Finite Group Databasebeta

← #13169280

Subgroups — #13169280

Order 1728.

1462 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89042C211yesmaximal, characteristic
2C3⋊D3×C4⋊Dic3C212yesmaximal, complemented
3#11723404C211yesmaximal, characteristic, complemented
4C3×#63900C211yesmaximal, characteristic, complemented
5C3×#63849C212yesmaximal, complemented
6#11722944C212yesmaximal, complemented
7#11722938C212yesmaximal, complemented
8#11722934C214yesmaximal, complemented
9#10594511—31nomaximal
10#10594510—32nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2212yes—
13C2×#6296C2211yescharacteristic
14C2×#6295C2212yes—
15C2×#6295C2211yescharacteristic
16#89042C418yescomplemented
17C2×Dic3×C3⋊D3C2214yes—
18C2×C3×#624C2212yes—
19C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
20C32×C4⋊Dic3C2211yescharacteristic
21C32×C4⋊Dic3C2212yes—
22#88807C2214yes—
23#88803C2212yes—
24Dic3×C3⋊Dic3C2212yes—
25C3×#3724C2212yes—
26C3×#3722C2214yes—
27C3×Dic32C2214yes—
28C2×#3798—32no—
29C2×#3798—31no—
30#64101—31no—
31C4⋊C4×C3⋊D3—32no—
32C3×#892—32no—
33C3×#892—32no—
34C3×D3×C4⋊C4—32no—
35C4×#627—32no—
36C4×#626—34no—
37#63902—32no—
38#63900D311yescharacteristic, complemented
39D3×C4⋊Dic3—32no—
40D3×C4⋊Dic3—32no—
41D3×C4⋊Dic3—62no—
42#63881—34no—
43#63880—34no—
44#63879—34no—
45#63849D312yescomplemented
46#63841—32no—
47#63841—61no—
48#63837—34no—
49C22×C3×C3⋊D3C2311yescharacteristic
50C22×C3×C3⋊D3C2312yes—
51C2×C33×C4C2311yescharacteristic
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C2×C3×C3⋊Dic3C2312yes—
54C2×C3×C3⋊Dic3C2311yescharacteristic
55C3×C4×C3⋊D3C2×C414yescomplemented
56C3×C4×C3⋊D3C2×C418yescomplemented
57C2×C32×Dic3C2314yes—
58C2×C32×Dic3C2312yes—
59#6296C2×C414yescomplemented
60#6295C2×C418yescomplemented
61#6295C2×C414yescomplemented
62Dic3×C3⋊D3—44no—
63C3×#624—42no—
64#5523—92no—
65#5504—91no—
66C2×C4×C3⋊D3—34no—
67C2×C4×C3⋊D3C2×D312yes—
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×C4×D3—32no—
70C2×C3×C4×D3—32no—
71C2×C3×C4×D3—32no—
72C2×C3×C4×D3—34no—
73C2×C3×C4×D3—61no—
74C2×C3×C4×D3—62no—
75C2×#627—32no—
76C2×#627—31no—
77C2×#626—32no—
78C2×#626—32no—
79C2×#626—32no—
80C2×#625—32no—
81C2×#625—31no—
82C2×#624C2×D312yes—
83C2×D3×Dic3—34no—
84C2×D3×Dic3—34no—
85C2×D3×Dic3—64no—
86#3798—316no—
87#3798—38no—
88C32×C4⋊C4—32no—
89C32×C4⋊C4—31no—
90#3752—32no—
91C4⋊(C3⋊Dic3)—32no—
92C4⋊(C3⋊Dic3)—31no—
93C4⋊(C3⋊Dic3)—61no—
94C4⋊(C3⋊Dic3)—62no—
95C4⋊(C3⋊Dic3)—121no—
96#3750—34no—
97C4×C3⋊Dic3—32no—
98C3×(C2×D3)⋊C4—34no—
99C3×(C2×D3)⋊C4—34no—
100C3×C4⋊Dic3—21no—
101C3×C4⋊Dic3—22no—
102C3×C4⋊Dic3C2×D312yes—
103C3×C4⋊Dic3C2×D312yes—
104C3×C4⋊Dic3C2×D312yes—
105C3×Dic3⋊C4—34no—
106C3×C4×Dic3—34no—
107C3×C4×Dic3—34no—
108#3724C2×D312yes—
109Dic3⋊Dic3—34no—
110Dic3⋊Dic3—34no—
111Dic3⋊Dic3—64no—
112#3722C2×D314yes—
113(C2×D3)⋊Dic3—34no—
114(C2×D3)⋊Dic3—62no—
115Dic32—34no—
116Dic32—62no—
117Dic32C2×D314yes—
118C22×C33C2411yescharacteristic
119C2×C3×C3⋊D3—22no—
120C2×C3×C3⋊D3—21no—
121C2×C3×C3⋊D3—22no—
122C2×C3×C3⋊D3—21no—
123C2×C3×C3⋊D3C22×C414yes—
124C2×C3×C3⋊D3C22×C412yes—
125C33×C4C22×C412yes—
126C3⋊(C3⋊Dic3)—22no—
127C3×C3⋊Dic3C22×C414yes—
128C3×C3⋊Dic3C22×C412yes—
129C32×Dic3—24no—
130C32×Dic3—22no—
131C2×#235—91no—
132C2×#235—92no—
133C3×#117—92no—
134C2×C3×C4⋊C4—91no—
135C4×#67—94no—
136#926—91no—
137C2×C4⋊Dic3—92no—
138#893—92no—
139#892—32no—
140#892—32no—
141#892—92no—
142#892—181no—
143D3×C4⋊C4—32no—
144D3×C4⋊C4—92no—
145D3×C4⋊C4—92no—
146D3×C4⋊C4—182no—
147#887—92no—
148#67⋊C4—94no—
149#67⋊C4—94no—
150C4×D12—94no—
151C4×Dic6—94no—
152C22×C3⋊D3C22×D312yes—
153C22×C3⋊D3C22×D311yescharacteristic
154C22×C3×D3—32no—
155C22×C3×D3—32no—
156C22×C3×D3—32no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C2×C32×C4—21no—
160C2×C32×C4—22no—
161C2×C32×C4—32no—
162C2×C32×C4—34no—
163C2×C32×C4—41no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D312yes—
166C2×C3⋊Dic3—34no—
167C2×C3⋊Dic3—32no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—64no—
170C2×C3⋊Dic3—122no—
171C2×C3⋊Dic3C22×D312yes—
172C2×C3⋊Dic3C22×D311yescharacteristic
173C4×C3⋊D3—124no—
174C4×C3⋊D3C4×D318yescomplemented
175C4×C3⋊D3C4×D314yescomplemented
176C2×C3×Dic3—22no—
177C2×C3×Dic3—24no—
178C2×C3×Dic3—32no—
179C2×C3×Dic3—32no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—62no—
183C2×C3×Dic3C22×D314yes—
184C2×C3×Dic3C22×D314yes—
185C2×C3×Dic3C22×D314yes—
186C3×C4×D3—38no—
187C3×C4×D3—38no—
188C3×C4×D3—38no—
189C3×C4×D3—68no—
190C3×C4×D3—64no—
191C3×C4×D3—124no—
192#627—38no—
193#627—34no—
194#626—38no—
195#626—38no—
196#626—38no—
197#625—38no—
198#625—34no—
199#624—42no—
200D3×Dic3—124no—
201D3×Dic3—124no—
202D3×Dic3—128no—
203#520—271noSylow
204C2×C33C23×C411yescharacteristic
205C2×C33H(5,𝔽2)11yescharacteristic
206C2×C332−1+411yescharacteristic
207C3×C3⋊D3—24no—
208C3×C3⋊D3—22no—
209C22×C3×C4—92no—
210C22×C3×C4—92no—
211C22×C3×C4—91no—
212C2×#67—92no—
213C2×#67—92no—
214C2×#67—92no—
215C22×Dic3—94no—
216#235—98no—
217#235—916no—
218C2×D12—92no—
219C2×D12—91no—
220C2×C4×D3—32no—
221C2×C4×D3—32no—
222C2×C4×D3—32no—
223C2×C4×D3—34no—
224C2×C4×D3—62no—
225C2×C4×D3—61no—
226C2×C4×D3—94no—
227C2×C4×D3—94no—
228C2×C4×D3—184no—
229C2×Dic6—92no—
230C2×Dic6—91no—
231C3×C4⋊C4—32no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—32no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—92no—
237C3×C22⋊C4—94no—
238C3×C42—94no—
239#217—92no—
240(C2×D3)⋊C4—34no—
241(C2×D3)⋊C4—34no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—182no—
244C4⋊Dic3—91no—
245C4⋊Dic3—92no—
246C4⋊Dic3—94no—
247C4⋊Dic3—182no—
248C4⋊Dic3—181no—
249C4⋊Dic3—361no—
250C4⋊Dic3D3212yes—
251C4⋊Dic3D3211yescharacteristic
252Dic3⋊C4—34no—
253Dic3⋊C4—94no—
254Dic3⋊C4—94no—
255Dic3⋊C4—184no—
256C4×Dic3—34no—
257C4×Dic3—34no—
258C4×Dic3—94no—
259C4×Dic3—92no—
260C4×Dic3—182no—
261C22×C32—21no—
262C22×C32—22no—
263C22×C32—41no—
264C22×C32C23×D311yescharacteristic
265C22×C32C23×D312yes—
266C2×C3⋊D3—22no—
267C2×C3⋊D3—22no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3⋊D3C2×C4×D314yes—
271C2×C3⋊D3C2×C4×D312yes—
272C2×C3×D3—34no—
273C2×C3×D3—34no—
274C2×C3×D3—34no—
275C2×C3×D3—64no—
276C2×C3×D3—62no—
277C2×C3×D3—64no—
278C2×C3×D3—64no—
279C2×C3×D3—62no—
280C2×C3×D3—62no—
281C2×C3×D3—62no—
282C2×C3×D3—62no—
283C2×C3×D3—62no—
284C2×C3×D3—62no—
285C2×C3×D3—121no—
286C2×C3×D3—121no—
287C32×C4—24no—
288C32×C4—22no—
289C32×C4—42no—
290C32×C4—62no—
291C32×C4—64no—
292C32×C4C2×C4×D312yes—
293C32×C4C2×C4×D314yes—
294C3⋊Dic3—62no—
295C3⋊Dic3—64no—
296C3⋊Dic3—124no—
297C3⋊Dic3—122no—
298C3⋊Dic3—242no—
299C3⋊Dic3C2×C4×D312yes—
300C3⋊Dic3C2×C4×D314yes—
301C3×Dic3—24no—
302C3×Dic3—24no—
303C3×Dic3—24no—
304C3×Dic3—34no—
305C3×Dic3—34no—
306C3×Dic3—34no—
307C3×Dic3—44no—
308C3×Dic3—42no—
309C3×Dic3—62no—
310C3×Dic3—64no—
311C2×#41—271no—
312C4×Q8—272no—
313C4×D4—274no—
314C4×D4—272no—
315#117—92no—
316#117—271no—
317C2×C4⋊C4—91no—
318C2×C4⋊C4—272no—
319C33#52011yescharacteristic, complemented, Sylow
320C23×C3—91no—
321C23×C3—92no—
322C22×D3—32no—
323C22×D3—32no—
324C22×D3—32no—
325C22×D3—62no—
326C22×D3—61no—
327C2×C3×C4—22no—
328C2×C3×C4—21no—
329C2×C3×C4—34no—
330C2×C3×C4—34no—
331C2×C3×C4—34no—
332C2×C3×C4—41no—
333C2×C3×C4—62no—
334C2×C3×C4—64no—
335C2×C3×C4—91no—
336C2×C3×C4—92no—
337C2×C3×C4—98no—
338C2×C3×C4—94no—
339C2×C3×C4—362no—
340C2×C3×C4C2×D3211yescharacteristic
341C2×C3×C4C2×D3212yes—
342#67—98no—
343#67—98no—
344#67—98no—
345C2×Dic3—32no—
346C2×Dic3—32no—
347C2×Dic3—32no—
348C2×Dic3—61no—
349C2×Dic3—62no—
350C2×Dic3—92no—
351C2×Dic3—94no—
352C2×Dic3—182no—
353C2×Dic3—184no—
354C2×Dic3—362no—
355C2×Dic3—364no—
356C2×Dic3C2×D3212yes—
357C2×Dic3C2×D3214yes—
358D12—98no—
359D12—94no—
360C4×D3—38no—
361C4×D3—38no—
362C4×D3—38no—
363C4×D3—68no—
364C4×D3—64no—
365C4×D3—124no—
366C4×D3—364no—
367C4×D3—364no—
368C4×D3—368no—
369Dic6—98no—
370Dic6—94no—
371C2×C32—22no—
372C2×C32—21no—
373C2×C32—21no—
374C2×C32—22no—
375C2×C32—22no—
376C2×C32—21no—
377C2×C32—41no—
378C2×C32—41no—
379C2×C32—41no—
380C2×C32C22×C4×D311yescharacteristic
381C2×C32C22×C4×D312yes—
382C2×C32#100411yescharacteristic
383C2×C32#100711yescharacteristic
384C2×C32#100912yes—
385C2×C32#101012yes—
386C3⋊D3—24no—
387C3⋊D3—22no—
388C3×D3—68no—
389C3×D3—64no—
390C3×D3—64no—
391C3×D3—64no—
392C3×D3—122no—
393#41—278no—
394C2×Q8—271no—
395C2×D4—272no—
396C2×D4—271no—
397C22×C4—92no—
398C22×C4—91no—
399C22×C4—92no—
400C22×C4—274no—
401C4⋊C4—32no—
402C4⋊C4—31no—
403C4⋊C4—92no—
404C4⋊C4—274no—
405C4⋊C4—271no—
406C22⋊C4—94no—
407C22⋊C4—272no—
408C42—94no—
409C42—272no—
410C22×C3—22no—
411C22×C3—21no—
412C22×C3—41no—
413C22×C3—94no—
414C22×C3—92no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—182no—
418C22×C3—181no—
419C22×C3C22×D3211yescharacteristic
420C22×C3C22×D3212yes—
421C2×D3—34no—
422C2×D3—34no—
423C2×D3—34no—
424C2×D3—64no—
425C2×D3—62no—
426C2×D3—64no—
427C2×D3—64no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—62no—
432C2×D3—62no—
433C2×D3—62no—
434C2×D3—121no—
435C2×D3—121no—
436C3×C4—24no—
437C3×C4—22no—
438C3×C4—42no—
439C3×C4—64no—
440C3×C4—64no—
441C3×C4—64no—
442C3×C4—94no—
443C3×C4—92no—
444C3×C4—124no—
445C3×C4—122no—
446C3×C4C4×D3214yes—
447C3×C4C4×D3212yes—
448Dic3—22no—
449Dic3—24no—
450Dic3—34no—
451Dic3—34no—
452Dic3—34no—
453Dic3—62no—
454Dic3—64no—
455Dic3—184no—
456Dic3—182no—
457Dic3—364no—
458Dic3—362no—
459Dic3—722no—
460C32—22no—
461C32—21no—
462C32—41no—
463C32#550411yescharacteristic, complemented
464C32#552312yescomplemented
465C23—92no—
466C23—91no—
467Q8—274no—
468D4—278no—
469D4—274no—
470C2×C4—32no—
471C2×C4—34no—
472C2×C4—92no—
473C2×C4—91no—
474C2×C4—94no—
475C2×C4—98no—
476C2×C4—272no—
477C2×C4—362no—
478C2×C4—1084no—
479C2×C4D3311yescharacteristic
480C2×C3—22no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—22no—
484C2×C3—22no—
485C2×C3—21no—
486C2×C3—41no—
487C2×C3—41no—
488C2×C3—41no—
489C2×C3—184no—
490C2×C3—182no—
491C2×C3C2×C4×D3212yes—
492C2×C3C2×C4×D3211yescharacteristic
493C2×C3#6430911yescharacteristic
494C2×C3#6431412yes—
495C2×C3#6431512yes—
496C2×C3#6432011yescharacteristic
497D3—68no—
498D3—64no—
499D3—64no—
500D3—64no—
501D3—122no—
502C22—94no—
503C22—92no—
504C22—181no—
505C22—182no—
506C22—182no—
507C22—181no—
508C22C2×D3311yescharacteristic
509C4—64no—
510C4—62no—
511C4—94no—
512C4—92no—
513C4—542no—
514C4#8894912yes—
515C3—22no—
516C3—21no—
517C3—41no—
518C3#1059451012yescomplemented
519C3#1059451111yescharacteristic, complemented
520C2—184no—
521C2—182no—
522C2#1172438611yescharacteristic
523C2#1172439411yescharacteristic
524C2C2×#8894911yescharacteristic
subgroupG/H
center, FrattiniC22C2×D33
commutatorC2×C33C23×C4
FittingC2×C33×C4C23
socle, cosocleC22×C33C24
prime radicalC2×#89042C2
hypercenterC2×C4D33
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C3⋊D3×C4⋊Dic3C2semidirect
3. #11723404C2semidirect
4. C3×#63900C2semidirect
5. C3×#63849C2semidirect
6. #11722944C2semidirect
7. #11722938C2semidirect
8. #11722934C2semidirect
16. #89042C4semidirect
38. #63900D3semidirect
45. #63849D3semidirect
55. C3×C4×C3⋊D3C2×C4semidirect
56. C3×C4×C3⋊D3C2×C4semidirect
59. #6296C2×C4semidirect
60. #6295C2×C4semidirect
61. #6295C2×C4semidirect
174. C4×C3⋊D3C4×D3semidirect
175. C4×C3⋊D3C4×D3semidirect
319. C33#520semidirect
463. C32#5504semidirect
464. C32#5523semidirect
518. C3#10594510semidirect
519. C3#10594511semidirect

Maximal subgroups

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

subgroupindex
C2×#890422
C3⋊D3×C4⋊Dic32
#117234042
C3×#639002
C3×#638492
#117229442
#117229382
#117229342
#105945113
#105945103