Finite Group Databasebeta

← C2×D3×#3718 #13170593

Subgroups — C2×D3×#3718 #13170593

Order 1728.

1250 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D3×Dic6C212yesmaximal
2C2×#88812C211yesmaximal, characteristic, complemented
3C2×#88798C212yesmaximal, complemented
4C2×D3×C3⋊(C3⋊C8)C211yesmaximal, characteristic
5C2×C3×#3718C211yesmaximal, characteristic, complemented
6D3×#3718C218yesmaximal, direct product factor
7C2×D3×#216—32nomaximal
8C22×#3718—31nomaximal
9C2×C3×#634C2212yes—
10C2×C32×C4×D3C2211yescharacteristic
11C2×C32×Dic6C2212yes—
12C2×C3×#627—22no—
13C2×C3×D3×Dic3—22no—
14C3×D3×Dic6—24no—
15C3×D3×Dic6C2218yes—
16C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18#88812C2214yescomplemented
19#88798C2218yescomplemented
20D3×C3⋊(C3⋊C8)C2214yes—
21C3×#3718C2214yescomplemented
22C2×C3×D3×Q8—32no—
23C22×C3×Dic6—32no—
24C2×D3×Dic6D312yes—
25C2×#3756—32no—
26C22×C3⋊(C3⋊C8)—31no—
27C2×C3×#216—32no—
28D3×#216—316no—
29C2×#3719—32no—
30C2×#3718—32no—
31C2×#3718—32no—
32C2×#3718—31no—
33C2×#3718—38no—
34C2×#3718—62no—
35C2×#3718D311yescharacteristic, direct product factor
36C2×D3×C3⋊C8—32no—
37C2×D3×C3⋊C8—61no—
38C22×C32×D3D411yescharacteristic
39C2×C33×C4C2311yescharacteristic
40C2×C3×C3⋊Dic3—22no—
41C3×#634—22no—
42C3×#634C2314yes—
43C2×C32×Dic3—22no—
44C2×C32×Dic3D411yescharacteristic
45C32×C4×D3D412yes—
46C32×C4×D3C2312yes—
47C32×Dic6—22no—
48C32×Dic6C2314yes—
49C3×#627—28no—
50C3×D3×Dic3—28no—
51C3⋊(C3⋊(C3⋊C8))C2312yes—
52C3×C3⋊(C3⋊C8)C2312yes—
53C22×#216—92no—
54C2×D3×Q16—91no—
55C2×C32×Q8—32no—
56C22×C32×C4—31no—
57C2×#634C2×D312yes—
58C22×C3×Dic3—62no—
59C3×D3×Q8—38no—
60C3×D3×Q8—64no—
61C2×C3×C4×D3—21no—
62C2×C3×C4×D3—62no—
63C2×C3×C4×D3C2×D312yes—
64C2×C3×Dic6—22no—
65C2×C3×Dic6—32no—
66C2×C3×Dic6—32no—
67C2×C3×Dic6—38no—
68C2×C3×Dic6—64no—
69C2×C3×Dic6—62no—
70C2×C3×Dic6—62no—
71C2×C3×Dic6—62no—
72C2×C3×Dic6C2×D312yes—
73C2×C3×Dic6C2×D312yes—
74C2×#627—22no—
75C2×D3×Dic3—22no—
76D3×Dic6—24no—
77D3×Dic6C2×D318yes—
78#3756—38no—
79C2×C3⋊(C3⋊C8)—32no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—34no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—62no—
84C2×C3⋊(C3⋊C8)—121no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×#216—38no—
87C2×C3×C3⋊C8—32no—
88C2×C3×C3⋊C8—61no—
89#3719—38no—
90#3718—38no—
91#3718—38no—
92#3718—34no—
93#3718—68no—
94#3718C2×D314yesdirect product factor
95D3×C3⋊C8—38no—
96D3×C3⋊C8—64no—
97C22×C33C2×D411yescharacteristic
98C2×C32×D3Q1614yescomplemented
99C2×C32×D3C2×D412yes—
100C33×C4C2×D411yescharacteristic
101C33×C4C2411yescharacteristic
102C3×C3⋊Dic3—24no—
103C32×Dic3—24no—
104C32×Dic3C2×D412yes—
105C22×C3×Q8—92no—
106C2×D3×Q8—32no—
107C22×Dic6—32no—
108C2×C3×Q16—91no—
109C2×#216—32no—
110C2×#216—92no—
111C2×#216—916no—
112C2×#216—92no—
113C2×#216—92no—
114C2×#216—92no—
115C2×#216—182no—
116C22×C3⋊C8—92no—
117C22×C3⋊C8—181no—
118D3×Q16—98no—
119C2×Dic12—91no—
120C2×D3×C8—91no—
121C23×C32—31no—
122C22×C3×D3—21no—
123C22×C3×D3#6712yes—
124C32×Q8—34no—
125C32×Q8—62no—
126C2×C32×C4—21no—
127C2×C32×C4—22no—
128C2×C32×C4—31no—
129C2×C32×C4—32no—
130C2×C32×C4—32no—
131C2×C32×C4—41no—
132C2×C32×C4—62no—
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D312yes—
135C2×C3⋊Dic3—22no—
136#634—22no—
137#634C22×D314yes—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—68no—
143C2×C3×Dic3—62no—
144C2×C3×Dic3—62no—
145C2×C3×Dic3—122no—
146C2×C3×Dic3#6712yes—
147C3×C4×D3—22no—
148C3×C4×D3—22no—
149C3×C4×D3—68no—
150C3×C4×D3#6714yes—
151C3×C4×D3C22×D314yes—
152C3×Dic6—22no—
153C3×Dic6—22no—
154C3×Dic6—24no—
155C3×Dic6—34no—
156C3×Dic6—34no—
157C3×Dic6—42no—
158C3×Dic6—64no—
159C3×Dic6—68no—
160C3×Dic6—62no—
161C3×Dic6—62no—
162C3×Dic6—68no—
163C3×Dic6—122no—
164C3×Dic6C22×D314yes—
165C3×Dic6C22×D314yes—
166#627—28no—
167D3×Dic3—28no—
168C3⋊(C3⋊C8)—34no—
169C3⋊(C3⋊C8)—32no—
170C3⋊(C3⋊C8)—62no—
171C3⋊(C3⋊C8)—64no—
172C3⋊(C3⋊C8)—122no—
173C3⋊(C3⋊C8)C22×D312yes—
174C3×C3⋊C8—34no—
175C3×C3⋊C8—62no—
176C22×Q16—271noSylow
177C2×C33C2×Q1612yescomplemented
178C2×C33C22×D411yescharacteristic
179C32×D3C2×Q1614yescomplemented
180C2×C3×Q8—32no—
181C2×C3×Q8—32no—
182C2×C3×Q8—62no—
183C2×C3×Q8—98no—
184C2×C3×Q8—92no—
185C2×C3×Q8—182no—
186C2×C3×Q8—184no—
187C22×C3×C4—32no—
188C22×C3×C4—61no—
189C22×C3×C4—182no—
190C22×Dic3—62no—
191D3×Q8—38no—
192D3×Q8—64no—
193C2×C4×D3—62no—
194C2×C4×D3D3211yescharacteristic
195C2×Dic6—32no—
196C2×Dic6—32no—
197C2×Dic6—38no—
198C2×Dic6—62no—
199C2×Dic6—64no—
200C2×Dic6—62no—
201C2×Dic6—62no—
202C2×Dic6D3212yes—
203C3×Q16—94no—
204C2×C3×C8—91no—
205#216—38no—
206#216—98no—
207#216—98no—
208#216—98no—
209#216—98no—
210#216—188no—
211C2×C3⋊C8—32no—
212C2×C3⋊C8—61no—
213C2×C3⋊C8—98no—
214C2×C3⋊C8—91no—
215C2×C3⋊C8—92no—
216C2×C3⋊C8—184no—
217C2×C3⋊C8—181no—
218C2×C3⋊C8—182no—
219C2×C3⋊C8—361no—
220Dic12—94no—
221D3×C8—94no—
222C22×C32—21no—
223C22×C32—22no—
224C22×C32—34no—
225C22×C32—32no—
226C22×C32—41no—
227C22×C32D3×D411yescharacteristic
228C22×C32C2×#6712yes—
229C2×C3×D3—24no—
230C2×C3×D3—22no—
231C2×C3×D3#21618yescomplemented
232C2×C3×D3C2×#6714yes—
233C32×C4—21no—
234C32×C4—22no—
235C32×C4—21no—
236C32×C4—22no—
237C32×C4—32no—
238C32×C4—41no—
239C32×C4—41no—
240C32×C4—64no—
241C32×C4D3×D411yescharacteristic
242C32×C4C2×#6712yes—
243C32×C4C23×D311yescharacteristic
244C32×C4C23×D312yes—
245C3⋊Dic3—24no—
246C3×Dic3—24no—
247C3×Dic3—24no—
248C3×Dic3—22no—
249C3×Dic3—44no—
250C3×Dic3—64no—
251C3×Dic3—64no—
252C3×Dic3—124no—
253C3×Dic3C2×#6714yes—
254C22×Q8—92no—
255C2×Q16—91no—
256C2×Q16—271no—
257C2×Q16—278no—
258C2×Q16—272no—
259C22×C8—271no—
260C33C22×Q1611yescharacteristic, complemented, Sylow
261C23×C3—32no—
262C23×C3—61no—
263C22×D3#62511yescharacteristic
264C3×Q8—34no—
265C3×Q8—34no—
266C3×Q8—64no—
267C3×Q8—62no—
268C3×Q8—62no—
269C3×Q8—94no—
270C3×Q8—122no—
271C3×Q8—188no—
272C3×Q8—182no—
273C2×C3×C4—21no—
274C2×C3×C4—22no—
275C2×C3×C4—32no—
276C2×C3×C4—34no—
277C2×C3×C4—34no—
278C2×C3×C4—41no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—61no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—122no—
285C2×C3×C4—188no—
286C2×C3×C4—182no—
287C2×C3×C4C2×D3211yescharacteristic
288C2×C3×C4C2×D3212yes—
289C2×Dic3—22no—
290C2×Dic3—68no—
291C2×Dic3—62no—
292C2×Dic3—62no—
293C2×Dic3—122no—
294C2×Dic3#62511yescharacteristic
295C4×D3—68no—
296C4×D3#62512yes—
297C4×D3C2×D3212yes—
298Dic6—22no—
299Dic6—34no—
300Dic6—34no—
301Dic6—64no—
302Dic6—68no—
303Dic6—62no—
304Dic6—62no—
305Dic6—68no—
306Dic6—122no—
307Dic6C2×D3214yes—
308C3×C8—92no—
309C3⋊C8—34no—
310C3⋊C8—62no—
311C3⋊C8—92no—
312C3⋊C8—94no—
313C3⋊C8—182no—
314C3⋊C8—184no—
315C3⋊C8—362no—
316C2×C32—21no—
317C2×C32—22no—
318C2×C32—22no—
319C2×C32—24no—
320C2×C32—34no—
321C2×C32—42no—
322C2×C32—41no—
323C2×C32D3×Q1612yescomplemented
324C2×C32C2×#21614yescomplemented
325C2×C32C2×D3×D411yescharacteristic
326C2×C32C22×#6712yes—
327C3×D3—24no—
328C3×D3C2×#21618yescomplemented
329C2×Q8—32no—
330C2×Q8—98no—
331C2×Q8—92no—
332C2×Q8—182no—
333C2×Q8—184no—
334C22×C4—31no—
335C22×C4—182no—
336Q16—94no—
337Q16—274no—
338Q16—278no—
339C2×C8—91no—
340C2×C8—274no—
341C2×C8—271no—
342C22×C3—22no—
343C22×C3—21no—
344C22×C3—34no—
345C22×C3—38no—
346C22×C3—41no—
347C22×C3—62no—
348C22×C3—64no—
349C22×C3C2×#62511yescharacteristic
350C22×C3D3×#6712yes—
351C2×D3#371814yesdirect product factor
352C2×D3C2×#62512yes—
353C3×C4—21no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—21no—
357C3×C4—34no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—62no—
361C3×C4—64no—
362C3×C4—64no—
363C3×C4—124no—
364C3×C4—184no—
365C3×C4C2×#62511yescharacteristic
366C3×C4D3×#6712yes—
367C3×C4C22×D3211yescharacteristic
368C3×C4C22×D3212yes—
369Dic3—24no—
370Dic3—64no—
371Dic3—64no—
372Dic3—124no—
373Dic3C2×#62512yes—
374C32—21no—
375C32—22no—
376C32—41no—
377C32C2×D3×Q1611yescharacteristic, complemented
378C32C22×#21612yescomplemented
379C23—31no—
380Q8—34no—
381Q8—62no—
382Q8—94no—
383Q8—188no—
384Q8—182no—
385C2×C4—31no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—62no—
389C2×C4—188no—
390C2×C4—182no—
391C2×C4D3311yescharacteristic
392C8—92no—
393C8—272no—
394C2×C3—22no—
395C2×C3—24no—
396C2×C3—22no—
397C2×C3—21no—
398C2×C3—38no—
399C2×C3—42no—
400C2×C3—41no—
401C2×C3—64no—
402C2×C3C2×#371812yescomplemented
403C2×C3D3×#21614yescomplemented
404C2×C3C22×#62511yescharacteristic
405C2×C3C2×D3×#6712yes—
406D3C2×#371814yesdirect product factor
407C22—34no—
408C22—32no—
409C22D3×#62511yescharacteristic
410C4—32no—
411C4—64no—
412C4—184no—
413C4D3×#62511yescharacteristic
414C4C2×D3311yescharacteristic
415C3—21no—
416C3—22no—
417C3—41no—
418C3C22×#371811yescharacteristic, complemented
419C3C2×D3×#21612yescomplemented
420C2—34no—
421C2D3×#371812yesdirect product factor
422C2C2×D3×#62511yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, cosocleC33×C4C24
FrattiniC4C2×D33
FittingC2×C33×C4C23
socleC22×C33C2×D4
prime radicalC22×C32×D3D4
hypercenterC2×C4D33
hypocenterC33C22×Q16
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88812C2semidirect
3. C2×#88798C2semidirect
5. C2×C3×#3718C2semidirect
6. D3×#3718C2direct (N × H)
18. #88812C22semidirect
19. #88798C22semidirect
21. C3×#3718C22semidirect
35. C2×#3718D3direct (N × H)
94. #3718C2×D3direct (N × H)
98. C2×C32×D3Q16semidirect
177. C2×C33C2×Q16semidirect
179. C32×D3C2×Q16semidirect
231. C2×C3×D3#216semidirect
260. C33C22×Q16semidirect
323. C2×C32D3×Q16semidirect
324. C2×C32C2×#216semidirect
328. C3×D3C2×#216semidirect
351. C2×D3#3718direct (N × H)
377. C32C2×D3×Q16semidirect
378. C32C22×#216semidirect
402. C2×C3C2×#3718semidirect
403. C2×C3D3×#216semidirect
406. D3C2×#3718direct (N × H)
418. C3C22×#3718semidirect
419. C3C2×D3×#216semidirect
421. C2D3×#3718direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D3×Dic62
C2×#888122
C2×#887982
C2×D3×C3⋊(C3⋊C8)2
C2×C3×#37182
D3×#37182
C2×D3×#2163
C22×#37183