Finite Group Databasebeta

← #13176529

Subgroups — #13176529

Order 1728.

2332 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
1C2×#89026C211yesmaximal, characteristic
2#11723933C211yesmaximal, characteristic, complemented
3C3×#64102C211yesmaximal, characteristic, complemented
4C32×#893C211yesmaximal, characteristic, complemented
5C3⋊D3×(C2×D3)⋊C4C212yesmaximal, complemented
6D3×#3752C212yesmaximal, complemented
7#11723508C212yesmaximal, complemented
8C4×#6290C212yesmaximal, complemented
9D12×C3⋊Dic3C211yesmaximal, characteristic, complemented
10Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
11#11723452C211yesmaximal, characteristic, complemented
12#10596016—31nomaximal
13#10594606—34nomaximal
14C22×D3×C3⋊D3C2212yes—
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×C4×C3⋊D3C2212yescomplemented
18C2×C32×D12C2211yescharacteristic
19C2×C32×C4×D3C2212yescomplemented
20C2×#6290C2212yes—
21C2×Dic3×C3⋊D3C2212yescomplemented
22C2×D3×C3⋊Dic3C2212yescomplemented
23#89026C418yescomplemented
24C33×C4⋊C4C2211yescharacteristic, complemented
25#88859C2212yes—
26C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic, complemented
27C4×C3⋊(C3⋊Dic3)C2212yescomplemented
28C3×#3752C2212yescomplemented
29C3×C4×C3⋊Dic3C2211yescharacteristic, complemented
30C32×(C2×D3)⋊C4C2212yescomplemented
31C32×C4×Dic3C2211yescharacteristic, complemented
32#88806C2211yescharacteristic, complemented
33#88806C2211yescharacteristic, complemented
34#88804C2211yescharacteristic
35#88804C2211yescharacteristic
36#88803C2212yescomplemented
37(C2×D3)⋊(C3⋊Dic3)C2212yescomplemented
38C2×D4×C3⋊D3—31no—
39C2×#3802—34no—
40C32×C4×D4—31no—
41D4×C3⋊Dic3—31no—
42C4×#638—32no—
43C2×#3752—32no—
44#64102D311yescharacteristic, complemented
45#64101—31no—
46#64101—34no—
47#64101—64no—
48#64090—32no—
49C22⋊C4×C3⋊D3—32no—
50C4×C4⋊(C3⋊D3)—31no—
51#64081—31no—
52C3×#893—34no—
53C3×#893D314yescomplemented
54D3×(C2×D3)⋊C4—38no—
55D3×(C2×D3)⋊C4—38no—
56#63902—38no—
57C4×#625—38no—
58Dic3×D12—34no—
59Dic3×D12—34no—
60#63874—34no—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C22×C3×C3⋊D3C2312yes—
63C22×C32×D3C2312yes—
64C2×D3×C3⋊D3—42no—
65C2×D3×C3⋊D3—42no—
66C2×D3×C3⋊D3C2×C418yescomplemented
67C2×C33×C4C2312yes—
68C2×C33×C4C2311yescharacteristic
69C2×C3⋊(C3⋊Dic3)C2312yes—
70C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
71C4×C3⋊(C3⋊D3)C2×C414yescomplemented
72C2×C3×C3⋊Dic3C2311yescharacteristic
73C2×C3×C3⋊Dic3C2311yescharacteristic
74C3×C4⋊(C3⋊D3)C2×C414yescomplemented
75C3×C4×C3⋊D3—42no—
76C2×C32×Dic3C2311yescharacteristic
77C2×C32×Dic3C2311yescharacteristic
78C32×D12C2×C414yescomplemented
79C32×C4×D3—42no—
80#6290C2×C418yescomplemented
81Dic3×C3⋊D3—42no—
82D3×C3⋊Dic3—42no—
83#5545—94no—
84#5545—91no—
85C23×C3⋊D3—32no—
86C22×D32—38no—
87C2×C32×D4—31no—
88C22×C32×C4—32no—
89C2×#638—32no—
90C22×C3⋊Dic3—32no—
91D4×C3⋊D3—38no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—34no—
95C2×C4×C3⋊D3—32no—
96C2×C4×C3⋊D3—64no—
97C2×C4×C3⋊D3C2×D312yescomplemented
98C2×C3×D12—34no—
99C2×C3×D12C2×D314yes—
100C2×C3×C4×D3—38no—
101C2×C3×C4×D3C2×D318yescomplemented
102C2×#625—38no—
103C2×D3×Dic3—38no—
104C2×D3×Dic3—38no—
105#3802—332no—
106C32×C4⋊C4—24no—
107C32×C4⋊C4C2×D314yescomplemented
108C32×C4⋊C4C2×D311yescharacteristic, complemented
109C32×C22⋊C4—32no—
110C32×C42—31no—
111#3757—32no—
112#3752—32no—
113#3752—32no—
114#3752—38no—
115#3752—31no—
116#3752—31no—
117#3752—68no—
118#3752C2×D312yescomplemented
119C4⋊(C3⋊Dic3)—31no—
120C4⋊(C3⋊Dic3)—34no—
121C4⋊(C3⋊Dic3)—64no—
122#3750—31no—
123#3750—31no—
124C4×C3⋊Dic3—32no—
125C4×C3⋊Dic3—38no—
126C4×C3⋊Dic3—68no—
127C4×C3⋊Dic3C2×D311yescharacteristic, complemented
128C3×(C2×D3)⋊C4—38no—
129C3×(C2×D3)⋊C4C2×D318yescomplemented
130C3×C4×Dic3—34no—
131C3×C4×Dic3C2×D314yescomplemented
132#3724—34no—
133#3724—34no—
134#3722—34no—
135#3722—34no—
136(C2×D3)⋊Dic3—38no—
137(C2×D3)⋊Dic3—38no—
138C22×C33C2411yescharacteristic
139C2×C3⋊(C3⋊D3)—21no—
140C2×C3⋊(C3⋊D3)—21no—
141C2×C3⋊(C3⋊D3)C22×C412yes—
142C2×C3×C3⋊D3—22no—
143C2×C3×C3⋊D3—22no—
144C2×C3×C3⋊D3C22×C414yes—
145C2×C32×D3—22no—
146C2×C32×D3—22no—
147C2×C32×D3C22×C414yes—
148D3×C3⋊D3—48no—
149C33×C4—22no—
150C33×C4C22×C412yes—
151C3⋊(C3⋊Dic3)—22no—
152C3⋊(C3⋊Dic3)C22×C412yes—
153C3×C3⋊Dic3—21no—
154C3×C3⋊Dic3—21no—
155C32×Dic3—21no—
156C32×Dic3—21no—
157C2×D3×D4—94no—
158C2×D3×D4—91no—
159C3×C4×D4—34no—
160C3×C4×D4—91no—
161D4×Dic3—94no—
162D4×Dic3—91no—
163C4×#67—98no—
164C4×#67—92no—
165C2×(C2×D3)⋊C4—98no—
166C2×(C2×D3)⋊C4—92no—
167#893—34no—
168#893C3⋊D311yescharacteristic, complemented
169#892—94no—
170#892—91no—
171#892—184no—
172#67⋊C4—98no—
173#67⋊C4—92no—
174D3×C22⋊C4—98no—
175D3×C22⋊C4—92no—
176C4×D12—94no—
177C4×D12—91no—
178#872—94no—
179#872—91no—
180C23×C32—32no—
181C22×C3⋊D3—31no—
182C22×C3⋊D3—38no—
183C22×C3⋊D3—34no—
184C22×C3⋊D3—64no—
185C22×C3⋊D3—122no—
186C22×C3⋊D3—122no—
187C22×C3⋊D3C22×D312yes—
188C22×C3×D3—38no—
189C22×C3×D3C22×D318yes—
190C2×D32—332no—
191C2×D32—128no—
192C2×D32—128no—
193C32×D4—34no—
194C2×C32×C4—24no—
195C2×C32×C4—28no—
196C2×C32×C4—31no—
197C2×C32×C4—31no—
198C2×C32×C4—122no—
199C2×C32×C4C22×D318yes—
200C2×C32×C4C22×D314yes—
201C2×C32×C4C22×D312yes—
202C2×C32×C4C22×D311yescharacteristic
203#638—38no—
204C2×C3⋊Dic3—31no—
205C2×C3⋊Dic3—32no—
206C2×C3⋊Dic3—34no—
207C2×C3⋊Dic3—38no—
208C2×C3⋊Dic3—64no—
209C2×C3⋊Dic3—68no—
210C2×C3⋊Dic3—122no—
211C2×C3⋊Dic3C22×D311yescharacteristic
212C2×C3⋊Dic3C22×D311yescharacteristic
213C4⋊(C3⋊D3)C4×D314yescomplemented
214C4×C3⋊D3—34no—
215C4×C3⋊D3—316no—
216C4×C3⋊D3—42no—
217C4×C3⋊D3—616no—
218C4×C3⋊D3—122no—
219C2×C3×Dic3—34no—
220C2×C3×Dic3—34no—
221C2×C3×Dic3C22×D314yes—
222C2×C3×Dic3C22×D314yes—
223C3×D12—316no—
224C3×D12C4×D3116yescomplemented
225C3×C4×D3—48no—
226C3×C4×D3—128no—
227#625—332no—
228D3×Dic3—128no—
229D3×Dic3—128no—
230#518—271noSylow
231C2×C33C23×C411yescharacteristic
232C2×C33H(5,𝔽2)11yescharacteristic
233C2×C33H(5,𝔽2)11yescharacteristic
234C3⋊(C3⋊D3)—22no—
235C3×C3⋊D3—24no—
236C32×D3—24no—
237C23×D3—98no—
238C23×D3—92no—
239C2×C3×D4—34no—
240C2×C3×D4—91no—
241C22×C3×C4—38no—
242C22×C3×C4—92no—
243C2×#67—98no—
244C2×#67—92no—
245C22×Dic3—98no—
246C22×Dic3—92no—
247D3×D4—932no—
248D3×D4—98no—
249C2×D12—34no—
250C2×D12C2×C3⋊D311yescharacteristic
251C2×C4×D3—38no—
252C2×C4×D3—94no—
253C2×C4×D3—91no—
254C2×C4×D3—98no—
255C2×C4×D3—92no—
256C2×C4×D3—184no—
257C2×C4×D3C2×C3⋊D312yescomplemented
258C3×C4⋊C4—24no—
259C3×C4⋊C4D3214yescomplemented
260C3×C4⋊C4C2×C3⋊D311yescharacteristic, complemented
261C3×C22⋊C4—38no—
262C3×C22⋊C4—92no—
263C3×C42—34no—
264C3×C42—91no—
265#217—98no—
266#217—92no—
267(C2×D3)⋊C4—38no—
268(C2×D3)⋊C4—94no—
269(C2×D3)⋊C4—94no—
270(C2×D3)⋊C4—98no—
271(C2×D3)⋊C4—92no—
272(C2×D3)⋊C4—98no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—91no—
275(C2×D3)⋊C4—91no—
276(C2×D3)⋊C4—188no—
277(C2×D3)⋊C4C2×C3⋊D312yescomplemented
278C4⋊Dic3—94no—
279C4⋊Dic3—91no—
280C4⋊Dic3—184no—
281Dic3⋊C4—94no—
282Dic3⋊C4—94no—
283Dic3⋊C4—91no—
284Dic3⋊C4—91no—
285C4×Dic3—34no—
286C4×Dic3—98no—
287C4×Dic3—92no—
288C4×Dic3—188no—
289C4×Dic3C2×C3⋊D311yescharacteristic, complemented
290C22×C32—24no—
291C22×C32—34no—
292C22×C32—62no—
293C22×C32—62no—
294C22×C32C23×D314yes—
295C22×C32C23×D311yescharacteristic
296C2×C3⋊D3—22no—
297C2×C3⋊D3—22no—
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—
307C2×C3⋊D3—128no—
308C2×C3⋊D3C2×C4×D314yes—
309C2×C3×D3—28no—
310C2×C3×D3—28no—
311C2×C3×D3—316no—
312C2×C3×D3—68no—
313C2×C3×D3—68no—
314C2×C3×D3C2×C4×D3116yes—
315D32—1232no—
316C32×C4—22no—
317C32×C4—28no—
318C32×C4—28no—
319C32×C4—48no—
320C32×C4—61no—
321C32×C4—61no—
322C32×C4C2×C4×D318yes—
323C32×C4C2×C4×D312yes—
324C3⋊Dic3—21no—
325C3⋊Dic3—21no—
326C3⋊Dic3—32no—
327C3⋊Dic3—38no—
328C3⋊Dic3—68no—
329C3⋊Dic3—62no—
330C3⋊Dic3—68no—
331C3⋊Dic3—128no—
332C3×Dic3—24no—
333C3×Dic3—24no—
334C3×Dic3—64no—
335C3×Dic3—64no—
336C22×D4—271no—
337C4×D4—31no—
338C4×D4—91no—
339C4×D4—271no—
340C4×D4—271no—
341C4×D4—272no—
342C4×D4—272no—
343#117—271no—
344#117—271no—
345C2×C22⋊C4—272no—
346C2×C22⋊C4—272no—
347C33#51811yescharacteristic, complemented, Sylow
348C23×C3—38no—
349C23×C3—92no—
350C22×D3—38no—
351C22×D3—932no—
352C22×D3—94no—
353C22×D3—91no—
354C22×D3—98no—
355C22×D3—184no—
356C22×D3—362no—
357C22×D3—362no—
358C22×D3—368no—
359C22×D3—368no—
360C22×D3C22×C3⋊D312yes—
361C3×D4—316no—
362C3×D4—94no—
363C2×C3×C4—24no—
364C2×C3×C4—28no—
365C2×C3×C4—34no—
366C2×C3×C4—34no—
367C2×C3×C4—91no—
368C2×C3×C4—91no—
369C2×C3×C4—128no—
370C2×C3×C4—362no—
371C2×C3×C4C2×D3218yes—
372C2×C3×C4C2×D3214yes—
373C2×C3×C4C22×C3⋊D312yes—
374C2×C3×C4C22×C3⋊D311yescharacteristic
375#67—932no—
376#67—98no—
377C2×Dic3—34no—
378C2×Dic3—34no—
379C2×Dic3—94no—
380C2×Dic3—91no—
381C2×Dic3—98no—
382C2×Dic3—92no—
383C2×Dic3—184no—
384C2×Dic3—188no—
385C2×Dic3—362no—
386C2×Dic3—368no—
387C2×Dic3C22×C3⋊D311yescharacteristic
388C2×Dic3C22×C3⋊D311yescharacteristic
389D12—316no—
390D12C4×C3⋊D314yescomplemented
391C4×D3—42no—
392C4×D3—916no—
393C4×D3—94no—
394C4×D3—128no—
395C4×D3—1816no—
396C4×D3—368no—
397C4×D3—362no—
398C2×C32—24no—
399C2×C32—24no—
400C2×C32—24no—
401C2×C32—64no—
402C2×C32C22×C4×D314yes—
403C2×C32C22×C4×D311yescharacteristic
404C2×C32#100414yes—
405C2×C32#100411yescharacteristic
406C2×C32#100914yes—
407C2×C32#100911yescharacteristic
408C3⋊D3—24no—
409C3⋊D3—62no—
410C3⋊D3—68no—
411C3⋊D3—128no—
412C3×D3—216no—
413C3×D3—616no—
414C24—272no—
415C2×D4—31no—
416C2×D4—91no—
417C2×D4—278no—
418C2×D4—272no—
419C22×C4—32no—
420C22×C4—92no—
421C22×C4—272no—
422C22×C4—272no—
423C22×C4—271no—
424C4⋊C4—271no—
425C4⋊C4—271no—
426C4⋊C4—271no—
427C4⋊C4D3×C3⋊D311yescharacteristic, complemented
428C22⋊C4—32no—
429C22⋊C4—92no—
430C22⋊C4—271no—
431C22⋊C4—271no—
432C22⋊C4—272no—
433C22⋊C4—272no—
434C22⋊C4—272no—
435C42—31no—
436C42—91no—
437C42—272no—
438C22×C3—24no—
439C22×C3—316no—
440C22×C3—68no—
441C22×C3—68no—
442C22×C3—94no—
443C22×C3—182no—
444C22×C3—182no—
445C22×C3C22×D3214yes—
446C22×C3C23×C3⋊D311yescharacteristic
447C2×D3—22no—
448C2×D3—22no—
449C2×D3—316no—
450C2×D3—68no—
451C2×D3—68no—
452C2×D3—98no—
453C2×D3—92no—
454C2×D3—188no—
455C2×D3—184no—
456C2×D3—184no—
457C2×D3—181no—
458C2×D3—181no—
459C2×D3—364no—
460C2×D3—364no—
461C2×D3—368no—
462C2×D3—3632no—
463C2×D3C2×C4×C3⋊D314yes—
464C3×C4—28no—
465C3×C4—22no—
466C3×C4—28no—
467C3×C4—48no—
468C3×C4—64no—
469C3×C4—64no—
470C3×C4—181no—
471C3×C4—181no—
472C3×C4C4×D3218yes—
473C3×C4C2×C4×C3⋊D312yes—
474Dic3—21no—
475Dic3—21no—
476Dic3—64no—
477Dic3—64no—
478Dic3—98no—
479Dic3—92no—
480Dic3—188no—
481Dic3—188no—
482Dic3—182no—
483Dic3—368no—
484C32—24no—
485C32#554514yescomplemented
486C32#554511yescharacteristic, complemented
487C23—32no—
488C23—92no—
489C23—278no—
490C23—271no—
491C23—1082no—
492C23—1082no—
493D4—34no—
494D4—94no—
495D4—278no—
496C2×C4—31no—
497C2×C4—31no—
498C2×C4—91no—
499C2×C4—91no—
500C2×C4—122no—
501C2×C4—271no—
502C2×C4—274no—
503C2×C4—272no—
504C2×C4—362no—
505C2×C4—1082no—
506C2×C4—1082no—
507C2×C4C2×D3×C3⋊D312yes—
508C2×C4C2×D3×C3⋊D311yescharacteristic
509C2×C3—24no—
510C2×C3—24no—
511C2×C3—24no—
512C2×C3—616no—
513C2×C3—184no—
514C2×C3C2×C4×D3214yes—
515C2×C3#6431314yes—
516C2×C3#6432014yes—
517C2×C3C22×C4×C3⋊D311yescharacteristic
518C2×C3#6436111yescharacteristic
519C2×C3#6436611yescharacteristic
520D3—24no—
521D3—616no—
522D3—188no—
523D3—182no—
524D3—368no—
525C22—34no—
526C22—62no—
527C22—62no—
528C22—94no—
529C22—182no—
530C22—182no—
531C22—272no—
532C22—541no—
533C22—541no—
534C22—1088no—
535C22C22×D3×C3⋊D311yescharacteristic
536C4—22no—
537C4—61no—
538C4—61no—
539C4—181no—
540C4—181no—
541C4—272no—
542C4—542no—
543C4C4×D3×C3⋊D312yes—
544C3—24no—
545C3#1059460614yescomplemented
546C3#1059601611yescharacteristic, complemented
547C2—64no—
548C2—184no—
549C2—542no—
550C2C2×C4×D3×C3⋊D311yescharacteristic
551C2#1172458711yescharacteristic
552C2#1172459711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutatorC2×C33C23×C4
FittingC33×C4⋊C4C22
socle, cosocleC22×C33C24
prime radicalC2×#89026C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#518
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #11723933C2semidirect
3. C3×#64102C2semidirect
4. C32×#893C2semidirect
5. C3⋊D3×(C2×D3)⋊C4C2semidirect
6. D3×#3752C2semidirect
7. #11723508C2semidirect
8. C4×#6290C2semidirect
9. D12×C3⋊Dic3C2semidirect
10. Dic3×C4⋊(C3⋊D3)C2semidirect
11. #11723452C2semidirect
17. C2×C3×C4×C3⋊D3C22semidirect
19. C2×C32×C4×D3C22semidirect
21. C2×Dic3×C3⋊D3C22semidirect
22. C2×D3×C3⋊Dic3C22semidirect
23. #89026C4semidirect
24. C33×C4⋊C4C22semidirect
26. C4⋊(C3⋊(C3⋊Dic3))C22semidirect
27. C4×C3⋊(C3⋊Dic3)C22semidirect
28. C3×#3752C22semidirect
29. C3×C4×C3⋊Dic3C22semidirect
30. C32×(C2×D3)⋊C4C22semidirect
31. C32×C4×Dic3C22semidirect
32. #88806C22semidirect
33. #88806C22semidirect
36. #88803C22semidirect
37. (C2×D3)⋊(C3⋊Dic3)C22semidirect
44. #64102D3semidirect
53. C3×#893D3semidirect
66. C2×D3×C3⋊D3C2×C4semidirect
71. C4×C3⋊(C3⋊D3)C2×C4semidirect
74. C3×C4⋊(C3⋊D3)C2×C4semidirect
78. C32×D12C2×C4semidirect
80. #6290C2×C4semidirect
97. C2×C4×C3⋊D3C2×D3semidirect
101. C2×C3×C4×D3C2×D3semidirect
107. C32×C4⋊C4C2×D3semidirect
108. C32×C4⋊C4C2×D3semidirect
118. #3752C2×D3semidirect
127. C4×C3⋊Dic3C2×D3semidirect
129. C3×(C2×D3)⋊C4C2×D3semidirect
131. C3×C4×Dic3C2×D3semidirect
168. #893C3⋊D3semidirect
213. C4⋊(C3⋊D3)C4×D3semidirect
224. C3×D12C4×D3semidirect
257. C2×C4×D3C2×C3⋊D3semidirect
259. C3×C4⋊C4D32semidirect
260. C3×C4⋊C4C2×C3⋊D3semidirect
277. (C2×D3)⋊C4C2×C3⋊D3semidirect
289. C4×Dic3C2×C3⋊D3semidirect
347. C33#518semidirect
390. D12C4×C3⋊D3semidirect
427. C4⋊C4D3×C3⋊D3semidirect
485. C32#5545semidirect
486. C32#5545semidirect
545. C3#10594606semidirect
546. C3#10596016semidirect

Maximal subgroups

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

subgroupindex
C2×#890262
#117239332
C3×#641022
C32×#8932
C3⋊D3×(C2×D3)⋊C42
D3×#37522
#117235082
C4×#62902
D12×C3⋊Dic32
Dic3×C4⋊(C3⋊D3)2
#117234522
#105960163
#105946063