Finite Group Databasebeta

← #13175380

Subgroups — #13175380

Order 1728.

1764 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C42×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C3×C4×#634C211yesmaximal, characteristic, complemented
3C32×C4×Dic6C211yesmaximal, characteristic, complemented
4#11723536C211yesmaximal, characteristic, complemented
5C4×#6293C212yesmaximal, complemented
6#11723456C211yesmaximal, characteristic
7#11723452C212yesmaximal
8#11723451C211yesmaximal, characteristic, complemented
9#11723449C211yesmaximal, characteristic
10#11723448C212yesmaximal, complemented
11#11723442C212yesmaximal
12#10596049—31nomaximal
13#10593882—34nomaximal
14C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
15C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×C3×#634C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6293C2212yes—
20C33×C42C2211yescharacteristic
21C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×Dic3⋊C4C2212yes—
29C32×C4×Dic3C2212yes—
30#88807C2212yes—
31#88806C2212yes—
32#88806C2212yes—
33#88806C2212yes—
34Dic3⋊(C3⋊Dic3)C2212yes—
35#88804C2212yes—
36#88804C2212yes—
37C32×C4×Q8—31no—
38#64155—31no—
39#64153—31no—
40#64106—31no—
41#64105—32no—
42#64098—32no—
43#64097—31no—
44#64081—32no—
45C42×C3⋊D3—31no—
46C42×C3⋊D3—34no—
47C42×C3⋊D3—64no—
48C4×#634—32no—
49C4×#634D311yescharacteristic, complemented
50C3×C4×Dic6—34no—
51C3×C4×Dic6D314yescomplemented
52#63919—34no—
53C4×#627—38no—
54#63874—38no—
55#63873—34no—
56#63871—34no—
57#63871—34no—
58#63870—38no—
59#63865—38no—
60C22×C3⋊(C3⋊D3)C2311yescharacteristic
61C2×C33×C4C2311yescharacteristic
62C2×C33×C4C2311yescharacteristic
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
67C4×C3⋊(C3⋊D3)—22no—
68C4×C3⋊(C3⋊D3)—22no—
69C4×C3⋊(C3⋊D3)Q814yescomplemented
70C2×C3×C3⋊Dic3C2312yes—
71C2×C3×C3⋊Dic3C2312yes—
72C3×#634—22no—
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2312yes—
75C32×Dic6—22no—
76#6293—24no—
77#5578—94no—
78#5578—91no—
79C2×C32×Q8—31no—
80C2×C4×C3⋊D3—31no—
81C2×C4×C3⋊D3—34no—
82C2×C4×C3⋊D3—31no—
83C2×C4×C3⋊D3—31no—
84C2×C4×C3⋊D3—34no—
85C2×C4×C3⋊D3—34no—
86C2×C4×C3⋊D3—64no—
87C2×C4×C3⋊D3—64no—
88C2×C4×C3⋊D3—64no—
89C2×#634—32no—
90C2×#634C2×D311yescharacteristic
91C2×C3×Dic6—34no—
92C2×C3×Dic6C2×D314yes—
93C2×#627—38no—
94C32×C4⋊C4—31no—
95C32×C4⋊C4—32no—
96C32×C42—24no—
97C32×C42—32no—
98C32×C42C2×D314yes—
99C32×C42C2×D311yescharacteristic
100#3752—32no—
101#3752—32no—
102C4⋊(C3⋊Dic3)—32no—
103C4⋊(C3⋊Dic3)C2×D311yescharacteristic
104#3750—32no—
105#3750—32no—
106#3750—32no—
107#3750—32no—
108#3750C2×D312yes—
109C4×C3⋊Dic3—31no—
110C4×C3⋊Dic3—31no—
111C4×C3⋊Dic3—31no—
112C4×C3⋊Dic3—34no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—34no—
115C4×C3⋊Dic3—64no—
116C4×C3⋊Dic3—64no—
117C4×C3⋊Dic3—64no—
118C4×C3⋊Dic3C2×D312yes—
119C3×C4⋊Dic3—34no—
120C3×C4⋊Dic3C2×D314yes—
121C3×Dic3⋊C4—38no—
122C3×Dic3⋊C4C2×D318yes—
123C3×C4×Dic3—38no—
124C3×C4×Dic3C2×D318yes—
125#3724—38no—
126#3724—38no—
127#3724—38no—
128Dic3⋊Dic3—38no—
129Dic3⋊Dic3—38no—
130#3722—38no—
131#3722—38no—
132C22×C33C2411yescharacteristic
133C2×C3⋊(C3⋊D3)—21no—
134C2×C3⋊(C3⋊D3)—21no—
135C2×C3⋊(C3⋊D3)C2×Q812yes—
136C33×C4—21no—
137C33×C4C2×Q812yes—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)C2×Q812yes—
141C3⋊(C3⋊Dic3)#4112yes—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—22no—
144C32×Dic3—22no—
145C32×Dic3—22no—
146C3×C4×Q8—34no—
147C3×C4×Q8—91no—
148#946—94no—
149#946—91no—
150#944—94no—
151#944—91no—
152#897—94no—
153#897—91no—
154#896—98no—
155#896—92no—
156#889—98no—
157#889—92no—
158#888—94no—
159#888—91no—
160#872—98no—
161#872—92no—
162C42×D3—94no—
163C42×D3—91no—
164C42×D3—184no—
165C4×Dic6—34no—
166C4×Dic6—98no—
167C4×Dic6—92no—
168C4×Dic6C3⋊D311yescharacteristic, complemented
169C22×C3⋊D3—31no—
170C22×C3⋊D3—34no—
171C22×C3⋊D3—64no—
172C32×Q8—62no—
173C2×C32×C4—24no—
174C2×C32×C4—24no—
175C2×C32×C4—24no—
176C2×C32×C4—32no—
177C2×C32×C4—32no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D314yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D311yescharacteristic
182C2×C32×C4C22×D311yescharacteristic
183C2×C32×C4C22×D311yescharacteristic
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—31no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—34no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—64no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D312yes—
195C4×C3⋊D3—34no—
196C4×C3⋊D3—316no—
197C4×C3⋊D3—616no—
198C4×C3⋊D3—62no—
199C4×C3⋊D3—68no—
200C4×C3⋊D3—616no—
201C4×C3⋊D3—616no—
202C4×C3⋊D3—68no—
203C4×C3⋊D3—62no—
204#634—22no—
205#634—64no—
206C2×C3×Dic3—38no—
207C2×C3×Dic3—38no—
208C2×C3×Dic3C22×D318yes—
209C2×C3×Dic3C22×D318yes—
210C3×Dic6—28no—
211C3×Dic6—68no—
212#627—616no—
213#533—271noSylow
214C2×C33C22×Q811yescharacteristic
215C2×C33C2×#4111yescharacteristic
216C2×C33C2×#4111yescharacteristic
217C3⋊(C3⋊D3)—22no—
218C2×C3×Q8—34no—
219C2×C3×Q8—91no—
220C2×C4×D3—94no—
221C2×C4×D3—91no—
222C2×C4×D3—94no—
223C2×C4×D3—94no—
224C2×C4×D3—91no—
225C2×C4×D3—91no—
226C2×C4×D3—184no—
227C2×C4×D3—184no—
228C2×C4×D3—184no—
229C2×Dic6—34no—
230C2×Dic6—98no—
231C2×Dic6—92no—
232C2×Dic6C2×C3⋊D311yescharacteristic
233C3×C4⋊C4—34no—
234C3×C4⋊C4—38no—
235C3×C4⋊C4—91no—
236C3×C4⋊C4—92no—
237C3×C42—24no—
238C3×C42—38no—
239C3×C42—92no—
240C3×C42D3214yes—
241C3×C42C2×C3⋊D311yescharacteristic
242(C2×D3)⋊C4—98no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246C4⋊Dic3—34no—
247C4⋊Dic3—98no—
248C4⋊Dic3—92no—
249C4⋊Dic3C2×C3⋊D311yescharacteristic
250Dic3⋊C4—38no—
251Dic3⋊C4—98no—
252Dic3⋊C4—98no—
253Dic3⋊C4—98no—
254Dic3⋊C4—98no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259Dic3⋊C4C2×C3⋊D312yes—
260C4×Dic3—38no—
261C4×Dic3—94no—
262C4×Dic3—94no—
263C4×Dic3—94no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—184no—
268C4×Dic3—184no—
269C4×Dic3—184no—
270C4×Dic3C2×C3⋊D312yes—
271C22×C32—24no—
272C22×C32C23×D314yes—
273C22×C32C23×D311yescharacteristic
274C2×C3⋊D3—32no—
275C2×C3⋊D3—38no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3—68no—
278C2×C3⋊D3—68no—
279C2×C3⋊D3—64no—
280C2×C3⋊D3—61no—
281C2×C3⋊D3—64no—
282C2×C3⋊D3—61no—
283C32×C4—28no—
284C32×C4—24no—
285C32×C4—21no—
286C32×C4—28no—
287C32×C4—28no—
288C32×C4—62no—
289C32×C4—62no—
290C32×C4#23518yes—
291C32×C4#23512yes—
292C32×C4D3×Q818yes—
293C32×C4D3×Q812yes—
294C3⋊Dic3—22no—
295C3⋊Dic3—22no—
296C3⋊Dic3—32no—
297C3⋊Dic3—32no—
298C3⋊Dic3—38no—
299C3⋊Dic3—38no—
300C3⋊Dic3—68no—
301C3⋊Dic3—68no—
302C3⋊Dic3—68no—
303C3⋊Dic3—64no—
304C3⋊Dic3—61no—
305C3×Dic3—28no—
306C3×Dic3—28no—
307C3×Dic3—68no—
308C3×Dic3—68no—
309C4⋊Q8—271no—
310C4⋊Q8—271no—
311#125—272no—
312C22⋊Q8—272no—
313C22⋊Q8—271no—
314C22⋊Q8—271no—
315C4×Q8—31no—
316C4×Q8—91no—
317C4×Q8—272no—
318#117—272no—
319C2×C42—271no—
320C33#53311yescharacteristic, complemented, Sylow
321C22×D3—94no—
322C22×D3—91no—
323C22×D3—184no—
324C3×Q8—68no—
325C3×Q8—182no—
326C2×C3×C4—24no—
327C2×C3×C4—24no—
328C2×C3×C4—24no—
329C2×C3×C4—38no—
330C2×C3×C4—38no—
331C2×C3×C4—92no—
332C2×C3×C4—92no—
333C2×C3×C4C2×D3214yes—
334C2×C3×C4C2×D3214yes—
335C2×C3×C4C2×D3214yes—
336C2×C3×C4C22×C3⋊D311yescharacteristic
337C2×C3×C4C22×C3⋊D311yescharacteristic
338C2×C3×C4C22×C3⋊D311yescharacteristic
339C2×Dic3—38no—
340C2×Dic3—38no—
341C2×Dic3—94no—
342C2×Dic3—94no—
343C2×Dic3—91no—
344C2×Dic3—91no—
345C2×Dic3—94no—
346C2×Dic3—91no—
347C2×Dic3—184no—
348C2×Dic3—184no—
349C2×Dic3—184no—
350C2×Dic3C22×C3⋊D312yes—
351C2×Dic3C22×C3⋊D312yes—
352C4×D3—916no—
353C4×D3—94no—
354C4×D3—1816no—
355C4×D3—188no—
356C4×D3—182no—
357C4×D3—1816no—
358C4×D3—1816no—
359C4×D3—188no—
360C4×D3—182no—
361Dic6—22no—
362Dic6—68no—
363Dic6—184no—
364Dic6—1816no—
365C2×C32—24no—
366C2×C32—24no—
367C2×C32—24no—
368C2×C32C2×#23514yes—
369C2×C32C2×#23511yescharacteristic
370C2×C32C2×D3×Q814yes—
371C2×C32C2×D3×Q811yescharacteristic
372C2×C32D3×#4114yes—
373C2×C32D3×#4111yescharacteristic
374C3⋊D3—616no—
375C3⋊D3—68no—
376C3⋊D3—62no—
377C2×Q8—31no—
378C2×Q8—91no—
379C2×Q8—272no—
380C22×C4—271no—
381C22×C4—271no—
382C22×C4—271no—
383C4⋊C4—32no—
384C4⋊C4—31no—
385C4⋊C4—92no—
386C4⋊C4—91no—
387C4⋊C4—272no—
388C4⋊C4—272no—
389C4⋊C4—272no—
390C4⋊C4—272no—
391C4⋊C4—272no—
392C22⋊C4—272no—
393C22⋊C4—272no—
394C42—32no—
395C42—92no—
396C42—271no—
397C42—271no—
398C42—271no—
399C42D3×C3⋊D311yescharacteristic
400C22×C3—24no—
401C22×C3C22×D3214yes—
402C22×C3C23×C3⋊D311yescharacteristic
403C2×D3—98no—
404C2×D3—92no—
405C2×D3—188no—
406C2×D3—188no—
407C2×D3—188no—
408C2×D3—184no—
409C2×D3—184no—
410C2×D3—181no—
411C2×D3—181no—
412C3×C4—28no—
413C3×C4—24no—
414C3×C4—21no—
415C3×C4—28no—
416C3×C4—28no—
417C3×C4—68no—
418C3×C4—68no—
419C3×C4—182no—
420C3×C4—182no—
421C3×C4#379718yes—
422C3×C4#379818yes—
423C3×C4#382812yes—
424C3×C4Q8×C3⋊D312yes—
425Dic3—22no—
426Dic3—22no—
427Dic3—68no—
428Dic3—68no—
429Dic3—98no—
430Dic3—98no—
431Dic3—92no—
432Dic3—92no—
433Dic3—188no—
434Dic3—188no—
435Dic3—188no—
436Dic3—184no—
437Dic3—181no—
438C32—24no—
439C32#557814yescomplemented
440C32#557811yescharacteristic, complemented
441C23—271no—
442Q8—62no—
443Q8—182no—
444Q8—544no—
445C2×C4—32no—
446C2×C4—32no—
447C2×C4—92no—
448C2×C4—92no—
449C2×C4—271no—
450C2×C4—271no—
451C2×C4—274no—
452C2×C4—271no—
453C2×C4—542no—
454C2×C4—542no—
455C2×C4C2×D3×C3⋊D311yescharacteristic
456C2×C4C2×D3×C3⋊D311yescharacteristic
457C2×C4C2×D3×C3⋊D311yescharacteristic
458C2×C3—24no—
459C2×C3—24no—
460C2×C3—24no—
461C2×C3C2×#379714yes—
462C2×C3C2×#379814yes—
463C2×C3#6430614yes—
464C2×C3C2×#382811yescharacteristic
465C2×C3C2×Q8×C3⋊D311yescharacteristic
466C2×C3#41×C3⋊D311yescharacteristic
467D3—1816no—
468D3—188no—
469D3—182no—
470C22—272no—
471C22—541no—
472C22—541no—
473C22C22×D3×C3⋊D311yescharacteristic
474C4—21no—
475C4—62no—
476C4—62no—
477C4—182no—
478C4—182no—
479C4—272no—
480C4—272no—
481C4—541no—
482C4#8901912yes—
483C4#8902012yes—
484C3—24no—
485C3#1059388214yescomplemented
486C3#1059604911yescharacteristic, complemented
487C2—542no—
488C2C2×#8901911yescharacteristic
489C2C2×#8902011yescharacteristic
490C2#1172457811yescharacteristic
subgroupG/H
centerC2×C4C2×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC33×C42C22
prime radicalC22×C3⋊(C3⋊D3)C23
hypercenterC42D3×C3⋊D3
hypocenterC33#533
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
2. C3×C4×#634C2semidirect
3. C32×C4×Dic6C2semidirect
4. #11723536C2semidirect
5. C4×#6293C2semidirect
8. #11723451C2semidirect
10. #11723448C2semidirect
49. C4×#634D3semidirect
51. C3×C4×Dic6D3semidirect
69. C4×C3⋊(C3⋊D3)Q8semidirect
168. C4×Dic6C3⋊D3semidirect
320. C33#533semidirect
439. C32#5578semidirect
440. C32#5578semidirect
485. C3#10593882semidirect
486. C3#10596049semidirect

Maximal subgroups

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

subgroupindex
C42×C3⋊(C3⋊D3)2
C3×C4×#6342
C32×C4×Dic62
#117235362
C4×#62932
#117234562
#117234522
#117234512
#117234492
#117234482
#117234422
#105960493
#105938823