Finite Group Databasebeta

← #13180985

Subgroups — #13180985

Order 1728.

996 conjugacy classes of subgroups.

The subgroup lattice has 554 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3⋊D3×#237C211yesmaximal, characteristic
2D4×C3⋊#157C211yesmaximal, characteristic, complemented
3C3×D4×#157C211yesmaximal, characteristic, complemented
4#67⋊#157C212yesmaximal, complemented
5#67×#157C212yesmaximal, complemented
6C2×Dic3⋊#157C212yesmaximal, complemented
7#11724261C212yesmaximal, complemented
8D3×C4⋊#157C211yesmaximal, characteristic, complemented
9#11724246C211yesmaximal, characteristic, complemented
10#11724245C211yesmaximal, characteristic, complemented
11Dic6×#157C211yesmaximal, characteristic, complemented
12#10596650—31nomaximal
13C22×C3⋊#157C2212yes—
14C22×C3×#157C2212yes—
15C2×D3×#157C2212yes—
16#89078C411yescharacteristic, complemented
17C3×D4×C3⋊D3C2211yescharacteristic
18C32×#237C411yescharacteristic, complemented
19C3⋊D3×#67C2212yes—
20#89031C412yescomplemented
21C2×Dic3×C3⋊D3C2212yes—
22#89029C412yescomplemented
23C4×D3×C3⋊D3C2211yescharacteristic
24#89021C411yescharacteristic, complemented
25#89017C411yescharacteristic, complemented
26C3⋊D3×Dic6C2211yescharacteristic
27#88994C2212yes—
28C4⋊(C3⋊#157)C2211yescharacteristic
29C4×C3⋊#157C2211yescharacteristic
30C3×C22⋊#157C2212yes—
31C3×C4⋊#157C2211yescharacteristic
32C3×C4×#157C2211yescharacteristic
33Dic3⋊#157C2212yes—
34Dic3⋊#157C2212yes—
35Dic3⋊#157C2211yescharacteristic
36Dic3⋊#157C2211yescharacteristic
37Dic3⋊#157C2212yes—
38(C2×D3)⋊#157C2211yescharacteristic
39(C2×D3)⋊#157C2211yescharacteristic
40Dic3×#157C2212yes—
41Dic3×#157C2212yes—
42#41×C3⋊D3—31no—
43D3×#237—62no—
44Q8×#157—31no—
45Q8×#157—31no—
46D4×#157—31no—
47D4×#157—32no—
48D4×#157—32no—
49D4×#157D311yescharacteristic, complemented
50#64286—32no—
51#64286—31no—
52C2×C4⋊#157—31no—
53C2×C4⋊#157—32no—
54C22×C3×C3⋊D3C2312yes—
55C2×D3×C3⋊D3C2311yescharacteristic
56C2×C3⋊#157—42no—
57C2×C3⋊#157C2312yes—
58C2×C3⋊#157C2311yescharacteristic
59C2×C3⋊#157C2311yescharacteristic
60C2×C3×#157—42no—
61C2×C3×#157C2312yes—
62C2×C3×#157C2311yescharacteristic
63C2×C3×#157C2311yescharacteristic
64D3×#157—42no—
65C33×D4C2×C411yescharacteristic, complemented
66#6312C2×C412yescomplemented
67C2×C3⋊(C3⋊Dic3)C2×C412yescomplemented
68C4×C3⋊(C3⋊D3)C2×C411yescharacteristic, complemented
69#6308C2×C411yescharacteristic, complemented
70C3×#638C2×C412yescomplemented
71C3×C4⋊(C3⋊D3)C2×C411yescharacteristic, complemented
72C3×C4×C3⋊D3C2311yescharacteristic
73C32×#67C2×C412yescomplemented
74C2×C32×Dic3C2×C412yescomplemented
75C32×C4×D3C2×C411yescharacteristic, complemented
76C32×Dic6C2×C411yescharacteristic, complemented
77#6293C2×C412yescomplemented
78#6292C2×C412yescomplemented
79#6290C2×C412yescomplemented
80#6289C2×C411yescharacteristic, complemented
81Dic3×C3⋊D3C2×C412yescomplemented
82Dic3×C3⋊D3C2×C412yescomplemented
83Dic3×C3⋊D3C2312yes—
84Dic3×C3⋊D3C2311yescharacteristic
85D3×C3⋊Dic3C2×C411yescharacteristic, complemented
86#5546—91nomaximal
87C22×#157—32no—
88C22×#157—32no—
89C22×#157C2×D312yes—
90C32×#41—31no—
91#3832—31no—
92Q8×C3⋊D3—31no—
93#3830—31no—
94#3830—32no—
95#3830—62no—
96#3830—122no—
97D4×C3⋊D3—32no—
98D4×C3⋊D3C2×D311yescharacteristic
99#3828—32no—
100#3828—31no—
101C2×C4×C3⋊D3—32no—
102C2×C4×C3⋊D3—31no—
103C3×#237—22no—
104C3×D3×D4—62no—
105D3×#67—64no—
106#3805—64no—
107#3804—64no—
108C2×D3×Dic3—64no—
109C4×D32—62no—
110#3796—62no—
111#3795—62no—
112D3×Dic6—62no—
113C22⋊#157—32no—
114C22⋊#157—31no—
115C22⋊#157—31no—
116C22⋊#157C2×D312yes—
117C4⋊#157—32no—
118C4⋊#157—31no—
119C4⋊#157—32no—
120C4⋊#157—32no—
121C4⋊#157—31no—
122C4⋊#157—31no—
123C4⋊#157C2×D311yescharacteristic
124C4×#157—32no—
125C4×#157—32no—
126C4×#157—31no—
127C4×#157C2×D311yescharacteristic
128C22×C33C22×C412yes—
129C2×C3⋊(C3⋊D3)C22×C411yescharacteristic
130C2×C3×C3⋊D3—22no—
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3C22×C412yes—
133C2×C3×C3⋊D3C2411yescharacteristic
134C2×C32×D3C22×C411yescharacteristic
135D3×C3⋊D3—21no—
136D3×C3⋊D3—21no—
137C3⋊#157—22no—
138C3⋊#157—21no—
139C3⋊#157—21no—
140C3×#157—22no—
141C3×#157—21no—
142C3×#157—21no—
143C33×C4C22×C411yescharacteristic
144C3⋊(C3⋊Dic3)C22×C411yescharacteristic
145C3⋊(C3⋊Dic3)C22×C412yes—
146C3×C3⋊Dic3C22×C411yescharacteristic
147C32×Dic3C22×C411yescharacteristic
148C32×Dic3C22×C412yes—
149D3×#41—182no—
150C2×#237—91no—
151C3×C4×D4—91no—
152D4×Dic3—91no—
153C4×#67—92no—
154C2×Dic3⋊C4—92no—
155D3×C4⋊C4—91no—
156#887—91no—
157#67⋊C4—92no—
158#877—92no—
159#872—91no—
160C4×Dic6—91no—
161C22×C3⋊D3—31no—
162C22×C3⋊D3C22×D312yes—
163C22×C3×D3—64no—
164C2×D32—62no—
165C2×#157—31no—
166C2×#157—32no—
167C2×#157—31no—
168C2×#157—42no—
169C2×#157—122no—
170C2×#157—122no—
171C2×#157C22×D312yes—
172C2×#157C22×D311yescharacteristic
173C2×#157C22×D311yescharacteristic
174C32×Q8—31no—
175C32×D4—22no—
176C32×D4—32no—
177C32×D4—42no—
178C32×D4C4×D311yescharacteristic, complemented
179C2×C32×C4—32no—
180C2×C32×C4—31no—
181#638—32no—
182#638—32no—
183#638—64no—
184#638—124no—
185#638C4×D312yescomplemented
186C2×C3⋊Dic3—32no—
187C2×C3⋊Dic3—31no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—124no—
190C4⋊(C3⋊D3)—32no—
191C4⋊(C3⋊D3)C4×D311yescharacteristic, complemented
192C4×C3⋊D3—32no—
193C4×C3⋊D3—32no—
194C4×C3⋊D3—32no—
195C4×C3⋊D3—31no—
196C4×C3⋊D3—31no—
197C4×C3⋊D3—31no—
198C4×C3⋊D3—62no—
199C4×C3⋊D3—122no—
200C4×C3⋊D3C22×D311yescharacteristic
201#634—31no—
202#634—32no—
203#634—62no—
204#634—122no—
205C3×#67—24no—
206C3×#67—64no—
207C2×C3×Dic3—24no—
208C3×D12—62no—
209C3×C4×D3—22no—
210C3×C4×D3—62no—
211C3×Dic6—22no—
212#627—64no—
213#626—64no—
214#625—64no—
215#624—62no—
216D3×Dic3—62no—
217D3×Dic3—64no—
218D3×Dic3—64no—
219D3×Dic3—64no—
220D3×Dic3—62no—
221#520—271noSylow
222C2×C33—22no—
223C2×C33C23×C411yescharacteristic
224C3⋊(C3⋊D3)—21no—
225C3×C3⋊D3—22no—
226C3×C3⋊D3H(5,𝔽2)11yescharacteristic
227C3×C3⋊D32−1+411yescharacteristic
228C32×D3—21no—
229C3×#41—62no—
230C2×C3×D4—91no—
231C22×C3×C4—92no—
232C2×#67—92no—
233C22×Dic3—92no—
234C22×Dic3—92no—
235#239—182no—
236D3×Q8—182no—
237#237—91no—
238#237—92no—
239#237—92no—
240#237—91no—
241#237—91no—
242#237—184no—
243#237—182no—
244#237—362no—
245#237#15711yescharacteristic, complemented
246D3×D4—62no—
247D3×D4—184no—
248#235—182no—
249#235—184no—
250C2×C4×D3—91no—
251C2×C4×D3—92no—
252C2×C4×D3—182no—
253C2×C4×D3—184no—
254C2×Dic6—91no—
255C3×C4⋊C4—91no—
256C3×C22⋊C4—92no—
257C3×C42—91no—
258#217—92no—
259(C2×D3)⋊C4—91no—
260(C2×D3)⋊C4—91no—
261C4⋊Dic3—91no—
262Dic3⋊C4—92no—
263Dic3⋊C4—91no—
264Dic3⋊C4—91no—
265Dic3⋊C4—92no—
266Dic3⋊C4—92no—
267C4×Dic3—92no—
268C4×Dic3—92no—
269C4×Dic3—91no—
270C22×C32—24no—
271C22×C32—31no—
272C22×C32—44no—
273C22×C32C2×C4×D312yes—
274C2×C3⋊D3—22no—
275C2×C3⋊D3—22no—
276C2×C3⋊D3—31no—
277C2×C3⋊D3—62no—
278C2×C3⋊D3—61no—
279C2×C3⋊D3—61no—
280C2×C3⋊D3—122no—
281C2×C3⋊D3C2×C4×D312yes—
282C2×C3⋊D3C23×D311yescharacteristic
283C2×C3×D3—22no—
284C2×C3×D3—62no—
285C2×C3×D3—64no—
286C2×C3×D3—124no—
287C2×C3×D3—124no—
288D32—122no—
289D32—122no—
290#157—21no—
291#157—22no—
292#157—21no—
293#157—62no—
294#157—61no—
295#157—61no—
296C32×C4—22no—
297C32×C4—32no—
298C32×C4—31no—
299C32×C4—42no—
300C32×C4C2×C4×D311yescharacteristic
301C3⋊Dic3—32no—
302C3⋊Dic3—31no—
303C3⋊Dic3—62no—
304C3⋊Dic3—64no—
305C3⋊Dic3—124no—
306C3⋊Dic3—122no—
307C3⋊Dic3C2×C4×D311yescharacteristic
308C3×Dic3—22no—
309C3×Dic3—24no—
310C3×Dic3—62no—
311C2×#41—271no—
312C4×Q8—271no—
313C4×Q8—271no—
314C4×D4—91no—
315C4×D4—272no—
316C4×D4—272no—
317C4×D4—271no—
318#117—271no—
319#117—272no—
320C2×C4⋊C4—271no—
321C2×C4⋊C4—272no—
322C33#52011yescharacteristic, complemented, Sylow
323C23×C3—92no—
324C22×D3—64no—
325C22×D3—91no—
326C22×D3—182no—
327C3×Q8—62no—
328C3×D4—22no—
329C3×D4—42no—
330C3×D4—64no—
331C3×D4—92no—
332C3×D4—91no—
333C3×D4C2×#15711yescharacteristic, complemented
334C2×C3×C4—62no—
335C2×C3×C4—64no—
336C2×C3×C4—91no—
337C2×C3×C4—91no—
338C2×C3×C4—92no—
339C2×C3×C4—91no—
340C2×C3×C4—362no—
341#67—64no—
342#67—92no—
343#67—92no—
344#67—92no—
345#67—184no—
346#67—184no—
347#67—364no—
348#67C2×#15712yescomplemented
349C2×Dic3—92no—
350C2×Dic3—92no—
351C2×Dic3—91no—
352C2×Dic3—92no—
353C2×Dic3—92no—
354C2×Dic3—91no—
355C2×Dic3—91no—
356C2×Dic3—92no—
357C2×Dic3—182no—
358C2×Dic3—184no—
359C2×Dic3—364no—
360C2×Dic3—362no—
361C2×Dic3C2×#15712yescomplemented
362D12—62no—
363D12—184no—
364C4×D3—62no—
365C4×D3—91no—
366C4×D3—91no—
367C4×D3—91no—
368C4×D3—182no—
369C4×D3—182no—
370C4×D3—182no—
371C4×D3—184no—
372C4×D3—184no—
373C4×D3—184no—
374C4×D3—362no—
375C4×D3—362no—
376C4×D3C2×#15711yescharacteristic, complemented
377Dic6—91no—
378Dic6—92no—
379Dic6—184no—
380Dic6—182no—
381Dic6—362no—
382Dic6C2×#15711yescharacteristic, complemented
383C2×C32—22no—
384C2×C32—22no—
385C2×C32—42no—
386C2×C32—44no—
387C2×C32—61no—
388C2×C32—84no—
389C2×C32C22×C4×D311yescharacteristic
390C3⋊D3—22no—
391C3⋊D3—61no—
392C3⋊D3—122no—
393C3⋊D3—242no—
394C3⋊D3#100411yescharacteristic
395C3⋊D3#101011yescharacteristic
396C3×D3—42no—
397C3×D3—62no—
398C3×D3—62no—
399C3×D3—124no—
400#41—31no—
401#41—272no—
402#41—271no—
403#41—272no—
404#41—271no—
405#41—271no—
406C2×Q8—271no—
407C2×D4—91no—
408C2×D4—272no—
409C22×C4—92no—
410C22×C4—271no—
411C22×C4—272no—
412C22×C4—272no—
413C22×C4—272no—
414C4⋊C4—91no—
415C4⋊C4—271no—
416C4⋊C4—272no—
417C4⋊C4—272no—
418C4⋊C4—272no—
419C4⋊C4—271no—
420C4⋊C4—271no—
421C22⋊C4—92no—
422C22⋊C4—271no—
423C22⋊C4—271no—
424C22⋊C4—272no—
425C42—91no—
426C42—271no—
427C42—272no—
428C42—272no—
429C22×C3—24no—
430C22×C3—44no—
431C22×C3—62no—
432C22×C3—91no—
433C22×C3—92no—
434C22×C3—182no—
435C22×C3—182no—
436C22×C3C22×#15712yes—
437C2×D3—62no—
438C2×D3—64no—
439C2×D3—91no—
440C2×D3—124no—
441C2×D3—124no—
442C2×D3—182no—
443C2×D3—181no—
444C2×D3—181no—
445C2×D3—362no—
446C2×D3—362no—
447C2×D3—362no—
448C2×D3C22×#15711yescharacteristic
449C3×C4—22no—
450C3×C4—42no—
451C3×C4—64no—
452C3×C4—62no—
453C3×C4—91no—
454C3×C4—182no—
455C3×C4—181no—
456C3×C4—181no—
457C3×C4C22×#15711yescharacteristic
458Dic3—62no—
459Dic3—92no—
460Dic3—91no—
461Dic3—184no—
462Dic3—182no—
463Dic3—182no—
464Dic3—181no—
465Dic3—181no—
466Dic3—364no—
467Dic3—362no—
468Dic3C22×#15712yes—
469Dic3C22×#15711yescharacteristic
470C32—22no—
471C32—42no—
472C32#554611yescharacteristic, complemented
473C23—92no—
474C23—271no—
475Q8—31no—
476Q8—272no—
477Q8—271no—
478D4—32no—
479D4—92no—
480D4—91no—
481D4—272no—
482D4—272no—
483D4—272no—
484D4D3×#15711yescharacteristic, complemented
485C2×C4—32no—
486C2×C4—31no—
487C2×C4—91no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—92no—
491C2×C4—271no—
492C2×C4—272no—
493C2×C4—271no—
494C2×C4—272no—
495C2×C4—272no—
496C2×C4—271no—
497C2×C4—272no—
498C2×C4—271no—
499C2×C4—271no—
500C2×C4—271no—
501C2×C4—272no—
502C2×C4—362no—
503C2×C4—1082no—
504C2×C4—1082no—
505C2×C3—22no—
506C2×C3—22no—
507C2×C3—44no—
508C2×C3—42no—
509C2×C3—84no—
510C2×C3—91no—
511C2×C3—91no—
512C2×C3—122no—
513C2×C3—182no—
514C2×C3C23×#15711yescharacteristic
515D3—21no—
516D3—62no—
517D3—62no—
518D3—124no—
519D3—181no—
520D3—362no—
521D3—722no—
522C22—31no—
523C22—91no—
524C22—92no—
525C22—182no—
526C22—182no—
527C22—271no—
528C22—541no—
529C22—541no—
530C22C2×D3×#15712yes—
531C4—31no—
532C4—32no—
533C4—91no—
534C4—181no—
535C4—182no—
536C4—181no—
537C4—271no—
538C4—272no—
539C4—542no—
540C4—541no—
541C4—541no—
542C4C2×D3×#15711yescharacteristic
543C3—22no—
544C3—42no—
545C3#1059665011yescharacteristic, complemented
546C2—22no—
547C2—61no—
548C2—91no—
549C2—91no—
550C2—182no—
551C2—541no—
552C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC33×D4C2×C4
prime radicalC3⋊D3×#237C2
cosocleC2×C3×C3⋊D3C24
hypercenterD4D3×#157
hypocenterC33#520
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
2. D4×C3⋊#157C2semidirect
3. C3×D4×#157C2semidirect
4. #67⋊#157C2semidirect
5. #67×#157C2semidirect
6. C2×Dic3⋊#157C2semidirect
7. #11724261C2semidirect
8. D3×C4⋊#157C2semidirect
9. #11724246C2semidirect
10. #11724245C2semidirect
11. Dic6×#157C2semidirect
16. #89078C4semidirect
18. C32×#237C4semidirect
20. #89031C4semidirect
22. #89029C4semidirect
24. #89021C4semidirect
25. #89017C4semidirect
49. D4×#157D3semidirect
65. C33×D4C2×C4semidirect
66. #6312C2×C4semidirect
67. C2×C3⋊(C3⋊Dic3)C2×C4semidirect
68. C4×C3⋊(C3⋊D3)C2×C4semidirect
69. #6308C2×C4semidirect
70. C3×#638C2×C4semidirect
71. C3×C4⋊(C3⋊D3)C2×C4semidirect
73. C32×#67C2×C4semidirect
74. C2×C32×Dic3C2×C4semidirect
75. C32×C4×D3C2×C4semidirect
76. C32×Dic6C2×C4semidirect
77. #6293C2×C4semidirect
78. #6292C2×C4semidirect
79. #6290C2×C4semidirect
80. #6289C2×C4semidirect
81. Dic3×C3⋊D3C2×C4semidirect
82. Dic3×C3⋊D3C2×C4semidirect
85. D3×C3⋊Dic3C2×C4semidirect
178. C32×D4C4×D3semidirect
185. #638C4×D3semidirect
191. C4⋊(C3⋊D3)C4×D3semidirect
245. #237#157semidirect
322. C33#520semidirect
333. C3×D4C2×#157semidirect
348. #67C2×#157semidirect
361. C2×Dic3C2×#157semidirect
376. C4×D3C2×#157semidirect
382. Dic6C2×#157semidirect
472. C32#5546semidirect
484. D4D3×#157semidirect
545. C3#10596650semidirect

Maximal subgroups

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

subgroupindex
C3⋊D3×#2372
D4×C3⋊#1572
C3×D4×#1572
#67⋊#1572
#67×#1572
C2×Dic3⋊#1572
#117242612
D3×C4⋊#1572
#117242462
#117242452
Dic6×#1572
#105966503
#55469