Finite Group Databasebeta

← #13176464

Subgroups — #13176464

Order 1728.

1568 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11723929C211yesmaximal, characteristic, complemented
2C3×#64106C211yesmaximal, characteristic, complemented
3C32×#893C211yesmaximal, characteristic, complemented
4#11723528C212yesmaximal, complemented
5#11723522C211yesmaximal, characteristic
6#11723513C212yesmaximal, complemented
7#11723437C211yesmaximal, characteristic, complemented
8#11723434C211yesmaximal, characteristic, complemented
9#11723427C212yesmaximal, complemented
10#11723426C212yesmaximal, complemented
11Dic3×#634C211yesmaximal, characteristic, complemented
12#10596032—31nomaximal
13#10594592—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4×C3⋊D3C2211yescharacteristic
16C2×C3×#634C2211yescharacteristic
17C2×C32×D12C2211yescharacteristic
18C2×C32×C4×D3C2212yes—
19C2×#6290C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3752C2212yes—
25C3×C4⋊(C3⋊Dic3)C2212yes—
26C32×(C2×D3)⋊C4C2212yes—
27C32×C4×Dic3C2211yescharacteristic
28#88807C2212yes—
29#88807C2211yescharacteristic
30#88807C2211yescharacteristic
31#88803C2211yescharacteristic
32#88803C2211yescharacteristic
33(C2×D3)⋊(C3⋊Dic3)C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2212yes—
35(C2×D3)⋊(C3⋊Dic3)C2212yes—
36Dic3×C3⋊Dic3C2212yes—
37C32×C4×D4—31no—
38#64147—31no—
39#64145—31no—
40#64138—32no—
41#64133—32no—
42#64106D311yescharacteristic, complemented
43#64097—31no—
44#64097—34no—
45#64097—64no—
46#64094—32no—
47#64088—32no—
48#64081—31no—
49C4×#634—31no—
50C3×#897—34no—
51C3×#893D314yescomplemented
52#63913—38no—
53#63908—34no—
54#63905—38no—
55#63863—34no—
56#63858—38no—
57#63857—38no—
58#63855—34no—
59Dic3×Dic6—34no—
60C22×C3×C3⋊D3C2311yescharacteristic
61C22×C32×D3C2312yes—
62C2×C33×C4C2312yes—
63C2×C33×C4C2311yescharacteristic
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67#6308—42no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C3×C4×C3⋊D3—22no—
71C3×#634—22no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2311yescharacteristic
74C32×D12—22no—
75C32×C4×D3—42no—
76#6290—42no—
77#5643—91no—
78#5561—94no—
79C2×C32×D4—31no—
80C22×C32×C4—32no—
81C2×#638—32no—
82C2×C4×C3⋊D3C2×D311yescharacteristic
83C2×#634—32no—
84C2×#634—38no—
85C2×#634—68no—
86C2×#634C2×D311yescharacteristic
87C2×C3×D12C2×D314yes—
88C2×C3×C4×D3—34no—
89C2×C3×C4×D3C2×D318yes—
90C2×C3×Dic6—34no—
91C2×#625—38no—
92C32×C4⋊C4—24no—
93C32×C4⋊C4C2×D314yes—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C22⋊C4—32no—
96C32×C42—31no—
97#3757—32no—
98#3757—32no—
99#3757—32no—
100#3752—31no—
101#3752—31no—
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—34no—
105C4⋊(C3⋊Dic3)—64no—
106C4⋊(C3⋊Dic3)C2×D312yes—
107#3750—32no—
108#3750—38no—
109#3750—32no—
110#3750—31no—
111#3750—31no—
112#3750—68no—
113C4×C3⋊Dic3—31no—
114C4×C3⋊Dic3—34no—
115C4×C3⋊Dic3—32no—
116C4×C3⋊Dic3—64no—
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4C2×D318yes—
119C3×C4⋊Dic3—38no—
120C3×C4×Dic3C2×D314yes—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—34no—
124(C2×D3)⋊Dic3—38no—
125(C2×D3)⋊Dic3—38no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—34no—
128(C2×D3)⋊Dic3—34no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—21no—
132C2×C3×C3⋊D3—21no—
133C2×C3×C3⋊D3—21no—
134C2×C32×D3—22no—
135C2×C32×D3—22no—
136C2×C32×D3—22no—
137C33×C4—22no—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)—22no—
141C3⋊(C3⋊Dic3)—21no—
142C3×C3⋊Dic3—22no—
143C3×C3⋊Dic3—21no—
144C32×Dic3—21no—
145C32×Dic3—21no—
146C3×C22⋊Q8—91no—
147C3×C4×D4—34no—
148#947—91no—
149Q8×Dic3—91no—
150#938—94no—
151#936—94no—
152#930—91no—
153#929—98no—
154#926—91no—
155#924—98no—
156#898—92no—
157#897—34no—
158#897—92no—
159#893C3⋊D311yescharacteristic, complemented
160#888—94no—
161#888—91no—
162#888—184no—
163#885—98no—
164#885—92no—
165#883—92no—
166#879—98no—
167#872—94no—
168C4×Dic6—94no—
169C23×C32—32no—
170C22×C3⋊D3C22×D311yescharacteristic
171C22×C3×D3—34no—
172C22×C3×D3C22×D318yes—
173C32×D4—62no—
174C2×C32×C4—24no—
175C2×C32×C4—28no—
176C2×C32×C4—31no—
177C2×C32×C4—31no—
178C2×C32×C4—122no—
179C2×C32×C4C22×D318yes—
180C2×C32×C4C22×D314yes—
181C2×C32×C4C22×D312yes—
182C2×C32×C4C22×D311yescharacteristic
183#638—122no—
184C2×C3⋊Dic3—31no—
185C2×C3⋊Dic3—32no—
186C2×C3⋊Dic3—31no—
187C2×C3⋊Dic3—38no—
188C2×C3⋊Dic3—34no—
189C2×C3⋊Dic3—34no—
190C2×C3⋊Dic3—64no—
191C2×C3⋊Dic3—64no—
192C2×C3⋊Dic3—68no—
193C2×C3⋊Dic3C22×D312yes—
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4×C3⋊D3—22no—
196#634—22no—
197#634—122no—
198#634—128no—
199#634—1216no—
200C2×C3×Dic3—38no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3C22×D314yes—
203C2×C3×Dic3C22×D314yes—
204C3×D12—28no—
205C3×C4×D3—48no—
206C3×C4×D3—68no—
207C3×Dic6—68no—
208#625—128no—
209#542—271noSylow
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C2×C332−1+411yescharacteristic
213C3×C3⋊D3—41no—
214C32×D3—42no—
215C2×C3×Q8—91no—
216C2×C3×D4—34no—
217C22×C3×C4—38no—
218C22×C3×C4—91no—
219C2×#67—98no—
220C2×#67—92no—
221C2×D12C2×C3⋊D311yescharacteristic
222C2×C4×D3—34no—
223C2×C4×D3C2×C3⋊D312yes—
224C2×Dic6—34no—
225C2×Dic6—98no—
226C2×Dic6—92no—
227C2×Dic6—188no—
228C3×C4⋊C4—24no—
229C3×C4⋊C4—92no—
230C3×C4⋊C4D3214yes—
231C3×C4⋊C4C2×C3⋊D311yescharacteristic
232C3×C22⋊C4—38no—
233C3×C22⋊C4—92no—
234C3×C42—34no—
235#217—98no—
236#217—98no—
237#217—98no—
238#217—91no—
239#217—91no—
240(C2×D3)⋊C4—38no—
241(C2×D3)⋊C4—94no—
242(C2×D3)⋊C4—94no—
243(C2×D3)⋊C4—92no—
244(C2×D3)⋊C4—92no—
245(C2×D3)⋊C4—92no—
246(C2×D3)⋊C4C2×C3⋊D312yes—
247C4⋊Dic3—38no—
248C4⋊Dic3—94no—
249C4⋊Dic3—91no—
250C4⋊Dic3—92no—
251C4⋊Dic3—91no—
252C4⋊Dic3—91no—
253C4⋊Dic3—184no—
254Dic3⋊C4—98no—
255Dic3⋊C4—94no—
256Dic3⋊C4—94no—
257Dic3⋊C4—98no—
258Dic3⋊C4—92no—
259Dic3⋊C4—188no—
260C4×Dic3—98no—
261C4×Dic3—94no—
262C4×Dic3—91no—
263C4×Dic3—92no—
264C4×Dic3—184no—
265C4×Dic3C2×C3⋊D311yescharacteristic
266C22×C32—24no—
267C22×C32—62no—
268C22×C32—62no—
269C22×C32—62no—
270C22×C32C23×D314yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—21no—
273C2×C3⋊D3—21no—
274C2×C3⋊D3—21no—
275C2×C3×D3—28no—
276C2×C3×D3—28no—
277C2×C3×D3—28no—
278C2×C3×D3—64no—
279C2×C3×D3—64no—
280C2×C3×D3—64no—
281C32×C4—22no—
282C32×C4—28no—
283C32×C4—28no—
284C32×C4—48no—
285C32×C4—61no—
286C32×C4—61no—
287C32×C4#23518yes—
288C32×C4#23712yes—
289C3⋊Dic3—22no—
290C3⋊Dic3—21no—
291C3⋊Dic3—68no—
292C3⋊Dic3—68no—
293C3⋊Dic3—64no—
294C3⋊Dic3—61no—
295C3⋊Dic3—61no—
296C3⋊Dic3—62no—
297C3⋊Dic3—64no—
298C3⋊Dic3—68no—
299C3⋊Dic3—128no—
300C3×Dic3—24no—
301C3×Dic3—24no—
302C3×Dic3—68no—
303C3×Dic3—64no—
304C4⋊Q8—271no—
305#126—272no—
306#124—272no—
307#124—271no—
308#123—272no—
309C22⋊Q8—91no—
310C22⋊Q8—272no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—31no—
314#117—271no—
315C33#54211yescharacteristic, complemented, Sylow
316C23×C3—38no—
317C23×C3—91no—
318C22×D3—34no—
319C22×D3C22×C3⋊D312yes—
320C3×Q8—182no—
321C3×D4—68no—
322C2×C3×C4—24no—
323C2×C3×C4—28no—
324C2×C3×C4—34no—
325C2×C3×C4—34no—
326C2×C3×C4—92no—
327C2×C3×C4—91no—
328C2×C3×C4—128no—
329C2×C3×C4—182no—
330C2×C3×C4C2×D3218yes—
331C2×C3×C4C2×D3214yes—
332C2×C3×C4C22×C3⋊D312yes—
333C2×C3×C4C22×C3⋊D311yescharacteristic
334#67—362no—
335#67—368no—
336C2×Dic3—38no—
337C2×Dic3—34no—
338C2×Dic3—94no—
339C2×Dic3—98no—
340C2×Dic3—94no—
341C2×Dic3—91no—
342C2×Dic3—92no—
343C2×Dic3—91no—
344C2×Dic3—184no—
345C2×Dic3—184no—
346C2×Dic3—188no—
347C2×Dic3C22×C3⋊D311yescharacteristic
348C2×Dic3C22×C3⋊D311yescharacteristic
349D12—22no—
350C4×D3—42no—
351C4×D3—68no—
352Dic6—68no—
353Dic6—368no—
354Dic6—362no—
355Dic6—3616no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—122no—
360C2×C32C2×#23514yes—
361C2×C32C2×#23711yescharacteristic
362C2×C32#100414yes—
363C2×C32#100711yescharacteristic
364C2×C32#100911yescharacteristic
365C2×C32#101014yes—
366C3⋊D3—41no—
367C3×D3—48no—
368C3×D3—124no—
369C2×Q8—91no—
370C2×Q8—272no—
371C2×D4—31no—
372C2×D4—272no—
373C22×C4—32no—
374C22×C4—91no—
375C4⋊C4—92no—
376C4⋊C4—271no—
377C4⋊C4—271no—
378C4⋊C4—272no—
379C4⋊C4—271no—
380C4⋊C4—272no—
381C4⋊C4D3×C3⋊D311yescharacteristic
382C22⋊C4—32no—
383C22⋊C4—92no—
384C22⋊C4—271no—
385C22⋊C4—271no—
386C22⋊C4—272no—
387C22⋊C4—272no—
388C22⋊C4—272no—
389C42—31no—
390C42—272no—
391C42—271no—
392C22×C3—24no—
393C22×C3—68no—
394C22×C3—68no—
395C22×C3—68no—
396C22×C3—181no—
397C22×C3—181no—
398C22×C3—181no—
399C22×C3C22×D3214yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—22no—
402C2×D3—22no—
403C2×D3—22no—
404C2×D3—64no—
405C2×D3—64no—
406C2×D3—64no—
407C3×C4—28no—
408C3×C4—22no—
409C3×C4—28no—
410C3×C4—48no—
411C3×C4—64no—
412C3×C4—64no—
413C3×C4—181no—
414C3×C4—182no—
415C3×C4#379518yes—
416C3×C4#382812yes—
417Dic3—21no—
418Dic3—21no—
419Dic3—68no—
420Dic3—64no—
421Dic3—188no—
422Dic3—188no—
423Dic3—184no—
424Dic3—188no—
425Dic3—184no—
426Dic3—181no—
427Dic3—182no—
428Dic3—181no—
429Dic3—368no—
430C32—24no—
431C32#556114yescomplemented
432C32#564311yescharacteristic, complemented
433C23—32no—
434C23—91no—
435Q8—182no—
436Q8—1082no—
437D4—62no—
438D4—1082no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—92no—
442C2×C4—91no—
443C2×C4—122no—
444C2×C4—182no—
445C2×C4—271no—
446C2×C4—271no—
447C2×C4—272no—
448C2×C4C2×D3×C3⋊D312yes—
449C2×C4C2×D3×C3⋊D311yescharacteristic
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—128no—
454C2×C3—361no—
455C2×C3C2×#379514yes—
456C2×C3#6431414yes—
457C2×C3#6431514yes—
458C2×C3C2×#382811yescharacteristic
459C2×C3#6436111yescharacteristic
460C2×C3#6436711yescharacteristic
461D3—42no—
462D3—124no—
463C22—62no—
464C22—62no—
465C22—62no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—22no—
471C4—61no—
472C4—61no—
473C4—181no—
474C4—182no—
475C4—542no—
476C4—541no—
477C4—541no—
478C4#8902112yes—
479C3—24no—
480C3#1059459214yescomplemented
481C3#1059603211yescharacteristic, complemented
482C2—122no—
483C2—361no—
484C2C2×#8902111yescharacteristic
485C2#1172458811yescharacteristic
486C2#1172459011yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radical#11723522C2
hypercenterC4⋊C4D3×C3⋊D3
hypocenterC33#542
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. #11723929C2semidirect
2. C3×#64106C2semidirect
3. C32×#893C2semidirect
4. #11723528C2semidirect
6. #11723513C2semidirect
7. #11723437C2semidirect
8. #11723434C2semidirect
9. #11723427C2semidirect
10. #11723426C2semidirect
11. Dic3×#634C2semidirect
42. #64106D3semidirect
51. C3×#893D3semidirect
159. #893C3⋊D3semidirect
315. C33#542semidirect
431. C32#5561semidirect
432. C32#5643semidirect
480. C3#10594592semidirect
481. C3#10596032semidirect

Maximal subgroups

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

subgroupindex
#117239292
C3×#641062
C32×#8932
#117235282
#117235222
#117235132
#117234372
#117234342
#117234272
#117234262
Dic3×#6342
#105960323
#105945923