#634—21no—
56C2×C32×Dic3—21no—
57C2×C32×Dic3D411yescharacteristic
58C2×C32×Dic3D411yescharacteristic
59C32×Dic6—21no—
60C32×Dic6—21no—
61Dic3×C3⋊D3—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5162—91no—
65#5083—91no—
66#4841—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×C4×C3⋊D3—61no—
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×Dic6—21no—
75C2×C3×Dic6—31no—
76C2×C3×Dic6—31no—
77C2×C3×Dic6—61no—
78C2×C3×Dic6C2×D311yescharacteristic
79C2×C3×Dic6C2×D311yescharacteristic
80C2×D3×Dic3—62no—
81C2×D3×Dic3—61no—
82C2×D3×Dic3—61no—
83C2×C32×C8—31no—
84C32×C42—31no—
85#3752—61no—
86C4⋊(C3⋊Dic3)—31no—
87C4⋊(C3⋊Dic3)—31no—
88C4⋊(C3⋊Dic3)—31no—
89C4⋊(C3⋊Dic3)—61no—
90C4⋊(C3⋊Dic3)—61no—
91C4⋊(C3⋊Dic3)—61no—
92C4⋊(C3⋊Dic3)—121no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#215—62no—
95C3×C4⋊Dic3—61no—
96C3×C4⋊Dic3—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D311yescharacteristic
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109Dic3⋊Dic3—21no—
110Dic3⋊Dic3—21no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—61no—
113(C2×D3)⋊Dic3—61no—
114#3717—22no—
115#3717—62no—
116#3716—62no—
117#3715—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C33×C4C2×D411yescharacteristic
123C33×C4#4111yescharacteristic
124C3⋊(C3⋊Dic3)—41no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—21no—
127C32×Dic3—41no—
128C32×Dic3SD1612yescomplemented
129C2×C3×SD16—91no—
130C3×C4⋊Q8—91no—
131#944—31no—
132C2×#215—31no—
133C2×#215—91no—
134D4×Dic3—91no—
135C2×#214—91no—
136C2×#204—31no—
137C2×#204—91no—
138C4×D12—91no—
139C4×D12—91no—
140C4×D12—181no—
141#869—31no—
142C3×C4⋊C8—91no—
143#835—91no—
144#835—91no—
145#835—91no—
146#835—181no—
147#816—91no—
148#816—91no—
149#816—91no—
150#816—181no—
151Dic3⋊C8—31no—
152C4⋊(C3⋊C8)—31no—
153C22×C3⋊D3—21no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C32×Q8—61no—
158C32×Q8—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—31no—
163C2×C32×C4—31no—
164C2×C32×C4—41no—
165C2×C32×C4—61no—
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—62no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—121no—
175C2×C3⋊Dic3—121no—
176C2×C3⋊Dic3—122no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)#6712yes—
179C4×C3⋊D3—122no—
180#634—21no—
181#634—21no—
182C2×C3×Dic3—21no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—21no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—41no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—121no—
190C2×C3×Dic3D1211yescharacteristic
191C2×C3×Dic3D1211yescharacteristic
192C2×C3×Dic3#6711yescharacteristic
193C2×C3×Dic3#6711yescharacteristic
194C3×D12—32no—
195C3×D12—32no—
196C3×D12—62no—
197C3×D12—62no—
198C3×D12—61no—
199C3×D12—61no—
200C3×Dic6—22no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—41no—
206C3×Dic6—62no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—121no—
212D3×Dic3—124no—
213D3×Dic3—122no—
214D3×Dic3—122no—
215C32×C8—61no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#461—271noSylow
224C2×C33#12111yescharacteristic
225C2×C33C2×SD1611yescharacteristic
226C2×C33#13711yescharacteristic
227C3×C3⋊D3—42no—
228C2×C3×Q8—31no—
229C2×C3×Q8—31no—
230C2×C3×Q8—61no—
231C2×C3×Q8—91no—
232C2×C3×D4—91no—
233C22×Dic3—181no—
234C2×D12—31no—
235C2×D12—31no—
236C2×D12—61no—
237C2×C4×D3—181no—
238C2×C4×D3—181no—
239C2×C4×D3—182no—
240C2×Dic6—31no—
241C2×Dic6—31no—
242C2×Dic6—61no—
243C2×Dic6D3211yescharacteristic
244C3×SD16—182no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—61no—
248C2×C3×C8—91no—
249C3×C4⋊C4—181no—
250C3×C4⋊C4—181no—
251C3×C42—31no—
252C3×C42—31no—
253C3×C42—61no—
254#217—181no—
255#215—62no—
256#215—182no—
257#214—182no—
258(C2×D3)⋊C4—181no—
259(C2×D3)⋊C4—181no—
260(C2×D3)⋊C4—182no—
261C4⋊Dic3—61no—
262C4⋊Dic3—61no—
263C4⋊Dic3—91no—
264C4⋊Dic3—91no—
265C4⋊Dic3—91no—
266C4⋊Dic3—181no—
267C4⋊Dic3—181no—
268C4⋊Dic3—181no—
269C4⋊Dic3—361no—
270Dic3⋊C4—61no—
271Dic3⋊C4—61no—
272C4×Dic3D3211yescharacteristic
273C2×C3⋊C8—31no—
274C2×C3⋊C8—31no—
275C2×C3⋊C8—61no—
276C2×C3⋊C8D3211yescharacteristic
277#204—62no—
278#204—182no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—21no—
282C22×C32—41no—
283C22×C32C2×D1211yescharacteristic
284C22×C32D3×D411yescharacteristic
285C22×C32C2×#6711yescharacteristic
286C2×C3⋊D3—22no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—41no—
289C2×C3×D3—64no—
290C2×C3×D3—62no—
291C2×C3×D3—62no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—121no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—32no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—61no—
308C32×C4—121no—
309C32×C4D3×D411yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4#23711yescharacteristic
312C32×C4#23711yescharacteristic
313C32×C4#23911yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C3⋊Dic3—41no—
316C3⋊Dic3—121no—
317C3⋊Dic3—122no—
318C3⋊Dic3—121no—
319C3⋊Dic3—121no—
320C3⋊Dic3—241no—
321C3⋊Dic3—241no—
322C3⋊Dic3—242no—
323C3×Dic3—21no—
324C3×Dic3—21no—
325C3×Dic3—22no—
326C3×Dic3—41no—
327C3×Dic3—42no—
328C3×Dic3—41no—
329C3×Dic3—41no—
330C3×Dic3—122no—
331C3×Dic3—121no—
332C3×Dic3—121no—
333C3×Dic3#20412yescomplemented
334C3×Dic3#21512yescomplemented
335C2×SD16—91no—
336C2×SD16—271no—
337C4⋊Q8—91no—
338C4×D4—271no—
339C4⋊C8—91no—
340#103—271no—
341#103—271no—
342C33#46111yescharacteristic, complemented, Sylow
343C23×C3—181no—
344C22×D3—62no—
345C22×D3—61no—
346C22×D3—61no—
347C3×Q8—62no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×Q8—181no—
354C3×Q8—181no—
355C3×D4—92no—
356C3×D4—181no—
357C2×C3×C4—21no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—31no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—41no—
365C2×C3×C4—61no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—121no—
370C2×C3×C4—181no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374C2×Dic3—21no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—121no—
378C2×Dic3—182no—
379C2×Dic3—181no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—361no—
383C2×Dic3—361no—
384C2×Dic3—362no—
385C2×Dic3—362no—
386C2×Dic3#62611yescharacteristic
387C2×Dic3#62611yescharacteristic
388D12—32no—
389D12—32no—
390D12—62no—
391D12—62no—
392D12—61no—
393D12—61no—
394C4×D3—362no—
395C4×D3—362no—
396C4×D3—364no—
397Dic6—21no—
398Dic6—21no—
399Dic6—62no—
400Dic6—61no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—121no—
405C3×C8—61no—
406C3×C8—61no—
407C3×C8—121no—
408C3×C8—181no—
409C3⋊C8—21no—
410C3⋊C8—61no—
411C3⋊C8—61no—
412C3⋊C8—121no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32#89511yescharacteristic
426C2×C32C2×#20411yescharacteristic
427C2×C32#90911yescharacteristic
428C2×C32D3×SD1611yescharacteristic
429C2×C32#91511yescharacteristic
430C2×C32#93811yescharacteristic
431C2×C32#93911yescharacteristic
432C2×C32C2×#21511yescharacteristic
433C2×C32#94211yescharacteristic
434C3⋊D3—42no—
435C3×D3—124no—
436C3×D3—122no—
437C3×D3—122no—
438C2×Q8—31no—
439C2×Q8—91no—
440C2×D4—91no—
441C22×C4—541no—
442SD16—182no—
443SD16—542no—
444C2×C8—31no—
445C2×C8—91no—
446C4⋊C4—181no—
447C4⋊C4—181no—
448C4⋊C4—271no—
449C22⋊C4—541no—
450C42—31no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—41no—
455C22×C3—182no—
456C22×C3—361no—
457C22×C3—361no—
458C22×C3D3×D1211yescharacteristic
459C22×C3C2×#62611yescharacteristic
460C22×C3D3×#6711yescharacteristic
461C2×D3—64no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—122no—
465C2×D3—122no—
466C2×D3—121no—
467C2×D3—121no—
468C2×D3—121no—
469C2×D3—121no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—21no—
473C3×C4—21no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—32no—
477C3×C4—32no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—62no—
481C3×C4—61no—
482C3×C4—61no—
483C3×C4—121no—
484C3×C4—122no—
485C3×C4—121no—
486C3×C4—121no—
487C3×C4—361no—
488C3×C4#379611yescharacteristic
489C3×C4#379611yescharacteristic
490C3×C4#380211yescharacteristic
491C3×C4#380511yescharacteristic
492C3×C4D3×#6711yescharacteristic
493C3×C4D3×#6711yescharacteristic
494Dic3—21no—
495Dic3—41no—
496Dic3—122no—
497Dic3—121no—
498Dic3—121no—
499Dic3—362no—
500Dic3—361no—
501Dic3—361no—
502Dic3—361no—
503Dic3—721no—
504Dic3—721no—
505Dic3—722no—
506Dic3#371712yescomplemented
507C32—21no—
508C32—21no—
509C32—21no—
510C32—41no—
511C32#484111yescharacteristic, complemented
512C32#508311yescharacteristic, complemented
513C32#516211yescharacteristic, complemented
514C23—181no—
515Q8—61no—
516Q8—61no—
517Q8—181no—
518Q8—181no—
519D4—92no—
520D4—181no—
521C2×C4—31no—
522C2×C4—31no—
523C2×C4—61no—
524C2×C4—181no—
525C2×C4—541no—
526C2×C4—1082no—
527C2×C4D3311yescharacteristic
528C8—61no—
529C8—181no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—41no—
540C2×C3—41no—
541C2×C3—41no—
542C2×C3—362no—
543C2×C3D3×#20411yescharacteristic
544C2×C3#6380011yescharacteristic
545C2×C3C2×#371711yescharacteristic
546C2×C3#6383311yescharacteristic
547C2×C3#6388611yescharacteristic
548C2×C3#6391611yescharacteristic
549C2×C3D3×#21511yescharacteristic
550C2×C3#6394611yescharacteristic
551C2×C3#6397011yescharacteristic
552D3—124no—
553D3—122no—
554D3—122no—
555C22—182no—
556C22—361no—
557C22—361no—
558C22D3×#62611yescharacteristic
559C4—32no—
560C4—61no—
561C4—121no—
562C4—361no—
563C4—1081no—
564C4#8895311yescharacteristic
565C4#8896411yescharacteristic
566C3—21no—
567C3—21no—
568C3—21no—
569C3—41no—
570C3#1059037911yescharacteristic, complemented
571C3#1059075211yescharacteristic, complemented
572C3#1059156411yescharacteristic, complemented
573C2—362no—
574C2D3×#371711yescharacteristic
575C2#1172288411yescharacteristic
576C2#1172296211yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#461
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. C3×#63866C2semidirect
5. #11722040C2semidirect
6. #11722038C2semidirect
7. C3×Dic3⋊(C3⋊C8)C2semidirect
32. #63866D3semidirect
41. Dic3⋊(C3⋊C8)D3semidirect
128. C32×Dic3SD16semidirect
333. C3×Dic3#204semidirect
334. C3×Dic3#215semidirect
342. C33#461semidirect
506. Dic3#3717semidirect
511. C32#4841semidirect
512. C32#5083semidirect
513. C32#5162semidirect
570. C3#10590379semidirect
571. C3#10590752semidirect
572. C3#10591564semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
Dic3×C4⋊(C3⋊D3)2
C3×#638662
C2×C3×#37172
#117220402
#117220382
C3×Dic3⋊(C3⋊C8)2
#105915643
#105907523
#105903793