Finite Group Databasebeta

← #13155537

Subgroups — #13155537

Order 1728.

574 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723423C211yesmaximal, characteristic
2#11722078C211yesmaximal, characteristic, complemented
3#11722021C211yesmaximal, characteristic, complemented
4#11722018C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
7C3×(C2×D3)⋊(C3⋊C8)C211yesmaximal, characteristic
8#10591062—31nomaximal
9#10590699—31nomaximal
10#10590698—31nomaximal
11C2×C32×C4×D3C2211yescharacteristic
12C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18Dic3⋊(C3⋊Dic3)C411yescharacteristic
19Dic3⋊(C3⋊Dic3)C411yescharacteristic
20(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
21(C2×D3)⋊(C3⋊Dic3)C411yescharacteristic
22#64135—31no—
23#63855—31no—
24#63855—31no—
25#63855—61no—
26#63642—31no—
27C8×C3⋊Dic3—31no—
28C3×#830—31no—
29C3×#830—31no—
30C3×(C2×D3)⋊C8—31no—
31C3×(C2×D3)⋊C8—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—31no—
34#63572—31no—
35#63572D311yescharacteristic, complemented
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—31no—
38(C2×D3)⋊(C3⋊C8)D311yescharacteristic
39(C2×D3)⋊(C3⋊C8)D311yescharacteristic
40#63555—31no—
41#63555—31no—
42#63555—61no—
43#63553—31no—
44#63553—31no—
45#63553—61no—
46C22×C32×D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
49C2×C3⋊(C3⋊Dic3)C2×C411yescharacteristic
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C32×Dic3C2×C411yescharacteristic
53C32×C4×D3—21no—
54C32×C4×D3—21no—
55C3×C3⋊(C3⋊C8)—21no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59#4840—91no—
60#4720—91no—
61#4720—91no—
62C22×C32×C4—31no—
63C2×C3×C4×D3—21no—
64C2×C3×C4×D3C2×D311yescharacteristic
65C2×C3×C4×D3C2×D311yescharacteristic
66C2×C32×C8—31no—
67C2×C32×C8—31no—
68#3757—31no—
69#3757—31no—
70C4⋊(C3⋊Dic3)—31no—
71C4⋊(C3⋊Dic3)—31no—
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⋊Dic3C2×D311yescharacteristic
80C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
82C3×C4×Dic3—31no—
83C3×C4×Dic3—31no—
84C3×C4×Dic3—61no—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—21no—
87C2×C3×C3⋊C8—31no—
88C2×C3×C3⋊C8—31no—
89C2×C3×C3⋊C8—31no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—61no—
92C2×C3×C3⋊C8—61no—
93C2×C3×C3⋊C8C2×D311yescharacteristic
94C2×C3×C3⋊C8C2×D311yescharacteristic
95C2×C3×C3⋊C8C2×D311yescharacteristic
96C2×C3×C3⋊C8C2×D311yescharacteristic
97Dic3⋊Dic3—31no—
98Dic3⋊Dic3—31no—
99Dic3⋊Dic3—31no—
100Dic3⋊Dic3—31no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103(C2×D3)⋊Dic3—31no—
104(C2×D3)⋊Dic3—31no—
105(C2×D3)⋊Dic3—31no—
106(C2×D3)⋊Dic3—31no—
107(C2×D3)⋊Dic3—61no—
108(C2×D3)⋊Dic3—61no—
109C22×C33C22×C411yescharacteristic
110C2×C32×D3—21no—
111C2×C32×D3—21no—
112C2×C32×D3—21no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3⋊(C3⋊Dic3)—21no—
116C3⋊(C3⋊Dic3)—21no—
117C3×C3⋊Dic3—21no—
118C3×C3⋊Dic3—21no—
119C32×Dic3—21no—
120#926—91no—
121#926—91no—
122#926—181no—
123#872—91no—
124C3×C4⋊C8—91no—
125C3×C22⋊C8—91no—
126C3×C22⋊C8—91no—
127#830—31no—
128#830—31no—
129(C2×D3)⋊C8—31no—
130(C2×D3)⋊C8—31no—
131#815—91no—
132#815—91no—
133#815—91no—
134#815—181no—
135Dic3⋊C8—31no—
136Dic3⋊C8—31no—
137Dic3⋊C8—91no—
138Dic3⋊C8—91no—
139C8×Dic3—91no—
140C8×Dic3—91no—
141C8×Dic3—91no—
142C8×Dic3—181no—
143C4⋊(C3⋊C8)—91no—
144C23×C32—31no—
145C22×C3×D3—21no—
146C22×C3×D3C4×D311yescharacteristic
147C22×C3×D3C4×D311yescharacteristic
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×C4C22×D311yescharacteristic
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—31no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—121no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3C4×D311yescharacteristic
173C2×C3⋊Dic3C4×D311yescharacteristic
174C2×C3×Dic3—21no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—61no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3C4×D311yescharacteristic
182C2×C3×Dic3C4×D311yescharacteristic
183C3×C4×D3—22no—
184C3×C4×D3—22no—
185C3×C4×D3—21no—
186C3×C4×D3—21no—
187C3×C4×D3—21no—
188C3×C4×D3—21no—
189C32×C8—61no—
190C32×C8—61no—
191C3⋊(C3⋊C8)—21no—
192C3⋊(C3⋊C8)—21no—
193C3×C3⋊C8—22no—
194C3×C3⋊C8—22no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—21no—
197C3×C3⋊C8—21no—
198C3×C3⋊C8—21no—
199C3×C3⋊C8—62no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205#433—271noSylow
206C2×C33#11711yescharacteristic
207C2×C33#13111yescharacteristic
208C2×C33#13111yescharacteristic
209C32×D3—41no—
210C22×C3×C4—31no—
211C22×C3×C4—31no—
212C22×C3×C4—61no—
213C2×C4×D3D3211yescharacteristic
214C2×C3×C8—31no—
215C2×C3×C8—31no—
216C2×C3×C8—31no—
217C2×C3×C8—31no—
218C2×C3×C8—61no—
219C2×C3×C8—61no—
220C2×C3×C8—91no—
221C2×C3×C8—91no—
222C3×C42—91no—
223#217—91no—
224#217—91no—
225#217—91no—
226#217—91no—
227#217—181no—
228#217—181no—
229(C2×D3)⋊C4—91no—
230(C2×D3)⋊C4—91no—
231C4⋊Dic3—91no—
232C4⋊Dic3—91no—
233C4⋊Dic3—91no—
234C4⋊Dic3—91no—
235C4⋊Dic3—181no—
236C4⋊Dic3—181no—
237Dic3⋊C4—91no—
238Dic3⋊C4—91no—
239C4×Dic3—31no—
240C4×Dic3—31no—
241C4×Dic3—61no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—91no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—181no—
248C4×Dic3—361no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8D3211yescharacteristic
256C2×C3⋊C8D3211yescharacteristic
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—61no—
262C22×C32—61no—
263C22×C32—61no—
264C22×C32C2×C4×D311yescharacteristic
265C22×C32C2×C4×D311yescharacteristic
266C22×C32C2×C4×D311yescharacteristic
267C2×C3×D3—22no—
268C2×C3×D3—21no—
269C2×C3×D3—21no—
270C2×C3×D3—21no—
271C2×C3×D3—21no—
272C2×C3×D3—21no—
273C2×C3×D3—21no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—21no—
282C32×C4—41no—
283C32×C4—41no—
284C32×C4—61no—
285C32×C4#23711yescharacteristic
286C32×C4#23711yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23711yescharacteristic
289C32×C4#23711yescharacteristic
290C32×C4#23911yescharacteristic
291C3⋊Dic3—21no—
292C3⋊Dic3—21no—
293C3⋊Dic3—61no—
294C3⋊Dic3—61no—
295C3⋊Dic3—62no—
296C3⋊Dic3—62no—
297C3⋊Dic3—61no—
298C3⋊Dic3—61no—
299C3⋊Dic3—61no—
300C3⋊Dic3—61no—
301C3⋊Dic3—121no—
302C3⋊Dic3—121no—
303C3⋊Dic3—121no—
304C3⋊Dic3—121no—
305C3⋊Dic3—122no—
306C3⋊Dic3—122no—
307C3×Dic3—22no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—61no—
311C3×Dic3—61no—
312C3×Dic3—61no—
313C3×Dic3—61no—
314C3×Dic3—121no—
315C3×Dic3—121no—
316#117—271no—
317C4⋊C8—91no—
318C4⋊C8—271no—
319C22⋊C8—91no—
320C22⋊C8—91no—
321#97—271no—
322C4×C8—271no—
323C33#43311yescharacteristic, complemented, Sylow
324C23×C3—31no—
325C23×C3—31no—
326C23×C3—61no—
327C22×D3#62411yescharacteristic
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—21no—
331C2×C3×C4—31no—
332C2×C3×C4—31no—
333C2×C3×C4—41no—
334C2×C3×C4—61no—
335C2×C3×C4—62no—
336C2×C3×C4—62no—
337C2×C3×C4—61no—
338C2×C3×C4—61no—
339C2×C3×C4—61no—
340C2×C3×C4—61no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4C2×D3211yescharacteristic
344C2×C3×C4C2×D3211yescharacteristic
345C2×C3×C4C2×D3211yescharacteristic
346C2×Dic3—31no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—31no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—91no—
357C2×Dic3—91no—
358C2×Dic3—181no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—181no—
363C2×Dic3—181no—
364C2×Dic3—361no—
365C2×Dic3—361no—
366C2×Dic3#62411yescharacteristic
367C4×D3—21no—
368C4×D3—21no—
369C3×C8—62no—
370C3×C8—62no—
371C3×C8—61no—
372C3×C8—61no—
373C3×C8—61no—
374C3×C8—61no—
375C3×C8—181no—
376C3×C8—181no—
377C3⋊C8—21no—
378C3⋊C8—21no—
379C3⋊C8—62no—
380C3⋊C8—62no—
381C3⋊C8—61no—
382C3⋊C8—61no—
383C3⋊C8—61no—
384C3⋊C8—61no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32—41no—
397C2×C32—121no—
398C2×C32#87711yescharacteristic
399C2×C32#87711yescharacteristic
400C2×C32#89211yescharacteristic
401C2×C32#90111yescharacteristic
402C2×C32#90111yescharacteristic
403C2×C32#90111yescharacteristic
404C2×C32#90711yescharacteristic
405C2×C32#90711yescharacteristic
406C2×C32#90711yescharacteristic
407C3×D3—42no—
408C3×D3—41no—
409C3×D3—41no—
410C22×C4—31no—
411C2×C8—31no—
412C2×C8—31no—
413C2×C8—91no—
414C2×C8—91no—
415C4⋊C4—271no—
416C4⋊C4—271no—
417C22⋊C4—271no—
418C22⋊C4—271no—
419C42—91no—
420C42—271no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—21no—
424C22×C3—41no—
425C22×C3—62no—
426C22×C3—61no—
427C22×C3—61no—
428C22×C3—61no—
429C22×C3—61no—
430C22×C3—61no—
431C22×C3—61no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3C4×D3211yescharacteristic
435C22×C3C4×D3211yescharacteristic
436C22×C3C2×#62411yescharacteristic
437C2×D3—21no—
438C2×D3—21no—
439C2×D3—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—41no—
447C3×C4—41no—
448C3×C4—62no—
449C3×C4—61no—
450C3×C4—61no—
451C3×C4—181no—
452C3×C4—181no—
453C3×C4#379611yescharacteristic
454C3×C4#379611yescharacteristic
455C3×C4#380511yescharacteristic
456C3×C4#380511yescharacteristic
457C3×C4#380511yescharacteristic
458C3×C4#380511yescharacteristic
459Dic3—21no—
460Dic3—61no—
461Dic3—61no—
462Dic3—61no—
463Dic3—61no—
464Dic3—121no—
465Dic3—121no—
466Dic3—182no—
467Dic3—182no—
468Dic3—181no—
469Dic3—181no—
470Dic3—181no—
471Dic3—181no—
472Dic3—181no—
473Dic3—181no—
474Dic3—361no—
475Dic3—361no—
476Dic3—361no—
477Dic3—361no—
478Dic3—362no—
479Dic3—362no—
480C32—21no—
481C32—21no—
482C32—21no—
483C32—41no—
484C32#472011yescharacteristic, complemented
485C32#472011yescharacteristic, complemented
486C32#484011yescharacteristic, complemented
487C23—31no—
488C2×C4—31no—
489C2×C4—61no—
490C2×C4—61no—
491C2×C4—91no—
492C2×C4—91no—
493C2×C4—271no—
494C2×C4—271no—
495C2×C4D3311yescharacteristic
496C8—61no—
497C8—61no—
498C8—181no—
499C8—181no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—41no—
510C2×C3—41no—
511C2×C3—41no—
512C2×C3—122no—
513C2×C3—121no—
514C2×C3—121no—
515C2×C3#6380311yescharacteristic
516C2×C3#6380411yescharacteristic
517C2×C3#6380411yescharacteristic
518C2×C3#6380511yescharacteristic
519C2×C3#6381711yescharacteristic
520C2×C3#6381711yescharacteristic
521C2×C3#6387811yescharacteristic
522C2×C3#6387811yescharacteristic
523C2×C3#6395711yescharacteristic
524D3—41no—
525C22—61no—
526C22—61no—
527C22—61no—
528C22D3×#62411yescharacteristic
529C4—61no—
530C4—181no—
531C4—181no—
532C4—541no—
533C4—541no—
534C4#8896311yescharacteristic
535C4#8896411yescharacteristic
536C3—21no—
537C3—21no—
538C3—21no—
539C3—41no—
540C3#1059069811yescharacteristic, complemented
541C3#1059069911yescharacteristic, complemented
542C3#1059106211yescharacteristic, complemented
543C2—121no—
544C2#1172282411yescharacteristic
545C2#1172282511yescharacteristic
546C2#1172291711yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C32×D3C2×C4
hypocenterC33#433
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11722078C2semidirect
3. #11722021C2semidirect
4. #11722018C2semidirect
5. C3×#63572C2semidirect
35. #63572D3semidirect
323. C33#433semidirect
484. C32#4720semidirect
485. C32#4720semidirect
486. C32#4840semidirect
540. C3#10590698semidirect
541. C3#10590699semidirect
542. C3#10591062semidirect

Maximal subgroups

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

subgroupindex
#117234232
#117220782
#117220212
#117220182
C3×#635722
C3×(C2×D3)⋊(C3⋊C8)2
#105910623
#105906993
#105906983