Finite Group Databasebeta

← #13175190

Subgroups — #13175190

Order 1728.

2020 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3⋊D3×Dic6C211yesmaximal, characteristic, complemented
2C2×#89014C211yesmaximal, characteristic, complemented
3C4×#6308C211yesmaximal, characteristic, complemented
4C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
5C32×#875C211yesmaximal, characteristic, complemented
6#11723530C212yesmaximal, complemented
7#11723522C211yesmaximal, characteristic, complemented
8#11723519C212yesmaximal, complemented
9#11723508C212yesmaximal, complemented
10#11723500C212yesmaximal, complemented
11#11723435C211yesmaximal, characteristic, complemented
12#10596148—31nomaximal
13#10593905—34nomaximal
14C2×#6308C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2211yescharacteristic
18C2×C32×D12C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293C2212yes—
21C2×#6290C2212yes—
22C2×#6290C2212yes—
23C2×Dic3×C3⋊D3C2212yes—
24C2×D3×C3⋊Dic3C2212yes—
25C3⋊D3×Dic6—24no—
26#89014—24no—
27C33×C42C2211yescharacteristic
28C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
29#88857C2212yes—
30C4×C3⋊(C3⋊Dic3)C2212yes—
31C3×#3752C2211yescharacteristic
32C3×#3752C2211yescharacteristic
33C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
34C32×(C2×D3)⋊C4C2212yes—
35C32×(C2×D3)⋊C4C2212yes—
36#88806C2212yes—
37#88803C2212yes—
38(C2×D3)⋊(C3⋊Dic3)C2212yes—
39C2×Q8×C3⋊D3—31no—
40C2×#3830—31no—
41C2×#3795—34no—
42C2×D3×Dic6—34no—
43C32×#124—31no—
44#64155—31no—
45#64147—31no—
46#64094—32no—
47#64093—32no—
48#64090—32no—
49#64087—32no—
50C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
51C4×#634—31no—
52C4×#634—34no—
53C4×#634—64no—
54C3×#875D314yescomplemented
55C3×C4×D12—34no—
56#63915—38no—
57#63908—34no—
58#63902—38no—
59#63899—38no—
60#63885—38no—
61#63856—34no—
62C22×C3×C3⋊D3C2311yescharacteristic
63C22×C3×C3⋊D3C2311yescharacteristic
64C22×C32×D3C2312yes—
65C2×C33×C4C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3⋊(C3⋊Dic3)C2312yes—
69C2×C3⋊(C3⋊Dic3)C2312yes—
70#6308D414yescomplemented
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C3×C4⋊(C3⋊D3)—22no—
74C3×C4×C3⋊D3—22no—
75C3×C4×C3⋊D3—22no—
76C2×C32×Dic3C2312yes—
77C32×D12—22no—
78C32×Dic6—22no—
79#6293—24no—
80#6290—24no—
81#6290—24no—
82Dic3×C3⋊D3—24no—
83D3×C3⋊Dic3—24no—
84#5677—94no—
85#5552—91no—
86C2×C32×Q8—31no—
87C2×C32×D4—31no—
88C2×#638—32no—
89C2×#638—32no—
90C22×C3⋊Dic3—32no—
91Q8×C3⋊D3—64no—
92#3830—64no—
93C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
94C2×C4×C3⋊D3—32no—
95C2×C4×C3⋊D3C2×D311yescharacteristic
96C2×C4×C3⋊D3C2×D311yescharacteristic
97C2×#634—31no—
98C2×#634—34no—
99C2×#634—32no—
100C2×#634—64no—
101C2×C3×D12—34no—
102C2×C3×D12C2×D314yes—
103C2×C3×C4×D3—34no—
104C2×C3×C4×D3—34no—
105C2×C3×Dic6C2×D314yes—
106C2×#627—38no—
107C2×#625—38no—
108C2×#625—38no—
109C2×D3×Dic3—38no—
110C2×D3×Dic3—38no—
111#3795—616no—
112D3×Dic6—616no—
113C32×C22⋊C4—32no—
114C32×C22⋊C4—32no—
115C32×C42—24no—
116C32×C42C2×D314yes—
117C32×C42C2×D311yescharacteristic
118#3757—32no—
119#3752—32no—
120#3752C2×D311yescharacteristic
121#3752C2×D311yescharacteristic
122C4⋊(C3⋊Dic3)—31no—
123C4⋊(C3⋊Dic3)—34no—
124C4⋊(C3⋊Dic3)—64no—
125C4⋊(C3⋊Dic3)C2×D311yescharacteristic
126#3750—32no—
127#3750—38no—
128#3750—32no—
129#3750—68no—
130C4×C3⋊Dic3—32no—
131C4×C3⋊Dic3—38no—
132C4×C3⋊Dic3—68no—
133C3×(C2×D3)⋊C4—34no—
134C3×(C2×D3)⋊C4—34no—
135C3×(C2×D3)⋊C4C2×D318yes—
136C3×(C2×D3)⋊C4C2×D318yes—
137C3×C4⋊Dic3—34no—
138#3724—38no—
139(C2×D3)⋊Dic3—38no—
140(C2×D3)⋊Dic3—38no—
141C22×C33C2411yescharacteristic
142C2×C3×C3⋊D3—21no—
143C2×C3×C3⋊D3—21no—
144C2×C3×C3⋊D3—21no—
145C2×C3×C3⋊D3—21no—
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3#4112yescomplemented
148C2×C32×D3—22no—
149C2×C32×D3—22no—
150C2×C32×D3—22no—
151C33×C4—21no—
152C33×C4—21no—
153C33×C4C2×D412yes—
154C3⋊(C3⋊Dic3)—22no—
155C3⋊(C3⋊Dic3)C2×D414yes—
156C3×C3⋊Dic3—21no—
157C3×C3⋊Dic3—21no—
158C32×Dic3—22no—
159C2×D3×Q8—94no—
160C2×#237—94no—
161C2×#235—91no—
162C22×Dic6—91no—
163C3×#124—34no—
164C3×C4×D4—91no—
165#946—94no—
166#939—92no—
167#938—94no—
168#930—91no—
169#924—91no—
170#896—92no—
171#885—98no—
172#885—92no—
173#884—98no—
174#67⋊C4—98no—
175#67⋊C4—92no—
176#878—98no—
177#875C3⋊D311yescharacteristic, complemented
178C4×D12—34no—
179C4×Dic6—94no—
180C4×Dic6—91no—
181C4×Dic6—184no—
182C23×C32—32no—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3⋊D3C22×D311yescharacteristic
185C22×C3×D3—34no—
186C22×C3×D3—34no—
187C22×C3×D3C22×D318yes—
188C32×Q8—62no—
189C32×D4—62no—
190C2×C32×C4—24no—
191C2×C32×C4—24no—
192C2×C32×C4—24no—
193C2×C32×C4—32no—
194C2×C32×C4C22×D314yes—
195C2×C32×C4C22×D314yes—
196C2×C32×C4C22×D314yes—
197C2×C32×C4C22×D311yescharacteristic
198C2×C32×C4C22×D311yescharacteristic
199C2×C32×C4C22×D311yescharacteristic
200#638—64no—
201#638—64no—
202C2×C3⋊Dic3—32no—
203C2×C3⋊Dic3—32no—
204C2×C3⋊Dic3—38no—
205C2×C3⋊Dic3—38no—
206C2×C3⋊Dic3—68no—
207C2×C3⋊Dic3—68no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3C22×D311yescharacteristic
210C2×C3⋊Dic3C22×D311yescharacteristic
211C4⋊(C3⋊D3)—22no—
212C4×C3⋊D3—22no—
213C4×C3⋊D3—22no—
214C4×C3⋊D3—64no—
215#634—34no—
216#634—316no—
217#634—616no—
218#634—64no—
219C2×C3×Dic3—34no—
220C2×C3×Dic3—34no—
221C2×C3×Dic3C22×D318yes—
222C3×D12—28no—
223C3×D12—68no—
224C3×C4×D3—68no—
225C3×C4×D3—68no—
226C3×Dic6—28no—
227#627—616no—
228#625—616no—
229#625—616no—
230D3×Dic3—616no—
231D3×Dic3—616no—
232#548—271noSylow
233C2×C33C22×D411yescharacteristic
234C2×C33C2×#4111yescharacteristic
235C2×C332−1+411yescharacteristic
236C3×C3⋊D3—22no—
237C3×C3⋊D3—41no—
238C32×D3—42no—
239C2×C3×Q8—34no—
240C2×C3×D4—34no—
241C2×C3×D4—91no—
242C22×C3×C4—91no—
243C22×C3×C4—91no—
244C2×#67—98no—
245C2×#67—98no—
246C2×#67—92no—
247C2×#67—92no—
248C22×Dic3—98no—
249C22×Dic3—92no—
250D3×Q8—1816no—
251#237—1816no—
252#235—184no—
253C2×D12—34no—
254C2×D12C2×C3⋊D311yescharacteristic
255C2×C4×D3—34no—
256C2×C4×D3—34no—
257C2×C4×D3—98no—
258C2×C4×D3—92no—
259C2×Dic6—94no—
260C2×Dic6—91no—
261C2×Dic6—98no—
262C2×Dic6—92no—
263C2×Dic6—184no—
264C2×Dic6—184no—
265C2×Dic6C2×C3⋊D311yescharacteristic
266C3×C4⋊C4—91no—
267C3×C22⋊C4—38no—
268C3×C22⋊C4—38no—
269C3×C22⋊C4—91no—
270C3×C22⋊C4—91no—
271C3×C42—24no—
272C3×C42D3214yes—
273C3×C42C2×C3⋊D311yescharacteristic
274#217—98no—
275#217—92no—
276(C2×D3)⋊C4—34no—
277(C2×D3)⋊C4—34no—
278(C2×D3)⋊C4—98no—
279(C2×D3)⋊C4—92no—
280(C2×D3)⋊C4C2×C3⋊D312yes—
281(C2×D3)⋊C4C2×C3⋊D312yes—
282C4⋊Dic3—34no—
283C4⋊Dic3—94no—
284C4⋊Dic3—91no—
285C4⋊Dic3—184no—
286Dic3⋊C4—98no—
287Dic3⋊C4—98no—
288Dic3⋊C4—92no—
289Dic3⋊C4—92no—
290Dic3⋊C4—188no—
291C4×Dic3—98no—
292C4×Dic3—92no—
293C4×Dic3—188no—
294C22×C32—24no—
295C22×C32—62no—
296C22×C32—62no—
297C22×C32—62no—
298C22×C32C23×D314yes—
299C22×C32C23×D311yescharacteristic
300C2×C3⋊D3—21no—
301C2×C3⋊D3—21no—
302C2×C3⋊D3—21no—
303C2×C3⋊D3—21no—
304C2×C3⋊D3—21no—
305C2×C3⋊D3#23512yescomplemented
306C2×C3×D3—28no—
307C2×C3×D3—28no—
308C2×C3×D3—28no—
309C2×C3×D3—38no—
310C2×C3×D3—64no—
311C2×C3×D3—64no—
312C2×C3×D3—64no—
313C2×C3×D3—64no—
314C2×C3×D3—64no—
315C32×C4—28no—
316C32×C4—21no—
317C32×C4—24no—
318C32×C4—21no—
319C32×C4—28no—
320C32×C4—24no—
321C32×C4—44no—
322C32×C4—62no—
323C32×C4D3×D418yes—
324C32×C4D3×D412yes—
325C3⋊Dic3—21no—
326C3⋊Dic3—21no—
327C3⋊Dic3—34no—
328C3⋊Dic3—316no—
329C3⋊Dic3—616no—
330C3⋊Dic3—62no—
331C3⋊Dic3—68no—
332C3⋊Dic3—128no—
333C3×Dic3—28no—
334C3×Dic3—64no—
335C3×Dic3—64no—
336C2×#41—271no—
337C22×Q8—271no—
338#124—31no—
339#124—272no—
340C22⋊Q8—272no—
341C22⋊Q8—271no—
342#121—272no—
343#121—271no—
344C4×Q8—271no—
345C4×D4—91no—
346C4×D4—272no—
347C33#54811yescharacteristic, complemented, Sylow
348C23×C3—38no—
349C23×C3—91no—
350C23×C3—91no—
351C22×D3—34no—
352C22×D3—34no—
353C22×D3C22×C3⋊D312yes—
354C3×Q8—68no—
355C3×D4—68no—
356C3×D4—182no—
357C2×C3×C4—24no—
358C2×C3×C4—24no—
359C2×C3×C4—24no—
360C2×C3×C4—38no—
361C2×C3×C4—91no—
362C2×C3×C4—91no—
363C2×C3×C4—182no—
364C2×C3×C4—182no—
365C2×C3×C4C2×D3214yes—
366C2×C3×C4C2×D3214yes—
367C2×C3×C4C2×D3214yes—
368C2×C3×C4C22×C3⋊D311yescharacteristic
369C2×C3×C4C22×C3⋊D311yescharacteristic
370C2×C3×C4C22×C3⋊D311yescharacteristic
371#67—184no—
372#67—1816no—
373#67—184no—
374#67—1816no—
375C2×Dic3—34no—
376C2×Dic3—34no—
377C2×Dic3—98no—
378C2×Dic3—92no—
379C2×Dic3—98no—
380C2×Dic3—92no—
381C2×Dic3—188no—
382C2×Dic3—188no—
383C2×Dic3—184no—
384C2×Dic3—1816no—
385C2×Dic3C22×C3⋊D312yes—
386D12—22no—
387D12—68no—
388C4×D3—68no—
389C4×D3—68no—
390C4×D3—1816no—
391C4×D3—184no—
392Dic6—22no—
393Dic6—916no—
394Dic6—94no—
395Dic6—1816no—
396Dic6—184no—
397Dic6—1816no—
398C2×C32—24no—
399C2×C32—24no—
400C2×C32—24no—
401C2×C32—122no—
402C2×C32C2×#23511yescharacteristic
403C2×C32C2×D3×D414yes—
404C2×C32C2×D3×D411yescharacteristic
405C2×C32D3×#4114yes—
406C2×C32#101014yes—
407C2×C32#101011yescharacteristic
408C3⋊D3—22no—
409C3⋊D3—41no—
410C3×D3—48no—
411C3×D3—68no—
412C3×D3—124no—
413#41—544no—
414C2×Q8—31no—
415C2×Q8—272no—
416C2×Q8—271no—
417C2×Q8—544no—
418C2×D4—31no—
419C2×D4—91no—
420C2×D4—272no—
421C2×D4—272no—
422C22×C4—91no—
423C22×C4—91no—
424C22×C4—272no—
425C22×C4—272no—
426C4⋊C4—91no—
427C4⋊C4—272no—
428C4⋊C4—272no—
429C4⋊C4—271no—
430C22⋊C4—32no—
431C22⋊C4—32no—
432C22⋊C4—91no—
433C22⋊C4—91no—
434C22⋊C4—272no—
435C22⋊C4—272no—
436C42—272no—
437C42D3×C3⋊D311yescharacteristic
438C22×C3—24no—
439C22×C3—68no—
440C22×C3—68no—
441C22×C3—68no—
442C22×C3—92no—
443C22×C3—181no—
444C22×C3—181no—
445C22×C3—181no—
446C22×C3—181no—
447C22×C3—181no—
448C22×C3C22×D3214yes—
449C22×C3C23×C3⋊D311yescharacteristic
450C2×D3—22no—
451C2×D3—22no—
452C2×D3—22no—
453C2×D3—38no—
454C2×D3—64no—
455C2×D3—64no—
456C2×D3—64no—
457C2×D3—64no—
458C2×D3—64no—
459C3×C4—28no—
460C3×C4—24no—
461C3×C4—24no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—28no—
465C3×C4—44no—
466C3×C4—68no—
467C3×C4—181no—
468C3×C4—181no—
469C3×C4#380218yes—
470C3×C4D4×C3⋊D312yes—
471Dic3—22no—
472Dic3—64no—
473Dic3—64no—
474Dic3—916no—
475Dic3—94no—
476Dic3—1816no—
477Dic3—188no—
478Dic3—182no—
479Dic3—368no—
480C32—24no—
481C32#555211yescharacteristic, complemented
482C32#567714yescomplemented
483C23—32no—
484C23—91no—
485C23—91no—
486Q8—62no—
487Q8—274no—
488Q8—544no—
489D4—62no—
490D4—182no—
491D4—544no—
492D4—544no—
493C2×C4—32no—
494C2×C4—91no—
495C2×C4—91no—
496C2×C4—182no—
497C2×C4—182no—
498C2×C4—272no—
499C2×C4—272no—
500C2×C4—544no—
501C2×C4—544no—
502C2×C4C2×D3×C3⋊D311yescharacteristic
503C2×C4C2×D3×C3⋊D311yescharacteristic
504C2×C4C2×D3×C3⋊D311yescharacteristic
505C2×C3—24no—
506C2×C3—24no—
507C2×C3—24no—
508C2×C3—128no—
509C2×C3—182no—
510C2×C3—361no—
511C2×C3#6429814yes—
512C2×C3C2×#380214yes—
513C2×C3D3×#23514yes—
514C2×C3C2×D4×C3⋊D311yescharacteristic
515C2×C3#41×C3⋊D311yescharacteristic
516C2×C3#6436711yescharacteristic
517D3—42no—
518D3—68no—
519D3—124no—
520C22—62no—
521C22—62no—
522C22—62no—
523C22—92no—
524C22—181no—
525C22—181no—
526C22—181no—
527C22—181no—
528C22—181no—
529C22C22×D3×C3⋊D311yescharacteristic
530C4—21no—
531C4—21no—
532C4—62no—
533C4—181no—
534C4—181no—
535C4—274no—
536C4—542no—
537C4#8902612yes—
538C3—24no—
539C3#1059390514yescomplemented
540C3#1059614811yescharacteristic, complemented
541C2—122no—
542C2—182no—
543C2—361no—
544C2#1172456311yescharacteristic
545C2C2×#8902611yescharacteristic
546C2C3⋊D3×#23511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
hypercenterC42D3×C3⋊D3
hypocenterC33#548
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C2×C3⋊D3×Dic6C2semidirect
2. C2×#89014C2semidirect
3. C4×#6308C2semidirect
4. C3×C4×C4⋊(C3⋊D3)C2semidirect
5. C32×#875C2semidirect
6. #11723530C2semidirect
7. #11723522C2semidirect
8. #11723519C2semidirect
9. #11723508C2semidirect
10. #11723500C2semidirect
11. #11723435C2semidirect
50. C4×C4⋊(C3⋊D3)D3semidirect
54. C3×#875D3semidirect
70. #6308D4semidirect
147. C2×C3×C3⋊D3#41semidirect
177. #875C3⋊D3semidirect
305. C2×C3⋊D3#235semidirect
347. C33#548semidirect
481. C32#5552semidirect
482. C32#5677semidirect
539. C3#10593905semidirect
540. C3#10596148semidirect

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×#890142
C4×#63082
C3×C4×C4⋊(C3⋊D3)2
C32×#8752
#117235302
#117235222
#117235192
#117235082
#117235002
#117234352
#105961483
#105939053