Finite Group Databasebeta

← C2×#11723369 #13176887

Subgroups — C2×#11723369 #13176887

Order 1728.

2252 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#89076C211yesmaximal, characteristic, complemented
2C2×C3×#3828C211yesmaximal, characteristic, complemented
3C22×C32×C3⋊C8C211yesmaximal, characteristic, complemented
4C2×#88800C211yesmaximal, characteristic, complemented
5C2×#88797C211yesmaximal, characteristic, complemented
6C2×#88796C211yesmaximal, characteristic, complemented
7#11723369C218yesmaximal, direct product factor
8C2×#88791C211yesmaximal, characteristic, complemented
9C2×#64113—31nomaximal
10C2×#63829—34nomaximal
11C22×C33×C4C2211yescharacteristic
12C2×#6312—21no—
13#89076—22no—
14#89076C2214yescomplemented
15C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
16C2×C4×C3⋊(C3⋊D3)—21no—
17C2×#6308C2211yescharacteristic
18C2×C3×#638—21no—
19C3×#3828—22no—
20C3×#3828C2214yescomplemented
21C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
22C2×C3×C4×C3⋊D3—21no—
23C2×C3×#634C2211yescharacteristic
24C2×C32×C3⋊C8C2211yescharacteristic
25C2×C32×C3⋊C8C2211yescharacteristic
26C2×C32×C3⋊C8C2214yescomplemented
27#88800C2214yescomplemented
28#88797C2214yescomplemented
29#88796C2214yescomplemented
30#88791C2214yescomplemented
31C2×#3828—31no—
32C2×#3828—34no—
33C2×#3828—64no—
34C2×#3828D311yescharacteristic, complemented
35C2×C3×#235—34no—
36C22×C32×C8—31no—
37C2×#3746—31no—
38#64113—38no—
39C2×#3745—31no—
40C2×#3744—31no—
41C2×#3744—31no—
42C22×C3×C3⋊C8D314yescomplemented
43C2×#3719—34no—
44C2×#3717—34no—
45#63829—332no—
46C2×#3716—34no—
47C2×#3714—34no—
48C23×C33D411yescharacteristic
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3×C3⋊D3—21no—
51C2×C33×C4D411yescharacteristic
52C2×C33×C4D412yes—
53C2×C33×C4C2311yescharacteristic
54C2×C33×C4C2312yes—
55#6312—24no—
56C2×C3⋊(C3⋊Dic3)—21no—
57C4⋊(C3⋊(C3⋊D3))—21no—
58C4⋊(C3⋊(C3⋊D3))C2312yes—
59C4×C3⋊(C3⋊D3)—24no—
60#6308—21no—
61#6308C2312yes—
62C3×#638—24no—
63C2×C3×C3⋊Dic3—21no—
64C3×C4⋊(C3⋊D3)—21no—
65C3×C4⋊(C3⋊D3)C2312yes—
66C3×C4×C3⋊D3—24no—
67C3×#634—21no—
68C3×#634C2312yes—
69C32×C3⋊C8C2312yes—
70C32×C3⋊C8C2312yes—
71C2×#950—91no—
72C2×#904—94no—
73C22×C32×C4—24no—
74C22×C32×C4C2×D314yes—
75C22×C32×C4C2×D311yescharacteristic
76C2×#638—21no—
77C2×#638—68no—
78C2×#638—64no—
79C2×#638—61no—
80#3828—22no—
81#3828—34no—
82#3828—316no—
83#3828—616no—
84#3828—616no—
85#3828—68no—
86#3828—62no—
87#3828C2×D314yescomplemented
88C2×C4⋊(C3⋊D3)—31no—
89C2×C4⋊(C3⋊D3)—34no—
90C2×C4⋊(C3⋊D3)—64no—
91C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
92C2×C4×C3⋊D3—21no—
93C2×C4×C3⋊D3—68no—
94C2×C4×C3⋊D3—64no—
95C2×C4×C3⋊D3—61no—
96C2×#634—31no—
97C2×#634—34no—
98C2×#634—64no—
99C2×#634C2×D311yescharacteristic
100C2×C3×#67—64no—
101C3×#235—316no—
102C3×#235—68no—
103C2×C3×D12—34no—
104C2×C3×C4×D3—64no—
105C2×C3×Dic6—34no—
106C2×C32×C8—31no—
107C2×C32×C8—31no—
108C2×C32×C8—34no—
109#3746—34no—
110#3745—34no—
111#3744—34no—
112#3744—34no—
113C2×C3×C3⋊C8C2×D314yes—
114C2×C3×C3⋊C8C2×D314yes—
115C2×C3×C3⋊C8C2×D3116yescomplemented
116#3719—316no—
117#3717—316no—
118#3716—316no—
119#3714—316no—
120C22×C33—22no—
121C22×C33C2×D412yes—
122C22×C33C2×D411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)—42no—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2×D412yes—
129C33×C4C2411yescharacteristic
130C3⋊(C3⋊Dic3)—22no—
131C3×C3⋊Dic3—22no—
132C2×C3×#41—91no—
133C2×#235—34no—
134C2×#235—94no—
135C2×#235—91no—
136C2×#235—184no—
137C22×C3×C8—34no—
138#950—98no—
139C2×#216—91no—
140C2×#215—91no—
141C2×#214—91no—
142C2×#213—91no—
143C22×C3⋊C8C3⋊D311yescharacteristic, complemented
144C2×Dic12—94no—
145#904—932no—
146C2×D24—94no—
147C2×#204—94no—
148C2×#204—94no—
149C23×C32—24no—
150C23×C32D1214yes—
151C23×C32#6711yescharacteristic
152C22×C3⋊D3—21no—
153C22×C3⋊D3—68no—
154C22×C3⋊D3—64no—
155C22×C3⋊D3—61no—
156C22×C3×D3—64no—
157C2×C32×C4—24no—
158C2×C32×C4—28no—
159C2×C32×C4—24no—
160C2×C32×C4—28no—
161C2×C32×C4D1214yes—
162C2×C32×C4D1218yes—
163C2×C32×C4#6711yescharacteristic
164C2×C32×C4#6712yes—
165C2×C32×C4C22×D314yes—
166C2×C32×C4C22×D318yes—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D312yes—
169#638—24no—
170#638—632no—
171#638—616no—
172#638—64no—
173C2×C3⋊Dic3—21no—
174C2×C3⋊Dic3—68no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—61no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—32no—
179C4⋊(C3⋊D3)—38no—
180C4⋊(C3⋊D3)—68no—
181C4⋊(C3⋊D3)—68no—
182C4⋊(C3⋊D3)—64no—
183C4⋊(C3⋊D3)—61no—
184C4⋊(C3⋊D3)C22×D312yes—
185C4×C3⋊D3—24no—
186C4×C3⋊D3—632no—
187C4×C3⋊D3—616no—
188C4×C3⋊D3—64no—
189#634—21no—
190#634—32no—
191#634—38no—
192#634—68no—
193#634—68no—
194#634—64no—
195#634—61no—
196#634C22×D312yes—
197C3×#67—616no—
198C2×C3×Dic3—64no—
199C3×D12—38no—
200C3×D12—64no—
201C3×C4×D3—616no—
202C3×Dic6—38no—
203C3×Dic6—64no—
204C32×C8—32no—
205C32×C8—32no—
206C3×C3⋊C8C22×D318yes—
207C3×C3⋊C8C22×D318yes—
208C2×#135—271noSylow
209C2×C33—22no—
210C2×C33#13512yescomplemented
211C2×C33C22×D411yescharacteristic
212C3⋊(C3⋊D3)—42no—
213C3×C3⋊D3—42no—
214C3×#41—94no—
215C3×#41—182no—
216C2×C3×Q8—91no—
217C2×C3×D4—91no—
218C2×C3×D4—181no—
219C22×C3×C4—24no—
220C22×C3×C4—181no—
221C22×C3×C4D3214yes—
222C22×C3×C4C2×C3⋊D311yescharacteristic
223C2×#67—64no—
224C2×#67—188no—
225C2×#67—184no—
226C2×#67—181no—
227#235—316no—
228#235—68no—
229#235—916no—
230#235—94no—
231#235—1816no—
232#235—1816no—
233#235—188no—
234#235—182no—
235C2×D12—34no—
236C2×D12—94no—
237C2×D12—91no—
238C2×D12—184no—
239C2×C4×D3—64no—
240C2×C4×D3—188no—
241C2×C4×D3—184no—
242C2×C4×D3—181no—
243C2×Dic6—34no—
244C2×Dic6—94no—
245C2×Dic6—91no—
246C2×Dic6—184no—
247C2×C3×C8—34no—
248C2×C3×C8—34no—
249C2×C3×C8—316no—
250#216—94no—
251#215—94no—
252#214—94no—
253#213—94no—
254C2×C3⋊C8C2×C3⋊D311yescharacteristic
255C2×C3⋊C8C2×C3⋊D311yescharacteristic
256C2×C3⋊C8C2×C3⋊D314yescomplemented
257Dic12—916no—
258D24—916no—
259#204—916no—
260#204—916no—
261C22×C32—22no—
262C22×C32—24no—
263C22×C32—28no—
264C22×C32—28no—
265C22×C32—48no—
266C22×C32C2×D1218yes—
267C22×C32C2×D1214yes—
268C22×C32C2×#6712yes—
269C22×C32C2×#6711yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3—42no—
272C2×C3⋊D3—616no—
273C2×C3⋊D3—68no—
274C2×C3⋊D3—62no—
275C2×C3⋊D3—1216no—
276C2×C3⋊D3—128no—
277C2×C3⋊D3—122no—
278C2×C3×D3—68no—
279C2×C3×D3—128no—
280C32×C4—24no—
281C32×C4—24no—
282C32×C4—28no—
283C32×C4C2×D1214yes—
284C32×C4C2×D1218yes—
285C32×C4C2×#6711yescharacteristic
286C32×C4C2×#6712yes—
287C32×C4C23×D314yes—
288C32×C4C23×D311yescharacteristic
289C3⋊Dic3—22no—
290C3⋊Dic3—616no—
291C3⋊Dic3—68no—
292C3⋊Dic3—62no—
293C3×Dic3—68no—
294C2×#41—91no—
295C2×#41—271no—
296#135—278no—
297C2×Q16—271no—
298C2×SD16—271no—
299C2×SD16—271no—
300C2×D8—271no—
301C22×C8—31no—
302C33C2×#13511yescharacteristic, complemented, Sylow
303C23×C3—24no—
304C23×C3—181no—
305C23×C3#62614yes—
306C23×C3C4⋊(C3⋊D3)11yescharacteristic
307C22×D3—64no—
308C22×D3—188no—
309C22×D3—184no—
310C22×D3—181no—
311C3×Q8—92no—
312C3×Q8—181no—
313C3×D4—92no—
314C3×D4—184no—
315C3×D4—181no—
316C2×C3×C4—28no—
317C2×C3×C4—24no—
318C2×C3×C4—24no—
319C2×C3×C4—28no—
320C2×C3×C4—181no—
321C2×C3×C4—184no—
322C2×C3×C4#62618yes—
323C2×C3×C4#62614yes—
324C2×C3×C4C4⋊(C3⋊D3)12yes—
325C2×C3×C4C4⋊(C3⋊D3)11yescharacteristic
326C2×C3×C4C2×D3214yes—
327C2×C3×C4C2×D3218yes—
328C2×C3×C4C22×C3⋊D311yescharacteristic
329C2×C3×C4C22×C3⋊D312yes—
330#67—616no—
331#67—1832no—
332#67—1816no—
333#67—184no—
334C2×Dic3—64no—
335C2×Dic3—188no—
336C2×Dic3—184no—
337C2×Dic3—181no—
338D12—38no—
339D12—64no—
340D12—98no—
341D12—92no—
342D12—188no—
343D12—188no—
344D12—184no—
345D12—181no—
346C4×D3—616no—
347C4×D3—1832no—
348C4×D3—1816no—
349C4×D3—184no—
350Dic6—38no—
351Dic6—64no—
352Dic6—98no—
353Dic6—92no—
354Dic6—188no—
355Dic6—188no—
356Dic6—184no—
357Dic6—181no—
358C3×C8—38no—
359C3×C8—38no—
360C3⋊C8C22×C3⋊D312yes—
361C3⋊C8C22×C3⋊D312yes—
362C2×C32—22no—
363C2×C32—24no—
364C2×C32—28no—
365C2×C32—28no—
366C2×C32—48no—
367C2×C32#90418yescomplemented
368C2×C32#95012yescomplemented
369C2×C32C22×D1214yes—
370C2×C32C22×#6711yescharacteristic
371C3⋊D3—42no—
372C3⋊D3—1216no—
373C3⋊D3—128no—
374C3⋊D3—122no—
375C3×D3—128no—
376#41—94no—
377#41—182no—
378#41—274no—
379#41—542no—
380C2×Q8—91no—
381C2×Q8—271no—
382C2×D4—91no—
383C2×D4—181no—
384C2×D4—271no—
385C2×D4—541no—
386C22×C4—181no—
387C22×C4—541no—
388C22×C4D3×C3⋊D311yescharacteristic
389Q16—274no—
390SD16—274no—
391SD16—274no—
392D8—274no—
393C2×C8—31no—
394C2×C8—31no—
395C2×C8—34no—
396C22×C3—28no—
397C22×C3—22no—
398C22×C3—24no—
399C22×C3—28no—
400C22×C3—48no—
401C22×C3—182no—
402C22×C3—362no—
403C22×C3C2×#62618yes—
404C22×C3C2×#62614yes—
405C22×C3C2×C4⋊(C3⋊D3)12yes—
406C22×C3C2×C4⋊(C3⋊D3)11yescharacteristic
407C2×D3—68no—
408C2×D3—128no—
409C2×D3—1816no—
410C2×D3—188no—
411C2×D3—182no—
412C2×D3—3616no—
413C2×D3—368no—
414C2×D3—362no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—28no—
418C3×C4—182no—
419C3×C4C2×#62614yes—
420C3×C4C2×#62618yes—
421C3×C4C2×C4⋊(C3⋊D3)11yescharacteristic
422C3×C4C2×C4⋊(C3⋊D3)12yes—
423C3×C4C22×D3214yes—
424C3×C4C23×C3⋊D311yescharacteristic
425Dic3—68no—
426Dic3—1816no—
427Dic3—188no—
428Dic3—182no—
429C32—24no—
430C32C2×#90414yescomplemented
431C32C2×#95011yescharacteristic, complemented
432C23—181no—
433C23—541no—
434C23#629211yescharacteristic
435Q8—92no—
436Q8—181no—
437Q8—272no—
438Q8—541no—
439D4—92no—
440D4—184no—
441D4—181no—
442D4—272no—
443D4—544no—
444D4—541no—
445C2×C4—181no—
446C2×C4—184no—
447C2×C4—541no—
448C2×C4—544no—
449C2×C4#629212yes—
450C2×C4#629211yescharacteristic
451C2×C4C2×D3×C3⋊D311yescharacteristic
452C2×C4C2×D3×C3⋊D312yes—
453C8—32no—
454C8—32no—
455C2×C3—22no—
456C2×C3—28no—
457C2×C3—24no—
458C2×C3—28no—
459C2×C3—48no—
460C2×C3—362no—
461C2×C3#6382918yescomplemented
462C2×C3#6411312yescomplemented
463C2×C3C22×#62614yes—
464C2×C3C22×C4⋊(C3⋊D3)11yescharacteristic
465D3—128no—
466D3—3616no—
467D3—362no—
468D3—368no—
469C22—22no—
470C22—182no—
471C22—362no—
472C22—542no—
473C22—1082no—
474C22C2×#629212yes—
475C22C2×#629211yescharacteristic
476C4—182no—
477C4—542no—
478C4C2×#629212yes—
479C4C2×#629211yescharacteristic
480C4C22×D3×C3⋊D311yescharacteristic
481C3—24no—
482C3C2×#6382914yescomplemented
483C3C2×#6411311yescharacteristic, complemented
484C2—22no—
485C2—362no—
486C2—1082no—
487C2#1172336912yesdirect product factor
488C2C22×#629211yescharacteristic
subgroupG/H
centerC2×C4#6292
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC22×C33×C4C22
socleC22×C33C2×D4
hypercenterC22×C4D3×C3⋊D3
hypocenterC33C2×#135
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
1. C2×#89076C2semidirect
2. C2×C3×#3828C2semidirect
3. C22×C32×C3⋊C8C2semidirect
4. C2×#88800C2semidirect
5. C2×#88797C2semidirect
6. C2×#88796C2semidirect
7. #11723369C2direct (N × H)
8. C2×#88791C2semidirect
14. #89076C22semidirect
20. C3×#3828C22semidirect
26. C2×C32×C3⋊C8C22semidirect
27. #88800C22semidirect
28. #88797C22semidirect
29. #88796C22semidirect
30. #88791C22semidirect
34. C2×#3828D3semidirect
42. C22×C3×C3⋊C8D3semidirect
87. #3828C2×D3semidirect
115. C2×C3×C3⋊C8C2×D3semidirect
143. C22×C3⋊C8C3⋊D3semidirect
210. C2×C33#135semidirect
256. C2×C3⋊C8C2×C3⋊D3semidirect
302. C33C2×#135semidirect
367. C2×C32#904semidirect
368. C2×C32#950semidirect
430. C32C2×#904semidirect
431. C32C2×#950semidirect
461. C2×C3#63829semidirect
462. C2×C3#64113semidirect
482. C3C2×#63829semidirect
483. C3C2×#64113semidirect
487. C2#11723369direct (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×#38282
C22×C32×C3⋊C82
C2×#888002
C2×#887972
C2×#887962
#117233692
C2×#887912
C2×#641133
C2×#638293