Finite Group Databasebeta

← C2×#11723442 #13177267

Subgroups — C2×#11723442 #13177267

Order 1728.

2876 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C22×#6293C211yesmaximal, characteristic, complemented
3C2×C3×#3750C211yesmaximal, characteristic, complemented
4C2×C32×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#88806C211yesmaximal, characteristic, complemented
6C2×#88804C211yesmaximal, characteristic
7C2×#88804C211yesmaximal, characteristic
8#11723442C218yesmaximal, direct product factor
9C2×#64105—31nomaximal
10C2×#63865—34nomaximal
11C23×C3⋊(C3⋊D3)C2211yescharacteristic
12C22×C33×C4C2211yescharacteristic
13C22×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C2×C4×C3⋊(C3⋊D3)—24no—
15C2×C4×C3⋊(C3⋊D3)C2214yes—
16C22×C3×C3⋊Dic3C2211yescharacteristic
17C22×C3×C3⋊Dic3C2211yescharacteristic
18C22×C32×Dic3C2211yescharacteristic
19C22×C32×Dic3C2211yescharacteristic
20C2×#6293—24no—
21C2×#6293C2214yescomplemented
22C3×#3750C2214yescomplemented
23C32×Dic3⋊C4C2214yescomplemented
24#88806C2214yescomplemented
25#88804C2214yes—
26#88804C2214yes—
27C22×C4×C3⋊D3—31no—
28C22×C4×C3⋊D3—34no—
29C22×C4×C3⋊D3—64no—
30C22×#634—31no—
31C22×#627—34no—
32C2×C32×C4⋊C4—31no—
33C2×#3752—31no—
34C2×#3752—31no—
35C2×#3750—31no—
36C2×#3750D311yescharacteristic, complemented
37#64105—38no—
38C2×C3×Dic3⋊C4—34no—
39C2×C3×Dic3⋊C4D314yescomplemented
40C2×#3724—34no—
41C2×#3722—34no—
42C2×#3722—34no—
43#63865—332no—
44C23×C33C2311yescharacteristic
45C22×C3⋊(C3⋊D3)—22no—
46C22×C3⋊(C3⋊D3)—22no—
47C22×C3⋊(C3⋊D3)Q814yescomplemented
48C22×C3⋊(C3⋊D3)C2312yes—
49C2×C33×C4—22no—
50C2×C33×C4C2312yes—
51C2×C3⋊(C3⋊Dic3)D414yes—
52C2×C3⋊(C3⋊Dic3)C2312yes—
53C4×C3⋊(C3⋊D3)—28no—
54C2×C3×C3⋊Dic3—22no—
55C2×C3×C3⋊Dic3—22no—
56C2×C3×C3⋊Dic3C2312yes—
57C2×C3×C3⋊Dic3C2312yes—
58C2×C32×Dic3—22no—
59C2×C32×Dic3—22no—
60C2×C32×Dic3C2312yes—
61C2×C32×Dic3C2312yes—
62#6293—28no—
63C2×#896—94no—
64C2×#896—91no—
65C23×C3⋊D3—31no—
66C23×C3⋊D3—34no—
67C23×C3⋊D3—64no—
68C22×C32×C4—24no—
69C22×C32×C4—31no—
70C22×C32×C4—31no—
71C22×C32×C4C2×D314yes—
72C22×C32×C4C2×D311yescharacteristic
73C22×C3⋊Dic3—31no—
74C22×C3⋊Dic3—34no—
75C22×C3⋊Dic3—64no—
76C22×C3⋊Dic3C2×D311yescharacteristic
77C22×C3⋊Dic3C2×D311yescharacteristic
78C2×C4×C3⋊D3—34no—
79C2×C4×C3⋊D3—316no—
80C2×C4×C3⋊D3—64no—
81C2×C4×C3⋊D3—616no—
82C2×C4×C3⋊D3—616no—
83C2×C4×C3⋊D3—632no—
84C2×#634—34no—
85C2×#634—64no—
86C22×C3×Dic3—34no—
87C22×C3×Dic3—34no—
88C22×C3×Dic3C2×D314yes—
89C22×C3×Dic3C2×D314yes—
90C2×#627—316no—
91C2×#627—616no—
92C32×C4⋊C4—34no—
93#3752—34no—
94#3752—34no—
95#3750—34no—
96#3750C2×D314yescomplemented
97C3×Dic3⋊C4—316no—
98C3×Dic3⋊C4C2×D3116yescomplemented
99#3724—316no—
100#3722—316no—
101#3722—316no—
102C22×C33C2×D412yes—
103C22×C33C2×Q812yes—
104C22×C33#4112yes—
105C22×C33C2411yescharacteristic
106C2×C3⋊(C3⋊D3)—28no—
107C2×C3⋊(C3⋊D3)—22no—
108C2×C3⋊(C3⋊D3)—22no—
109C2×C3⋊(C3⋊D3)C2×Q814yescomplemented
110C33×C4—22no—
111C3⋊(C3⋊Dic3)C2×D414yes—
112C3×C3⋊Dic3—22no—
113C3×C3⋊Dic3—22no—
114C32×Dic3—22no—
115C32×Dic3—22no—
116C22×C4×D3—94no—
117C22×C4×D3—91no—
118C22×C4×D3—184no—
119C22×Dic6—94no—
120C22×Dic6—91no—
121C2×C3×C4⋊C4—34no—
122C2×C3×C4⋊C4—91no—
123C2×(C2×D3)⋊C4—94no—
124C2×(C2×D3)⋊C4—94no—
125C2×(C2×D3)⋊C4—91no—
126C2×(C2×D3)⋊C4—91no—
127C2×Dic3⋊C4—34no—
128C2×Dic3⋊C4—94no—
129C2×Dic3⋊C4—91no—
130C2×Dic3⋊C4C3⋊D311yescharacteristic, complemented
131#896—932no—
132#896—98no—
133C23×C32—24no—
134C23×C32C22×D314yes—
135C23×C32C22×D311yescharacteristic
136C22×C3⋊D3—34no—
137C22×C3⋊D3—32no—
138C22×C3⋊D3—316no—
139C22×C3⋊D3—38no—
140C22×C3⋊D3—616no—
141C22×C3⋊D3—68no—
142C22×C3⋊D3—616no—
143C22×C3⋊D3—616no—
144C22×C3⋊D3—68no—
145C22×C3⋊D3—62no—
146C22×C3⋊D3—68no—
147C22×C3⋊D3—62no—
148C2×C32×C4—216no—
149C2×C32×C4—28no—
150C2×C32×C4—22no—
151C2×C32×C4—28no—
152C2×C32×C4—32no—
153C2×C32×C4—32no—
154C2×C32×C4—62no—
155C2×C32×C4—62no—
156C2×C32×C4C22×D318yes—
157C2×C32×C4C22×D312yes—
158C2×C3⋊Dic3—22no—
159C2×C3⋊Dic3—22no—
160C2×C3⋊Dic3—34no—
161C2×C3⋊Dic3—32no—
162C2×C3⋊Dic3—316no—
163C2×C3⋊Dic3—38no—
164C2×C3⋊Dic3—616no—
165C2×C3⋊Dic3—68no—
166C2×C3⋊Dic3C22×D312yes—
167C2×C3⋊Dic3C22×D312yes—
168C4×C3⋊D3—68no—
169C4×C3⋊D3—632no—
170C4×C3⋊D3—664no—
171#634—68no—
172C2×C3×Dic3—28no—
173C2×C3×Dic3—28no—
174C2×C3×Dic3—38no—
175C2×C3×Dic3—38no—
176C2×C3×Dic3—68no—
177C2×C3×Dic3—68no—
178C2×C3×Dic3C22×D318yes—
179C2×C3×Dic3C22×D318yes—
180#627—632no—
181C2×C22⋊Q8—271noSylow
182C2×C33C22⋊Q814yescomplemented
183C2×C33C22×D411yescharacteristic
184C2×C33C22×Q811yescharacteristic
185C2×C33C2×#4111yescharacteristic
186C3⋊(C3⋊D3)—24no—
187C23×D3—94no—
188C23×D3—91no—
189C23×D3—184no—
190C22×C3×C4—24no—
191C22×C3×C4—34no—
192C22×C3×C4—34no—
193C22×C3×C4—91no—
194C22×C3×C4—91no—
195C22×C3×C4D3214yes—
196C22×C3×C4C2×C3⋊D311yescharacteristic
197C22×Dic3—34no—
198C22×Dic3—34no—
199C22×Dic3—94no—
200C22×Dic3—91no—
201C22×Dic3—184no—
202C22×Dic3C2×C3⋊D311yescharacteristic
203C22×Dic3C2×C3⋊D311yescharacteristic
204C2×C4×D3—916no—
205C2×C4×D3—94no—
206C2×C4×D3—1816no—
207C2×C4×D3—1816no—
208C2×C4×D3—184no—
209C2×C4×D3—1832no—
210C2×Dic6—916no—
211C2×Dic6—94no—
212C2×Dic6—184no—
213C2×Dic6—1816no—
214C3×C4⋊C4—316no—
215C3×C4⋊C4—94no—
216(C2×D3)⋊C4—916no—
217(C2×D3)⋊C4—916no—
218(C2×D3)⋊C4—94no—
219(C2×D3)⋊C4—94no—
220Dic3⋊C4—316no—
221Dic3⋊C4—916no—
222Dic3⋊C4—94no—
223Dic3⋊C4C2×C3⋊D314yescomplemented
224C22×C32—24no—
225C22×C32—28no—
226C22×C32—28no—
227C22×C32—28no—
228C22×C32#23518yes—
229C22×C32#23512yes—
230C22×C32D3×D418yes—
231C22×C32D3×D412yes—
232C22×C32D3×Q818yes—
233C22×C32D3×Q812yes—
234C22×C32C23×D314yes—
235C22×C32C23×D311yescharacteristic
236C2×C3⋊D3—34no—
237C2×C3⋊D3—316no—
238C2×C3⋊D3—616no—
239C2×C3⋊D3—616no—
240C2×C3⋊D3—616no—
241C2×C3⋊D3—664no—
242C2×C3⋊D3—632no—
243C2×C3⋊D3—68no—
244C2×C3⋊D3—68no—
245C2×C3⋊D3—62no—
246C2×C3⋊D3—68no—
247C2×C3⋊D3—62no—
248C32×C4—216no—
249C32×C4—28no—
250C32×C4—22no—
251C32×C4—62no—
252C32×C4—62no—
253C3⋊Dic3—22no—
254C3⋊Dic3—22no—
255C3⋊Dic3—34no—
256C3⋊Dic3—316no—
257C3⋊Dic3—616no—
258C3×Dic3—28no—
259C3×Dic3—28no—
260C3×Dic3—68no—
261C3×Dic3—68no—
262C22×Q8—271no—
263C23×C4—271no—
264C22⋊Q8—278no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—91no—
267C2×C4⋊C4—271no—
268C2×C22⋊C4—271no—
269C2×C22⋊C4—271no—
270C33C2×C22⋊Q811yescharacteristic, complemented, Sylow
271C23×C3—24no—
272C23×C3C2×D3214yes—
273C23×C3C22×C3⋊D311yescharacteristic
274C22×D3—98no—
275C22×D3—92no—
276C22×D3—916no—
277C22×D3—94no—
278C22×D3—188no—
279C22×D3—1816no—
280C22×D3—1816no—
281C22×D3—1816no—
282C22×D3—188no—
283C22×D3—188no—
284C22×D3—182no—
285C22×D3—182no—
286C2×C3×C4—216no—
287C2×C3×C4—28no—
288C2×C3×C4—22no—
289C2×C3×C4—28no—
290C2×C3×C4—38no—
291C2×C3×C4—38no—
292C2×C3×C4—68no—
293C2×C3×C4—68no—
294C2×C3×C4—92no—
295C2×C3×C4—92no—
296C2×C3×C4—182no—
297C2×C3×C4—182no—
298C2×C3×C4C2×D3218yes—
299C2×C3×C4C22×C3⋊D312yes—
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—38no—
303C2×Dic3—38no—
304C2×Dic3—68no—
305C2×Dic3—68no—
306C2×Dic3—98no—
307C2×Dic3—92no—
308C2×Dic3—916no—
309C2×Dic3—94no—
310C2×Dic3—188no—
311C2×Dic3—1816no—
312C2×Dic3C22×C3⋊D312yes—
313C2×Dic3C22×C3⋊D312yes—
314C4×D3—1832no—
315C4×D3—188no—
316C4×D3—1864no—
317Dic6—188no—
318Dic6—1832no—
319C2×C32—24no—
320C2×C32—24no—
321C2×C32—24no—
322C2×C32—216no—
323C2×C32#896116yescomplemented
324C2×C32#89614yescomplemented
325C2×C32C2×#23514yes—
326C2×C32C2×#23511yescharacteristic
327C2×C32C2×D3×D414yes—
328C2×C32C2×D3×D411yescharacteristic
329C2×C32C2×D3×Q814yes—
330C2×C32C2×D3×Q811yescharacteristic
331C3⋊D3—632no—
332C3⋊D3—616no—
333C3⋊D3—64no—
334C24—271no—
335C2×Q8—274no—
336C2×Q8—544no—
337C22×C4—31no—
338C22×C4—31no—
339C22×C4—91no—
340C22×C4—91no—
341C22×C4—271no—
342C22×C4—274no—
343C22×C4—544no—
344C22×C4D3×C3⋊D311yescharacteristic
345C4⋊C4—34no—
346C4⋊C4—94no—
347C4⋊C4—274no—
348C22⋊C4—274no—
349C22⋊C4—274no—
350C22×C3—24no—
351C22×C3—28no—
352C22×C3—28no—
353C22×C3—28no—
354C22×C3#379718yes—
355C22×C3#379818yes—
356C22×C3#381118yes—
357C22×C3#382812yes—
358C22×C3D4×C3⋊D312yes—
359C22×C3Q8×C3⋊D312yes—
360C22×C3C22×D3214yes—
361C22×C3C23×C3⋊D311yescharacteristic
362C2×D3—916no—
363C2×D3—94no—
364C2×D3—1816no—
365C2×D3—1816no—
366C2×D3—1816no—
367C2×D3—1864no—
368C2×D3—188no—
369C2×D3—188no—
370C2×D3—182no—
371C2×D3—182no—
372C2×D3—188no—
373C2×D3—1832no—
374C3×C4—216no—
375C3×C4—28no—
376C3×C4—22no—
377C3×C4—68no—
378C3×C4—68no—
379C3×C4—182no—
380C3×C4—182no—
381Dic3—22no—
382Dic3—22no—
383Dic3—68no—
384Dic3—68no—
385Dic3—916no—
386Dic3—94no—
387Dic3—1816no—
388C32—24no—
389C32C2×#89614yescomplemented
390C32C2×#89611yescharacteristic, complemented
391C23—272no—
392C23—274no—
393C23—542no—
394C23—542no—
395C23C2×D3×C3⋊D311yescharacteristic
396Q8—548no—
397C2×C4—22no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—92no—
403C2×C4—92no—
404C2×C4—182no—
405C2×C4—182no—
406C2×C4—272no—
407C2×C4—274no—
408C2×C4—548no—
409C2×C4C2×D3×C3⋊D312yes—
410C2×C3—24no—
411C2×C3—24no—
412C2×C3—24no—
413C2×C3—216no—
414C2×C3#63865116yescomplemented
415C2×C3#6410514yescomplemented
416C2×C3C2×#379714yes—
417C2×C3C2×#379814yes—
418C2×C3C2×#381114yes—
419C2×C3C2×#382811yescharacteristic
420C2×C3C2×D4×C3⋊D311yescharacteristic
421C2×C3C2×Q8×C3⋊D311yescharacteristic
422D3—1832no—
423D3—1816no—
424D3—184no—
425C22—274no—
426C22—542no—
427C22—542no—
428C22—548no—
429C22#8901912yes—
430C22#8902012yes—
431C22#8903912yes—
432C22C22×D3×C3⋊D311yescharacteristic
433C4—22no—
434C4—62no—
435C4—62no—
436C4—182no—
437C4—182no—
438C4—274no—
439C3—24no—
440C3C2×#6386514yescomplemented
441C3C2×#6410511yescharacteristic, complemented
442C2—544no—
443C2#1172344214yesdirect product factor
444C2C2×#8901911yescharacteristic
445C2C2×#8902011yescharacteristic
446C2C2×#8903911yescharacteristic
subgroupG/H
centerC23C2×D3×C3⋊D3
commutator, cosocleC22×C33C24
FrattiniC22C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC23×C33C23
prime radicalC23×C3⋊(C3⋊D3)C22
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×C22⋊Q8
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C23 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C22×#6293C2semidirect
3. C2×C3×#3750C2semidirect
4. C2×C32×Dic3⋊C4C2semidirect
5. C2×#88806C2semidirect
8. #11723442C2direct (N × H)
21. C2×#6293C22semidirect
22. C3×#3750C22semidirect
23. C32×Dic3⋊C4C22semidirect
24. #88806C22semidirect
36. C2×#3750D3semidirect
39. C2×C3×Dic3⋊C4D3semidirect
47. C22×C3⋊(C3⋊D3)Q8semidirect
96. #3750C2×D3semidirect
98. C3×Dic3⋊C4C2×D3semidirect
109. C2×C3⋊(C3⋊D3)C2×Q8semidirect
130. C2×Dic3⋊C4C3⋊D3semidirect
182. C2×C33C22⋊Q8semidirect
223. Dic3⋊C4C2×C3⋊D3semidirect
270. C33C2×C22⋊Q8semidirect
323. C2×C32#896semidirect
324. C2×C32#896semidirect
389. C32C2×#896semidirect
390. C32C2×#896semidirect
414. C2×C3#63865semidirect
415. C2×C3#64105semidirect
440. C3C2×#63865semidirect
441. C3C2×#64105semidirect
443. C2#11723442direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C3⋊(C3⋊D3)2
C22×#62932
C2×C3×#37502
C2×C32×Dic3⋊C42
C2×#888062
C2×#888042
#117234422
C2×#641053
C2×#638653