Finite Group Databasebeta

← #13177667

Subgroups — #13177667

Order 1728.

1744 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D4×C3⋊(C3⋊Dic3)C211yesmaximal, characteristic
2C3×D4×C3⋊Dic3C211yesmaximal, characteristic
3C32×D4×Dic3C211yesmaximal, characteristic
4#11723642C212yesmaximal, complemented
5#11723641C212yesmaximal, complemented
6#11723640C212yesmaximal, complemented
7C2×#88807C212yesmaximal, complemented
8#11723535C211yesmaximal, characteristic, complemented
9C4×#6293C211yesmaximal, characteristic, complemented
10#11723454C211yesmaximal, characteristic, complemented
11#11723450C211yesmaximal, characteristic, complemented
12#10596010—31nomaximal
13#10595272—34nomaximal
14C2×C33×D4C2211yescharacteristic
15C22×C3⋊(C3⋊Dic3)C2212yes—
16C22×C3×C3⋊Dic3C2212yes—
17C22×C32×Dic3C2212yes—
18C2×#6293C2211yescharacteristic
19C2×#6293C2212yes—
20#88864C2212yes—
21C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
22C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
23C3×#3757C2212yes—
24C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
25C3×C4×C3⋊Dic3C2211yescharacteristic
26C32×#217C2212yes—
27C32×C4⋊Dic3C2211yescharacteristic
28C32×C4×Dic3C2211yescharacteristic
29#88807C2211yescharacteristic
30#88807C2212yes—
31#88807C2212yes—
32#88807C2212yes—
33#88806C2211yescharacteristic
34#88806C2212yes—
35Dic3⋊(C3⋊Dic3)C2211yescharacteristic
36Dic3⋊(C3⋊Dic3)C2212yes—
37C32×C4×D4—31no—
38D4×C3⋊Dic3—31no—
39D4×C3⋊Dic3—34no—
40D4×C3⋊Dic3—64no—
41D4×C3⋊Dic3D311yescharacteristic
42#64133—32no—
43C2×#3750—32no—
44#64099—31no—
45#64097—31no—
46#64087—32no—
47#64087—32no—
48#64079—31no—
49C4×#634—31no—
50C3×D4×Dic3—34no—
51C3×D4×Dic3D314yes—
52#63982—38no—
53C2×Dic3⋊Dic3—38no—
54#63964—38no—
55#63964—38no—
56#63918—34no—
57C4×#627—34no—
58#63872—34no—
59#63869—34no—
60C23×C33C2312yes—
61C33×D4Q814yescomplemented
62C2×C33×C4C2311yescharacteristic
63C2×C3⋊(C3⋊Dic3)—42no—
64C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
65C2×C3⋊(C3⋊Dic3)C2311yescharacteristic
66C2×C3⋊(C3⋊Dic3)C2312yes—
67C2×C3×C3⋊Dic3—24no—
68C2×C3×C3⋊Dic3C2311yescharacteristic
69C2×C3×C3⋊Dic3C2312yes—
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C32×Dic3—42no—
72C2×C32×Dic3C2311yescharacteristic
73C2×C32×Dic3C2312yes—
74C2×C32×Dic3C2311yescharacteristic
75#6293—22no—
76#6293—42no—
77#5543—91no—
78#5539—94no—
79C2×C32×D4—24no—
80C2×C32×D4C2×D314yes—
81C2×C32×D4C2×D311yescharacteristic
82C22×C32×C4—32no—
83C22×C3⋊Dic3—32no—
84C22×C3⋊Dic3—38no—
85C22×C3⋊Dic3—68no—
86C22×C3⋊Dic3C2×D312yes—
87C2×#634—31no—
88C2×#634—32no—
89C22×C3×Dic3—38no—
90C22×C3×Dic3C2×D318yes—
91C2×#627—34no—
92C2×#627—38no—
93C32×C4⋊C4—31no—
94C32×C22⋊C4—32no—
95C32×C42—31no—
96#3757—32no—
97#3757—38no—
98#3757—68no—
99#3757C2×D312yes—
100C4⋊(C3⋊Dic3)—31no—
101C4⋊(C3⋊Dic3)—34no—
102C4⋊(C3⋊Dic3)—31no—
103C4⋊(C3⋊Dic3)—32no—
104C4⋊(C3⋊Dic3)—64no—
105C4⋊(C3⋊Dic3)C2×D311yescharacteristic
106#3750—31no—
107#3750—31no—
108#3750—32no—
109#3750—32no—
110#3750—32no—
111#3750—32no—
112C4×C3⋊Dic3—31no—
113C4×C3⋊Dic3—34no—
114C4×C3⋊Dic3—64no—
115C4×C3⋊Dic3C2×D311yescharacteristic
116C3×#217—38no—
117C3×#217C2×D318yes—
118C3×C4⋊Dic3—34no—
119C3×C4⋊Dic3C2×D314yes—
120C3×C4×Dic3—34no—
121C3×C4×Dic3C2×D314yes—
122#3724—38no—
123#3724—34no—
124Dic3⋊Dic3—38no—
125Dic3⋊Dic3—38no—
126Dic3⋊Dic3—38no—
127Dic3⋊Dic3—34no—
128Dic3⋊Dic3—34no—
129Dic3⋊Dic3—38no—
130C22×C33—22no—
131C22×C33—22no—
132C22×C33C2×Q814yes—
133C22×C33C2411yescharacteristic
134C33×C4C2×Q812yes—
135C3⋊(C3⋊Dic3)—21no—
136C3⋊(C3⋊Dic3)—21no—
137C3⋊(C3⋊Dic3)—22no—
138C3×C3⋊Dic3—22no—
139C3×C3⋊Dic3—21no—
140C3×C3⋊Dic3#4112yes—
141C32×Dic3—21no—
142C32×Dic3—22no—
143C32×Dic3—21no—
144C3×C4×D4—34no—
145C3×C4×D4—91no—
146D4×Dic3—34no—
147D4×Dic3—94no—
148D4×Dic3—91no—
149D4×Dic3—184no—
150D4×Dic3C3⋊D311yescharacteristic
151C2×C4⋊Dic3—92no—
152#924—98no—
153#924—92no—
154C2×Dic3⋊C4—98no—
155#890—94no—
156#890—91no—
157#890—91no—
158#888—94no—
159#878—98no—
160#878—98no—
161#878—92no—
162#878—92no—
163#870—94no—
164#869—91no—
165C4×Dic6—94no—
166C4×Dic6—91no—
167C23×C32—28no—
168C23×C32C22×D318yes—
169C23×C32C22×D312yes—
170C32×D4—216no—
171C32×D4Dic6116yescomplemented
172C32×D4Dic614yescomplemented
173C2×C32×C4—24no—
174C2×C32×C4—31no—
175C2×C32×C4—31no—
176C2×C32×C4—32no—
177C2×C32×C4—122no—
178C2×C32×C4C22×D314yes—
179C2×C32×C4C22×D311yescharacteristic
180C2×C3⋊Dic3—24no—
181C2×C3⋊Dic3—31no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—32no—
184C2×C3⋊Dic3—34no—
185C2×C3⋊Dic3—34no—
186C2×C3⋊Dic3—38no—
187C2×C3⋊Dic3—64no—
188C2×C3⋊Dic3—64no—
189C2×C3⋊Dic3—68no—
190C2×C3⋊Dic3—122no—
191C2×C3⋊Dic3—128no—
192C2×C3⋊Dic3—1216no—
193C2×C3⋊Dic3C22×D311yescharacteristic
194C2×C3⋊Dic3C22×D312yes—
195C2×C3⋊Dic3C22×D311yescharacteristic
196#634—62no—
197#634—122no—
198C2×C3×Dic3—34no—
199C2×C3×Dic3—38no—
200C2×C3×Dic3—34no—
201C2×C3×Dic3—48no—
202C2×C3×Dic3—616no—
203C2×C3×Dic3C22×D314yes—
204C2×C3×Dic3C22×D318yes—
205C2×C3×Dic3C22×D314yes—
206#627—68no—
207#627—128no—
208#554—271noSylow
209C2×C33—24no—
210C2×C33C22×Q811yescharacteristic
211C2×C33C2×#4111yescharacteristic
212C2×C33H(5,𝔽2)11yescharacteristic
213C2×C3×D4—24no—
214C2×C3×D4D3214yes—
215C2×C3×D4C2×C3⋊D311yescharacteristic
216C22×C3×C4—38no—
217C22×C3×C4—92no—
218C22×Dic3—38no—
219C22×Dic3—98no—
220C22×Dic3—92no—
221C22×Dic3—188no—
222C22×Dic3C2×C3⋊D312yes—
223C2×Dic6—94no—
224C2×Dic6—91no—
225C2×Dic6—98no—
226C2×Dic6—92no—
227C3×C4⋊C4—34no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—38no—
230C3×C22⋊C4—92no—
231C3×C42—34no—
232C3×C42—91no—
233#217—38no—
234#217—98no—
235#217—92no—
236#217—188no—
237#217C2×C3⋊D312yes—
238C4⋊Dic3—34no—
239C4⋊Dic3—94no—
240C4⋊Dic3—98no—
241C4⋊Dic3—94no—
242C4⋊Dic3—91no—
243C4⋊Dic3—91no—
244C4⋊Dic3—92no—
245C4⋊Dic3—92no—
246C4⋊Dic3—92no—
247C4⋊Dic3—184no—
248C4⋊Dic3C2×C3⋊D311yescharacteristic
249Dic3⋊C4—94no—
250Dic3⋊C4—94no—
251Dic3⋊C4—98no—
252Dic3⋊C4—98no—
253Dic3⋊C4—98no—
254Dic3⋊C4—98no—
255Dic3⋊C4—91no—
256Dic3⋊C4—91no—
257Dic3⋊C4—92no—
258Dic3⋊C4—92no—
259C4×Dic3—34no—
260C4×Dic3—94no—
261C4×Dic3—91no—
262C4×Dic3—184no—
263C4×Dic3C2×C3⋊D311yescharacteristic
264C22×C32—22no—
265C22×C32—22no—
266C22×C32—24no—
267C22×C32—216no—
268C22×C32—28no—
269C22×C32—28no—
270C22×C32—48no—
271C22×C32—48no—
272C22×C32C2×Dic6116yes—
273C22×C32C2×Dic614yes—
274C22×C32C23×D314yes—
275C22×C32C23×D311yescharacteristic
276C32×C4—28no—
277C32×C4—61no—
278C32×C4—62no—
279C32×C4—61no—
280C32×C4C2×Dic618yes—
281C32×C4C2×Dic612yes—
282C3⋊Dic3—22no—
283C3⋊Dic3—21no—
284C3⋊Dic3—68no—
285C3⋊Dic3—61no—
286C3⋊Dic3—64no—
287C3⋊Dic3—62no—
288C3⋊Dic3—61no—
289C3⋊Dic3—68no—
290C3⋊Dic3—64no—
291C3⋊Dic3—124no—
292C3⋊Dic3—128no—
293C3⋊Dic3#23712yes—
294C3×Dic3—28no—
295C3×Dic3—24no—
296C3×Dic3—24no—
297C3×Dic3—38no—
298C3×Dic3—64no—
299C3×Dic3—68no—
300C4⋊Q8—271no—
301#125—271no—
302#125—271no—
303C22⋊Q8—272no—
304C22⋊Q8—272no—
305C22⋊Q8—272no—
306C4×Q8—271no—
307C4×D4—31no—
308C4×D4—91no—
309C4×D4—271no—
310C2×C4⋊C4—272no—
311C33#55411yescharacteristic, complemented, Sylow
312C23×C3—28no—
313C23×C3C2×D3218yes—
314C23×C3C22×C3⋊D312yes—
315C3×D4—216no—
316C3×D4#627116yescomplemented
317C3×D4#63414yescomplemented
318C2×C3×C4—24no—
319C2×C3×C4—34no—
320C2×C3×C4—34no—
321C2×C3×C4—38no—
322C2×C3×C4—91no—
323C2×C3×C4—92no—
324C2×C3×C4—91no—
325C2×C3×C4—128no—
326C2×C3×C4—184no—
327C2×C3×C4C2×D3214yes—
328C2×C3×C4C22×C3⋊D311yescharacteristic
329C2×Dic3—34no—
330C2×Dic3—34no—
331C2×Dic3—38no—
332C2×Dic3—42no—
333C2×Dic3—616no—
334C2×Dic3—94no—
335C2×Dic3—94no—
336C2×Dic3—98no—
337C2×Dic3—91no—
338C2×Dic3—91no—
339C2×Dic3—92no—
340C2×Dic3—184no—
341C2×Dic3—184no—
342C2×Dic3—188no—
343C2×Dic3—368no—
344C2×Dic3—362no—
345C2×Dic3—3616no—
346C2×Dic3C22×C3⋊D311yescharacteristic
347C2×Dic3C22×C3⋊D311yescharacteristic
348C2×Dic3C22×C3⋊D312yes—
349Dic6—182no—
350Dic6—188no—
351Dic6—362no—
352Dic6—368no—
353C2×C32—24no—
354C2×C32—24no—
355C2×C32—24no—
356C2×C32—24no—
357C2×C32—216no—
358C2×C32—416no—
359C2×C32C22×Dic614yes—
360C2×C32C22×Dic611yescharacteristic
361C2×C32C2×#23711yescharacteristic
362C2×C32#100414yes—
363C2×C32D3×#4114yes—
364C2×C32#100911yescharacteristic
365C2×Q8—271no—
366C2×Q8—272no—
367C2×D4D3×C3⋊D311yescharacteristic
368C22×C4—32no—
369C22×C4—92no—
370C22×C4—272no—
371C4⋊C4—31no—
372C4⋊C4—91no—
373C4⋊C4—271no—
374C4⋊C4—272no—
375C4⋊C4—272no—
376C4⋊C4—272no—
377C4⋊C4—271no—
378C4⋊C4—271no—
379C4⋊C4—272no—
380C4⋊C4—272no—
381C4⋊C4—271no—
382C22⋊C4—32no—
383C22⋊C4—92no—
384C22⋊C4—272no—
385C42—31no—
386C42—91no—
387C42—271no—
388C22×C3—28no—
389C22×C3—28no—
390C22×C3—22no—
391C22×C3—22no—
392C22×C3—24no—
393C22×C3—216no—
394C22×C3—48no—
395C22×C3—48no—
396C22×C3C2×#627116yes—
397C22×C3C2×#63414yes—
398C22×C3C22×D3214yes—
399C22×C3C23×C3⋊D311yescharacteristic
400C3×C4—28no—
401C3×C4—64no—
402C3×C4—68no—
403C3×C4—64no—
404C3×C4—92no—
405C3×C4—181no—
406C3×C4—182no—
407C3×C4C2×#62718yes—
408C3×C4C2×#63412yes—
409Dic3—22no—
410Dic3—21no—
411Dic3—21no—
412Dic3—38no—
413Dic3—68no—
414Dic3—64no—
415Dic3—188no—
416Dic3—184no—
417Dic3—188no—
418Dic3—184no—
419Dic3—182no—
420Dic3—181no—
421Dic3—181no—
422Dic3—368no—
423Dic3—364no—
424C32—24no—
425C32#553914yescomplemented
426C32#554311yescharacteristic, complemented
427C23C2×D3×C3⋊D312yes—
428Q8—542no—
429Q8—1082no—
430D4#629314yescomplemented
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—32no—
434C2×C4—91no—
435C2×C4—91no—
436C2×C4—92no—
437C2×C4—122no—
438C2×C4—184no—
439C2×C4—271no—
440C2×C4—271no—
441C2×C4—272no—
442C2×C4—1082no—
443C2×C4C2×D3×C3⋊D311yescharacteristic
444C2×C3—24no—
445C2×C3—216no—
446C2×C3—24no—
447C2×C3—24no—
448C2×C3—24no—
449C2×C3—416no—
450C2×C3D3×#23714yes—
451C2×C3#6431414yes—
452C2×C3C22×#62714yes—
453C2×C3C22×#63411yescharacteristic
454C2×C3#6436111yescharacteristic
455C2×C3#41×C3⋊D311yescharacteristic
456C22—22no—
457C22—22no—
458C22C2×#629314yes—
459C22C22×D3×C3⋊D311yescharacteristic
460C4—61no—
461C4—61no—
462C4—62no—
463C4—92no—
464C4—182no—
465C4—181no—
466C4—541no—
467C4—541no—
468C4—542no—
469C4C2×#629312yes—
470C3—24no—
471C3#1059527214yescomplemented
472C3#1059601011yescharacteristic, complemented
473C2—24no—
474C2C3⋊D3×#23711yescharacteristic
475C2#1172458811yescharacteristic
476C2C22×#629311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×C3⋊D3
commutator, socle, cosocleC22×C33C24
Fitting, prime radicalC2×C33×D4C22
hypercenterC2×D4D3×C3⋊D3
hypocenterC33#554
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
4. #11723642C2semidirect
5. #11723641C2semidirect
6. #11723640C2semidirect
7. C2×#88807C2semidirect
8. #11723535C2semidirect
9. C4×#6293C2semidirect
10. #11723454C2semidirect
11. #11723450C2semidirect
61. C33×D4Q8semidirect
171. C32×D4Dic6semidirect
172. C32×D4Dic6semidirect
311. C33#554semidirect
316. C3×D4#627semidirect
317. C3×D4#634semidirect
425. C32#5539semidirect
426. C32#5543semidirect
430. D4#6293semidirect
471. C3#10595272semidirect
472. C3#10596010semidirect

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×D4×C3⋊Dic32
C32×D4×Dic32
#117236422
#117236412
#117236402
C2×#888072
#117235352
C4×#62932
#117234542
#117234502
#105960103
#105952723