Finite Group Databasebeta

← #13177670

Subgroups — #13177670

Order 1728.

1792 conjugacy classes of subgroups.

The subgroup lattice has 508 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic, complemented
2C3×#64149C211yesmaximal, characteristic
3C32×#936C211yesmaximal, characteristic, complemented
4#11723626C212yesmaximal, complemented
5#11723621C212yesmaximal, complemented
6#11723609C212yesmaximal, complemented
7#11723591C212yesmaximal, complemented
8C4×#6293C211yesmaximal, characteristic, complemented
9Dic3×C4⋊(C3⋊D3)C211yesmaximal, characteristic, complemented
10#11723436C211yesmaximal, characteristic, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596152—31nomaximal
13#10595199—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C2×C3×#638C2212yes—
17C2×C3×#638C2212yes—
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6293C2211yescharacteristic
21C2×Dic3×C3⋊D3C2212yes—
22#88864C2212yes—
23C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
24C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×#217C2212yes—
28C32×C4×Dic3C2211yescharacteristic
29#88807C2211yescharacteristic
30#88806C2211yescharacteristic
31Dic3⋊(C3⋊Dic3)C2211yescharacteristic
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88803C2212yes—
34#88803C2212yes—
35#88803C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×#124—31no—
38#64156—31no—
39Q8×C3⋊Dic3—31no—
40#64149D311yescharacteristic
41D4×C3⋊Dic3—31no—
42D4×C3⋊Dic3—34no—
43D4×C3⋊Dic3—64no—
44#64095—32no—
45#64094—32no—
46#64092—32no—
47#64090—32no—
48C4×C4⋊(C3⋊D3)—31no—
49C4×#634—31no—
50C3×#940—34no—
51C3×#936D314yescomplemented
52#67⋊Dic3—38no—
53#63969—38no—
54#63962—38no—
55#63959—38no—
56C4×#627—34no—
57Dic3×D12—34no—
58#63863—34no—
59Dic3×Dic6—34no—
60C23×C33C2312yes—
61C22×C3×C3⋊D3C2312yes—
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)—42no—
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C2×C3⋊(C3⋊Dic3)C2312yes—
68C3×#638—24no—
69C3×#638—42no—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2311yescharacteristic
72C3×C4⋊(C3⋊D3)—22no—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2311yescharacteristic
75C2×C32×Dic3C2312yes—
76C32×Dic6—22no—
77#6293—22no—
78Dic3×C3⋊D3—42no—
79#5713—91no—
80#5681—94no—
81C2×C32×Q8—31no—
82C2×C32×D4—24no—
83C2×C32×D4C2×D314yes—
84C2×C32×D4C2×D311yescharacteristic
85C2×#638C2×D312yes—
86C2×#638C2×D312yes—
87C22×C3⋊Dic3—32no—
88C22×C3⋊Dic3—38no—
89C22×C3⋊Dic3—68no—
90C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
91C2×C4×C3⋊D3—32no—
92C2×#634—31no—
93C2×C3×#67—38no—
94C2×C3×#67—38no—
95C2×C3×D12—34no—
96C2×C3×Dic6C2×D314yes—
97C2×#627—34no—
98C2×D3×Dic3—38no—
99C32×C22⋊C4—32no—
100C32×C22⋊C4—32no—
101C32×C42—31no—
102#3757—32no—
103#3757—38no—
104#3757—68no—
105#3752—32no—
106#3752—32no—
107#3752—32no—
108C4⋊(C3⋊Dic3)—31no—
109C4⋊(C3⋊Dic3)—34no—
110C4⋊(C3⋊Dic3)—31no—
111C4⋊(C3⋊Dic3)—32no—
112C4⋊(C3⋊Dic3)—64no—
113#3750—31no—
114#3750—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3—34no—
117C4×C3⋊Dic3—32no—
118C4×C3⋊Dic3—64no—
119C4×C3⋊Dic3C2×D311yescharacteristic
120C3×#217C2×D318yes—
121C3×#217C2×D318yes—
122C3×C4×Dic3—34no—
123C3×C4×Dic3C2×D314yes—
124#3724—34no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—34no—
127Dic3⋊Dic3—34no—
128(C2×D3)⋊Dic3—38no—
129(C2×D3)⋊Dic3—38no—
130(C2×D3)⋊Dic3—38no—
131Dic32—38no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33—22no—
135C22×C33C2411yescharacteristic
136C2×C3×C3⋊D3—22no—
137C2×C3×C3⋊D3—22no—
138C2×C3×C3⋊D3—22no—
139C33×C4#4112yes—
140C3⋊(C3⋊Dic3)—21no—
141C3⋊(C3⋊Dic3)—21no—
142C3⋊(C3⋊Dic3)—22no—
143C3×C3⋊Dic3—21no—
144C3×C3⋊Dic3#4112yes—
145C32×Dic3—21no—
146C32×Dic3—22no—
147C32×Dic3—21no—
148C3×#127—91no—
149C3×#124—34no—
150#947—94no—
151Q8×Dic3—94no—
152#940—34no—
153#936—91no—
154#936—92no—
155#936C3⋊D311yescharacteristic, complemented
156#935—92no—
157#935—92no—
158D4×Dic3—94no—
159D4×Dic3—91no—
160D4×Dic3—92no—
161D4×Dic3—91no—
162D4×Dic3—184no—
163#886—98no—
164#885—98no—
165#883—98no—
166#67⋊C4—98no—
167C4×D12—94no—
168C4×Dic6—94no—
169C4×Dic6—91no—
170C4×Dic6—91no—
171C23×C32—28no—
172C23×C32C22×D318yes—
173C23×C32C22×D312yes—
174C22×C3⋊D3C22×D312yes—
175C22×C3×D3—38no—
176C32×Q8—62no—
177C32×D4—216no—
178C32×D4—28no—
179C32×D4—22no—
180C2×C32×C4—24no—
181C2×C32×C4—32no—
182C2×C32×C4—31no—
183C2×C32×C4—31no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186#638—24no—
187#638—42no—
188C2×C3⋊Dic3—31no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—38no—
192C2×C3⋊Dic3—34no—
193C2×C3⋊Dic3—34no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3—64no—
196C2×C3⋊Dic3—68no—
197C2×C3⋊Dic3—122no—
198C2×C3⋊Dic3—128no—
199C2×C3⋊Dic3—1216no—
200C2×C3⋊Dic3C22×D311yescharacteristic
201C2×C3⋊Dic3C22×D311yescharacteristic
202C4⋊(C3⋊D3)—22no—
203C4×C3⋊D3—122no—
204#634—62no—
205C3×#67—616no—
206C3×#67—128no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3—34no—
209C2×C3×Dic3C22×D318yes—
210C2×C3×Dic3C22×D314yes—
211C2×C3×Dic3C22×D314yes—
212C3×D12—68no—
213C3×Dic6—28no—
214#627—68no—
215D3×Dic3—128no—
216#559—271noSylow
217C2×C33—42no—
218C2×C33C2×#4111yescharacteristic
219C2×C33C2×#4111yescharacteristic
220C2×C33H(5,𝔽2)11yescharacteristic
221C3×C3⋊D3—42no—
222C2×C3×Q8—34no—
223C2×C3×D4—24no—
224C2×C3×D4—92no—
225C2×C3×D4—92no—
226C2×C3×D4—91no—
227C2×C3×D4D3214yes—
228C2×C3×D4C2×C3⋊D311yescharacteristic
229C2×#67—38no—
230C2×#67—38no—
231C22×Dic3—98no—
232C22×Dic3—92no—
233C22×Dic3—92no—
234C22×Dic3—188no—
235C2×D12—34no—
236C2×C4×D3—98no—
237C2×Dic6—94no—
238C2×Dic6—91no—
239C2×Dic6C2×C3⋊D311yescharacteristic
240C3×C22⋊C4—38no—
241C3×C22⋊C4—38no—
242C3×C42—34no—
243C3×C42—91no—
244#217—98no—
245#217—92no—
246#217—92no—
247#217—92no—
248#217—92no—
249#217—188no—
250#217C2×C3⋊D312yes—
251#217C2×C3⋊D312yes—
252(C2×D3)⋊C4—98no—
253(C2×D3)⋊C4—98no—
254(C2×D3)⋊C4—98no—
255C4⋊Dic3—94no—
256C4⋊Dic3—98no—
257C4⋊Dic3—94no—
258C4⋊Dic3—91no—
259C4⋊Dic3—91no—
260C4⋊Dic3—184no—
261Dic3⋊C4—94no—
262Dic3⋊C4—94no—
263Dic3⋊C4—92no—
264Dic3⋊C4—91no—
265Dic3⋊C4—91no—
266C4×Dic3—34no—
267C4×Dic3—98no—
268C4×Dic3—94no—
269C4×Dic3—91no—
270C4×Dic3—92no—
271C4×Dic3—184no—
272C4×Dic3C2×C3⋊D311yescharacteristic
273C22×C32—216no—
274C22×C32—22no—
275C22×C32—28no—
276C22×C32—22no—
277C22×C32—22no—
278C22×C32—24no—
279C22×C32—28no—
280C22×C32—28no—
281C22×C32—48no—
282C22×C32—48no—
283C22×C32C23×D314yes—
284C22×C32C23×D311yescharacteristic
285C2×C3⋊D3—22no—
286C2×C3⋊D3—22no—
287C2×C3⋊D3—22no—
288C2×C3×D3—68no—
289C2×C3×D3—68no—
290C2×C3×D3—68no—
291C32×C4—28no—
292C32×C4—61no—
293C32×C4—62no—
294C32×C4—61no—
295C32×C4#23718yes—
296C32×C4#23712yes—
297C3⋊Dic3—21no—
298C3⋊Dic3—68no—
299C3⋊Dic3—61no—
300C3⋊Dic3—62no—
301C3⋊Dic3—61no—
302C3⋊Dic3—64no—
303C3⋊Dic3—64no—
304C3⋊Dic3—68no—
305C3⋊Dic3—124no—
306C3⋊Dic3—128no—
307C3⋊Dic3#23712yes—
308C3×Dic3—28no—
309C3×Dic3—24no—
310C3×Dic3—24no—
311C3×Dic3—38no—
312C3×Dic3—64no—
313#127—91no—
314#124—31no—
315#124—272no—
316#124—271no—
317#123—272no—
318#123—272no—
319C4×Q8—271no—
320C4×Q8—271no—
321C4×D4—271no—
322C4×D4—272no—
323C4×D4—271no—
324C33#55911yescharacteristic, complemented, Sylow
325C23×C3—28no—
326C23×C3—92no—
327C23×C3C2×D3218yes—
328C23×C3C22×C3⋊D312yes—
329C22×D3—38no—
330C3×Q8—68no—
331C3×D4—216no—
332C3×D4—28no—
333C3×D4—22no—
334C3×D4—184no—
335C3×D4—182no—
336C3×D4—362no—
337C2×C3×C4—24no—
338C2×C3×C4—38no—
339C2×C3×C4—34no—
340C2×C3×C4—34no—
341C2×C3×C4—91no—
342C2×C3×C4—91no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345#67—616no—
346#67—128no—
347C2×Dic3—34no—
348C2×Dic3—34no—
349C2×Dic3—94no—
350C2×Dic3—94no—
351C2×Dic3—98no—
352C2×Dic3—92no—
353C2×Dic3—91no—
354C2×Dic3—91no—
355C2×Dic3—184no—
356C2×Dic3—184no—
357C2×Dic3—188no—
358C2×Dic3—362no—
359C2×Dic3—368no—
360C2×Dic3—362no—
361C2×Dic3—3616no—
362C2×Dic3C22×C3⋊D312yes—
363C2×Dic3C22×C3⋊D311yescharacteristic
364C2×Dic3C22×C3⋊D311yescharacteristic
365D12—68no—
366C4×D3—368no—
367Dic6—22no—
368Dic6—182no—
369Dic6—188no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—24no—
373C2×C32—416no—
374C2×C32—48no—
375C2×C32—42no—
376C2×C32C2×#23714yes—
377C2×C32C2×#23711yescharacteristic
378C2×C32C2×#23711yescharacteristic
379C2×C32#100411yescharacteristic
380C2×C32D3×#4114yes—
381C2×C32#100914yes—
382C3⋊D3—42no—
383C3×D3—128no—
384C2×Q8—31no—
385C2×Q8—271no—
386C2×D4—92no—
387C2×D4—92no—
388C2×D4—91no—
389C2×D4D3×C3⋊D311yescharacteristic
390C22×C4—272no—
391C22×C4—272no—
392C4⋊C4—271no—
393C4⋊C4—271no—
394C4⋊C4—271no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—32no—
399C22⋊C4—272no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C22⋊C4—272no—
403C42—31no—
404C42—91no—
405C42—272no—
406C42—271no—
407C22×C3—216no—
408C22×C3—28no—
409C22×C3—28no—
410C22×C3—28no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—24no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3—182no—
418C22×C3—182no—
419C22×C3—182no—
420C22×C3C22×D3214yes—
421C22×C3C23×C3⋊D311yescharacteristic
422C2×D3—68no—
423C2×D3—68no—
424C2×D3—68no—
425C3×C4—28no—
426C3×C4—64no—
427C3×C4—64no—
428C3×C4—68no—
429C3×C4—92no—
430C3×C4—181no—
431C3×C4#380518yes—
432C3×C4#383012yes—
433Dic3—22no—
434Dic3—21no—
435Dic3—21no—
436Dic3—38no—
437Dic3—64no—
438Dic3—188no—
439Dic3—184no—
440Dic3—188no—
441Dic3—184no—
442Dic3—181no—
443Dic3—181no—
444Dic3—182no—
445Dic3—368no—
446Dic3—364no—
447C32—24no—
448C32#568114yescomplemented
449C32#571311yescharacteristic, complemented
450C23—92no—
451C23C2×D3×C3⋊D312yes—
452Q8—62no—
453Q8—542no—
454D4—22no—
455D4—184no—
456D4—182no—
457D4—362no—
458C2×C4—31no—
459C2×C4—31no—
460C2×C4—32no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—271no—
464C2×C4—271no—
465C2×C4—272no—
466C2×C4—1082no—
467C2×C4—1082no—
468C2×C4C2×D3×C3⋊D311yescharacteristic
469C2×C3—24no—
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—416no—
473C2×C3—42no—
474C2×C3—48no—
475C2×C3—362no—
476C2×C3D3×#23714yes—
477C2×C3#6431414yes—
478C2×C3C2×#380514yes—
479C2×C3C2×#383011yescharacteristic
480C2×C3#41×C3⋊D311yescharacteristic
481C2×C3#6436611yescharacteristic
482D3—128no—
483C22—22no—
484C22—22no—
485C22—22no—
486C22—182no—
487C22—182no—
488C22—182no—
489C22C22×D3×C3⋊D311yescharacteristic
490C4—61no—
491C4—61no—
492C4—62no—
493C4—92no—
494C4—181no—
495C4—541no—
496C4—541no—
497C4—542no—
498C4#8902912yes—
499C3—24no—
500C3#1059519914yescomplemented
501C3#1059615211yescharacteristic, complemented
502C2—42no—
503C2—362no—
504C2C3⋊D3×#23711yescharacteristic
505C2#1172458911yescharacteristic
506C2C2×#8902911yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radicalC3×#64149C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#559
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. D4×C3⋊(C3⋊Dic3)C2semidirect
3. C32×#936C2semidirect
4. #11723626C2semidirect
5. #11723621C2semidirect
6. #11723609C2semidirect
7. #11723591C2semidirect
8. C4×#6293C2semidirect
9. Dic3×C4⋊(C3⋊D3)C2semidirect
10. #11723436C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
51. C3×#936D3semidirect
155. #936C3⋊D3semidirect
324. C33#559semidirect
448. C32#5681semidirect
449. C32#5713semidirect
500. C3#10595199semidirect
501. C3#10596152semidirect

Maximal subgroups

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

subgroupindex
D4×C3⋊(C3⋊Dic3)2
C3×#641492
C32×#9362
#117236262
#117236212
#117236092
#117235912
C4×#62932
Dic3×C4⋊(C3⋊D3)2
#117234362
Dic6×C3⋊Dic32
#105961523
#105951993