Finite Group Databasebeta

← #13156004

Subgroups — #13156004

Order 1728.

696 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#88811C211yesmaximal, characteristic
2#11723443C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4C3×#63894C211yesmaximal, characteristic
5#11722041C211yesmaximal, characteristic, complemented
6C3×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
7C3×#63567C211yesmaximal, characteristic, complemented
8#10590760—31nomaximal
9#10590559—31nomaximal
10#10590365—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C3×#624—21no—
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4×Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20Dic3⋊(C3⋊Dic3)—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#88796—22no—
23C3×#3722—21no—
24C2×#3744—31no—
25#64078—31no—
26C3×#893—31no—
27C3×C4×D12—31no—
28#63894D311yescharacteristic
29#63866—31no—
30#63866—31no—
31#63866—61no—
32C2×#3717—31no—
33C2×#3716—31no—
34C2×#3716—31no—
35C2×#3716—31no—
36C2×#3716—61no—
37C2×#3715—31no—
38#63637—31no—
39C3×#816—31no—
40C3×Dic3⋊C8—31no—
41C3×#810—31no—
42C3×C4⋊(C3⋊C8)—31no—
43Dic3⋊(C3⋊C8)D311yescharacteristic, complemented
44#63567—31no—
45#63567—31no—
46#63567—61no—
47#63567D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51#6308—21no—
52#6308—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×C4⋊(C3⋊D3)D412yes—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D411yescharacteristic
60C2×C32×Dic3D411yescharacteristic
61C3×#624—42no—
62C3×C3⋊(C3⋊C8)—21no—
63C32×C3⋊C8—21no—
64#5019—91no—
65#4841—91no—
66#4818—91no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×#634—31no—
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—61no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—121no—
75C2×#634C2×D311yescharacteristic
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×C4×D3—61no—
80C2×C3×C4×D3—61no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×#624—21no—
85C2×C32×C8—31no—
86C32×C4⋊C4—31no—
87C32×C42—31no—
88C4⋊(C3⋊Dic3)—61no—
89C4⋊(C3⋊Dic3)—61no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91#3744—62no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×C4⋊Dic3—21no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
97C3×C4×Dic3—21no—
98C3×C4×Dic3C2×D311yescharacteristic
99C3×C4×Dic3C2×D311yescharacteristic
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—61no—
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C2×C3×C3⋊C8C2×D311yescharacteristic
106Dic3⋊Dic3—62no—
107Dic3⋊Dic3—62no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—61no—
112#3722—21no—
113#3717—62no—
114#3716—62no—
115#3716—62no—
116#3716—64no—
117#3716—62no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—22no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3⋊(C3⋊Dic3)—41no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—21no—
128C32×Dic3—41no—
129C32×Dic3SD1612yescomplemented
130C3×C4×D4—91no—
131#944—91no—
132C2×#215—91no—
133C2×#214—91no—
134C2×#214—91no—
135C2×#204—91no—
136C2×#204—91no—
137C2×#204—91no—
138C2×#204—181no—
139#893—31no—
140C4×D12—31no—
141#869—91no—
142#869—91no—
143#869—181no—
144C3×C4⋊C8—91no—
145C3×#103—91no—
146#816—31no—
147#816—91no—
148Dic3⋊C8—31no—
149#810—31no—
150#810—91no—
151#810—91no—
152#810—181no—
153C4⋊(C3⋊C8)—31no—
154C22×C3⋊D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—31no—
162C2×C32×C4—31no—
163C2×C32×C4—41no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—62no—
171C2×C3⋊Dic3—61no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—121no—
174C2×C3⋊Dic3—121no—
175C2×C3⋊Dic3—122no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)D1212yes—
178#634—21no—
179#634—21no—
180#634—61no—
181#634—61no—
182#634—62no—
183#634—62no—
184#634—61no—
185#634—61no—
186#634—61no—
187#634—61no—
188#634—121no—
189#634—121no—
190#634—121no—
191#634—121no—
192#634—122no—
193#634—122no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—41no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3D1211yescharacteristic
203C2×C3×Dic3D1211yescharacteristic
204C2×C3×Dic3#6711yescharacteristic
205C2×C3×Dic3#6711yescharacteristic
206C3×D12—32no—
207C3×D12—32no—
208C3×D12—62no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—121no—
212C3×C4×D3—122no—
213C3×C4×D3—122no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—121no—
219C3×Dic6—121no—
220#624—42no—
221C32×C8—61no—
222C3⋊(C3⋊C8)—21no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—61no—
227C3×C3⋊C8—61no—
228C3×C3⋊C8—121no—
229#461—271noSylow
230C2×C33#12111yescharacteristic
231C2×C33C2×SD1611yescharacteristic
232C2×C33#13711yescharacteristic
233C3×C3⋊D3—42no—
234C2×C3×Q8—91no—
235C2×C3×D4—91no—
236C22×C3×C4—181no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—61no—
240C2×C4×D3—61no—
241C2×C4×D3—61no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C2×Dic6—91no—
246C2×Dic6—91no—
247C2×Dic6—91no—
248C2×Dic6—181no—
249C2×Dic6—181no—
250C2×Dic6—181no—
251C2×Dic6—361no—
252C2×C3×C8—31no—
253C2×C3×C8—31no—
254C2×C3×C8—61no—
255C2×C3×C8—91no—
256C3×C4⋊C4—31no—
257C3×C4⋊C4—31no—
258C3×C4⋊C4—61no—
259C3×C22⋊C4—181no—
260C3×C42—31no—
261C3×C42—31no—
262C3×C42—61no—
263#215—182no—
264#214—182no—
265#214—182no—
266(C2×D3)⋊C4—61no—
267(C2×D3)⋊C4—61no—
268C4⋊Dic3—181no—
269C4⋊Dic3—181no—
270C4⋊Dic3—181no—
271C4⋊Dic3—181no—
272C4⋊Dic3—182no—
273C4⋊Dic3—182no—
274C4⋊Dic3D3211yescharacteristic
275Dic3⋊C4—181no—
276Dic3⋊C4—181no—
277C4×Dic3D3211yescharacteristic
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—61no—
281C2×C3⋊C8D3211yescharacteristic
282#204—182no—
283#204—182no—
284#204—184no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32C2×D1211yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—22no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—41no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—241no—
304C2×C3×D3—241no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—32no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—61no—
315C32×C4—121no—
316C32×C4C2×D1211yescharacteristic
317C32×C4#23511yescharacteristic
318C32×C4#23511yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4D3×D411yescharacteristic
321C32×C4#23911yescharacteristic
322C3⋊Dic3—41no—
323C3⋊Dic3—122no—
324C3⋊Dic3—122no—
325C3⋊Dic3—122no—
326C3⋊Dic3—124no—
327C3⋊Dic3—121no—
328C3⋊Dic3—121no—
329C3⋊Dic3—121no—
330C3×Dic3—22no—
331C3×Dic3—21no—
332C3×Dic3—21no—
333C3×Dic3—22no—
334C3×Dic3—41no—
335C3×Dic3—41no—
336C3×Dic3—81no—
337C3×Dic3—122no—
338C3×Dic3—121no—
339C3×Dic3—121no—
340C3×Dic3#20412yescomplemented
341C3×Dic3#21412yescomplemented
342C2×SD16—271no—
343C2×SD16—271no—
344C4⋊Q8—271no—
345C4×D4—91no—
346C4⋊C8—91no—
347#103—91no—
348#103—271no—
349C33#46111yescharacteristic, complemented, Sylow
350C23×C3—181no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—181no—
355C3×Q8—181no—
356C3×D4—92no—
357C3×D4—181no—
358C2×C3×C4—21no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—31no—
362C2×C3×C4—31no—
363C2×C3×C4—31no—
364C2×C3×C4—31no—
365C2×C3×C4—41no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—61no—
370C2×C3×C4—121no—
371C2×C3×C4—181no—
372C2×C3×C4—362no—
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×Dic3—21no—
377C2×Dic3—62no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—182no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—181no—
384C2×Dic3—361no—
385C2×Dic3—361no—
386C2×Dic3—362no—
387C2×Dic3#62611yescharacteristic
388C2×Dic3#62611yescharacteristic
389D12—32no—
390D12—32no—
391D12—62no—
392D12—61no—
393D12—61no—
394D12—121no—
395C4×D3—122no—
396C4×D3—122no—
397Dic6—61no—
398Dic6—61no—
399Dic6—61no—
400Dic6—61no—
401Dic6—121no—
402Dic6—121no—
403Dic6—182no—
404Dic6—182no—
405Dic6—181no—
406Dic6—181no—
407Dic6—181no—
408Dic6—181no—
409Dic6—181no—
410Dic6—181no—
411Dic6—361no—
412Dic6—361no—
413Dic6—361no—
414Dic6—361no—
415Dic6—362no—
416Dic6—362no—
417C3×C8—62no—
418C3×C8—61no—
419C3×C8—61no—
420C3×C8—181no—
421C3⋊C8—21no—
422C3⋊C8—61no—
423C3⋊C8—61no—
424C3⋊C8—121no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—41no—
435C2×C32—41no—
436C2×C32—41no—
437C2×C32#88411yescharacteristic
438C2×C32#89511yescharacteristic
439C2×C32C2×#20411yescharacteristic
440C2×C32#90911yescharacteristic
441C2×C32D3×SD1611yescharacteristic
442C2×C32#91811yescharacteristic
443C2×C32#93011yescharacteristic
444C2×C32C2×#21411yescharacteristic
445C2×C32#95111yescharacteristic
446C3⋊D3—42no—
447C3×D3—122no—
448C3×D3—122no—
449C3×D3—242no—
450C2×Q8—91no—
451C2×Q8—271no—
452C2×D4—91no—
453C22×C4—181no—
454SD16—542no—
455SD16—542no—
456C2×C8—31no—
457C2×C8—91no—
458C4⋊C4—31no—
459C4⋊C4—541no—
460C4⋊C4—541no—
461C22⋊C4—181no—
462C42—31no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—182no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3D3×D1211yescharacteristic
471C22×C3C2×#62611yescharacteristic
472C22×C3D3×#6711yescharacteristic
473C2×D3—62no—
474C2×D3—62no—
475C2×D3—122no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—121no—
480C2×D3—241no—
481C2×D3—241no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—32no—
489C3×C4—32no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—62no—
493C3×C4—62no—
494C3×C4—61no—
495C3×C4—61no—
496C3×C4—121no—
497C3×C4—121no—
498C3×C4—241no—
499C3×C4—361no—
500C3×C4#379811yescharacteristic
501C3×C4#379911yescharacteristic
502C3×C4#379911yescharacteristic
503C3×C4D3×D1211yescharacteristic
504C3×C4D3×D1211yescharacteristic
505C3×C4#381111yescharacteristic
506Dic3—21no—
507Dic3—41no—
508Dic3—122no—
509Dic3—121no—
510Dic3—121no—
511Dic3—362no—
512Dic3—362no—
513Dic3—362no—
514Dic3—364no—
515Dic3—361no—
516Dic3—361no—
517Dic3—361no—
518Dic3#371612yescomplemented
519C32—21no—
520C32—21no—
521C32—21no—
522C32—41no—
523C32#481811yescharacteristic, complemented
524C32#484111yescharacteristic, complemented
525C32#501911yescharacteristic, complemented
526C23—181no—
527Q8—181no—
528Q8—181no—
529Q8—541no—
530Q8—541no—
531D4—92no—
532D4—181no—
533C2×C4—31no—
534C2×C4—31no—
535C2×C4—61no—
536C2×C4—181no—
537C2×C4—362no—
538C2×C4—541no—
539C2×C4D3311yescharacteristic
540C8—61no—
541C8—181no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—41no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—362no—
555C2×C3D3×#20411yescharacteristic
556C2×C3#6380111yescharacteristic
557C2×C3C2×#371611yescharacteristic
558C2×C3#6383111yescharacteristic
559C2×C3#6390611yescharacteristic
560C2×C3#6390911yescharacteristic
561C2×C3#6391011yescharacteristic
562C2×C3D3×#21411yescharacteristic
563C2×C3#6394311yescharacteristic
564D3—122no—
565D3—122no—
566D3—242no—
567C22—182no—
568C22—361no—
569C22—361no—
570C22D3×#62611yescharacteristic
571C4—32no—
572C4—61no—
573C4—121no—
574C4—361no—
575C4—1081no—
576C4#8895511yescharacteristic
577C4#8896511yescharacteristic
578C3—21no—
579C3—21no—
580C3—21no—
581C3—41no—
582C3#1059036511yescharacteristic, complemented
583C3#1059055911yescharacteristic, complemented
584C3#1059076011yescharacteristic, complemented
585C2—362no—
586C2D3×#371611yescharacteristic
587C2#1172288711yescharacteristic
588C2#1172296111yescharacteristic
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
2. #11723443C2semidirect
5. #11722041C2semidirect
6. C3×Dic3⋊(C3⋊C8)C2semidirect
7. C3×#63567C2semidirect
43. Dic3⋊(C3⋊C8)D3semidirect
47. #63567D3semidirect
129. C32×Dic3SD16semidirect
340. C3×Dic3#204semidirect
341. C3×Dic3#214semidirect
349. C33#461semidirect
518. Dic3#3716semidirect
523. C32#4818semidirect
524. C32#4841semidirect
525. C32#5019semidirect
582. C3#10590365semidirect
583. C3#10590559semidirect
584. C3#10590760semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234432
C2×#887962
C3×#638942
#117220412
C3×Dic3⋊(C3⋊C8)2
C3×#635672
#105907603
#105905593
#105903653