Finite Group Databasebeta

← #13176454

Subgroups — #13176454

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×#64102C211yesmaximal, characteristic, complemented
3C32×#897C211yesmaximal, characteristic, complemented
4#11723528C212yesmaximal, complemented
5#11723518C212yesmaximal, complemented
6#11723515C211yesmaximal, characteristic
7#11723436C211yesmaximal, characteristic, complemented
8#11723431C212yesmaximal, complemented
9#11723423C211yesmaximal, characteristic, complemented
10#11723407C212yesmaximal, complemented
11Dic6×C3⋊Dic3C211yesmaximal, characteristic, complemented
12#10596114—31nomaximal
13#10594592—34nomaximal
14C2×#6308C2212yes—
15C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
16C2×C3×C4×C3⋊D3C2212yes—
17C2×C32×C4×D3C2211yescharacteristic
18C2×C32×Dic6C2211yescharacteristic
19C2×#6290C2212yes—
20C33×C4⋊C4C2211yescharacteristic
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22#88857C2212yes—
23C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
24C3×#3752C2212yes—
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×(C2×D3)⋊C4C2212yes—
27C32×C4⋊Dic3C2212yes—
28Dic3⋊(C3⋊Dic3)C2212yes—
29Dic3⋊(C3⋊Dic3)C2211yescharacteristic
30Dic3⋊(C3⋊Dic3)C2211yescharacteristic
31#88803C2212yes—
32#88803C2212yes—
33#88803C2212yes—
34(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
35(C2×D3)⋊(C3⋊Dic3)C2211yescharacteristic
36Dic3×C3⋊Dic3C2212yes—
37C32×C22⋊Q8—31no—
38#64156—31no—
39Q8×C3⋊Dic3—31no—
40#64139—31no—
41#64135—31no—
42#64107—32no—
43#64106—32no—
44#64102D311yescharacteristic, complemented
45#64097—31no—
46#64097—34no—
47#64097—64no—
48#64094—32no—
49#64092—32no—
50C3×#897D314yescomplemented
51C3×#893—34no—
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⋊D3C2312yes—
61C22×C32×D3C2311yescharacteristic
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⋊Dic3C2311yescharacteristic
70C3×C4⋊(C3⋊D3)—22no—
71C3×C4×C3⋊D3—42no—
72C2×C32×Dic3C2312yes—
73C2×C32×Dic3C2311yescharacteristic
74C32×C4×D3—22no—
75C32×Dic6—22no—
76#6290—42no—
77#5643—94no—
78#5561—91no—
79C2×C32×Q8—31no—
80C22×C32×C4—31no—
81C2×#638—32no—
82C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
83C2×C4×C3⋊D3C2×D312yes—
84C2×#634—32no—
85C2×#634—38no—
86C2×#634—68no—
87C2×C3×D12—34no—
88C2×C3×C4×D3—38no—
89C2×C3×C4×D3C2×D314yes—
90C2×C3×Dic6C2×D314yes—
91C2×#625—38no—
92C32×C4⋊C4—24no—
93C32×C4⋊C4—32no—
94C32×C4⋊C4C2×D314yes—
95C32×C4⋊C4C2×D311yescharacteristic
96C32×C22⋊C4—32no—
97#3757—31no—
98#3757—31no—
99#3752—32no—
100#3752—32no—
101#3752—32no—
102#3752C2×D312yes—
103C4⋊(C3⋊Dic3)—31no—
104C4⋊(C3⋊Dic3)—34no—
105C4⋊(C3⋊Dic3)—32no—
106C4⋊(C3⋊Dic3)—31no—
107C4⋊(C3⋊Dic3)—31no—
108C4⋊(C3⋊Dic3)—64no—
109#3750—32no—
110#3750—38no—
111#3750—68no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—32no—
115C4×C3⋊Dic3—64no—
116C4×C3⋊Dic3C2×D311yescharacteristic
117C3×(C2×D3)⋊C4—38no—
118C3×(C2×D3)⋊C4C2×D318yes—
119C3×C4⋊Dic3C2×D318yes—
120C3×C4×Dic3—34no—
121Dic3⋊Dic3—38no—
122Dic3⋊Dic3—34no—
123Dic3⋊Dic3—34no—
124(C2×D3)⋊Dic3—34no—
125(C2×D3)⋊Dic3—34no—
126(C2×D3)⋊Dic3—38no—
127(C2×D3)⋊Dic3—38no—
128(C2×D3)⋊Dic3—38no—
129Dic32—38no—
130C22×C33C2411yescharacteristic
131C2×C3×C3⋊D3—22no—
132C2×C3×C3⋊D3—22no—
133C2×C3×C3⋊D3—22no—
134C2×C32×D3—21no—
135C2×C32×D3—21no—
136C2×C32×D3—21no—
137C33×C4—22no—
138C33×C4#4112yes—
139C3⋊(C3⋊Dic3)—21no—
140C3⋊(C3⋊Dic3)—22no—
141C3⋊(C3⋊Dic3)—21no—
142C3×C3⋊Dic3—21no—
143C3×C3⋊Dic3—21no—
144C32×Dic3—22no—
145C32×Dic3—21no—
146C3×C22⋊Q8—34no—
147C3×C4×D4—91no—
148#947—94no—
149Q8×Dic3—94no—
150#938—91no—
151#936—91no—
152#930—94no—
153#929—92no—
154#926—94no—
155#924—92no—
156#898—98no—
157#897—98no—
158#897C3⋊D311yescharacteristic, complemented
159#893—34no—
160#888—94no—
161#888—91no—
162#888—184no—
163#885—98no—
164#885—92no—
165#883—98no—
166#879—92no—
167#872—91no—
168C4×Dic6—91no—
169C23×C32—31no—
170C22×C3⋊D3C22×D312yes—
171C22×C3×D3—38no—
172C22×C3×D3C22×D314yes—
173C32×Q8—62no—
174C2×C32×C4—24no—
175C2×C32×C4—28no—
176C2×C32×C4—32no—
177C2×C32×C4—31no—
178C2×C32×C4—62no—
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×D311yescharacteristic
194C2×C3⋊Dic3C22×D311yescharacteristic
195C4⋊(C3⋊D3)—22no—
196C4×C3⋊D3—42no—
197#634—122no—
198#634—128no—
199#634—1216no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—34no—
202C2×C3×Dic3C22×D318yes—
203C2×C3×Dic3C22×D314yes—
204C3×D12—68no—
205C3×C4×D3—28no—
206C3×C4×D3—128no—
207C3×Dic6—28no—
208#625—128no—
209#542—271noSylow
210C2×C33C2×#4111yescharacteristic
211C2×C33H(5,𝔽2)11yescharacteristic
212C2×C332−1+411yescharacteristic
213C3×C3⋊D3—42no—
214C32×D3—41no—
215C2×C3×Q8—34no—
216C2×C3×D4—91no—
217C22×C3×C4—34no—
218C22×C3×C4—92no—
219C2×#67—98no—
220C2×#67—92no—
221C2×D12—34no—
222C2×C4×D3—38no—
223C2×C4×D3C2×C3⋊D311yescharacteristic
224C2×Dic6—98no—
225C2×Dic6—92no—
226C2×Dic6—188no—
227C2×Dic6C2×C3⋊D311yescharacteristic
228C3×C4⋊C4—24no—
229C3×C4⋊C4—38no—
230C3×C4⋊C4D3214yes—
231C3×C4⋊C4C2×C3⋊D311yescharacteristic
232C3×C22⋊C4—38no—
233C3×C22⋊C4—92no—
234C3×C42—91no—
235#217—94no—
236#217—94no—
237#217—92no—
238#217—92no—
239#217—92no—
240(C2×D3)⋊C4—38no—
241(C2×D3)⋊C4—98no—
242(C2×D3)⋊C4—98no—
243(C2×D3)⋊C4—98no—
244(C2×D3)⋊C4—91no—
245(C2×D3)⋊C4—91no—
246(C2×D3)⋊C4C2×C3⋊D312yes—
247C4⋊Dic3—98no—
248C4⋊Dic3—94no—
249C4⋊Dic3—94no—
250C4⋊Dic3—94no—
251C4⋊Dic3—91no—
252C4⋊Dic3—184no—
253C4⋊Dic3C2×C3⋊D312yes—
254Dic3⋊C4—98no—
255Dic3⋊C4—92no—
256Dic3⋊C4—92no—
257Dic3⋊C4—91no—
258Dic3⋊C4—91no—
259Dic3⋊C4—188no—
260C4×Dic3—34no—
261C4×Dic3—98no—
262C4×Dic3—94no—
263C4×Dic3—91no—
264C4×Dic3—92no—
265C4×Dic3—184no—
266C22×C32—24no—
267C22×C32—61no—
268C22×C32—61no—
269C22×C32—61no—
270C22×C32C23×D314yes—
271C22×C32C23×D311yescharacteristic
272C2×C3⋊D3—22no—
273C2×C3⋊D3—22no—
274C2×C3⋊D3—22no—
275C2×C3×D3—24no—
276C2×C3×D3—24no—
277C2×C3×D3—24no—
278C2×C3×D3—68no—
279C2×C3×D3—68no—
280C2×C3×D3—68no—
281C32×C4—22no—
282C32×C4—28no—
283C32×C4—28no—
284C32×C4—48no—
285C32×C4—61no—
286C32×C4—62no—
287C32×C4#23512yes—
288C32×C4#23718yes—
289C3⋊Dic3—21no—
290C3⋊Dic3—21no—
291C3⋊Dic3—68no—
292C3⋊Dic3—68no—
293C3⋊Dic3—61no—
294C3⋊Dic3—62no—
295C3⋊Dic3—64no—
296C3⋊Dic3—64no—
297C3⋊Dic3—61no—
298C3⋊Dic3—68no—
299C3⋊Dic3—128no—
300C3×Dic3—28no—
301C3×Dic3—24no—
302C3×Dic3—64no—
303C3×Dic3—64no—
304C4⋊Q8—271no—
305#126—272no—
306#124—272no—
307#124—271no—
308#123—272no—
309C22⋊Q8—31no—
310C22⋊Q8—272no—
311#121—271no—
312C4×Q8—271no—
313C4×D4—91no—
314#117—271no—
315C33#54211yescharacteristic, complemented, Sylow
316C23×C3—34no—
317C23×C3—92no—
318C22×D3—38no—
319C22×D3C22×C3⋊D311yescharacteristic
320C3×Q8—68no—
321C3×D4—182no—
322C2×C3×C4—24no—
323C2×C3×C4—28no—
324C2×C3×C4—38no—
325C2×C3×C4—34no—
326C2×C3×C4—68no—
327C2×C3×C4—91no—
328C2×C3×C4—91no—
329C2×C3×C4—362no—
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—34no—
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⋊D312yes—
348C2×Dic3C22×C3⋊D311yescharacteristic
349D12—68no—
350C4×D3—22no—
351C4×D3—128no—
352Dic6—22no—
353Dic6—368no—
354Dic6—362no—
355Dic6—3616no—
356C2×C32—24no—
357C2×C32—24no—
358C2×C32—24no—
359C2×C32—121no—
360C2×C32C2×#23511yescharacteristic
361C2×C32C2×#23714yes—
362C2×C32#100411yescharacteristic
363C2×C32#100714yes—
364C2×C32#100914yes—
365C2×C32#101011yescharacteristic
366C3⋊D3—42no—
367C3×D3—44no—
368C3×D3—128no—
369C2×Q8—31no—
370C2×Q8—272no—
371C2×D4—91no—
372C2×D4—272no—
373C22×C4—31no—
374C22×C4—92no—
375C4⋊C4—32no—
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—272no—
385C22⋊C4—272no—
386C22⋊C4—272no—
387C22⋊C4—271no—
388C22⋊C4—271no—
389C42—91no—
390C42—272no—
391C42—271no—
392C22×C3—24no—
393C22×C3—64no—
394C22×C3—64no—
395C22×C3—64no—
396C22×C3—182no—
397C22×C3—182no—
398C22×C3—182no—
399C22×C3C22×D3214yes—
400C22×C3C23×C3⋊D311yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—21no—
404C2×D3—68no—
405C2×D3—68no—
406C2×D3—68no—
407C3×C4—28no—
408C3×C4—22no—
409C3×C4—28no—
410C3×C4—48no—
411C3×C4—64no—
412C3×C4—68no—
413C3×C4—181no—
414C3×C4—181no—
415C3×C4#379518yes—
416C3×C4#383012yes—
417Dic3—22no—
418Dic3—21no—
419Dic3—64no—
420Dic3—64no—
421Dic3—188no—
422Dic3—188no—
423Dic3—184no—
424Dic3—188no—
425Dic3—184no—
426Dic3—181no—
427Dic3—181no—
428Dic3—182no—
429Dic3—368no—
430C32—24no—
431C32#556111yescharacteristic, complemented
432C32#564314yescomplemented
433C23—31no—
434C23—92no—
435Q8—62no—
436Q8—1082no—
437D4—182no—
438D4—1082no—
439C2×C4—32no—
440C2×C4—31no—
441C2×C4—62no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—271no—
445C2×C4—271no—
446C2×C4—272no—
447C2×C4—362no—
448C2×C4C2×D3×C3⋊D312yes—
449C2×C4C2×D3×C3⋊D311yescharacteristic
450C2×C3—24no—
451C2×C3—24no—
452C2×C3—24no—
453C2×C3—124no—
454C2×C3—362no—
455C2×C3C2×#379514yes—
456C2×C3#6431414yes—
457C2×C3#6431514yes—
458C2×C3C2×#383011yescharacteristic
459C2×C3#6436411yescharacteristic
460C2×C3#6436611yescharacteristic
461D3—41no—
462D3—128no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—182no—
467C22—182no—
468C22—182no—
469C22C22×D3×C3⋊D311yescharacteristic
470C4—22no—
471C4—61no—
472C4—62no—
473C4—181no—
474C4—181no—
475C4—542no—
476C4—541no—
477C4—541no—
478C4#8901412yes—
479C3—24no—
480C3#1059459214yescomplemented
481C3#1059611411yescharacteristic, complemented
482C2—121no—
483C2—362no—
484C2C2×#8901411yescharacteristic
485C2#1172458911yescharacteristic
486C2#1172459211yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radical#11723515C2
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×#64102C2semidirect
3. C32×#897C2semidirect
4. #11723528C2semidirect
5. #11723518C2semidirect
7. #11723436C2semidirect
8. #11723431C2semidirect
9. #11723423C2semidirect
10. #11723407C2semidirect
11. Dic6×C3⋊Dic3C2semidirect
44. #64102D3semidirect
50. C3×#897D3semidirect
158. #897C3⋊D3semidirect
315. C33#542semidirect
431. C32#5561semidirect
432. C32#5643semidirect
480. C3#10594592semidirect
481. C3#10596114semidirect

Maximal subgroups

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

subgroupindex
#117239292
C3×#641022
C32×#8972
#117235282
#117235182
#117235152
#117234362
#117234312
#117234232
#117234072
Dic6×C3⋊Dic32
#105961143
#105945923