Finite Group Databasebeta

← #13156011

Subgroups — #13156011

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
1#11723691C211yesmaximal, characteristic
2#11723455C211yesmaximal, characteristic, complemented
3C2×#88796C211yesmaximal, characteristic
4C2×C3×#3717C211yesmaximal, characteristic
5#11722042C211yesmaximal, characteristic, complemented
6C3×#63572C211yesmaximal, characteristic, complemented
7C3×#63566C211yesmaximal, characteristic, complemented
8#10591569—31nomaximal
9#10590760—31nomaximal
10#10590752—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C32×Dic6C2211yescharacteristic
14C2×#6294—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88807—21no—
21#88807—21no—
22#88796—22no—
23C3×#3717—22no—
24#64153—31no—
25C2×#3744—31no—
26C2×C3×#215—31no—
27C2×C3×#204—31no—
28#63898—31no—
29#63894—31no—
30Dic3×D12—31no—
31#63866—31no—
32#63866—31no—
33#63866—61no—
34C2×#3717D311yescharacteristic
35C2×#3716—31no—
36C2×#3716—31no—
37C2×#3716—61no—
38#63643—31no—
39C3×#835—31no—
40C3×#816—31no—
41C3×Dic3⋊C8—31no—
42C3×Dic3⋊C8—31no—
43#63572D311yescharacteristic, complemented
44#63567—31no—
45#63567—31no—
46#63567—61no—
47#63566D311yescharacteristic, 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—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C2×C3×C3⋊Dic3D411yescharacteristic
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)D412yes—
58C2×C32×Dic3—21no—
59C32×Dic6—21no—
60C32×Dic6—21no—
61#6294—42no—
62C32×C3⋊C8—21no—
63C32×C3⋊C8—21no—
64#5162—91no—
65#4841—91no—
66#4818—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—121no—
76C2×C3×D12—31no—
77C2×C3×D12—31no—
78C2×C3×D12—61no—
79C2×C3×Dic6—21no—
80C2×C3×Dic6C2×D311yescharacteristic
81C2×C3×Dic6C2×D311yescharacteristic
82C2×#624—61no—
83C2×D3×Dic3—61no—
84C2×D3×Dic3—61no—
85C2×C32×C8—31no—
86C2×C32×C8—31no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3750—61no—
89#3750—61no—
90C4×C3⋊Dic3C2×D311yescharacteristic
91#3744—62no—
92C3×#215—62no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—61no—
96C3×C4×Dic3—31no—
97C3×C4×Dic3—31no—
98C3×C4×Dic3—61no—
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—21no—
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105C3×#204—62no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—61no—
110Dic3⋊Dic3—62no—
111Dic3⋊Dic3—62no—
112#3722—61no—
113(C2×D3)⋊Dic3—61no—
114(C2×D3)⋊Dic3—61no—
115#3717—22no—
116#3716—62no—
117#3716—62no—
118#3716—64no—
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—21no—
127C3×C3⋊Dic3—41no—
128C3×C3⋊Dic3SD1612yescomplemented
129C32×Dic3—41no—
130C2×C3×SD16—91no—
131#944—91no—
132#944—91no—
133#944—181no—
134C2×#215—31no—
135D4×Dic3—91no—
136C2×#214—91no—
137C2×#204—31no—
138C2×#204—91no—
139C2×#204—91no—
140C2×#204—181no—
141#893—91no—
142C4×D12—91no—
143#869—91no—
144C3×C4⋊C8—91no—
145C3×#103—91no—
146#835—31no—
147#816—31no—
148#816—91no—
149#816—91no—
150#816—181no—
151Dic3⋊C8—31no—
152Dic3⋊C8—31no—
153#810—91no—
154C22×C3⋊D3—21no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C22×C3×D3—121no—
158C32×Q8—61no—
159C32×Q8—61no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
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—
176C2×C3⋊Dic3D1211yescharacteristic
177C2×C3⋊Dic3D1211yescharacteristic
178C4⋊(C3⋊D3)—21no—
179C4⋊(C3⋊D3)#6712yes—
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—22no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—31no—
198C2×C3×Dic3—31no—
199C2×C3×Dic3—31no—
200C2×C3×Dic3—31no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—61no—
204C2×C3×Dic3—61no—
205C2×C3×Dic3—121no—
206C3×D12—32no—
207C3×D12—32no—
208C3×D12—62no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—121no—
212C3×Dic6—21no—
213C3×Dic6—21no—
214C3×Dic6—21no—
215C3×Dic6—21no—
216C3×Dic6—41no—
217C3×Dic6—41no—
218#624—122no—
219D3×Dic3—122no—
220D3×Dic3—122no—
221C32×C8—61no—
222C32×C8—61no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8—21no—
225C3×C3⋊C8—21no—
226C3×C3⋊C8—21no—
227C3×C3⋊C8—21no—
228C3×C3⋊C8—41no—
229#461—271noSylow
230C2×C33#12111yescharacteristic
231C2×C33C2×SD1611yescharacteristic
232C2×C33#13711yescharacteristic
233C3×C3⋊D3—42no—
234C2×C3×Q8—31no—
235C2×C3×Q8—31no—
236C2×C3×Q8—61no—
237C2×C3×D4—91no—
238C22×Dic3—181no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×Dic6—91no—
245C2×Dic6—91no—
246C2×Dic6—91no—
247C2×Dic6—181no—
248C2×Dic6—181no—
249C2×Dic6—181no—
250C2×Dic6—361no—
251C2×Dic6D3211yescharacteristic
252C3×SD16—182no—
253C2×C3×C8—31no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—61no—
258C2×C3×C8—61no—
259C3×C4⋊C4—91no—
260C3×C42—91no—
261#217—181no—
262#215—62no—
263#214—182no—
264(C2×D3)⋊C4—181no—
265(C2×D3)⋊C4—181no—
266C4⋊Dic3—31no—
267C4⋊Dic3—31no—
268C4⋊Dic3—61no—
269C4⋊Dic3—181no—
270C4⋊Dic3—181no—
271Dic3⋊C4—181no—
272Dic3⋊C4—181no—
273Dic3⋊C4—181no—
274Dic3⋊C4—181no—
275Dic3⋊C4—182no—
276Dic3⋊C4—182no—
277C4×Dic3—31no—
278C4×Dic3—31no—
279C4×Dic3—61no—
280C2×C3⋊C8D3211yescharacteristic
281C2×C3⋊C8D3211yescharacteristic
282#204—62no—
283#204—182no—
284#204—182no—
285#204—184no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32C2×D1211yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32D3×D411yescharacteristic
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—41no—
312C32×C4—41no—
313C32×C4—121no—
314C32×C4#23511yescharacteristic
315C32×C4D3×D411yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4#23711yescharacteristic
318C32×C4#23911yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C3⋊Dic3—21no—
321C3⋊Dic3—41no—
322C3⋊Dic3—122no—
323C3⋊Dic3—122no—
324C3⋊Dic3—122no—
325C3⋊Dic3—124no—
326C3⋊Dic3—121no—
327C3⋊Dic3—121no—
328C3⋊Dic3—121no—
329C3⋊Dic3#20412yescomplemented
330C3×Dic3—32no—
331C3×Dic3—32no—
332C3×Dic3—42no—
333C3×Dic3—41no—
334C3×Dic3—41no—
335C3×Dic3—62no—
336C3×Dic3—62no—
337C3×Dic3—61no—
338C3×Dic3—61no—
339C3×Dic3—121no—
340C3×Dic3—121no—
341C3×Dic3—241no—
342C2×SD16—91no—
343C2×SD16—271no—
344C4⋊Q8—271no—
345C4×D4—271no—
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—61no—
355C3×Q8—61no—
356C3×Q8—61no—
357C3×Q8—61no—
358C3×Q8—121no—
359C3×Q8—121no—
360C3×D4—92no—
361C3×D4—181no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—62no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—91no—
370C2×C3×C4—91no—
371C2×C3×C4—181no—
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—31no—
377C2×Dic3—31no—
378C2×Dic3—31no—
379C2×Dic3—31no—
380C2×Dic3—61no—
381C2×Dic3—61no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—121no—
385C2×Dic3—182no—
386C2×Dic3—181no—
387C2×Dic3—181no—
388C2×Dic3—181no—
389C2×Dic3—361no—
390C2×Dic3—361no—
391C2×Dic3—362no—
392C2×Dic3—362no—
393D12—32no—
394D12—32no—
395D12—62no—
396D12—61no—
397D12—61no—
398D12—121no—
399C4×D3—362no—
400C4×D3—362no—
401Dic6—21no—
402Dic6—21no—
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—61no—
421C3×C8—61no—
422C3×C8—121no—
423C3⋊C8—21no—
424C3⋊C8—21no—
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×C32D3×SD1611yescharacteristic
443C2×C32#91511yescharacteristic
444C2×C32#91811yescharacteristic
445C2×C32#93911yescharacteristic
446C3⋊D3—42no—
447C3×D3—122no—
448C3×D3—122no—
449C3×D3—242no—
450C2×Q8—31no—
451C2×Q8—271no—
452C2×D4—91no—
453C22×C4—541no—
454SD16—182no—
455SD16—542no—
456C2×C8—31no—
457C2×C8—31no—
458C4⋊C4—91no—
459C4⋊C4—541no—
460C4⋊C4—541no—
461C22⋊C4—541no—
462C42—91no—
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×C3D3×D1211yescharacteristic
472C22×C3#381111yescharacteristic
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—41no—
489C3×C4—41no—
490C3×C4—92no—
491C3×C4—122no—
492C3×C4—121no—
493C3×C4—121no—
494C3×C4—181no—
495C3×C4—361no—
496C3×C4#379611yescharacteristic
497C3×C4#379911yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4D3×#6711yescharacteristic
500C3×C4D3×#6711yescharacteristic
501C3×C4#381111yescharacteristic
502Dic3—32no—
503Dic3—32no—
504Dic3—41no—
505Dic3—62no—
506Dic3—62no—
507Dic3—61no—
508Dic3—61no—
509Dic3—121no—
510Dic3—121no—
511Dic3—241no—
512Dic3—362no—
513Dic3—362no—
514Dic3—362no—
515Dic3—364no—
516Dic3—361no—
517Dic3—361no—
518Dic3—361no—
519C32—21no—
520C32—21no—
521C32—21no—
522C32—41no—
523C32#481811yescharacteristic, complemented
524C32#484111yescharacteristic, complemented
525C32#516211yescharacteristic, complemented
526C23—181no—
527Q8—61no—
528Q8—61no—
529Q8—541no—
530Q8—541no—
531D4—92no—
532D4—181no—
533C2×C4—61no—
534C2×C4—91no—
535C2×C4—91no—
536C2×C4—181no—
537C2×C4—541no—
538C2×C4—1082no—
539C2×C4D3311yescharacteristic
540C8—61no—
541C8—61no—
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×C3D3×#20411yescharacteristic
557C2×C3#6380011yescharacteristic
558C2×C3#6380111yescharacteristic
559C2×C3#6388611yescharacteristic
560C2×C3#6391011yescharacteristic
561C2×C3#6394011yescharacteristic
562C2×C3#6395111yescharacteristic
563C2×C3#6397111yescharacteristic
564D3—122no—
565D3—122no—
566D3—242no—
567C22—182no—
568C22—361no—
569C22—361no—
570C22#8895511yescharacteristic
571C4—92no—
572C4—121no—
573C4—181no—
574C4—361no—
575C4—1081no—
576C4#8895211yescharacteristic
577C4#8896211yescharacteristic
578C3—21no—
579C3—21no—
580C3—21no—
581C3—41no—
582C3#1059075211yescharacteristic, complemented
583C3#1059076011yescharacteristic, complemented
584C3#1059156911yescharacteristic, complemented
585C2—362no—
586C2#1172284311yescharacteristic
587C2#1172289611yescharacteristic
588C2#1172296611yescharacteristic
subgroupG/H
centerC22#88955
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. #11723455C2semidirect
5. #11722042C2semidirect
6. C3×#63572C2semidirect
7. C3×#63566C2semidirect
43. #63572D3semidirect
47. #63566D3semidirect
128. C3×C3⋊Dic3SD16semidirect
329. C3⋊Dic3#204semidirect
349. C33#461semidirect
523. C32#4818semidirect
524. C32#4841semidirect
525. C32#5162semidirect
582. C3#10590752semidirect
583. C3#10590760semidirect
584. C3#10591569semidirect

Maximal subgroups

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

subgroupindex
#117236912
#117234552
C2×#887962
C2×C3×#37172
#117220422
C3×#635722
C3×#635662
#105915693
#105907603
#105907523