Finite Group Databasebeta

← #13177469

Subgroups — #13177469

Order 1728.

1780 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723979C211yesmaximal, characteristic
2C3×#64145C211yesmaximal, characteristic, complemented
3C32×D4×Dic3C211yesmaximal, characteristic, complemented
4#11723628C212yesmaximal, complemented
5#11723618C212yesmaximal, complemented
6#11723596C212yesmaximal, complemented
7#11723594C212yesmaximal, complemented
8C4×#6289C211yesmaximal, characteristic, complemented
9#11723449C211yesmaximal, characteristic, complemented
10#11723447C211yesmaximal, characteristic, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12#10596028—31nomaximal
13#10595191—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C2×#6312C2212yes—
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C22×C32×Dic3C2212yes—
19C2×#6289C2211yescharacteristic
20C2×#6289C2211yescharacteristic
21#88864C2212yes—
22C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
23C3×#3757C2212yes—
24C3×#3757C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2212yes—
30#88806C2212yes—
31Dic3⋊(C3⋊Dic3)C2212yes—
32Dic3⋊(C3⋊Dic3)C2212yes—
33#88804C2212yes—
34Dic3×C3⋊Dic3C2211yescharacteristic
35Dic3×C3⋊Dic3C2212yes—
36Dic3×C3⋊Dic3C2211yescharacteristic
37C32×C4×D4—31no—
38#64147—31no—
39#64147—34no—
40#64147—64no—
41#64145D311yescharacteristic, complemented
42C4×#638—32no—
43#64135—32no—
44#64106—31no—
45#64099—31no—
46#64092—32no—
47#64088—32no—
48C42×C3⋊D3—31no—
49C4×#634—31no—
50C3×#936—34no—
51C3×D4×Dic3D314yescomplemented
52#63975—38no—
53#63956—38no—
54#63955—38no—
55#63952—38no—
56C4×#624—34no—
57#63871—34no—
58#63869—34no—
59Dic3×Dic6—34no—
60C23×C33C2312yes—
61C22×C3⋊(C3⋊D3)C2311yescharacteristic
62C33×D4—22no—
63C2×C33×C4C2311yescharacteristic
64#6312—24no—
65C2×C3⋊(C3⋊Dic3)C2312yes—
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C4×C3⋊(C3⋊D3)—22no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2312yes—
71C3×#634—22no—
72C2×C32×Dic3—24no—
73C2×C32×Dic3C2311yescharacteristic
74C2×C32×Dic3C2312yes—
75C2×C32×Dic3C2311yescharacteristic
76#6289—22no—
77#6289—22no—
78#5680—91no—
79#5557—94no—
80C2×C32×D4—24no—
81C2×C32×D4C2×D314yes—
82C2×C32×D4C2×D311yescharacteristic
83C22×C32×C4—32no—
84C2×#638—32no—
85C2×#638—38no—
86C2×#638—68no—
87C2×C4×C3⋊D3—31no—
88C2×C4×C3⋊D3—34no—
89C2×C4×C3⋊D3—31no—
90C2×C4×C3⋊D3—31no—
91C2×C4×C3⋊D3—64no—
92C2×#634C2×D311yescharacteristic
93C22×C3×Dic3C2×D318yes—
94C2×C3×Dic6—34no—
95C2×#624—34no—
96C2×#624—34no—
97C32×C4⋊C4—31no—
98C32×C22⋊C4—32no—
99C32×C42—31no—
100#3757—32no—
101#3757—38no—
102#3757—68no—
103#3757C2×D312yes—
104#3757C2×D312yes—
105#3752—32no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—34no—
108C4⋊(C3⋊Dic3)—32no—
109C4⋊(C3⋊Dic3)—32no—
110C4⋊(C3⋊Dic3)—64no—
111#3750—32no—
112#3750—32no—
113C4×C3⋊Dic3—32no—
114C4×C3⋊Dic3—31no—
115C4×C3⋊Dic3—31no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×#217—38no—
118C3×#217—38no—
119C3×#217C2×D318yes—
120C3×C4⋊Dic3C2×D314yes—
121C3×C4×Dic3—34no—
122C3×C4×Dic3C2×D314yes—
123#3724—38no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127#3722—38no—
128Dic32—38no—
129Dic32—34no—
130Dic32—34no—
131C22×C33—22no—
132C22×C33—22no—
133C22×C33—22no—
134C22×C33C2411yescharacteristic
135C2×C3⋊(C3⋊D3)—21no—
136C2×C3⋊(C3⋊D3)—21no—
137C2×C3⋊(C3⋊D3)#4112yes—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)—22no—
141C3×C3⋊Dic3—21no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—21no—
145C32×Dic3—22no—
146C32×Dic3#4112yes—
147C3×#124—91no—
148C3×C4×D4—34no—
149#946—91no—
150Q8×Dic3—91no—
151#938—94no—
152#938—91no—
153#938—184no—
154#936—34no—
155D4×Dic3C3⋊D311yescharacteristic, complemented
156C4×#67—98no—
157#926—98no—
158#897—94no—
159#890—94no—
160#883—98no—
161#883—92no—
162#67⋊C4—92no—
163#879—98no—
164#879—92no—
165#877—92no—
166C42×D3—94no—
167C42×D3—91no—
168#870—91no—
169C4×Dic6—94no—
170C23×C32—28no—
171C23×C32C22×D318yes—
172C23×C32C22×D312yes—
173C22×C3⋊D3—31no—
174C22×C3⋊D3—34no—
175C22×C3⋊D3—64no—
176C32×D4—22no—
177C32×D4—28no—
178C32×D4—48no—
179C2×C32×C4—24no—
180C2×C32×C4—31no—
181C2×C32×C4—31no—
182C2×C32×C4—32no—
183C2×C32×C4—64no—
184C2×C32×C4C22×D314yes—
185C2×C32×C4C22×D311yescharacteristic
186#638—64no—
187#638—616no—
188#638—1216no—
189C2×C3⋊Dic3—31no—
190C2×C3⋊Dic3—32no—
191C2×C3⋊Dic3—34no—
192C2×C3⋊Dic3—38no—
193C2×C3⋊Dic3—68no—
194C2×C3⋊Dic3—64no—
195C2×C3⋊Dic3C22×D311yescharacteristic
196C2×C3⋊Dic3C22×D311yescharacteristic
197C2×C3⋊Dic3C22×D312yes—
198C4×C3⋊D3—62no—
199C4×C3⋊D3—62no—
200C4×C3⋊D3—62no—
201C4×C3⋊D3—68no—
202C4×C3⋊D3—128no—
203#634—22no—
204C2×C3×Dic3—216no—
205C2×C3×Dic3—38no—
206C2×C3×Dic3—34no—
207C2×C3×Dic3—34no—
208C2×C3×Dic3C22×D314yes—
209C2×C3×Dic3C22×D318yes—
210C2×C3×Dic3C22×D314yes—
211C3×Dic6—68no—
212#624—68no—
213#624—68no—
214#529—271noSylow
215C2×C33—42no—
216C2×C33C2×#4111yescharacteristic
217C2×C33C2×#4111yescharacteristic
218C2×C33C2×#4111yescharacteristic
219C3⋊(C3⋊D3)—22no—
220C2×C3×Q8—91no—
221C2×C3×D4—24no—
222C2×C3×D4D3214yes—
223C2×C3×D4C2×C3⋊D311yescharacteristic
224C22×C3×C4—38no—
225C2×#67—98no—
226C2×#67—92no—
227C2×#67—188no—
228C22×Dic3C2×C3⋊D312yes—
229C2×C4×D3—94no—
230C2×C4×D3—94no—
231C2×C4×D3—94no—
232C2×C4×D3—91no—
233C2×C4×D3—91no—
234C2×C4×D3—91no—
235C2×C4×D3—184no—
236C2×Dic6—34no—
237C3×C4⋊C4—34no—
238C3×C22⋊C4—38no—
239C3×C22⋊C4—92no—
240C3×C22⋊C4—92no—
241C3×C42—34no—
242C3×C42—91no—
243#217—38no—
244#217—38no—
245#217—98no—
246#217—92no—
247#217—188no—
248#217C2×C3⋊D312yes—
249(C2×D3)⋊C4—98no—
250(C2×D3)⋊C4—92no—
251C4⋊Dic3—98no—
252C4⋊Dic3—98no—
253C4⋊Dic3—94no—
254C4⋊Dic3—91no—
255C4⋊Dic3—92no—
256C4⋊Dic3—184no—
257C4⋊Dic3C2×C3⋊D311yescharacteristic
258Dic3⋊C4—98no—
259Dic3⋊C4—98no—
260Dic3⋊C4—92no—
261Dic3⋊C4—92no—
262Dic3⋊C4—92no—
263C4×Dic3—34no—
264C4×Dic3—98no—
265C4×Dic3—94no—
266C4×Dic3—94no—
267C4×Dic3—91no—
268C4×Dic3—91no—
269C4×Dic3—92no—
270C4×Dic3C2×C3⋊D311yescharacteristic
271C22×C32—22no—
272C22×C32—22no—
273C22×C32—22no—
274C22×C32—24no—
275C22×C32—28no—
276C22×C32—28no—
277C22×C32—28no—
278C22×C32—48no—
279C22×C32—48no—
280C22×C32—48no—
281C22×C32C23×D314yes—
282C22×C32C23×D311yescharacteristic
283C2×C3⋊D3—32no—
284C2×C3⋊D3—38no—
285C2×C3⋊D3—68no—
286C2×C3⋊D3—61no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—64no—
289C2×C3⋊D3—64no—
290C2×C3⋊D3—124no—
291C2×C3⋊D3—124no—
292C32×C4—28no—
293C32×C4—32no—
294C32×C4—61no—
295C32×C4—62no—
296C32×C4#23518yes—
297C32×C4#23712yes—
298C3⋊Dic3—22no—
299C3⋊Dic3—21no—
300C3⋊Dic3—21no—
301C3⋊Dic3—61no—
302C3⋊Dic3—62no—
303C3⋊Dic3—64no—
304C3⋊Dic3—68no—
305C3⋊Dic3—124no—
306C3⋊Dic3—128no—
307C3×Dic3—28no—
308C3×Dic3—24no—
309C3×Dic3—68no—
310C3×Dic3—64no—
311C3×Dic3—64no—
312C3×Dic3#23718yes—
313#126—272no—
314#125—271no—
315#124—91no—
316#123—272no—
317C22⋊Q8—271no—
318#121—271no—
319C4×Q8—271no—
320C4×D4—31no—
321C4×D4—272no—
322#117—272no—
323C2×C42—271no—
324C33#52911yescharacteristic, complemented, Sylow
325C23×C3—28no—
326C23×C3C2×D3218yes—
327C23×C3C22×C3⋊D312yes—
328C22×D3—94no—
329C22×D3—91no—
330C22×D3—184no—
331C3×Q8—182no—
332C3×D4—28no—
333C3×D4—22no—
334C3×D4—48no—
335C2×C3×C4—24no—
336C2×C3×C4—34no—
337C2×C3×C4—34no—
338C2×C3×C4—38no—
339C2×C3×C4—616no—
340C2×C3×C4—91no—
341C2×C3×C4—91no—
342C2×C3×C4—92no—
343C2×C3×C4C2×D3214yes—
344C2×C3×C4C22×C3⋊D311yescharacteristic
345#67—184no—
346#67—1816no—
347#67—3616no—
348C2×Dic3—24no—
349C2×Dic3—34no—
350C2×Dic3—34no—
351C2×Dic3—38no—
352C2×Dic3—94no—
353C2×Dic3—98no—
354C2×Dic3—91no—
355C2×Dic3—92no—
356C2×Dic3—188no—
357C2×Dic3—184no—
358C2×Dic3C22×C3⋊D311yescharacteristic
359C2×Dic3C22×C3⋊D312yes—
360C2×Dic3C22×C3⋊D311yescharacteristic
361C4×D3—182no—
362C4×D3—188no—
363C4×D3—188no—
364C4×D3—188no—
365C4×D3—182no—
366C4×D3—182no—
367C4×D3—368no—
368Dic6—68no—
369C2×C32—24no—
370C2×C32—24no—
371C2×C32—24no—
372C2×C32—42no—
373C2×C32—48no—
374C2×C32—88no—
375C2×C32C2×#23514yes—
376C2×C32C2×#23714yes—
377C2×C32C2×#23711yescharacteristic
378C2×C32D3×#4114yes—
379C2×C32D3×#4111yescharacteristic
380C2×C32D3×#4111yescharacteristic
381C3⋊D3—62no—
382C3⋊D3—68no—
383C3⋊D3—128no—
384C2×Q8—91no—
385C2×D4—272no—
386C2×D4D3×C3⋊D311yescharacteristic
387C22×C4—32no—
388C22×C4—271no—
389C22×C4—271no—
390C22×C4—271no—
391C4⋊C4—31no—
392C4⋊C4—272no—
393C4⋊C4—272no—
394C4⋊C4—272no—
395C4⋊C4—272no—
396C4⋊C4—271no—
397C22⋊C4—32no—
398C22⋊C4—92no—
399C22⋊C4—92no—
400C22⋊C4—272no—
401C22⋊C4—272no—
402C42—31no—
403C42—91no—
404C42—271no—
405C42—271no—
406C42—272no—
407C22×C3—28no—
408C22×C3—28no—
409C22×C3—28no—
410C22×C3—22no—
411C22×C3—22no—
412C22×C3—22no—
413C22×C3—24no—
414C22×C3—48no—
415C22×C3—48no—
416C22×C3—48no—
417C22×C3C22×D3214yes—
418C22×C3C23×C3⋊D311yescharacteristic
419C2×D3—98no—
420C2×D3—92no—
421C2×D3—188no—
422C2×D3—184no—
423C2×D3—184no—
424C2×D3—181no—
425C2×D3—181no—
426C2×D3—364no—
427C2×D3—364no—
428C3×C4—28no—
429C3×C4—38no—
430C3×C4—68no—
431C3×C4—64no—
432C3×C4—182no—
433C3×C4—181no—
434C3×C4—181no—
435C3×C4#380418yes—
436C3×C4#382812yes—
437Dic3—22no—
438Dic3—21no—
439Dic3—64no—
440Dic3—64no—
441Dic3—68no—
442Dic3—188no—
443Dic3—184no—
444Dic3—181no—
445Dic3—182no—
446Dic3—364no—
447Dic3—368no—
448Dic3#383012yes—
449C32—24no—
450C32#555714yescomplemented
451C32#568011yescharacteristic, complemented
452C23—271no—
453C23C2×D3×C3⋊D312yes—
454Q8—182no—
455D4—22no—
456D4—544no—
457C2×C4—31no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—64no—
461C2×C4—91no—
462C2×C4—91no—
463C2×C4—92no—
464C2×C4—271no—
465C2×C4—272no—
466C2×C4—542no—
467C2×C4—542no—
468C2×C4—542no—
469C2×C4C2×D3×C3⋊D311yescharacteristic
470C2×C3—24no—
471C2×C3—24no—
472C2×C3—24no—
473C2×C3—48no—
474C2×C3—42no—
475C2×C3—88no—
476C2×C3D3×#23714yes—
477C2×C3#6431214yes—
478C2×C3C2×#380414yes—
479C2×C3C2×#382811yescharacteristic
480C2×C3C2×#383011yescharacteristic
481C2×C3#41×C3⋊D311yescharacteristic
482D3—188no—
483D3—182no—
484D3—368no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—272no—
489C22—541no—
490C22—541no—
491C22C22×D3×C3⋊D311yescharacteristic
492C4—32no—
493C4—62no—
494C4—61no—
495C4—181no—
496C4—181no—
497C4—182no—
498C4—542no—
499C4—541no—
500C4#8903112yes—
501C3—24no—
502C3#1059519114yescomplemented
503C3#1059602811yescharacteristic, complemented
504C2—42no—
505C2—542no—
506C2D3×#383011yescharacteristic
507C2#1172458611yescharacteristic
508C2C2×#8903111yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC2×C33×D4C22
prime radical#11723979C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#529
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
2. C3×#64145C2semidirect
3. C32×D4×Dic3C2semidirect
4. #11723628C2semidirect
5. #11723618C2semidirect
6. #11723596C2semidirect
7. #11723594C2semidirect
8. C4×#6289C2semidirect
9. #11723449C2semidirect
10. #11723447C2semidirect
11. Dic3×#634C2semidirect
41. #64145D3semidirect
51. C3×D4×Dic3D3semidirect
155. D4×Dic3C3⋊D3semidirect
324. C33#529semidirect
450. C32#5557semidirect
451. C32#5680semidirect
502. C3#10595191semidirect
503. C3#10596028semidirect

Maximal subgroups

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

subgroupindex
#117239792
C3×#641452
C32×D4×Dic32
#117236282
#117236182
#117235962
#117235942
C4×#62892
#117234492
#117234472
Dic3×#6342
#105960283
#105951913