Finite Group Databasebeta

← #13168319

Subgroups — #13168319

Order 1728.

1554 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1113C211yesmaximal, characteristic
2C2×C3×#3797C211yesmaximal, characteristic, complemented
3C3⋊D3×C4⋊Dic3C211yesmaximal, characteristic, complemented
4#11723435C212yesmaximal, complemented
5C3×#63871C212yesmaximal, complemented
6#11722979C214yesmaximal, complemented
7#11722911C212yesmaximal, complemented
8#11722905C212yesmaximal
9D3×#946—32nomaximal
10#10594631—31nomaximal
11C22×#1113C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×C3×C4×C3⋊D3C2211yescharacteristic
14C2×C32×Dic6C2212yescomplemented
15C2×#6294C2211yescharacteristic
16C2×#6294C2212yes—
17C4×#1113—24no—
18C2×Dic3×C3⋊D3C2212yes—
19C2×C3×#627C2212yescomplemented
20C2×C3×#624C2212yes—
21C3×#3797—24no—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
23C32×C4⋊Dic3C2211yescharacteristic, complemented
24#88807C2212yescomplemented
25#88806C2214yescomplemented
26#88803C2214yes—
27C3×Dic3⋊Dic3C2214yescomplemented
28C3×#3722C2214yes—
29C2×C3×D3×Q8—32no—
30C2×C4×D32—31no—
31C2×C4×D32—32no—
32C2×#3797D311yescharacteristic, complemented
33#64155—32no—
34C4⋊C4×C3⋊D3—31no—
35C3×#946—32no—
36C3×#897—32no—
37#63921—32no—
38D3×(C2×D3)⋊C4—34no—
39#63900—32no—
40#63899—34no—
41D3×C4⋊Dic3—32no—
42D3×C4⋊Dic3—61no—
43#63888—34no—
44#63887—34no—
45D3×Dic3⋊C4—34no—
46#63871D312yescomplemented
47#63856—32no—
48#63856—32no—
49#63856—62no—
50C22×C3×C3⋊D3C2312yes—
51C22×C3×C3⋊D3C2311yescharacteristic
52C2×#1113—22no—
53C2×#1113—22no—
54C2×#1113Q814yescomplemented
55C2×C33×C4C2311yescharacteristic
56C2×C3⋊(C3⋊Dic3)C2312yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C3×C3⋊Dic3C2311yescharacteristic
59C3×C4×C3⋊D3—24no—
60C3×C4×C3⋊D3D414yes—
61C2×C32×Dic3C2314yes—
62C2×C32×Dic3C2312yes—
63C32×Dic6—24no—
64#6294—22no—
65#6294—24no—
66Dic3×C3⋊D3—24no—
67C3×#627—24no—
68C3×#624—24no—
69C2×#946—91no—
70D3×C22⋊Q8—92no—
71C22×D32—32no—
72C22×D32—31no—
73C2×C32×Q8—32no—
74C2×C4×C3⋊D3—32no—
75C2×C4×C3⋊D3C2×D312yes—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C3×D3×Q8—68no—
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—32no—
80C2×C3×C4×D3—32no—
81C2×C3×C4×D3—34no—
82C2×C3×C4×D3—61no—
83C2×C3×C4×D3—62no—
84C2×C3×Dic6—22no—
85C2×C3×Dic6—34no—
86C2×C3×Dic6C2×D312yescomplemented
87C2×C3×Dic6C2×D312yescomplemented
88C2×#627C2×D312yescomplemented
89C2×#624—32no—
90C2×#624—31no—
91C2×#624C2×D312yes—
92C2×D3×Dic3—32no—
93C2×D3×Dic3—32no—
94C2×D3×Dic3—32no—
95C2×D3×Dic3—34no—
96C2×D3×Dic3—62no—
97C4×D32—68no—
98C4×D32—64no—
99#3797—24no—
100C32×C4⋊C4—31no—
101#3752—34no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—62no—
105C4⋊(C3⋊Dic3)—61no—
106C4⋊(C3⋊Dic3)—121no—
107#3750—32no—
108#3750—34no—
109C3×(C2×D3)⋊C4—34no—
110C3×(C2×D3)⋊C4—34no—
111C3×C4⋊Dic3—21no—
112C3×C4⋊Dic3—34no—
113C3×C4⋊Dic3C2×D312yescomplemented
114C3×Dic3⋊C4—34no—
115#3724—34no—
116#3724—34no—
117#3724—64no—
118Dic3⋊Dic3—34no—
119Dic3⋊Dic3—62no—
120Dic3⋊Dic3C2×D314yescomplemented
121#3722C2×D314yes—
122(C2×D3)⋊Dic3—34no—
123(C2×D3)⋊Dic3—34no—
124(C2×D3)⋊Dic3—64no—
125C22×C33C2411yescharacteristic
126C2×C3×C3⋊D3—22no—
127C2×C3×C3⋊D3—22no—
128C2×C3×C3⋊D3C2×D412yes—
129C2×C3×C3⋊D3C2×Q812yes—
130C2×C3×C3⋊D3C2×Q814yescomplemented
131C2×C3×C3⋊D3#4112yes—
132#1113—28no—
133C33×C4C2×D412yes—
134C3⋊(C3⋊Dic3)—22no—
135C3×C3⋊Dic3—22no—
136C3×C3⋊Dic3C2×D412yes—
137C32×Dic3—24no—
138C32×Dic3—22no—
139C22×C3×Q8—91no—
140C2×D3×Q8—32no—
141C22×C4×D3—91no—
142C22×C4×D3—92no—
143C3×C22⋊Q8—92no—
144#946—32no—
145#946—94no—
146#946—92no—
147#946—92no—
148#946—182no—
149C2×(C2×D3)⋊C4—92no—
150C2×C4⋊Dic3—91no—
151#924—92no—
152C2×Dic3⋊C4—92no—
153#897—32no—
154#896—94no—
155D3×C4⋊C4—94no—
156D3×C4⋊C4—92no—
157D3×C4⋊C4—181no—
158D3×C22⋊C4—94no—
159#878—94no—
160C22×C3⋊D3C22×D312yes—
161C22×C3⋊D3C22×D311yescharacteristic
162C22×C3×D3—32no—
163C22×C3×D3—32no—
164C22×C3×D3—32no—
165C22×C3×D3—61no—
166C22×C3×D3—62no—
167C2×D32—34no—
168C2×D32—38no—
169C2×D32—64no—
170C2×D32—64no—
171C2×D32—62no—
172C2×D32—62no—
173C32×Q8—64no—
174C2×C32×C4—21no—
175C2×C32×C4—22no—
176C2×C32×C4—34no—
177C2×C32×C4—32no—
178C2×C32×C4—41no—
179C2×C32×C4C22×D312yes—
180C2×C32×C4C22×D311yescharacteristic
181C2×C3⋊Dic3—32no—
182C2×C3⋊Dic3—34no—
183C2×C3⋊Dic3—64no—
184C2×C3⋊Dic3—62no—
185C2×C3⋊Dic3—122no—
186C2×C3⋊Dic3C22×D312yes—
187C2×C3⋊Dic3C22×D311yescharacteristic
188C4×C3⋊D3—24no—
189C4×C3⋊D3—64no—
190C4×C3⋊D3#6714yes—
191C2×C3×Dic3—24no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—32no—
194C2×C3×Dic3—32no—
195C2×C3×Dic3—32no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—62no—
198C2×C3×Dic3C22×D314yes—
199C2×C3×Dic3C22×D314yes—
200C2×C3×Dic3C22×D314yes—
201C3×C4×D3—38no—
202C3×C4×D3—64no—
203C3×C4×D3—68no—
204C3×C4×D3—64no—
205C3×C4×D3—64no—
206C3×C4×D3—124no—
207C3×Dic6—24no—
208C3×Dic6—24no—
209C3×Dic6—44no—
210C3×Dic6—68no—
211#627—24no—
212#624—24no—
213#624—64no—
214#624—62no—
215D3×Dic3—68no—
216D3×Dic3—64no—
217D3×Dic3—64no—
218D3×Dic3—64no—
219D3×Dic3—124no—
220C2×C22⋊Q8—271noSylow
221C2×C33C22×D411yescharacteristic
222C2×C33C22×Q811yescharacteristic
223C2×C33C2×#4111yescharacteristic
224C3×C3⋊D3—24no—
225C3×C3⋊D3C22⋊Q814yescomplemented
226C23×D3—92no—
227C23×D3—91no—
228C2×C3×Q8—32no—
229C2×C3×Q8—32no—
230C2×C3×Q8—62no—
231C2×C3×Q8—92no—
232C2×C3×Q8—184no—
233C22×C3×C4—92no—
234C22×C3×C4—91no—
235C22×C3×C4—92no—
236C22×Dic3—92no—
237C22×Dic3—91no—
238C22×Dic3—92no—
239D3×Q8—68no—
240C2×C4×D3—32no—
241C2×C4×D3—32no—
242C2×C4×D3—32no—
243C2×C4×D3—34no—
244C2×C4×D3—61no—
245C2×C4×D3—62no—
246C2×C4×D3—92no—
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×C4×D3—94no—
250C2×C4×D3—182no—
251C2×C4×D3—188no—
252C2×C4×D3—184no—
253C2×Dic6—34no—
254C2×Dic6D3212yescomplemented
255C3×C4⋊C4—32no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4—94no—
258C3×C22⋊C4—94no—
259#217—94no—
260(C2×D3)⋊C4—34no—
261(C2×D3)⋊C4—34no—
262(C2×D3)⋊C4—94no—
263(C2×D3)⋊C4—94no—
264(C2×D3)⋊C4—184no—
265C4⋊Dic3—34no—
266C4⋊Dic3—92no—
267C4⋊Dic3—91no—
268C4⋊Dic3—92no—
269C4⋊Dic3—181no—
270C4⋊Dic3—182no—
271C4⋊Dic3—361no—
272C4⋊Dic3D3211yescharacteristic, complemented
273Dic3⋊C4—34no—
274Dic3⋊C4—94no—
275Dic3⋊C4—94no—
276Dic3⋊C4—94no—
277Dic3⋊C4—94no—
278Dic3⋊C4—182no—
279Dic3⋊C4—184no—
280C22×C32—21no—
281C22×C32—22no—
282C22×C32—41no—
283C22×C32C23×D312yes—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3D3×Q812yes—
288C2×C3⋊D3D3×Q814yescomplemented
289C2×C3⋊D3#23912yes—
290C2×C3⋊D3C2×#6712yes—
291C2×C3×D3—34no—
292C2×C3×D3—34no—
293C2×C3×D3—34no—
294C2×C3×D3—34no—
295C2×C3×D3—34no—
296C2×C3×D3—62no—
297C2×C3×D3—62no—
298C2×C3×D3—62no—
299C2×C3×D3—64no—
300C2×C3×D3—62no—
301C2×C3×D3—62no—
302C2×C3×D3—62no—
303C2×C3×D3—62no—
304C2×C3×D3—122no—
305C2×C3×D3—122no—
306D32—616no—
307D32—68no—
308C32×C4—22no—
309C32×C4—24no—
310C32×C4—42no—
311C32×C4—64no—
312C32×C4—62no—
313C32×C4D3×D414yes—
314C32×C4C2×#6712yes—
315C3⋊Dic3—22no—
316C3⋊Dic3—62no—
317C3⋊Dic3—64no—
318C3⋊Dic3—122no—
319C3⋊Dic3—124no—
320C3⋊Dic3—242no—
321C3⋊Dic3C2×#6712yes—
322C3×Dic3—24no—
323C3×Dic3—24no—
324C3×Dic3—24no—
325C3×Dic3—34no—
326C3×Dic3—42no—
327C3×Dic3—44no—
328C3×Dic3—62no—
329C3×Dic3—62no—
330C3×Dic3—62no—
331C3×Dic3—122no—
332C22×Q8—91no—
333C23×C4—271no—
334C22⋊Q8—92no—
335C22⋊Q8—274no—
336C22⋊Q8—272no—
337C2×C4⋊C4—271no—
338C2×C4⋊C4—272no—
339C2×C22⋊C4—272no—
340C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
341C23×C3—91no—
342C23×C3—92no—
343C22×D3—32no—
344C22×D3—32no—
345C22×D3—32no—
346C22×D3—61no—
347C22×D3—62no—
348C22×D3—98no—
349C22×D3—94no—
350C22×D3—184no—
351C22×D3—184no—
352C22×D3—182no—
353C22×D3—182no—
354C3×Q8—64no—
355C3×Q8—64no—
356C3×Q8—124no—
357C3×Q8—184no—
358C2×C3×C4—21no—
359C2×C3×C4—22no—
360C2×C3×C4—34no—
361C2×C3×C4—34no—
362C2×C3×C4—34no—
363C2×C3×C4—41no—
364C2×C3×C4—64no—
365C2×C3×C4—62no—
366C2×C3×C4—91no—
367C2×C3×C4—94no—
368C2×C3×C4—92no—
369C2×C3×C4—184no—
370C2×C3×C4—184no—
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3212yes—
373C2×Dic3—32no—
374C2×Dic3—32no—
375C2×Dic3—32no—
376C2×Dic3—61no—
377C2×Dic3—62no—
378C2×Dic3—94no—
379C2×Dic3—92no—
380C2×Dic3—184no—
381C2×Dic3—182no—
382C2×Dic3—184no—
383C2×Dic3—184no—
384C2×Dic3—182no—
385C2×Dic3—362no—
386C2×Dic3C2×D3212yes—
387C2×Dic3C2×D3214yes—
388C4×D3—38no—
389C4×D3—64no—
390C4×D3—68no—
391C4×D3—64no—
392C4×D3—64no—
393C4×D3—124no—
394C4×D3—184no—
395C4×D3—184no—
396C4×D3—184no—
397C4×D3—188no—
398C4×D3—364no—
399Dic6—24no—
400Dic6—68no—
401C2×C32—21no—
402C2×C32—21no—
403C2×C32—21no—
404C2×C32—22no—
405C2×C32—22no—
406C2×C32—22no—
407C2×C32—41no—
408C2×C32—41no—
409C2×C32—41no—
410C2×C32C2×D3×D412yes—
411C2×C32C2×D3×Q811yescharacteristic
412C2×C32C2×D3×Q812yes—
413C2×C32C2×#23911yescharacteristic
414C2×C32D3×#4112yes—
415C2×C32C22×#6711yescharacteristic
416C3⋊D3—24no—
417C3⋊D3#94614yescomplemented
418C3×D3—38no—
419C3×D3—64no—
420C3×D3—64no—
421C3×D3—64no—
422C3×D3—124no—
423C24—271no—
424C2×Q8—32no—
425C2×Q8—92no—
426C2×Q8—184no—
427C22×C4—92no—
428C22×C4—91no—
429C22×C4—92no—
430C22×C4—272no—
431C22×C4—271no—
432C22×C4—272no—
433C22×C4—544no—
434C4⋊C4—31no—
435C4⋊C4—94no—
436C4⋊C4—274no—
437C4⋊C4—272no—
438C4⋊C4—271no—
439C22⋊C4—94no—
440C22⋊C4—274no—
441C22×C3—21no—
442C22×C3—22no—
443C22×C3—41no—
444C22×C3—92no—
445C22×C3—92no—
446C22×C3—92no—
447C22×C3—94no—
448C22×C3—182no—
449C22×C3—182no—
450C22×C3C22×D3211yescharacteristic
451C22×C3C22×D3212yes—
452C2×D3—34no—
453C2×D3—34no—
454C2×D3—34no—
455C2×D3—34no—
456C2×D3—34no—
457C2×D3—62no—
458C2×D3—62no—
459C2×D3—62no—
460C2×D3—64no—
461C2×D3—62no—
462C2×D3—62no—
463C2×D3—62no—
464C2×D3—62no—
465C2×D3—122no—
466C2×D3—122no—
467C2×D3—1816no—
468C2×D3—188no—
469C3×C4—22no—
470C3×C4—24no—
471C3×C4—42no—
472C3×C4—64no—
473C3×C4—64no—
474C3×C4—64no—
475C3×C4—92no—
476C3×C4—122no—
477C3×C4—124no—
478C3×C4—182no—
479C3×C4#380212yes—
480C3×C4D3×#6714yes—
481Dic3—22no—
482Dic3—24no—
483Dic3—34no—
484Dic3—62no—
485Dic3—62no—
486Dic3—62no—
487Dic3—122no—
488Dic3—182no—
489Dic3—184no—
490Dic3—362no—
491Dic3—364no—
492Dic3—722no—
493C32—21no—
494C32—22no—
495C32—41no—
496C32D3×C22⋊Q812yescomplemented
497C32C2×#94611yescharacteristic, complemented
498C23—91no—
499C23—92no—
500C23—274no—
501C23—542no—
502C23—542no—
503Q8—64no—
504Q8—184no—
505C2×C4—32no—
506C2×C4—34no—
507C2×C4—91no—
508C2×C4—94no—
509C2×C4—92no—
510C2×C4—184no—
511C2×C4—184no—
512C2×C4—272no—
513C2×C4—544no—
514C2×C4—544no—
515C2×C4—542no—
516C2×C4D3311yescharacteristic
517C2×C3—21no—
518C2×C3—22no—
519C2×C3—21no—
520C2×C3—22no—
521C2×C3—21no—
522C2×C3—22no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—94no—
527C2×C3—184no—
528C2×C3C2×#380211yescharacteristic
529C2×C3#6431211yescharacteristic
530C2×C3D32×Q811yescharacteristic
531C2×C3D32×Q812yes—
532C2×C3D3×#23912yes—
533C2×C3C2×D3×#6712yes—
534D3—38no—
535D3—64no—
536D3—64no—
537D3—64no—
538D3—124no—
539C22—92no—
540C22—92no—
541C22—92no—
542C22—94no—
543C22—182no—
544C22—182no—
545C22—548no—
546C22C2×D3311yescharacteristic
547C4—62no—
548C4—64no—
549C4—92no—
550C4—182no—
551C4—542no—
552C4#8895312yes—
553C3—21no—
554C3—22no—
555C3—41no—
556C3#1059463111yescharacteristic, complemented
557C3D3×#94612yescomplemented
558C2—94no—
559C2—184no—
560C2#1172437711yescharacteristic
561C2#1172437811yescharacteristic
562C2C2×#8895311yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutator, socle, cosocleC22×C33C24
FittingC2×C33×C4C23
prime radicalC22×#1113C22
hypercenterC2×C4D33
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3797C2semidirect
3. C3⋊D3×C4⋊Dic3C2semidirect
4. #11723435C2semidirect
5. C3×#63871C2semidirect
6. #11722979C2semidirect
7. #11722911C2semidirect
14. C2×C32×Dic6C22semidirect
19. C2×C3×#627C22semidirect
22. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
23. C32×C4⋊Dic3C22semidirect
24. #88807C22semidirect
25. #88806C22semidirect
27. C3×Dic3⋊Dic3C22semidirect
32. C2×#3797D3semidirect
46. #63871D3semidirect
54. C2×#1113Q8semidirect
86. C2×C3×Dic6C2×D3semidirect
87. C2×C3×Dic6C2×D3semidirect
88. C2×#627C2×D3semidirect
113. C3×C4⋊Dic3C2×D3semidirect
120. Dic3⋊Dic3C2×D3semidirect
130. C2×C3×C3⋊D3C2×Q8semidirect
225. C3×C3⋊D3C22⋊Q8semidirect
254. C2×Dic6D32semidirect
272. C4⋊Dic3D32semidirect
288. C2×C3⋊D3D3×Q8semidirect
340. C33C2×C22⋊Q8semidirect
417. C3⋊D3#946semidirect
496. C32D3×C22⋊Q8semidirect
497. C32C2×#946semidirect
556. C3#10594631semidirect
557. C3D3×#946semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11132
C2×C3×#37972
C3⋊D3×C4⋊Dic32
#117234352
C3×#638712
#117229792
#117229112
#117229052
D3×#9463
#105946313