Finite Group Databasebeta

← C2×#11722048 #13160380

Subgroups — C2×#11722048 #13160380

Order 1728.

1676 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic
2C2×C3×#3748C211yesmaximal, characteristic, complemented
3C2×C32×C4×Dic3C211yesmaximal, characteristic, complemented
4#11722048C214yesmaximal, direct product factor
5C2×#63632—31nomaximal
6C2×#63570—34nomaximal
7C22×C33×C4C2211yescharacteristic
8C2×#6312—21no—
9#89076—21no—
10#89076—21no—
11#89076C412yescomplemented
12#89076C2212yes—
13C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
14C2×C4×C3⋊(C3⋊D3)—21no—
15C2×#6308C411yescharacteristic, complemented
16C22×C32×Dic3—21no—
17C3×#3748—21no—
18C3×#3748C2212yescomplemented
19C2×C3×C3⋊(C3⋊C8)—21no—
20C32×C4×Dic3—21no—
21C32×C4×Dic3C2212yescomplemented
22C2×#3828—31no—
23C2×#3828—34no—
24C2×#3828—64no—
25C2×C32×C42—31no—
26C2×#3748D311yescharacteristic, complemented
27C2×C3×C4×Dic3D314yescomplemented
28C2×C3×#208—34no—
29#63632—34no—
30#63570—316no—
31C23×C33D411yescharacteristic
32C22×C3⋊(C3⋊D3)—21no—
33C2×C33×C4C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C2×C33×C4D411yescharacteristic
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39#6312—22no—
40#6312—22no—
41C2×C3⋊(C3⋊Dic3)—21no—
42C4⋊(C3⋊(C3⋊D3))—21no—
43C4⋊(C3⋊(C3⋊D3))C2×C412yescomplemented
44C4×C3⋊(C3⋊D3)—22no—
45C4×C3⋊(C3⋊D3)—22no—
46#6308—21no—
47#6308C2×C412yescomplemented
48C2×C32×Dic3—24no—
49C2×C32×Dic3—22no—
50C3×C3⋊(C3⋊C8)—22no—
51C2×#825—91no—
52C2×#805—94no—
53C22×C32×C4—24no—
54C22×C32×C4—61no—
55C22×C32×C4C2×D314yes—
56C22×C32×C4C2×D311yescharacteristic
57C2×#638—68no—
58C2×#638—61no—
59C2×#638—64no—
60#3828—32no—
61#3828—38no—
62#3828—32no—
63#3828—38no—
64#3828—68no—
65#3828—68no—
66#3828—68no—
67#3828—68no—
68#3828—61no—
69#3828—64no—
70#3828—64no—
71#3828—61no—
72C2×C4⋊(C3⋊D3)—31no—
73C2×C4⋊(C3⋊D3)—34no—
74C2×C4⋊(C3⋊D3)—64no—
75C2×C4×C3⋊D3—68no—
76C2×C4×C3⋊D3—64no—
77C2×C4×C3⋊D3—61no—
78C2×#634—31no—
79C2×#634—34no—
80C2×#634—64no—
81C22×C3×Dic3—24no—
82C32×C42—32no—
83C32×C42—61no—
84#3748—21no—
85#3748C2×D312yescomplemented
86C2×C3⋊(C3⋊C8)—21no—
87C3×C4×Dic3—24no—
88C3×C4×Dic3C2×D318yescomplemented
89C3×#208—38no—
90C3×#208—64no—
91C2×C3×C3⋊C8—64no—
92C22×C33—22no—
93C22×C33C22⋊C411yescharacteristic
94C22×C33C22⋊C411yescharacteristic
95C22×C33C2×D411yescharacteristic
96C2×C3⋊(C3⋊D3)—22no—
97C2×C3⋊(C3⋊D3)—42no—
98C33×C4C22⋊C411yescharacteristic
99C33×C4C22⋊C411yescharacteristic
100C33×C4C22×C411yescharacteristic
101C33×C4C2×D411yescharacteristic
102C3⋊(C3⋊Dic3)—22no—
103C32×Dic3—24no—
104C2×#235—94no—
105C2×#235—91no—
106C2×#235—184no—
107C2×C3×M16—91no—
108C2×C3×C42—34no—
109C2×C4×Dic3C3⋊D311yescharacteristic, complemented
110C2×#208—34no—
111#825—94no—
112#805—916no—
113C23×C32—24no—
114C23×C32D1211yescharacteristic
115C23×C32#6714yes—
116C22×C3⋊D3—68no—
117C22×C3⋊D3—64no—
118C22×C3⋊D3—61no—
119C2×C32×C4—24no—
120C2×C32×C4—24no—
121C2×C32×C4—24no—
122C2×C32×C4—24no—
123C2×C32×C4—24no—
124C2×C32×C4—24no—
125C2×C32×C4—64no—
126C2×C32×C4—62no—
127C2×C32×C4C4×D314yes—
128C2×C32×C4C4×D314yes—
129C2×C32×C4C4×D311yescharacteristic
130C2×C32×C4C4×D311yescharacteristic
131C2×C32×C4D1214yes—
132C2×C32×C4D1214yes—
133C2×C32×C4D1211yescharacteristic
134C2×C32×C4#6714yes—
135C2×C32×C4#6711yescharacteristic
136C2×C32×C4#6711yescharacteristic
137C2×C32×C4C22×D314yes—
138C2×C32×C4C22×D311yescharacteristic
139#638—616no—
140#638—616no—
141#638—68no—
142#638—62no—
143#638—62no—
144#638—68no—
145C2×C3⋊Dic3—68no—
146C2×C3⋊Dic3—64no—
147C2×C3⋊Dic3—61no—
148C4⋊(C3⋊D3)—32no—
149C4⋊(C3⋊D3)—38no—
150C4⋊(C3⋊D3)—68no—
151C4⋊(C3⋊D3)—68no—
152C4⋊(C3⋊D3)—64no—
153C4⋊(C3⋊D3)—61no—
154C4×C3⋊D3—616no—
155C4×C3⋊D3—616no—
156C4×C3⋊D3—68no—
157C4×C3⋊D3—62no—
158C4×C3⋊D3—68no—
159C4×C3⋊D3—62no—
160#634—32no—
161#634—38no—
162#634—68no—
163#634—68no—
164#634—64no—
165#634—61no—
166C2×C3×Dic3—216no—
167C2×C3×Dic3—28no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—68no—
170C2×#104—271noSylow
171C2×C33—21no—
172C2×C33—21no—
173C2×C33#10412yescomplemented
174C2×C33C2×C22⋊C411yescharacteristic
175C3⋊(C3⋊D3)—42no—
176C22×C3×C4—24no—
177C22×C3×C4—64no—
178C22×C3×C4D3214yes—
179C22×C3×C4C2×C3⋊D311yescharacteristic
180C2×#67—188no—
181C2×#67—184no—
182C2×#67—181no—
183C22×Dic3—21no—
184#235—98no—
185#235—92no—
186#235—98no—
187#235—92no—
188#235—188no—
189#235—188no—
190#235—188no—
191#235—188no—
192#235—184no—
193#235—184no—
194#235—181no—
195#235—181no—
196C2×D12—94no—
197C2×D12—91no—
198C2×D12—184no—
199C2×C4×D3—188no—
200C2×C4×D3—184no—
201C2×C4×D3—181no—
202C2×Dic6—94no—
203C2×Dic6—91no—
204C2×Dic6—184no—
205C3×M16—92no—
206C3×M16—181no—
207C2×C3×C8—181no—
208C3×C42—38no—
209C3×C42—64no—
210C4×Dic3—21no—
211C4×Dic3C2×C3⋊D312yescomplemented
212#208—38no—
213#208—64no—
214C2×C3⋊C8—64no—
215C22×C32—22no—
216C22×C32—24no—
217C22×C32—24no—
218C22×C32—24no—
219C22×C32—28no—
220C22×C32—48no—
221C22×C32(C2×D3)⋊C414yes—
222C22×C32(C2×D3)⋊C414yes—
223C22×C32(C2×D3)⋊C411yescharacteristic
224C22×C32(C2×D3)⋊C411yescharacteristic
225C22×C32C2×D1211yescharacteristic
226C22×C32C2×#6714yes—
227C2×C3⋊D3—616no—
228C2×C3⋊D3—68no—
229C2×C3⋊D3—62no—
230C2×C3⋊D3—1216no—
231C2×C3⋊D3—128no—
232C2×C3⋊D3—122no—
233C32×C4—24no—
234C32×C4—24no—
235C32×C4—24no—
236C32×C4—24no—
237C32×C4—64no—
238C32×C4(C2×D3)⋊C414yes—
239C32×C4(C2×D3)⋊C414yes—
240C32×C4(C2×D3)⋊C411yescharacteristic
241C32×C4(C2×D3)⋊C411yescharacteristic
242C32×C4C2×C4×D314yes—
243C32×C4C2×C4×D311yescharacteristic
244C32×C4C2×D1214yes—
245C32×C4C2×#6711yescharacteristic
246C3⋊Dic3—616no—
247C3⋊Dic3—68no—
248C3⋊Dic3—62no—
249C3×Dic3—216no—
250C2×#41—271no—
251C2×M16—91no—
252C2×C42—31no—
253#104—274no—
254C33C2×#10411yescharacteristic, complemented, Sylow
255C23×C3—24no—
256C23×C3#62614yes—
257C23×C3#63811yescharacteristic
258C22×D3—188no—
259C22×D3—184no—
260C22×D3—181no—
261C2×C3×C4—24no—
262C2×C3×C4—24no—
263C2×C3×C4—24no—
264C2×C3×C4—24no—
265C2×C3×C4—24no—
266C2×C3×C4—24no—
267C2×C3×C4—616no—
268C2×C3×C4—68no—
269C2×C3×C4#62414yes—
270C2×C3×C4#62414yes—
271C2×C3×C4#62614yes—
272C2×C3×C4#62614yes—
273C2×C3×C4#62614yes—
274C2×C3×C4C4×C3⋊D311yescharacteristic
275C2×C3×C4C4×C3⋊D311yescharacteristic
276C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
277C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
278C2×C3×C4#63811yescharacteristic
279C2×C3×C4C2×D3214yes—
280C2×C3×C4C22×C3⋊D311yescharacteristic
281#67—1816no—
282#67—1816no—
283#67—188no—
284#67—182no—
285#67—188no—
286#67—182no—
287C2×Dic3—24no—
288C2×Dic3—22no—
289C2×Dic3—188no—
290C2×Dic3—184no—
291C2×Dic3—181no—
292D12—98no—
293D12—92no—
294D12—188no—
295D12—188no—
296D12—184no—
297D12—181no—
298C4×D3—1816no—
299C4×D3—1816no—
300C4×D3—188no—
301C4×D3—182no—
302C4×D3—188no—
303C4×D3—182no—
304Dic6—98no—
305Dic6—92no—
306Dic6—188no—
307Dic6—188no—
308Dic6—184no—
309Dic6—181no—
310C3×C8—182no—
311C3⋊C8—68no—
312C2×C32—21no—
313C2×C32—21no—
314C2×C32—28no—
315C2×C32—24no—
316C2×C32—24no—
317C2×C32—24no—
318C2×C32—44no—
319C2×C32—44no—
320C2×C32#80518yescomplemented
321C2×C32#82512yescomplemented
322C2×C32C2×(C2×D3)⋊C414yes—
323C2×C32C2×(C2×D3)⋊C411yescharacteristic
324C3⋊D3—1216no—
325C3⋊D3—122no—
326C3⋊D3—128no—
327#41—272no—
328#41—272no—
329#41—541no—
330#41—541no—
331C2×Q8—271no—
332C2×D4—271no—
333C2×D4—541no—
334C22×C4—61no—
335C22×C4—541no—
336C22×C4D3×C3⋊D311yescharacteristic
337M16—92no—
338M16—181no—
339C2×C8—181no—
340C42—32no—
341C42—61no—
342C22×C3—28no—
343C22×C3—22no—
344C22×C3—24no—
345C22×C3—24no—
346C22×C3—24no—
347C22×C3—48no—
348C22×C3#372214yes—
349C22×C3#372214yes—
350C22×C3#375211yescharacteristic
351C22×C3#375211yescharacteristic
352C22×C3C2×#62614yes—
353C22×C3C2×#63811yescharacteristic
354C2×D3—1816no—
355C2×D3—182no—
356C2×D3—188no—
357C2×D3—3616no—
358C2×D3—368no—
359C2×D3—362no—
360C3×C4—24no—
361C3×C4—24no—
362C3×C4—24no—
363C3×C4—24no—
364C3×C4—616no—
365C3×C4#372214yes—
366C3×C4#372214yes—
367C3×C4#375211yescharacteristic
368C3×C4#375211yescharacteristic
369C3×C4C2×#62414yes—
370C3×C4C2×#62614yes—
371C3×C4C2×C4×C3⋊D311yescharacteristic
372C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
373Dic3—24no—
374Dic3—1816no—
375Dic3—182no—
376Dic3—188no—
377C32—24no—
378C32C2×#80514yescomplemented
379C32C2×#82511yescharacteristic, complemented
380C23—541no—
381C23#629111yescharacteristic
382Q8—272no—
383Q8—541no—
384D4—272no—
385D4—542no—
386D4—542no—
387D4—541no—
388C2×C4—64no—
389C2×C4—62no—
390C2×C4—541no—
391C2×C4—542no—
392C2×C4—542no—
393C2×C4#628911yescharacteristic
394C2×C4#628911yescharacteristic
395C2×C4#629111yescharacteristic
396C2×C4#629211yescharacteristic
397C2×C4#629211yescharacteristic
398C2×C4C2×D3×C3⋊D311yescharacteristic
399C8—182no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—24no—
403C2×C3—24no—
404C2×C3—28no—
405C2×C3—24no—
406C2×C3—44no—
407C2×C3—44no—
408C2×C3#6357018yescomplemented
409C2×C3#6363212yescomplemented
410C2×C3C2×#372214yes—
411C2×C3C2×#375211yescharacteristic
412D3—3616no—
413D3—362no—
414D3—368no—
415C22—22no—
416C22—542no—
417C22—1082no—
418C22#8880411yescharacteristic
419C22#8880411yescharacteristic
420C22C2×#629111yescharacteristic
421C4—64no—
422C4—542no—
423C4#8880411yescharacteristic
424C4#8880411yescharacteristic
425C4C2×#628911yescharacteristic
426C4C2×#629211yescharacteristic
427C3—24no—
428C3C2×#6357014yescomplemented
429C3C2×#6363211yescharacteristic, complemented
430C2—21no—
431C2—21no—
432C2—1082no—
433C2#1172204812yesdirect product factor
434C2C2×#8880411yescharacteristic
subgroupG/H
centerC2×C4#6292
commutatorC33×C4C22×C4
FrattiniC2×C4C2×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C22⋊C4
prime radicalC2×#89076C2
cosocleC2×C33×C4C23
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#104
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2×C4 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
2. C2×C3×#3748C2semidirect
3. C2×C32×C4×Dic3C2semidirect
4. #11722048C2direct (N × H)
11. #89076C4semidirect
13. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
15. C2×#6308C4semidirect
18. C3×#3748C22semidirect
21. C32×C4×Dic3C22semidirect
26. C2×#3748D3semidirect
27. C2×C3×C4×Dic3D3semidirect
43. C4⋊(C3⋊(C3⋊D3))C2×C4semidirect
47. #6308C2×C4semidirect
85. #3748C2×D3semidirect
88. C3×C4×Dic3C2×D3semidirect
109. C2×C4×Dic3C3⋊D3semidirect
173. C2×C33#104semidirect
211. C4×Dic3C2×C3⋊D3semidirect
254. C33C2×#104semidirect
320. C2×C32#805semidirect
321. C2×C32#825semidirect
378. C32C2×#805semidirect
379. C32C2×#825semidirect
408. C2×C3#63570semidirect
409. C2×C3#63632semidirect
428. C3C2×#63570semidirect
429. C3C2×#63632semidirect
433. C2#11722048direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890762
C2×C3×#37482
C2×C32×C4×Dic32
#117220482
C2×#636323
C2×#635703