Finite Group Databasebeta

← #13175358

Subgroups — #13175358

Order 1728.

1600 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723913C211yesmaximal, characteristic, complemented
2C3×C4×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723536C211yesmaximal, characteristic, complemented
5#11723532C211yesmaximal, characteristic
6#11723456C211yesmaximal, characteristic, complemented
7#11723438C212yesmaximal, complemented
8#11723436C211yesmaximal, characteristic, complemented
9#11723418C212yesmaximal, complemented
10#11723417C212yesmaximal, complemented
11#11723416C212yesmaximal, complemented
12#10596053—31nomaximal
13#10593876—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×Dic6C2211yescharacteristic
18C2×#6293C2212yes—
19C2×#6292C2212yes—
20C33×C42C2211yescharacteristic
21#88859C2211yescharacteristic
22#88859C2211yescharacteristic
23#88857C2211yescharacteristic
24#88857C2211yescharacteristic
25C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
26C3×#3752C2212yes—
27C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
28C32×C4⋊Dic3C2211yescharacteristic
29C32×Dic3⋊C4C2212yes—
30C32×C4×Dic3C2212yes—
31#88807C2212yes—
32#88806C2212yes—
33#88804C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36#88803C2212yes—
37C32×C4×Q8—31no—
38#64156—31no—
39#64155—31no—
40#64105—32no—
41#64104—31no—
42#64103—32no—
43#64097—31no—
44#64085—32no—
45#64084—32no—
46C4×C4⋊(C3⋊D3)D311yescharacteristic, complemented
47#64081—31no—
48#64081—34no—
49#64081—64no—
50C3×C4×D12—34no—
51C3×C4×Dic6D314yescomplemented
52#63919—34no—
53#63916—34no—
54#63871—34no—
55#63863—34no—
56#63859—38no—
57#63852—38no—
58#63851—38no—
59#63850—38no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C22×C3×C3⋊D3C2312yes—
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C33×C4C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
68C4×C3⋊(C3⋊D3)—22no—
69C2×C3×C3⋊Dic3C2312yes—
70C3×C4⋊(C3⋊D3)—22no—
71C3×C4×C3⋊D3—42no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2312yes—
74C32×Dic6—22no—
75#6293—42no—
76#6292—42no—
77#5582—94no—
78#5561—91no—
79C2×C32×Q8—31no—
80C2×C4⋊(C3⋊D3)—32no—
81C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—34no—
84C2×C4×C3⋊D3—64no—
85C2×C4×C3⋊D3C2×D312yes—
86C2×#634—32no—
87C2×C3×D12—34no—
88C2×C3×C4×D3—38no—
89C2×C3×Dic6C2×D314yes—
90C2×#627—38no—
91C2×#626—38no—
92C32×C4⋊C4—31no—
93C32×C4⋊C4—32no—
94C32×C42—24no—
95C32×C42—32no—
96C32×C42C2×D314yes—
97C32×C42C2×D311yescharacteristic
98#3752—32no—
99#3752—32no—
100#3752—32no—
101#3752—31no—
102#3752—31no—
103#3752—34no—
104#3752—34no—
105#3752—32no—
106#3752—64no—
107#3752—64no—
108#3752C2×D312yes—
109C4⋊(C3⋊Dic3)C2×D311yescharacteristic
110#3750—31no—
111#3750—31no—
112#3750—34no—
113#3750—34no—
114#3750—32no—
115#3750—32no—
116#3750—64no—
117#3750—64no—
118C4×C3⋊Dic3—31no—
119C4×C3⋊Dic3—34no—
120C4×C3⋊Dic3—64no—
121C3×(C2×D3)⋊C4—38no—
122C3×C4⋊Dic3—34no—
123C3×C4⋊Dic3C2×D314yes—
124C3×Dic3⋊C4C2×D318yes—
125C3×C4×Dic3C2×D318yes—
126#3724—38no—
127Dic3⋊Dic3—38no—
128#3722—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131(C2×D3)⋊Dic3—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—21no—
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C33×C4—21no—
140C33×C4—21no—
141C33×C4#4112yes—
142C3⋊(C3⋊Dic3)—21no—
143C3⋊(C3⋊Dic3)—21no—
144C3⋊(C3⋊Dic3)—21no—
145C3×C3⋊Dic3—22no—
146C32×Dic3—22no—
147C32×Dic3—22no—
148C3×C4×Q8—34no—
149C3×C4×D4—91no—
150#947—94no—
151#946—94no—
152#938—91no—
153#936—91no—
154#929—92no—
155#924—92no—
156#897—91no—
157#896—98no—
158#895—94no—
159#894—98no—
160#888—94no—
161#888—91no—
162#885—92no—
163#879—92no—
164#876—98no—
165#875—98no—
166C4×D12—34no—
167#872—94no—
168#872—91no—
169#872—184no—
170C4×Dic6C3⋊D311yescharacteristic, complemented
171C22×C3⋊D3—31no—
172C22×C3⋊D3—34no—
173C22×C3⋊D3—64no—
174C22×C3⋊D3C22×D312yes—
175C22×C3×D3—38no—
176C32×Q8—62no—
177C2×C32×C4—24no—
178C2×C32×C4—24no—
179C2×C32×C4—24no—
180C2×C32×C4—32no—
181C2×C32×C4—32no—
182C2×C32×C4C22×D314yes—
183C2×C32×C4C22×D314yes—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—31no—
191C2×C3⋊Dic3—34no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—64no—
197C2×C3⋊Dic3C22×D312yes—
198C4⋊(C3⋊D3)—22no—
199C4⋊(C3⋊D3)—122no—
200C4×C3⋊D3—42no—
201C4×C3⋊D3—616no—
202C4×C3⋊D3—68no—
203C4×C3⋊D3—62no—
204#634—122no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3C22×D318yes—
207C2×C3×Dic3C22×D318yes—
208C3×D12—68no—
209C3×C4×D3—128no—
210C3×Dic6—28no—
211#627—128no—
212#626—128no—
213#542—271noSylow
214C2×C33C2×#4111yescharacteristic
215C2×C33H(5,𝔽2)11yescharacteristic
216C2×C332−1+411yescharacteristic
217C3⋊(C3⋊D3)—41no—
218C3×C3⋊D3—42no—
219C2×C3×Q8—34no—
220C2×C3×D4—91no—
221C22×C3×C4—92no—
222C2×#67—92no—
223C2×D12—34no—
224C2×D12—98no—
225C2×C4×D3—38no—
226C2×C4×D3—94no—
227C2×C4×D3—91no—
228C2×C4×D3—184no—
229C2×Dic6—98no—
230C2×Dic6—92no—
231C2×Dic6C2×C3⋊D311yescharacteristic
232C3×C4⋊C4—34no—
233C3×C4⋊C4—38no—
234C3×C4⋊C4—91no—
235C3×C22⋊C4—92no—
236C3×C42—24no—
237C3×C42—38no—
238C3×C42D3214yes—
239C3×C42C2×C3⋊D311yescharacteristic
240#217—92no—
241#217—92no—
242#217—92no—
243(C2×D3)⋊C4—38no—
244(C2×D3)⋊C4—98no—
245(C2×D3)⋊C4—98no—
246(C2×D3)⋊C4—98no—
247(C2×D3)⋊C4—98no—
248(C2×D3)⋊C4—94no—
249(C2×D3)⋊C4—94no—
250(C2×D3)⋊C4—91no—
251(C2×D3)⋊C4—91no—
252(C2×D3)⋊C4—92no—
253(C2×D3)⋊C4—184no—
254(C2×D3)⋊C4—184no—
255C4⋊Dic3—34no—
256C4⋊Dic3—92no—
257C4⋊Dic3C2×C3⋊D311yescharacteristic
258Dic3⋊C4—98no—
259Dic3⋊C4—98no—
260Dic3⋊C4—94no—
261Dic3⋊C4—94no—
262Dic3⋊C4—91no—
263Dic3⋊C4—91no—
264Dic3⋊C4—92no—
265Dic3⋊C4—184no—
266Dic3⋊C4—184no—
267Dic3⋊C4C2×C3⋊D312yes—
268C4×Dic3—94no—
269C4×Dic3—91no—
270C4×Dic3—184no—
271C4×Dic3C2×C3⋊D312yes—
272C22×C32—24no—
273C22×C32C23×D314yes—
274C22×C32C23×D311yescharacteristic
275C2×C3⋊D3—22no—
276C2×C3⋊D3—22no—
277C2×C3⋊D3—22no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—64no—
284C2×C3⋊D3—64no—
285C2×C3⋊D3—124no—
286C2×C3⋊D3—124no—
287C2×C3×D3—68no—
288C2×C3×D3—68no—
289C2×C3×D3—68no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—28no—
293C32×C4—24no—
294C32×C4—24no—
295C32×C4—44no—
296C32×C4—44no—
297C32×C4—62no—
298C32×C4—62no—
299C32×C4#23518yes—
300C32×C4#23512yes—
301C3⋊Dic3—22no—
302C3⋊Dic3—68no—
303C3⋊Dic3—64no—
304C3⋊Dic3—61no—
305C3⋊Dic3—61no—
306C3⋊Dic3—61no—
307C3⋊Dic3—64no—
308C3⋊Dic3—64no—
309C3⋊Dic3—124no—
310C3⋊Dic3—124no—
311C3×Dic3—28no—
312C3×Dic3—28no—
313C3×Dic3—68no—
314C4⋊Q8—271no—
315#126—272no—
316#124—272no—
317#124—271no—
318#123—272no—
319C22⋊Q8—272no—
320C22⋊Q8—271no—
321#121—271no—
322C4×Q8—31no—
323C4×D4—91no—
324#117—271no—
325C33#54211yescharacteristic, complemented, Sylow
326C23×C3—92no—
327C22×D3—38no—
328C22×D3—94no—
329C22×D3—91no—
330C22×D3—184no—
331C3×Q8—68no—
332C3×D4—182no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—24no—
336C2×C3×C4—38no—
337C2×C3×C4—38no—
338C2×C3×C4—92no—
339C2×C3×C4—362no—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C2×D3214yes—
342C2×C3×C4C2×D3214yes—
343C2×C3×C4C22×C3⋊D311yescharacteristic
344C2×C3×C4C22×C3⋊D311yescharacteristic
345C2×C3×C4C22×C3⋊D311yescharacteristic
346#67—362no—
347C2×Dic3—38no—
348C2×Dic3—94no—
349C2×Dic3—94no—
350C2×Dic3—94no—
351C2×Dic3—91no—
352C2×Dic3—91no—
353C2×Dic3—91no—
354C2×Dic3—184no—
355C2×Dic3—184no—
356C2×Dic3—184no—
357C2×Dic3C22×C3⋊D312yes—
358C2×Dic3C22×C3⋊D312yes—
359D12—68no—
360D12—368no—
361C4×D3—128no—
362C4×D3—1816no—
363C4×D3—188no—
364C4×D3—182no—
365Dic6—22no—
366Dic6—362no—
367Dic6—368no—
368C2×C32—24no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32C2×#23514yes—
372C2×C32C2×#23511yescharacteristic
373C2×C32#100411yescharacteristic
374C2×C32#100714yes—
375C2×C32#100914yes—
376C2×C32#101011yescharacteristic
377C3⋊D3—42no—
378C3⋊D3—128no—
379C3⋊D3—121no—
380C3⋊D3—124no—
381C3×D3—128no—
382C2×Q8—31no—
383C2×Q8—272no—
384C2×D4—91no—
385C2×D4—272no—
386C22×C4—92no—
387C22×C4—271no—
388C4⋊C4—31no—
389C4⋊C4—32no—
390C4⋊C4—91no—
391C4⋊C4—272no—
392C4⋊C4—272no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C22⋊C4—92no—
396C22⋊C4—272no—
397C22⋊C4—272no—
398C22⋊C4—272no—
399C22⋊C4—272no—
400C22⋊C4—271no—
401C22⋊C4—271no—
402C42—32no—
403C42—271no—
404C42D3×C3⋊D311yescharacteristic
405C22×C3—24no—
406C22×C3—182no—
407C22×C3—182no—
408C22×C3—182no—
409C22×C3C22×D3214yes—
410C22×C3C23×C3⋊D311yescharacteristic
411C2×D3—68no—
412C2×D3—68no—
413C2×D3—68no—
414C2×D3—188no—
415C2×D3—184no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—181no—
419C2×D3—181no—
420C2×D3—181no—
421C2×D3—364no—
422C2×D3—364no—
423C3×C4—24no—
424C3×C4—24no—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—28no—
428C3×C4—44no—
429C3×C4—44no—
430C3×C4—68no—
431C3×C4—68no—
432C3×C4—182no—
433C3×C4#379818yes—
434C3×C4#382812yes—
435Dic3—22no—
436Dic3—22no—
437Dic3—68no—
438Dic3—188no—
439Dic3—184no—
440Dic3—184no—
441Dic3—184no—
442Dic3—181no—
443Dic3—181no—
444Dic3—181no—
445Dic3—364no—
446Dic3—364no—
447C32—24no—
448C32#556111yescharacteristic, complemented
449C32#558214yescomplemented
450C23—92no—
451C23—271no—
452Q8—62no—
453Q8—1082no—
454D4—182no—
455D4—1082no—
456C2×C4—32no—
457C2×C4—32no—
458C2×C4—92no—
459C2×C4—271no—
460C2×C4—271no—
461C2×C4—271no—
462C2×C4—362no—
463C2×C4—542no—
464C2×C4C2×D3×C3⋊D311yescharacteristic
465C2×C4C2×D3×C3⋊D311yescharacteristic
466C2×C4C2×D3×C3⋊D311yescharacteristic
467C2×C3—24no—
468C2×C3—24no—
469C2×C3—24no—
470C2×C3—362no—
471C2×C3#6429714yes—
472C2×C3C2×#379814yes—
473C2×C3#6430714yes—
474C2×C3C2×#382811yescharacteristic
475C2×C3#6436411yescharacteristic
476C2×C3#6436611yescharacteristic
477D3—128no—
478D3—368no—
479D3—364no—
480D3—361no—
481C22—182no—
482C22—182no—
483C22—182no—
484C22—541no—
485C22—541no—
486C22—541no—
487C22C22×D3×C3⋊D311yescharacteristic
488C4—21no—
489C4—21no—
490C4—62no—
491C4—62no—
492C4—182no—
493C4—541no—
494C4—541no—
495C4—541no—
496C4#8902012yes—
497C3—24no—
498C3#1059387614yescomplemented
499C3#1059605311yescharacteristic, complemented
500C2—362no—
501C2—1081no—
502C2#1172456611yescharacteristic
503C2C2×#8902011yescharacteristic
504C2#1172458011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C42C22
prime radical#11723532C2
hypercenterC42D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C42

Complements

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

NH ≅ G/Ntype
1. #11723913C2semidirect
2. C3×C4×C4⋊(C3⋊D3)C2semidirect
3. C32×C4×Dic6C2semidirect
4. #11723536C2semidirect
6. #11723456C2semidirect
7. #11723438C2semidirect
8. #11723436C2semidirect
9. #11723418C2semidirect
10. #11723417C2semidirect
11. #11723416C2semidirect
46. C4×C4⋊(C3⋊D3)D3semidirect
51. C3×C4×Dic6D3semidirect
170. C4×Dic6C3⋊D3semidirect
325. C33#542semidirect
448. C32#5561semidirect
449. C32#5582semidirect
498. C3#10593876semidirect
499. C3#10596053semidirect

Maximal subgroups

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

subgroupindex
#117239132
C3×C4×C4⋊(C3⋊D3)2
C32×C4×Dic62
#117235362
#117235322
#117234562
#117234382
#117234362
#117234182
#117234172
#117234162
#105960533
#105938763