Finite Group Databasebeta

← C2×#11723565 #13177712

Subgroups — C2×#11723565 #13177712

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×#89078C211yesmaximal, characteristic, complemented
2C2×#89017C211yesmaximal, characteristic, complemented
3C2×C3×#3753C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723565C218yesmaximal, direct product factor
6C2×#88799C211yesmaximal, characteristic, complemented
7C2×#88796C211yesmaximal, characteristic, complemented
8C2×#88785C211yesmaximal, characteristic, complemented
9C2×#64125—31nomaximal
10C2×#63936—34nomaximal
11C2×C33×D4C2211yescharacteristic
12C2×#6312—21no—
13C22×C3⋊(C3⋊Dic3)—21no—
14#89078—22no—
15#89078C2214yescomplemented
16C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
17C2×#6308C2211yescharacteristic
18C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
19C2×C32×Dic6C2211yescharacteristic
20C2×#6292—21no—
21C2×Dic3×C3⋊D3—21no—
22#89017—22no—
23#89017C2214yescomplemented
24C3×#3753C2214yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic
28#88799C2214yescomplemented
29#88796C2214yescomplemented
30#88785C2214yescomplemented
31C2×#3832—31no—
32C2×#3830—31no—
33C2×#3830—34no—
34C2×#3830—64no—
35C2×#3796—34no—
36C2×C32×SD16—31no—
37C2×#3756—31no—
38C2×#3753D311yescharacteristic, complemented
39#64125—38no—
40C2×#3744—31no—
41C2×C8×C3⋊D3—31no—
42C2×C3×#214D314yescomplemented
43C2×C3×#213—34no—
44#63936—332no—
45C2×#3719—34no—
46C2×#3716—34no—
47C2×#3710—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)D411yescharacteristic
50C22×C3×C3⋊D3—21no—
51C33×D4—21no—
52C33×D4C2312yes—
53C2×C33×C4C2311yescharacteristic
54#6312—24no—
55C2×C3⋊(C3⋊Dic3)—21no—
56C2×C3⋊(C3⋊Dic3)—24no—
57C2×C3⋊(C3⋊Dic3)D411yescharacteristic
58C4×C3⋊(C3⋊D3)D412yes—
59C4×C3⋊(C3⋊D3)C2312yes—
60#6308—21no—
61#6308C2312yes—
62C3×C4⋊(C3⋊D3)—21no—
63C3×C4⋊(C3⋊D3)C2312yes—
64C2×C32×Dic3—21no—
65C32×Dic6—21no—
66C32×Dic6C2312yes—
67#6292—24no—
68Dic3×C3⋊D3—24no—
69C3×C3⋊(C3⋊C8)C2312yes—
70C32×C3⋊C8C2312yes—
71C2×#916—94no—
72C2×#912—91no—
73C2×C32×Q8—31no—
74C2×C32×D4—24no—
75C2×C32×D4C2×D314yes—
76C2×C32×D4C2×D311yescharacteristic
77C2×#638—61no—
78C2×#638—64no—
79C2×#638—68no—
80C22×C3⋊Dic3—61no—
81C22×C3⋊Dic3—64no—
82C22×C3⋊Dic3—68no—
83#3832—34no—
84#3832—62no—
85#3830—34no—
86#3830—316no—
87#3830—62no—
88#3830—68no—
89#3830—616no—
90#3830—616no—
91C2×C4⋊(C3⋊D3)—61no—
92C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
93C2×C4×C3⋊D3—31no—
94C2×C4×C3⋊D3—34no—
95C2×C4×C3⋊D3—61no—
96C2×C4×C3⋊D3—64no—
97C2×#634—31no—
98C2×#634—34no—
99C2×#634—64no—
100C2×C3×D12—34no—
101C2×C3×Dic6C2×D314yes—
102C2×#626—64no—
103C2×D3×Dic3—64no—
104#3796—316no—
105#3796—68no—
106C32×SD16—34no—
107C2×C32×C8—31no—
108#3756—34no—
109#3753C2×D314yescomplemented
110C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
111#3744—34no—
112C8×C3⋊D3—34no—
113C3×#214C2×D3116yescomplemented
114C3×#213—316no—
115C2×C3×C3⋊C8—34no—
116C2×C3×C3⋊C8C2×D314yes—
117#3719—316no—
118#3716—316no—
119#3710—316no—
120C22×C33—22no—
121C22×C33—42no—
122C22×C33C2×D411yescharacteristic
123C2×C3⋊(C3⋊D3)—22no—
124C2×C3⋊(C3⋊D3)C2×D412yes—
125C2×C3×C3⋊D3—22no—
126C2×C3×C3⋊D3—42no—
127C33×C4C2×D411yescharacteristic
128C33×C4C2411yescharacteristic
129C3⋊(C3⋊Dic3)—22no—
130C3⋊(C3⋊Dic3)C2×D412yes—
131C32×Dic3—22no—
132C2×#239—94no—
133C2×#237—94no—
134C2×#237—91no—
135C2×#237—91no—
136C2×#237—184no—
137C2×C3×SD16—34no—
138C2×C3×D8—91no—
139C2×#216—94no—
140C2×#214—91no—
141C2×#214C3⋊D311yescharacteristic, complemented
142C2×#213—34no—
143#916—932no—
144#912—98no—
145C2×Dic12—91no—
146C2×#204—94no—
147C2×D3×C8—94no—
148C2×D3×C8—91no—
149C23×C32—28no—
150C23×C32—24no—
151C23×C32—21no—
152C22×C3⋊D3—21no—
153C22×C3⋊D3—31no—
154C22×C3⋊D3—34no—
155C22×C3⋊D3—64no—
156C22×C3×D3—64no—
157C32×Q8—32no—
158C32×Q8—61no—
159C32×D4—28no—
160C32×D4—24no—
161C32×D4—21no—
162C32×D4—28no—
163C32×D4C22×D318yes—
164C32×D4C22×D312yes—
165C2×C32×C4—24no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D314yes—
168C2×C32×C4C22×D311yescharacteristic
169#638—64no—
170#638—616no—
171#638—632no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—34no—
174C2×C3⋊Dic3—64no—
175C2×C3⋊Dic3—64no—
176C2×C3⋊Dic3—616no—
177C2×C3⋊Dic3—68no—
178C2×C3⋊Dic3—632no—
179C2×C3⋊Dic3—64no—
180C2×C3⋊Dic3—61no—
181C4⋊(C3⋊D3)—21no—
182C4⋊(C3⋊D3)—64no—
183C4⋊(C3⋊D3)C22×D312yes—
184C4×C3⋊D3—32no—
185C4×C3⋊D3—32no—
186C4×C3⋊D3—38no—
187C4×C3⋊D3—38no—
188C4×C3⋊D3—68no—
189C4×C3⋊D3—68no—
190C4×C3⋊D3—64no—
191#634—32no—
192#634—38no—
193#634—68no—
194#634—68no—
195#634—64no—
196#634—61no—
197C2×C3×Dic3—24no—
198C3×D12—38no—
199C3×D12—64no—
200C3×Dic6—24no—
201C3×Dic6C22×D318yes—
202#626—616no—
203D3×Dic3—616no—
204C32×C8—32no—
205C3⋊(C3⋊C8)C22×D312yes—
206C3×C3⋊C8—38no—
207C3×C3⋊C8C22×D318yes—
208C2×#135—271noSylow
209C2×C33—42no—
210C2×C33#13512yescomplemented
211C2×C33C22×D411yescharacteristic
212C3⋊(C3⋊D3)—22no—
213C3×C3⋊D3—42no—
214C2×C3×Q8—34no—
215C2×C3×D4—24no—
216C2×C3×D4—91no—
217C2×C3×D4D3214yes—
218C2×C3×D4C2×C3⋊D311yescharacteristic
219C2×#67—181no—
220C2×#67—184no—
221C2×#67—181no—
222C2×#67—188no—
223C22×Dic3—181no—
224C22×Dic3—184no—
225C22×Dic3—181no—
226C22×Dic3—188no—
227#239—916no—
228#239—188no—
229#237—916no—
230#237—94no—
231#237—94no—
232#237—1816no—
233#237—182no—
234#237—188no—
235#237—182no—
236#237—1816no—
237C2×D12—34no—
238C2×D12—184no—
239C2×C4×D3—94no—
240C2×C4×D3—91no—
241C2×C4×D3—184no—
242C2×C4×D3—184no—
243C2×Dic6—94no—
244C2×Dic6—91no—
245C2×Dic6—184no—
246C2×Dic6C2×C3⋊D311yescharacteristic
247C3×SD16—316no—
248C3×D8—94no—
249C2×C3×C8—34no—
250C2×C3×C8—91no—
251#216—916no—
252#214—94no—
253#214C2×C3⋊D314yescomplemented
254#213—316no—
255C2×C3⋊C8—34no—
256C2×C3⋊C8C2×C3⋊D311yescharacteristic
257Dic12—94no—
258#204—916no—
259D3×C8—916no—
260D3×C8—94no—
261C22×C32—216no—
262C22×C32—28no—
263C22×C32—22no—
264C22×C32—24no—
265C22×C32—416no—
266C22×C32—42no—
267C22×C32—48no—
268C22×C32D3×D414yes—
269C22×C32D3×D411yescharacteristic
270C2×C3⋊D3—22no—
271C2×C3⋊D3—32no—
272C2×C3⋊D3—38no—
273C2×C3⋊D3—42no—
274C2×C3⋊D3—68no—
275C2×C3⋊D3—62no—
276C2×C3⋊D3—68no—
277C2×C3⋊D3—128no—
278C2×C3×D3—68no—
279C2×C3×D3—128no—
280C32×C4—24no—
281C32×C4—24no—
282C32×C4—62no—
283C32×C4D3×D414yes—
284C32×C4D3×D411yescharacteristic
285C32×C4C23×D314yes—
286C32×C4C23×D311yescharacteristic
287C3⋊Dic3—32no—
288C3⋊Dic3—38no—
289C3⋊Dic3—68no—
290C3⋊Dic3—616no—
291C3⋊Dic3—68no—
292C3⋊Dic3—62no—
293C3×Dic3—28no—
294C2×#41—271no—
295C2×#41—271no—
296#135—278no—
297C2×Q16—271no—
298C2×SD16—31no—
299C2×SD16—271no—
300C2×D8—91no—
301C22×C8—271no—
302C33C2×#13511yescharacteristic, complemented, Sylow
303C23×C3—28no—
304C23×C3—24no—
305C23×C3—21no—
306C23×C3—181no—
307C22×D3—64no—
308C22×D3—94no—
309C22×D3—91no—
310C22×D3—184no—
311C3×Q8—38no—
312C3×Q8—64no—
313C3×D4—28no—
314C3×D4—24no—
315C3×D4—21no—
316C3×D4—28no—
317C3×D4—92no—
318C3×D4—181no—
319C3×D4C2×D3218yes—
320C3×D4C22×C3⋊D312yes—
321C2×C3×C4—24no—
322C2×C3×C4—64no—
323C2×C3×C4C2×D3214yes—
324C2×C3×C4C22×C3⋊D311yescharacteristic
325#67—184no—
326#67—1816no—
327#67—184no—
328#67—1832no—
329C2×Dic3—21no—
330C2×Dic3—94no—
331C2×Dic3—91no—
332C2×Dic3—184no—
333C2×Dic3—184no—
334C2×Dic3—1816no—
335C2×Dic3—184no—
336C2×Dic3—188no—
337C2×Dic3—1832no—
338C2×Dic3—184no—
339C2×Dic3—181no—
340D12—38no—
341D12—64no—
342D12—1816no—
343C4×D3—98no—
344C4×D3—92no—
345C4×D3—98no—
346C4×D3—92no—
347C4×D3—188no—
348C4×D3—188no—
349C4×D3—1816no—
350Dic6—21no—
351Dic6—98no—
352Dic6—92no—
353Dic6—188no—
354Dic6—188no—
355Dic6—184no—
356Dic6—181no—
357Dic6C22×C3⋊D312yes—
358C3×C8—38no—
359C3×C8—92no—
360C3⋊C8—38no—
361C3⋊C8C22×C3⋊D312yes—
362C2×C32—24no—
363C2×C32—28no—
364C2×C32—416no—
365C2×C32—42no—
366C2×C32—48no—
367C2×C32#91212yescomplemented
368C2×C32#91618yescomplemented
369C2×C32C2×D3×D414yes—
370C2×C32C2×D3×D411yescharacteristic
371C3⋊D3—42no—
372C3⋊D3—62no—
373C3⋊D3—68no—
374C3⋊D3—128no—
375C3×D3—128no—
376#41—274no—
377#41—274no—
378#41—542no—
379#41—542no—
380C2×Q8—31no—
381C2×Q8—271no—
382C2×D4—91no—
383C2×D4—541no—
384C2×D4—541no—
385C2×D4D3×C3⋊D311yescharacteristic
386C22×C4—271no—
387C22×C4—541no—
388C22×C4—541no—
389Q16—274no—
390SD16—34no—
391SD16—274no—
392D8—94no—
393C2×C8—31no—
394C2×C8—91no—
395C2×C8—274no—
396C22×C3—216no—
397C22×C3—28no—
398C22×C3—22no—
399C22×C3—24no—
400C22×C3—416no—
401C22×C3—42no—
402C22×C3—48no—
403C22×C3—182no—
404C22×C3—362no—
405C22×C3#381114yes—
406C22×C3D4×C3⋊D311yescharacteristic
407C2×D3—68no—
408C2×D3—98no—
409C2×D3—92no—
410C2×D3—128no—
411C2×D3—188no—
412C2×D3—182no—
413C2×D3—188no—
414C2×D3—368no—
415C3×C4—24no—
416C3×C4—24no—
417C3×C4—68no—
418C3×C4#381114yes—
419C3×C4D4×C3⋊D311yescharacteristic
420C3×C4C22×D3214yes—
421C3×C4C23×C3⋊D311yescharacteristic
422Dic3—22no—
423Dic3—98no—
424Dic3—92no—
425Dic3—188no—
426Dic3—1816no—
427Dic3—188no—
428Dic3—182no—
429C32—24no—
430C32C2×#91211yescharacteristic, complemented
431C32C2×#91614yescomplemented
432C23—21no—
433C23—181no—
434C23—271no—
435Q8—32no—
436Q8—61no—
437Q8—272no—
438Q8—541no—
439D4—21no—
440D4—92no—
441D4—181no—
442D4—544no—
443D4—544no—
444D4C2×D3×C3⋊D312yes—
445C2×C4—61no—
446C2×C4—271no—
447C2×C4—272no—
448C2×C4—272no—
449C2×C4—544no—
450C2×C4—544no—
451C2×C4—541no—
452C2×C4C2×D3×C3⋊D311yescharacteristic
453C8—32no—
454C8—92no—
455C2×C3—28no—
456C2×C3—24no—
457C2×C3—416no—
458C2×C3—48no—
459C2×C3—42no—
460C2×C3—362no—
461C2×C3#6393618yescomplemented
462C2×C3#6412512yescomplemented
463C2×C3C2×#381114yes—
464C2×C3C2×D4×C3⋊D311yescharacteristic
465D3—128no—
466D3—188no—
467D3—182no—
468D3—368no—
469C22—22no—
470C22—42no—
471C22—182no—
472C22—272no—
473C22—362no—
474C22—542no—
475C22#8903911yescharacteristic
476C4—62no—
477C4—272no—
478C4—542no—
479C4#8903911yescharacteristic
480C4C22×D3×C3⋊D311yescharacteristic
481C3—24no—
482C3C2×#6393614yescomplemented
483C3C2×#6412511yescharacteristic, complemented
484C2—42no—
485C2—362no—
486C2—542no—
487C2#1172356512yesdirect product factor
488C2C2×#8903911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C2×#135
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×#89078C2semidirect
2. C2×#89017C2semidirect
3. C2×C3×#3753C2semidirect
4. C2×C32×#214C2semidirect
5. #11723565C2direct (N × H)
6. C2×#88799C2semidirect
7. C2×#88796C2semidirect
8. C2×#88785C2semidirect
15. #89078C22semidirect
23. #89017C22semidirect
24. C3×#3753C22semidirect
26. C32×#214C22semidirect
28. #88799C22semidirect
29. #88796C22semidirect
30. #88785C22semidirect
38. C2×#3753D3semidirect
42. C2×C3×#214D3semidirect
109. #3753C2×D3semidirect
113. C3×#214C2×D3semidirect
141. C2×#214C3⋊D3semidirect
210. C2×C33#135semidirect
253. #214C2×C3⋊D3semidirect
302. C33C2×#135semidirect
367. C2×C32#912semidirect
368. C2×C32#916semidirect
430. C32C2×#912semidirect
431. C32C2×#916semidirect
461. C2×C3#63936semidirect
462. C2×C3#64125semidirect
482. C3C2×#63936semidirect
483. C3C2×#64125semidirect
487. C2#11723565direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#890782
C2×#890172
C2×C3×#37532
C2×C32×#2142
#117235652
C2×#887992
C2×#887962
C2×#887852
C2×#641253
C2×#639363