Finite Group Databasebeta

← #13155546

Subgroups — #13155546

Order 1728.

538 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723673C211yesmaximal, characteristic
2#11722051C211yesmaximal, characteristic, complemented
3#11722023C211yesmaximal, characteristic
4C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
5C3×#63572C211yesmaximal, characteristic, complemented
6C3×#63557C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8#10590698—31nomaximal
9#10590697—31nomaximal
10#10590647—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
13C3×C4×C3⋊Dic3C2211yescharacteristic
14C3×C4×C3⋊Dic3C2211yescharacteristic
15C2×C32×C3⋊C8C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88815C411yescharacteristic
19#88815C411yescharacteristic
20#88814C411yescharacteristic
21#88814C411yescharacteristic
22#63874—31no—
23#63855—31no—
24#63855—31no—
25#63647—31no—
26#63641—31no—
27C8×C3⋊Dic3—31no—
28C3×(C2×D3)⋊C8—31no—
29C3×(C2×D3)⋊C8—31no—
30C3×#815—31no—
31C3×#815—31no—
32C3×Dic3⋊C8—31no—
33C3×Dic3⋊C8—31no—
34#63572D311yescharacteristic, complemented
35Dic3⋊(C3⋊C8)—31no—
36Dic3⋊(C3⋊C8)—31no—
37Dic3⋊(C3⋊C8)—61no—
38#63557D311yescharacteristic
39(C2×D3)⋊(C3⋊C8)—31no—
40(C2×D3)⋊(C3⋊C8)—31no—
41(C2×D3)⋊(C3⋊C8)—61no—
42#63555D311yescharacteristic, complemented
43Dic3×C3⋊C8—31no—
44Dic3×C3⋊C8—31no—
45Dic3×C3⋊C8—61no—
46C22×C3×C3⋊D3C2×C411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3×C3⋊Dic3C2×C411yescharacteristic
49C2×C3×C3⋊Dic3C2×C411yescharacteristic
50C2×C3×C3⋊Dic3C2×C411yescharacteristic
51C2×C3×C3⋊Dic3C2×C411yescharacteristic
52C2×C3×C3⋊Dic3C2×C411yescharacteristic
53C3×C4×C3⋊D3—21no—
54C3×C4×C3⋊D3—21no—
55C3⋊(C3⋊(C3⋊C8))—21no—
56C32×C3⋊C8—21no—
57C32×C3⋊C8—21no—
58C32×C3⋊C8—21no—
59#4840—91no—
60#4720—91no—
61#4709—91no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×C4×D3—31no—
64C2×C3×C4×D3—31no—
65C2×C3×C4×D3—61no—
66C2×C32×C8—31no—
67C2×C32×C8—31no—
68C2×C32×C8—31no—
69C4×C3⋊Dic3C2×D311yescharacteristic
70C4×C3⋊Dic3C2×D311yescharacteristic
71C2×C3⋊(C3⋊C8)—31no—
72C2×C3⋊(C3⋊C8)—31no—
73C2×C3⋊(C3⋊C8)—31no—
74C2×C3⋊(C3⋊C8)—61no—
75C2×C3⋊(C3⋊C8)—61no—
76C2×C3⋊(C3⋊C8)—61no—
77C2×C3⋊(C3⋊C8)—121no—
78C3×C4×Dic3—31no—
79C3×C4×Dic3—31no—
80C3×C4×Dic3—31no—
81C3×C4×Dic3—31no—
82C3×C4×Dic3—61no—
83C3×C4×Dic3—61no—
84C2×C3×C3⋊C8—21no—
85C2×C3×C3⋊C8—21no—
86C2×C3×C3⋊C8—21no—
87C2×C3×C3⋊C8C2×D311yescharacteristic
88C2×C3×C3⋊C8C2×D311yescharacteristic
89C2×C3×C3⋊C8C2×D311yescharacteristic
90C2×C3×C3⋊C8C2×D311yescharacteristic
91C2×C3×C3⋊C8C2×D311yescharacteristic
92C2×C3×C3⋊C8C2×D311yescharacteristic
93#3724—31no—
94#3724—31no—
95Dic3⋊Dic3—31no—
96Dic3⋊Dic3—31no—
97Dic3⋊Dic3—31no—
98Dic3⋊Dic3—31no—
99#3722—31no—
100#3722—31no—
101(C2×D3)⋊Dic3—31no—
102(C2×D3)⋊Dic3—31no—
103(C2×D3)⋊Dic3—31no—
104(C2×D3)⋊Dic3—31no—
105C22×C33C22×C411yescharacteristic
106C2×C3×C3⋊D3—21no—
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C33×C4#4111yescharacteristic
110C33×C4#4111yescharacteristic
111C3×C3⋊Dic3—21no—
112C3×C3⋊Dic3—21no—
113C3×C3⋊Dic3—21no—
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—21no—
116#926—91no—
117#872—91no—
118#872—91no—
119C3×C4⋊C8—91no—
120C3×C22⋊C8—91no—
121C3×#97—91no—
122#830—91no—
123(C2×D3)⋊C8—31no—
124(C2×D3)⋊C8—31no—
125(C2×D3)⋊C8—91no—
126(C2×D3)⋊C8—91no—
127(C2×D3)⋊C8—181no—
128#815—31no—
129#815—31no—
130Dic3⋊C8—31no—
131Dic3⋊C8—31no—
132Dic3⋊C8—91no—
133Dic3⋊C8—91no—
134Dic3⋊C8—181no—
135C8×Dic3—91no—
136C8×Dic3—91no—
137C8×Dic3—181no—
138C4⋊(C3⋊C8)—91no—
139C4×C3⋊C8—91no—
140C22×C3⋊D3C4×D311yescharacteristic
141C22×C3×D3—31no—
142C22×C3×D3—31no—
143C22×C3×D3—61no—
144C2×C32×C4—21no—
145C2×C32×C4—21no—
146C2×C32×C4—21no—
147C2×C32×C4—41no—
148C2×C32×C4C22×D311yescharacteristic
149C2×C32×C4C22×D311yescharacteristic
150C2×C32×C4C22×D311yescharacteristic
151C2×C3⋊Dic3C4×D311yescharacteristic
152C2×C3⋊Dic3C4×D311yescharacteristic
153C2×C3⋊Dic3C4×D311yescharacteristic
154C2×C3⋊Dic3C4×D311yescharacteristic
155C2×C3⋊Dic3C4×D311yescharacteristic
156C4×C3⋊D3—21no—
157C4×C3⋊D3—21no—
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—31no—
165C2×C3×Dic3—31no—
166C2×C3×Dic3—31no—
167C2×C3×Dic3—31no—
168C2×C3×Dic3—61no—
169C2×C3×Dic3—61no—
170C2×C3×Dic3—61no—
171C2×C3×Dic3—61no—
172C2×C3×Dic3—61no—
173C3×C4×D3—61no—
174C3×C4×D3—61no—
175C3×C4×D3—61no—
176C3×C4×D3—61no—
177C3×C4×D3—121no—
178C3×C4×D3—121no—
179C32×C8—61no—
180C32×C8—61no—
181C32×C8—61no—
182C3⋊(C3⋊C8)—61no—
183C3⋊(C3⋊C8)—62no—
184C3⋊(C3⋊C8)—61no—
185C3⋊(C3⋊C8)—61no—
186C3⋊(C3⋊C8)—121no—
187C3⋊(C3⋊C8)—121no—
188C3⋊(C3⋊C8)—122no—
189C3×C3⋊C8—22no—
190C3×C3⋊C8—21no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8—21no—
193C3×C3⋊C8—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—41no—
197C3×C3⋊C8—41no—
198#433—271noSylow
199C2×C33#11711yescharacteristic
200C2×C33#13111yescharacteristic
201C2×C33#13111yescharacteristic
202C3×C3⋊D3—41no—
203C22×C3×C4—91no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—61no—
207C2×C3×C8—31no—
208C2×C3×C8—31no—
209C2×C3×C8—31no—
210C2×C3×C8—31no—
211C2×C3×C8—31no—
212C2×C3×C8—31no—
213C2×C3×C8—61no—
214C2×C3×C8—61no—
215C2×C3×C8—61no—
216C3×C42—91no—
217C3×C42—91no—
218#217—91no—
219#217—91no—
220(C2×D3)⋊C4—91no—
221(C2×D3)⋊C4—91no—
222(C2×D3)⋊C4—91no—
223(C2×D3)⋊C4—91no—
224C4⋊Dic3—91no—
225C4⋊Dic3—91no—
226Dic3⋊C4—91no—
227Dic3⋊C4—91no—
228Dic3⋊C4—91no—
229Dic3⋊C4—91no—
230C4×Dic3—31no—
231C4×Dic3—31no—
232C4×Dic3—31no—
233C4×Dic3—31no—
234C4×Dic3—61no—
235C4×Dic3—61no—
236C2×C3⋊C8—91no—
237C2×C3⋊C8—91no—
238C2×C3⋊C8—91no—
239C2×C3⋊C8—181no—
240C2×C3⋊C8—181no—
241C2×C3⋊C8—181no—
242C2×C3⋊C8—361no—
243C2×C3⋊C8D3211yescharacteristic
244C2×C3⋊C8D3211yescharacteristic
245C2×C3⋊C8D3211yescharacteristic
246C22×C32—21no—
247C22×C32—21no—
248C22×C32—21no—
249C22×C32—41no—
250C22×C32C2×C4×D311yescharacteristic
251C22×C32C2×C4×D311yescharacteristic
252C22×C32C2×C4×D311yescharacteristic
253C2×C3⋊D3—21no—
254C2×C3⋊D3—21no—
255C2×C3⋊D3—21no—
256C2×C3×D3—62no—
257C2×C3×D3—61no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—61no—
261C2×C3×D3—61no—
262C2×C3×D3—61no—
263C2×C3×D3—121no—
264C2×C3×D3—121no—
265C32×C4—21no—
266C32×C4—21no—
267C32×C4—21no—
268C32×C4—21no—
269C32×C4—21no—
270C32×C4—21no—
271C32×C4—41no—
272C32×C4—41no—
273C32×C4#23511yescharacteristic
274C32×C4#23511yescharacteristic
275C32×C4#23711yescharacteristic
276C32×C4#23711yescharacteristic
277C32×C4#23711yescharacteristic
278C32×C4#23911yescharacteristic
279C3⋊Dic3—21no—
280C3⋊Dic3—21no—
281C3⋊Dic3—21no—
282C3⋊Dic3—21no—
283C3⋊Dic3—21no—
284C3×Dic3—62no—
285C3×Dic3—62no—
286C3×Dic3—62no—
287C3×Dic3—61no—
288C3×Dic3—61no—
289C3×Dic3—61no—
290C3×Dic3—61no—
291C3×Dic3—61no—
292C3×Dic3—61no—
293C3×Dic3—61no—
294C3×Dic3—61no—
295C3×Dic3—61no—
296C3×Dic3—61no—
297C3×Dic3—121no—
298C3×Dic3—121no—
299#117—271no—
300C4⋊C8—91no—
301C4⋊C8—271no—
302C22⋊C8—91no—
303C22⋊C8—271no—
304#97—91no—
305C4×C8—271no—
306C33#43311yescharacteristic, complemented, Sylow
307C23×C3—91no—
308C22×D3—31no—
309C22×D3—31no—
310C22×D3—61no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—91no—
316C2×C3×C4—91no—
317C2×C3×C4—91no—
318C2×C3×C4—91no—
319C2×C3×C4—91no—
320C2×C3×C4—181no—
321C2×C3×C4—181no—
322C2×C3×C4C2×D3211yescharacteristic
323C2×C3×C4C2×D3211yescharacteristic
324C2×C3×C4C2×D3211yescharacteristic
325C2×Dic3—31no—
326C2×Dic3—31no—
327C2×Dic3—31no—
328C2×Dic3—31no—
329C2×Dic3—31no—
330C2×Dic3—31no—
331C2×Dic3—31no—
332C2×Dic3—31no—
333C2×Dic3—31no—
334C2×Dic3—31no—
335C2×Dic3—61no—
336C2×Dic3—61no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C4×D3—61no—
341C4×D3—61no—
342C4×D3—61no—
343C4×D3—61no—
344C4×D3—121no—
345C4×D3—121no—
346C3×C8—62no—
347C3×C8—61no—
348C3×C8—61no—
349C3×C8—61no—
350C3×C8—61no—
351C3×C8—61no—
352C3×C8—61no—
353C3×C8—121no—
354C3×C8—121no—
355C3⋊C8—21no—
356C3⋊C8—21no—
357C3⋊C8—21no—
358C3⋊C8—182no—
359C3⋊C8—181no—
360C3⋊C8—181no—
361C3⋊C8—181no—
362C3⋊C8—361no—
363C3⋊C8—361no—
364C3⋊C8—362no—
365C2×C32—21no—
366C2×C32—21no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32—21no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—41no—
375C2×C32—41no—
376C2×C32—41no—
377C2×C32#87211yescharacteristic
378C2×C32#87711yescharacteristic
379C2×C32#89211yescharacteristic
380C2×C32#90111yescharacteristic
381C2×C32#90111yescharacteristic
382C2×C32#90711yescharacteristic
383C2×C32#90711yescharacteristic
384C2×C32#90711yescharacteristic
385C2×C32#90711yescharacteristic
386C3⋊D3—41no—
387C3×D3—122no—
388C3×D3—121no—
389C3×D3—121no—
390C22×C4—91no—
391C2×C8—31no—
392C2×C8—31no—
393C2×C8—31no—
394C2×C8—271no—
395C4⋊C4—271no—
396C4⋊C4—271no—
397C22⋊C4—271no—
398C22⋊C4—271no—
399C42—91no—
400C42—91no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—41no—
405C22×C3—181no—
406C22×C3—181no—
407C22×C3—181no—
408C22×C3C4×D3211yescharacteristic
409C22×C3C4×D3211yescharacteristic
410C22×C3C4×D3211yescharacteristic
411C2×D3—62no—
412C2×D3—61no—
413C2×D3—61no—
414C2×D3—61no—
415C2×D3—61no—
416C2×D3—61no—
417C2×D3—61no—
418C2×D3—121no—
419C2×D3—121no—
420C3×C4—21no—
421C3×C4—21no—
422C3×C4—21no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—21no—
426C3×C4—41no—
427C3×C4—41no—
428C3×C4—181no—
429C3×C4—181no—
430C3×C4—181no—
431C3×C4—181no—
432C3×C4—181no—
433C3×C4#379511yescharacteristic
434C3×C4#379611yescharacteristic
435C3×C4#379911yescharacteristic
436C3×C4#380411yescharacteristic
437C3×C4#380411yescharacteristic
438C3×C4#380511yescharacteristic
439Dic3—62no—
440Dic3—62no—
441Dic3—62no—
442Dic3—61no—
443Dic3—61no—
444Dic3—61no—
445Dic3—61no—
446Dic3—61no—
447Dic3—61no—
448Dic3—61no—
449Dic3—61no—
450Dic3—61no—
451Dic3—61no—
452Dic3—121no—
453Dic3—121no—
454C32—21no—
455C32—21no—
456C32—21no—
457C32—41no—
458C32#470911yescharacteristic, complemented
459C32#472011yescharacteristic, complemented
460C32#484011yescharacteristic, complemented
461C23—91no—
462C2×C4—91no—
463C2×C4—91no—
464C2×C4—91no—
465C2×C4—91no—
466C2×C4—91no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4D3311yescharacteristic
470C8—61no—
471C8—61no—
472C8—61no—
473C8—541no—
474C2×C3—21no—
475C2×C3—21no—
476C2×C3—21no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—41no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—361no—
487C2×C3#6380311yescharacteristic
488C2×C3#6380411yescharacteristic
489C2×C3#6380411yescharacteristic
490C2×C3#6380511yescharacteristic
491C2×C3#6380511yescharacteristic
492C2×C3#6380511yescharacteristic
493C2×C3#6383611yescharacteristic
494C2×C3#6387711yescharacteristic
495C2×C3#6387811yescharacteristic
496D3—122no—
497D3—121no—
498D3—121no—
499C22—181no—
500C22—181no—
501C22—181no—
502C22#8894911yescharacteristic
503C4—181no—
504C4—181no—
505C4—181no—
506C4—181no—
507C4—181no—
508C4#8896111yescharacteristic
509C4#8896211yescharacteristic
510C3—21no—
511C3—21no—
512C3—21no—
513C3—41no—
514C3#1059064711yescharacteristic, complemented
515C3#1059069711yescharacteristic, complemented
516C3#1059069811yescharacteristic, complemented
517C2—361no—
518C2#1172282611yescharacteristic
519C2#1172282711yescharacteristic
520C2#1172292211yescharacteristic
subgroupG/H
center, Frattini, hypercenterC2×C4D33
commutator, socleC22×C33C22×C4
Fitting, cosocleC2×C33×C4C23
prime radicalC22×C3×C3⋊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. #11722051C2semidirect
4. C3⋊C8×C3⋊Dic3C2semidirect
5. C3×#63572C2semidirect
7. C3×#63555C2semidirect
34. #63572D3semidirect
42. #63555D3semidirect
306. C33#433semidirect
458. C32#4709semidirect
459. C32#4720semidirect
460. C32#4840semidirect
514. C3#10590647semidirect
515. C3#10590697semidirect
516. C3#10590698semidirect

Maximal subgroups

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

subgroupindex
#117236732
#117220512
#117220232
C3⋊C8×C3⋊Dic32
C3×#635722
C3×#635572
C3×#635552
#105906983
#105906973
#105906473