Finite Group Databasebeta

← #13177453

Subgroups — #13177453

Order 1728.

2516 conjugacy classes of subgroups.

The subgroup lattice has 534 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89078C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723622C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6#11723598C212yesmaximal, complemented
7C2×Dic3×C3⋊Dic3C212yesmaximal, complemented
8C4×#6289C211yesmaximal, characteristic, complemented
9#11723413C211yesmaximal, characteristic, complemented
10#11723406C211yesmaximal, characteristic, complemented
11#11723399C211yesmaximal, characteristic, complemented
12C4×#3830—31nomaximal
13#10595190—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C22×C3⋊(C3⋊Dic3)C2212yes—
17#89078C418yescomplemented
18C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
19C2×#6308C2211yescharacteristic
20C22×C3×C3⋊Dic3C2212yes—
21C22×C32×Dic3C2212yes—
22C2×#6289C2211yescharacteristic
23C2×#6289C2211yescharacteristic
24C3×#3757C2212yes—
25C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C32×#217C2212yes—
28C32×C4⋊Dic3C2211yescharacteristic
29C32×C4×Dic3C2211yescharacteristic
30#88806C2212yes—
31#88806C2212yes—
32#88804C2212yes—
33Dic3×C3⋊Dic3C2211yescharacteristic
34Dic3×C3⋊Dic3C2212yes—
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37Dic3×C3⋊Dic3C2211yescharacteristic
38C2×#3830—31no—
39C2×#3830—34no—
40C2×#3830—64no—
41C32×C4×D4—31no—
42D4×C3⋊Dic3D311yescharacteristic, complemented
43C4×#638—32no—
44C2×C4×C3⋊Dic3—32no—
45#64101—31no—
46#64096—31no—
47#64090—32no—
48#64086—32no—
49C42×C3⋊D3—31no—
50C4×#634—31no—
51C3×D4×Dic3—34no—
52C3×D4×Dic3D314yescomplemented
53#63957—38no—
54C2×Dic32—38no—
55#63952—38no—
56#63952—38no—
57C4×#624—34no—
58#63849—34no—
59#63844—34no—
60#63844—34no—
61C23×C33C2312yes—
62C22×C3⋊(C3⋊D3)C2311yescharacteristic
63C33×D4C2×C414yescomplemented
64C2×C33×C4C2311yescharacteristic
65#6312C2×C418yescomplemented
66C2×C3⋊(C3⋊Dic3)C2×C418yescomplemented
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C4×C3⋊(C3⋊D3)C2×C414yescomplemented
70#6308C2×C414yescomplemented
71C2×C3×C3⋊Dic3—24no—
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C32×Dic3—24no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2312yes—
78C2×C32×Dic3C2311yescharacteristic
79#6289—22no—
80#6289—22no—
81C4×#237—94no—
82C4×#237—91no—
83C2×C32×D4—24no—
84C2×C32×D4C2×D314yes—
85C2×C32×D4C2×D311yescharacteristic
86C22×C32×C4—32no—
87C2×#638—32no—
88C2×#638—38no—
89C2×#638—68no—
90C22×C3⋊Dic3—32no—
91C22×C3⋊Dic3—38no—
92C22×C3⋊Dic3—68no—
93C22×C3⋊Dic3C2×D312yes—
94#3830—38no—
95#3830—332no—
96#3830—632no—
97C2×C4×C3⋊D3—31no—
98C2×C4×C3⋊D3—34no—
99C2×C4×C3⋊D3—31no—
100C2×C4×C3⋊D3—31no—
101C2×C4×C3⋊D3—64no—
102C2×#634—31no—
103C2×#634—34no—
104C2×#634—64no—
105C22×C3×Dic3—38no—
106C22×C3×Dic3C2×D318yes—
107C2×#624—34no—
108C2×#624—34no—
109C32×C4⋊C4—31no—
110C32×C22⋊C4—32no—
111C32×C42—31no—
112#3757C2×D312yes—
113#3752—32no—
114C4⋊(C3⋊Dic3)C2×D311yescharacteristic
115#3750—32no—
116#3750—32no—
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—31no—
120C4×C3⋊Dic3—32no—
121C4×C3⋊Dic3—32no—
122C4×C3⋊Dic3C2×D311yescharacteristic
123C3×#217—38no—
124C3×#217C2×D318yes—
125C3×C4⋊Dic3—34no—
126C3×C4⋊Dic3C2×D314yes—
127C3×C4×Dic3—34no—
128C3×C4×Dic3C2×D314yes—
129#3724—38no—
130#3724—38no—
131#3722—38no—
132Dic32—38no—
133Dic32—38no—
134Dic32—38no—
135Dic32—34no—
136Dic32—34no—
137C22×C33—22no—
138C22×C33—22no—
139C22×C33C22×C414yes—
140C22×C33C2411yescharacteristic
141C2×C3⋊(C3⋊D3)—21no—
142C2×C3⋊(C3⋊D3)—21no—
143C2×C3⋊(C3⋊D3)C22×C412yes—
144C33×C4C22×C412yes—
145C3⋊(C3⋊Dic3)C22×C414yes—
146C3⋊(C3⋊Dic3)C22×C412yes—
147C3×C3⋊Dic3—21no—
148C3×C3⋊Dic3—22no—
149C3×C3⋊Dic3#4112yescomplemented
150C32×Dic3—21no—
151C32×Dic3—22no—
152C32×Dic3#4112yescomplemented
153C2×#237—94no—
154C2×#237—91no—
155C2×#237—184no—
156C3×C4×D4—34no—
157C3×C4×D4—91no—
158D4×Dic3—34no—
159D4×Dic3C3⋊D311yescharacteristic, complemented
160C4×#67—98no—
161C4×#67—92no—
162C2×C4×Dic3—98no—
163C2×C4×Dic3—92no—
164#892—94no—
165#892—91no—
166#887—94no—
167#887—91no—
168#67⋊C4—98no—
169#67⋊C4—92no—
170#877—98no—
171#877—92no—
172C42×D3—94no—
173C42×D3—91no—
174C4×Dic6—94no—
175C4×Dic6—91no—
176C23×C32—28no—
177C23×C32C22×D318yes—
178C23×C32C22×D312yes—
179C22×C3⋊D3—31no—
180C22×C3⋊D3—34no—
181C22×C3⋊D3—64no—
182C32×D4—216no—
183C32×D4C4×D3116yescomplemented
184C32×D4C4×D314yescomplemented
185C2×C32×C4—24no—
186C2×C32×C4—31no—
187C2×C32×C4—32no—
188C2×C32×C4—31no—
189C2×C32×C4—64no—
190C2×C32×C4C22×D314yes—
191C2×C32×C4C22×D311yescharacteristic
192#638—38no—
193#638—332no—
194#638—632no—
195C2×C3⋊Dic3—24no—
196C2×C3⋊Dic3—38no—
197C2×C3⋊Dic3—31no—
198C2×C3⋊Dic3—32no—
199C2×C3⋊Dic3—332no—
200C2×C3⋊Dic3—34no—
201C2×C3⋊Dic3—38no—
202C2×C3⋊Dic3—632no—
203C2×C3⋊Dic3—68no—
204C2×C3⋊Dic3—64no—
205C2×C3⋊Dic3C22×D312yes—
206C2×C3⋊Dic3C22×D311yescharacteristic
207C2×C3⋊Dic3C22×D311yescharacteristic
208C4×C3⋊D3—34no—
209C4×C3⋊D3—316no—
210C4×C3⋊D3—616no—
211C4×C3⋊D3—62no—
212C4×C3⋊D3—62no—
213#634—34no—
214#634—316no—
215#634—616no—
216C2×C3×Dic3—216no—
217C2×C3×Dic3—34no—
218C2×C3×Dic3—38no—
219C2×C3×Dic3—34no—
220C2×C3×Dic3—616no—
221C2×C3×Dic3C22×D318yes—
222C2×C3×Dic3C22×D314yes—
223C2×C3×Dic3C22×D314yes—
224#624—68no—
225#624—68no—
226C4×#41—271noSylow
227C2×C33—24no—
228C2×C33C23×C411yescharacteristic
229C2×C33C2×#4111yescharacteristic
230C2×C33C2×#4111yescharacteristic
231C3⋊(C3⋊D3)—22no—
232C2×C3×D4—24no—
233C2×C3×D4D3214yes—
234C2×C3×D4C2×C3⋊D311yescharacteristic
235C22×C3×C4—38no—
236C22×C3×C4—92no—
237C2×#67—98no—
238C2×#67—92no—
239C2×#67—188no—
240C22×Dic3—38no—
241C22×Dic3—98no—
242C22×Dic3—92no—
243C22×Dic3—188no—
244C22×Dic3C2×C3⋊D312yes—
245#237—932no—
246#237—98no—
247#237—1832no—
248C2×C4×D3—94no—
249C2×C4×D3—91no—
250C2×C4×D3—94no—
251C2×C4×D3—94no—
252C2×C4×D3—91no—
253C2×C4×D3—91no—
254C2×C4×D3—184no—
255C2×Dic6—94no—
256C2×Dic6—91no—
257C2×Dic6—184no—
258C3×C4⋊C4—34no—
259C3×C4⋊C4—91no—
260C3×C22⋊C4—38no—
261C3×C22⋊C4—92no—
262C3×C42—34no—
263C3×C42—91no—
264#217—38no—
265#217C2×C3⋊D312yes—
266(C2×D3)⋊C4—98no—
267(C2×D3)⋊C4—92no—
268C4⋊Dic3—34no—
269C4⋊Dic3C2×C3⋊D311yescharacteristic
270Dic3⋊C4—98no—
271Dic3⋊C4—98no—
272Dic3⋊C4—92no—
273Dic3⋊C4—92no—
274C4×Dic3—34no—
275C4×Dic3—98no—
276C4×Dic3—94no—
277C4×Dic3—94no—
278C4×Dic3—98no—
279C4×Dic3—98no—
280C4×Dic3—92no—
281C4×Dic3—91no—
282C4×Dic3—91no—
283C4×Dic3—92no—
284C4×Dic3—92no—
285C4×Dic3C2×C3⋊D311yescharacteristic
286C22×C32—22no—
287C22×C32—22no—
288C22×C32—24no—
289C22×C32—216no—
290C22×C32—28no—
291C22×C32—28no—
292C22×C32—48no—
293C22×C32—48no—
294C22×C32C2×C4×D3116yes—
295C22×C32C2×C4×D314yes—
296C22×C32C23×D314yes—
297C22×C32C23×D311yescharacteristic
298C2×C3⋊D3—32no—
299C2×C3⋊D3—38no—
300C2×C3⋊D3—68no—
301C2×C3⋊D3—61no—
302C2×C3⋊D3—61no—
303C2×C3⋊D3—64no—
304C2×C3⋊D3—64no—
305C2×C3⋊D3—124no—
306C2×C3⋊D3—124no—
307C32×C4—28no—
308C32×C4—32no—
309C32×C4—61no—
310C32×C4—62no—
311C32×C4C2×C4×D318yes—
312C32×C4C2×C4×D312yes—
313C3⋊Dic3—22no—
314C3⋊Dic3—21no—
315C3⋊Dic3—34no—
316C3⋊Dic3—32no—
317C3⋊Dic3—316no—
318C3⋊Dic3—38no—
319C3⋊Dic3—616no—
320C3⋊Dic3—68no—
321C3⋊Dic3#23712yescomplemented
322C3×Dic3—28no—
323C3×Dic3—24no—
324C3×Dic3—38no—
325C3×Dic3—64no—
326C3×Dic3—68no—
327C3×Dic3#23718yescomplemented
328C2×#41—271no—
329C4×Q8—271no—
330C4×Q8—271no—
331C4×D4—31no—
332C4×D4—91no—
333C4×D4—272no—
334C4×D4—272no—
335#117—272no—
336#117—271no—
337C2×C42—271no—
338C2×C42—272no—
339C33C4×#4111yescharacteristic, complemented, Sylow
340C23×C3—28no—
341C23×C3C2×D3218yes—
342C23×C3C22×C3⋊D312yes—
343C22×D3—94no—
344C22×D3—91no—
345C22×D3—184no—
346C3×D4—216no—
347C3×D4#624116yescomplemented
348C3×D4C4×C3⋊D314yescomplemented
349C2×C3×C4—24no—
350C2×C3×C4—34no—
351C2×C3×C4—38no—
352C2×C3×C4—34no—
353C2×C3×C4—616no—
354C2×C3×C4—91no—
355C2×C3×C4—92no—
356C2×C3×C4—91no—
357C2×C3×C4—184no—
358C2×C3×C4C2×D3214yes—
359C2×C3×C4C22×C3⋊D311yescharacteristic
360#67—932no—
361#67—98no—
362#67—1832no—
363C2×Dic3—24no—
364C2×Dic3—34no—
365C2×Dic3—34no—
366C2×Dic3—38no—
367C2×Dic3—616no—
368C2×Dic3—94no—
369C2×Dic3—98no—
370C2×Dic3—91no—
371C2×Dic3—92no—
372C2×Dic3—932no—
373C2×Dic3—98no—
374C2×Dic3—188no—
375C2×Dic3—184no—
376C2×Dic3—1832no—
377C2×Dic3C22×C3⋊D312yes—
378C2×Dic3C22×C3⋊D311yescharacteristic
379C2×Dic3C22×C3⋊D311yescharacteristic
380C4×D3—916no—
381C4×D3—94no—
382C4×D3—1816no—
383C4×D3—182no—
384C4×D3—188no—
385C4×D3—188no—
386C4×D3—182no—
387Dic6—916no—
388Dic6—94no—
389Dic6—1816no—
390C2×C32—24no—
391C2×C32—24no—
392C2×C32—24no—
393C2×C32—24no—
394C2×C32—216no—
395C2×C32—416no—
396C2×C32C22×C4×D314yes—
397C2×C32C22×C4×D311yescharacteristic
398C2×C32C2×#23714yes—
399C2×C32C2×#23711yescharacteristic
400C2×C32D3×#4114yes—
401C2×C32D3×#4111yescharacteristic
402C3⋊D3—62no—
403C3⋊D3—68no—
404C3⋊D3—128no—
405#41—278no—
406C2×Q8—271no—
407C2×D4—272no—
408C2×D4D3×C3⋊D311yescharacteristic
409C22×C4—32no—
410C22×C4—92no—
411C22×C4—271no—
412C22×C4—271no—
413C22×C4—272no—
414C22×C4—271no—
415C4⋊C4—31no—
416C4⋊C4—91no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C22⋊C4—32no—
420C22⋊C4—92no—
421C22⋊C4—272no—
422C42—31no—
423C42—91no—
424C42—271no—
425C42—271no—
426C42—272no—
427C42—272no—
428C42—272no—
429C22×C3—28no—
430C22×C3—28no—
431C22×C3—22no—
432C22×C3—22no—
433C22×C3—24no—
434C22×C3—216no—
435C22×C3—48no—
436C22×C3—48no—
437C22×C3C2×#624116yes—
438C22×C3C2×C4×C3⋊D314yes—
439C22×C3C22×D3214yes—
440C22×C3C23×C3⋊D311yescharacteristic
441C2×D3—98no—
442C2×D3—92no—
443C2×D3—188no—
444C2×D3—184no—
445C2×D3—184no—
446C2×D3—181no—
447C2×D3—181no—
448C2×D3—364no—
449C2×D3—364no—
450C3×C4—28no—
451C3×C4—38no—
452C3×C4—64no—
453C3×C4—68no—
454C3×C4—92no—
455C3×C4—181no—
456C3×C4—182no—
457C3×C4C2×#62418yes—
458C3×C4C2×C4×C3⋊D312yes—
459Dic3—22no—
460Dic3—21no—
461Dic3—38no—
462Dic3—64no—
463Dic3—68no—
464Dic3—916no—
465Dic3—98no—
466Dic3—94no—
467Dic3—92no—
468Dic3—1816no—
469Dic3—188no—
470Dic3#383012yescomplemented
471C32—24no—
472C32C4×#23714yescomplemented
473C32C4×#23711yescharacteristic, complemented
474C23—271no—
475C23C2×D3×C3⋊D312yes—
476Q8—274no—
477D4—278no—
478D4#628914yescomplemented
479C2×C4—31no—
480C2×C4—31no—
481C2×C4—32no—
482C2×C4—64no—
483C2×C4—91no—
484C2×C4—91no—
485C2×C4—92no—
486C2×C4—184no—
487C2×C4—271no—
488C2×C4—272no—
489C2×C4—278no—
490C2×C4—274no—
491C2×C4—542no—
492C2×C4—542no—
493C2×C4C2×D3×C3⋊D311yescharacteristic
494C2×C3—216no—
495C2×C3—24no—
496C2×C3—24no—
497C2×C3—24no—
498C2×C3—24no—
499C2×C3—416no—
500C2×C3D3×#23714yes—
501C2×C3D3×#23714yes—
502C2×C3C22×#62414yes—
503C2×C3C22×C4×C3⋊D311yescharacteristic
504C2×C3C2×#383011yescharacteristic
505C2×C3#41×C3⋊D311yescharacteristic
506D3—188no—
507D3—182no—
508D3—368no—
509C22—22no—
510C22—22no—
511C22—272no—
512C22—541no—
513C22—541no—
514C22C2×#628914yes—
515C22C22×D3×C3⋊D311yescharacteristic
516C4—32no—
517C4—62no—
518C4—61no—
519C4—92no—
520C4—181no—
521C4—182no—
522C4—272no—
523C4—274no—
524C4C2×#628912yes—
525C3—24no—
526C3#1059519014yescomplemented
527C3C4×#383011yescharacteristic, complemented
528C2—24no—
529C2—542no—
530C2D3×#383011yescharacteristic
531C2C3⋊D3×#23711yescharacteristic
532C2C22×#628911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC2×C33×D4C22
socle, cosocleC22×C33C24
prime radicalC2×#89078C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C4×#41
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×D4×C3⋊Dic3C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723622C2semidirect
5. #11723618C2semidirect
6. #11723598C2semidirect
7. C2×Dic3×C3⋊Dic3C2semidirect
8. C4×#6289C2semidirect
9. #11723413C2semidirect
10. #11723406C2semidirect
11. #11723399C2semidirect
17. #89078C4semidirect
42. D4×C3⋊Dic3D3semidirect
52. C3×D4×Dic3D3semidirect
63. C33×D4C2×C4semidirect
65. #6312C2×C4semidirect
66. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
69. C4×C3⋊(C3⋊D3)C2×C4semidirect
70. #6308C2×C4semidirect
149. C3×C3⋊Dic3#41semidirect
152. C32×Dic3#41semidirect
159. D4×Dic3C3⋊D3semidirect
183. C32×D4C4×D3semidirect
184. C32×D4C4×D3semidirect
321. C3⋊Dic3#237semidirect
327. C3×Dic3#237semidirect
339. C33C4×#41semidirect
347. C3×D4#624semidirect
348. C3×D4C4×C3⋊D3semidirect
470. Dic3#3830semidirect
472. C32C4×#237semidirect
473. C32C4×#237semidirect
478. D4#6289semidirect
526. C3#10595190semidirect
527. C3C4×#3830semidirect

Maximal subgroups

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

subgroupindex
C2×#890782
C3×D4×C3⋊Dic32
C32×D4×Dic32
#117236222
#117236182
#117235982
C2×Dic3×C3⋊Dic32
C4×#62892
#117234132
#117234062
#117233992
C4×#38303
#105951903