Finite Group Databasebeta

← #13161998

Subgroups — #13161998

Order 1728.

664 conjugacy classes of subgroups.

The subgroup lattice has 582 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64156C211yesmaximal, characteristic
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3#11723668C211yesmaximal, characteristic, complemented
4C2×#88811C211yesmaximal, characteristic
5C3×#63662C211yesmaximal, characteristic, complemented
6#11722078C211yesmaximal, characteristic, complemented
7#11722075C211yesmaximal, characteristic
8#10591579—31nomaximal
9#10591578—31nomaximal
10#10591577—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C3×#3756—22no—
15C3×#3752—21no—
16C3×#3752—21no—
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88815—21no—
22#88813—21no—
23#88811—22no—
24#64156D311yescharacteristic
25C2×#3756D311yescharacteristic, complemented
26C3×#947—31no—
27C3×#947—31no—
28C3×#947—61no—
29C2×C3×#216—31no—
30C2×C3×#216—31no—
31C2×C3×#216—61no—
32#63844—31no—
33#63843—31no—
34Dic3×Dic6—31no—
35C2×#3717—31no—
36C2×#3716—31no—
37C2×#3715—31no—
38#63662D311yescharacteristic, complemented
39C3×#835—31no—
40C3×#835—31no—
41C3×#835—61no—
42#63572—31no—
43Dic3⋊(C3⋊C8)—31no—
44Dic3⋊(C3⋊C8)—31no—
45#63564—31no—
46#63563—31no—
47#63562—31no—
48C22×C3×C3⋊D3—21no—
49C33×Q8—21no—
50C33×Q8—21no—
51C2×C33×C4—21no—
52C2×C33×C4C2311yescharacteristic
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D411yescharacteristic
56C2×C3×C3⋊Dic3D411yescharacteristic
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×#634—21no—
60C3×#634D412yes—
61C3×C3⋊(C3⋊C8)—21no—
62C3×C3⋊(C3⋊C8)—21no—
63#5185—91no—
64#5054—91no—
65#4798—91no—
66C2×C32×Q8—21no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—41no—
70C2×C32×Q8C2×D311yescharacteristic
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×#634C2×D311yescharacteristic
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81#3756—22no—
82#3752—21no—
83#3752—21no—
84C4⋊(C3⋊Dic3)C2×D311yescharacteristic
85C4×C3⋊Dic3C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
88C3×#216—62no—
89C3×#216—62no—
90C3×#216—64no—
91C3×(C2×D3)⋊C4—61no—
92C3×(C2×D3)⋊C4—61no—
93C3×(C2×D3)⋊C4—61no—
94C3×(C2×D3)⋊C4—61no—
95C3×(C2×D3)⋊C4—62no—
96C3×(C2×D3)⋊C4—62no—
97C3×C4⋊Dic3—31no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—61no—
100C3×C4×Dic3—31no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—61no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—61no—
108C2×C3×C3⋊C8—61no—
109#3724—61no—
110Dic3⋊Dic3—61no—
111Dic3⋊Dic3—61no—
112Dic32—61no—
113Dic32—61no—
114Dic32—61no—
115#3717—62no—
116#3716—62no—
117#3715—62no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C33×C4—41no—
123C33×C4C2×D411yescharacteristic
124C33×C4#4111yescharacteristic
125C3×C3⋊Dic3—22no—
126C3×C3⋊Dic3—21no—
127C3×C3⋊Dic3—21no—
128C3×C3⋊Dic3—41no—
129C2×C3×Q16—91no—
130C3×#124—91no—
131#947—31no—
132#947—31no—
133#947—61no—
134Q8×Dic3—91no—
135C2×#216—31no—
136C2×#216—31no—
137C2×#216—61no—
138C2×#215—91no—
139C2×#214—91no—
140C2×#204—91no—
141#887—91no—
142C4×Dic6—91no—
143C3×#103—91no—
144#835—31no—
145#835—31no—
146#835—61no—
147#832—91no—
148#821—91no—
149Dic3⋊C8—91no—
150Dic3⋊C8—91no—
151#809—91no—
152C4⋊(C3⋊C8)—91no—
153C22×C3⋊D3—21no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C32×Q8—21no—
158C32×Q8—21no—
159C32×Q8—22no—
160C32×Q8—22no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—21no—
164C32×Q8—21no—
165C32×Q8—41no—
166C32×Q8—41no—
167C32×Q8—41no—
168C32×Q8—41no—
169C32×Q8—42no—
170C32×Q8—42no—
171C2×C32×C4—21no—
172C2×C32×C4—22no—
173C2×C32×C4—21no—
174C2×C32×C4—21no—
175C2×C32×C4—21no—
176C2×C32×C4—21no—
177C2×C32×C4—21no—
178C2×C32×C4—41no—
179C2×C32×C4—41no—
180C2×C32×C4—41no—
181C2×C32×C4—42no—
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C32×C4C22×D311yescharacteristic
185C2×C3⋊Dic3—21no—
186C2×C3⋊Dic3—21no—
187C2×C3⋊Dic3#6711yescharacteristic
188C2×C3⋊Dic3#6711yescharacteristic
189C4⋊(C3⋊D3)—21no—
190C4⋊(C3⋊D3)—21no—
191#634—21no—
192#634#6712yes—
193C2×C3×Dic3—31no—
194C2×C3×Dic3—31no—
195C2×C3×Dic3—31no—
196C2×C3×Dic3—31no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C2×C3×Dic3—121no—
204C2×C3×Dic3—121no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—121no—
210C3×D12—121no—
211C3×Dic6—32no—
212C3×Dic6—32no—
213C3×Dic6—62no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—121no—
217C3⋊(C3⋊C8)—21no—
218C3⋊(C3⋊C8)—21no—
219C3×C3⋊C8—62no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—61no—
223C3×C3⋊C8—61no—
224C3×C3⋊C8—121no—
225#464—271noSylow
226C2×C33#12111yescharacteristic
227C2×C33#13511yescharacteristic
228C2×C33#13711yescharacteristic
229C3×C3⋊D3—81no—
230C2×C3×Q8—21no—
231C2×C3×Q8—21no—
232C2×C3×Q8—21no—
233C2×C3×Q8—41no—
234C2×C3×Q8—91no—
235C2×C3×Q8D3211yescharacteristic
236C2×C3×Q8D3211yescharacteristic
237C2×C3×Q8D3211yescharacteristic
238C2×C3×D4—91no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×Dic6—31no—
243C2×Dic6—31no—
244C2×Dic6—61no—
245C3×Q16—182no—
246C2×C3×C8—91no—
247C2×C3×C8—91no—
248C3×C4⋊C4—91no—
249C3×C22⋊C4—181no—
250C3×C22⋊C4—181no—
251C3×C42—91no—
252#216—62no—
253#216—62no—
254#216—64no—
255#215—182no—
256#214—182no—
257(C2×D3)⋊C4—61no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—61no—
260(C2×D3)⋊C4—61no—
261(C2×D3)⋊C4—62no—
262(C2×D3)⋊C4—62no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3—181no—
267Dic3⋊C4—181no—
268Dic3⋊C4—181no—
269C4×Dic3—31no—
270C4×Dic3—31no—
271C4×Dic3—61no—
272C4×Dic3—181no—
273C4×Dic3—181no—
274C4×Dic3—181no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—61no—
280C2×C3⋊C8—61no—
281#204—182no—
282C22×C32—21no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—41no—
286C22×C32D3×D411yescharacteristic
287C22×C32D3×D411yescharacteristic
288C22×C32C2×#6711yescharacteristic
289C2×C3⋊D3—41no—
290C2×C3⋊D3—41no—
291C2×C3⋊D3—41no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—42no—
310C32×C4—42no—
311C32×C4—42no—
312C32×C4—44no—
313C32×C4—41no—
314C32×C4—41no—
315C32×C4—41no—
316C32×C4#23511yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4#23711yescharacteristic
320C32×C4#23711yescharacteristic
321C32×C4C2×#6711yescharacteristic
322C3⋊Dic3—22no—
323C3⋊Dic3—21no—
324C3⋊Dic3—21no—
325C3⋊Dic3—41no—
326C3×Dic3—62no—
327C3×Dic3—61no—
328C3×Dic3—61no—
329C3×Dic3—61no—
330C3×Dic3—61no—
331C3×Dic3—62no—
332C3×Dic3—122no—
333C3×Dic3—121no—
334C3×Dic3—121no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337C3×Dic3—241no—
338C2×Q16—91no—
339C2×SD16—271no—
340#124—91no—
341C4×Q8—271no—
342C4⋊C8—271no—
343#103—91no—
344#102—271no—
345C33#46411yescharacteristic, complemented, Sylow
346C23×C3—181no—
347C22×D3—62no—
348C22×D3—61no—
349C22×D3—61no—
350C3×Q8—22no—
351C3×Q8—22no—
352C3×Q8—21no—
353C3×Q8—21no—
354C3×Q8—21no—
355C3×Q8—21no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—41no—
359C3×Q8—41no—
360C3×Q8—41no—
361C3×Q8—41no—
362C3×Q8—42no—
363C3×Q8—42no—
364C3×Q8—92no—
365C3×Q8—181no—
366C3×D4—181no—
367C3×D4—181no—
368C2×C3×C4—22no—
369C2×C3×C4—21no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—41no—
376C2×C3×C4—41no—
377C2×C3×C4—42no—
378C2×C3×C4—41no—
379C2×C3×C4—91no—
380C2×C3×C4—91no—
381C2×C3×C4—181no—
382C2×C3×C4—181no—
383C2×C3×C4C2×D3211yescharacteristic
384C2×C3×C4C2×D3211yescharacteristic
385C2×C3×C4C2×D3211yescharacteristic
386C2×Dic3—31no—
387C2×Dic3—31no—
388C2×Dic3—31no—
389C2×Dic3—31no—
390C2×Dic3—61no—
391C2×Dic3—61no—
392C2×Dic3—61no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—61no—
396C2×Dic3—121no—
397C2×Dic3—121no—
398D12—61no—
399D12—61no—
400D12—61no—
401D12—61no—
402D12—121no—
403D12—121no—
404Dic6—32no—
405Dic6—32no—
406Dic6—62no—
407Dic6—61no—
408Dic6—61no—
409Dic6—121no—
410C3×C8—181no—
411C3×C8—181no—
412C3⋊C8—62no—
413C3⋊C8—61no—
414C3⋊C8—61no—
415C3⋊C8—61no—
416C3⋊C8—61no—
417C3⋊C8—121no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—41no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32#88411yescharacteristic
431C2×C32#91611yescharacteristic
432C2×C32#91811yescharacteristic
433C2×C32#91811yescharacteristic
434C2×C32#91911yescharacteristic
435C2×C32#93811yescharacteristic
436C2×C32#93911yescharacteristic
437C2×C32#95011yescharacteristic
438C2×C32#95111yescharacteristic
439C3⋊D3—81no—
440C3×D3—242no—
441C3×D3—241no—
442C3×D3—241no—
443C2×Q8—91no—
444C2×Q8#111311yescharacteristic
445C2×D4—91no—
446Q16—182no—
447SD16—542no—
448C2×C8—91no—
449C2×C8—91no—
450C4⋊C4—91no—
451C4⋊C4—541no—
452C22⋊C4—181no—
453C22⋊C4—181no—
454C42—91no—
455C42—541no—
456C22×C3—21no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—41no—
460C22×C3—361no—
461C22×C3—361no—
462C22×C3—361no—
463C22×C3D3×#6711yescharacteristic
464C22×C3D3×#6711yescharacteristic
465C22×C3#381111yescharacteristic
466C2×D3—122no—
467C2×D3—122no—
468C2×D3—122no—
469C2×D3—121no—
470C2×D3—121no—
471C2×D3—121no—
472C2×D3—121no—
473C2×D3—121no—
474C2×D3—121no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—44no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—41no—
488C3×C4—41no—
489C3×C4—41no—
490C3×C4—181no—
491C3×C4—181no—
492C3×C4—182no—
493C3×C4—361no—
494C3×C4#380411yescharacteristic
495C3×C4#380411yescharacteristic
496C3×C4#380511yescharacteristic
497C3×C4D3×#6711yescharacteristic
498C3×C4D3×#6711yescharacteristic
499C3×C4#381111yescharacteristic
500Dic3—61no—
501Dic3—61no—
502Dic3—61no—
503Dic3—61no—
504Dic3—62no—
505Dic3—62no—
506Dic3—122no—
507Dic3—121no—
508Dic3—121no—
509Dic3—121no—
510Dic3—121no—
511Dic3—241no—
512C32—21no—
513C32—21no—
514C32—21no—
515C32—41no—
516C32#479811yescharacteristic, complemented
517C32#505411yescharacteristic, complemented
518C32#518511yescharacteristic, complemented
519C23—181no—
520Q8—21no—
521Q8—21no—
522Q8—92no—
523Q8—181no—
524D4—181no—
525D4—181no—
526C2×C4—21no—
527C2×C4—91no—
528C2×C4—91no—
529C2×C4—181no—
530C2×C4—181no—
531C2×C4C2×#111311yescharacteristic
532C8—181no—
533C8—181no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—41no—
544C2×C3—41no—
545C2×C3—41no—
546C2×C3—721no—
547C2×C3#6394311yescharacteristic
548C2×C3#6394311yescharacteristic
549C2×C3#6394511yescharacteristic
550C2×C3#6394711yescharacteristic
551C2×C3#6394911yescharacteristic
552C2×C3#6395011yescharacteristic
553C2×C3#6395811yescharacteristic
554C2×C3#6397011yescharacteristic
555C2×C3#6397111yescharacteristic
556D3—242no—
557D3—241no—
558D3—241no—
559C22—361no—
560C22—361no—
561C22—361no—
562C22#8904911yescharacteristic
563C4—41no—
564C4—181no—
565C4—181no—
566C4—182no—
567C4—361no—
568C4#8904611yescharacteristic
569C4#8904911yescharacteristic
570C3—21no—
571C3—21no—
572C3—21no—
573C3—41no—
574C3#1059157711yescharacteristic, complemented
575C3#1059157811yescharacteristic, complemented
576C3#1059157911yescharacteristic, complemented
577C2—721no—
578C2#1172371111yescharacteristic
579C2#1172371311yescharacteristic
580C2#1172372511yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypercenterC2×Q8#1113
hypocenterC33#464
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3756C2semidirect
3. #11723668C2semidirect
5. C3×#63662C2semidirect
6. #11722078C2semidirect
25. C2×#3756D3semidirect
38. #63662D3semidirect
345. C33#464semidirect
516. C32#4798semidirect
517. C32#5054semidirect
518. C32#5185semidirect
574. C3#10591577semidirect
575. C3#10591578semidirect
576. C3#10591579semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C3×#37562
#117236682
C2×#888112
C3×#636622
#117220782
#117220752
#105915793
#105915783
#105915773