Finite Group Databasebeta

← C2×#11723553 #13177664

Subgroups — C2×#11723553 #13177664

Order 1728.

2732 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×C3⋊(C3⋊D3)C211yesmaximal, characteristic
2C2×#89019C211yesmaximal, characteristic, complemented
3C2×C3×#3754C211yesmaximal, characteristic, complemented
4C2×C32×#214C211yesmaximal, characteristic, complemented
5#11723553C218yesmaximal, direct product factor
6C2×#88797C211yesmaximal, characteristic, complemented
7C2×#88795C211yesmaximal, characteristic, complemented
8C2×#88785C211yesmaximal, characteristic, complemented
9C2×SD16×C3⋊D3—31nomaximal
10C2×#63930—34nomaximal
11C23×C3⋊(C3⋊D3)—21no—
12C2×C33×D4C2211yescharacteristic
13C2×#6312—21no—
14D4×C3⋊(C3⋊D3)—22no—
15D4×C3⋊(C3⋊D3)C2214yes—
16C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
17C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
18C2×C3×#634C2211yescharacteristic, complemented
19C2×C32×Dic6C2211yescharacteristic, complemented
20C2×#6293—21no—
21C2×#6289—21no—
22#89019—22no—
23#89019C2214yescomplemented
24C3×#3754C2214yescomplemented
25C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
26C32×#214C2214yescomplemented
27C2×C32×C3⋊C8C2211yescharacteristic, complemented
28#88797C2214yescomplemented
29#88795C2214yescomplemented
30#88785C2214yescomplemented
31C2×Q8×C3⋊D3—31no—
32C2×D4×C3⋊D3—31no—
33C2×D4×C3⋊D3—34no—
34C2×D4×C3⋊D3—64no—
35C2×#3797—34no—
36C2×C32×SD16—31no—
37C2×#3755—31no—
38C2×#3754D311yescharacteristic, complemented
39SD16×C3⋊D3—38no—
40C2×#3744—31no—
41C2×C8×C3⋊D3—31no—
42C2×C3×#214—34no—
43C2×C3×#214D314yescomplemented
44#63930—332no—
45C2×#3717—34no—
46C2×#3717—34no—
47C2×#3710—34no—
48C23×C33—21no—
49C22×C3⋊(C3⋊D3)—21no—
50C22×C3⋊(C3⋊D3)—24no—
51C22×C3⋊(C3⋊D3)—44no—
52C22×C3⋊(C3⋊D3)D411yescharacteristic
53C33×D4—21no—
54C33×D4C2312yes—
55C2×C33×C4C2311yescharacteristic
56#6312—24no—
57C2×C3⋊(C3⋊Dic3)D411yescharacteristic
58C4⋊(C3⋊(C3⋊D3))—21no—
59C4⋊(C3⋊(C3⋊D3))C2312yes—
60C4×C3⋊(C3⋊D3)D412yes—
61C4×C3⋊(C3⋊D3)C2312yes—
62C2×C3×C3⋊Dic3—21no—
63C3×#634—21no—
64C3×#634C2312yescomplemented
65C2×C32×Dic3—21no—
66C32×Dic6—21no—
67C32×Dic6C2312yescomplemented
68#6293—24no—
69#6289—24no—
70C3×C3⋊(C3⋊C8)C2312yescomplemented
71C32×C3⋊C8C2312yescomplemented
72C2×D3×SD16—94no—
73C2×D3×SD16—91no—
74C23×C3⋊D3—61no—
75C23×C3⋊D3—64no—
76C23×C3⋊D3—68no—
77C2×C32×Q8—31no—
78C2×C32×D4—24no—
79C2×C32×D4C2×D314yes—
80C2×C32×D4C2×D311yescharacteristic
81C2×#638—61no—
82C2×#638—64no—
83C2×#638—68no—
84Q8×C3⋊D3—34no—
85Q8×C3⋊D3—62no—
86D4×C3⋊D3—34no—
87D4×C3⋊D3—316no—
88D4×C3⋊D3—62no—
89D4×C3⋊D3—68no—
90D4×C3⋊D3—616no—
91D4×C3⋊D3—616no—
92C2×C4⋊(C3⋊D3)—31no—
93C2×C4⋊(C3⋊D3)—34no—
94C2×C4⋊(C3⋊D3)—64no—
95C2×C4×C3⋊D3—31no—
96C2×C4×C3⋊D3—34no—
97C2×C4×C3⋊D3—61no—
98C2×C4×C3⋊D3—64no—
99C2×#634—61no—
100C2×#634C2×D311yescharacteristic, complemented
101C2×C3×Dic6—34no—
102C2×C3×Dic6C2×D314yescomplemented
103C2×#627—64no—
104C2×#624—64no—
105#3797—316no—
106#3797—68no—
107C32×SD16—34no—
108C2×C32×C8—31no—
109#3755—34no—
110#3754C2×D314yescomplemented
111C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
112#3744—34no—
113C8×C3⋊D3—34no—
114C3×#214—316no—
115C3×#214C2×D3116yescomplemented
116C2×C3×C3⋊C8—34no—
117C2×C3×C3⋊C8C2×D314yescomplemented
118#3717—316no—
119#3717—316no—
120#3710—316no—
121C22×C33—22no—
122C22×C33—42no—
123C22×C33C2×D411yescharacteristic
124C2×C3⋊(C3⋊D3)—22no—
125C2×C3⋊(C3⋊D3)—42no—
126C2×C3⋊(C3⋊D3)—48no—
127C2×C3⋊(C3⋊D3)SD1614yescomplemented
128C2×C3⋊(C3⋊D3)C2×D412yes—
129C33×C4C2×D411yescharacteristic
130C33×C4C2411yescharacteristic
131C3⋊(C3⋊Dic3)C2×D412yes—
132C3×C3⋊Dic3—22no—
133C32×Dic3—22no—
134C2×D3×Q8—94no—
135C2×D3×Q8—91no—
136C2×D3×D4—94no—
137C2×D3×D4—91no—
138C2×D3×D4—184no—
139C2×C3×SD16—34no—
140C2×C3×SD16—91no—
141C2×#215—94no—
142C2×#215—91no—
143C2×#214—34no—
144C2×#214C3⋊D311yescharacteristic, complemented
145D3×SD16—932no—
146D3×SD16—98no—
147C2×#204—94no—
148C2×#204—91no—
149C2×D3×C8—94no—
150C2×D3×C8—91no—
151C23×C32—28no—
152C23×C32—24no—
153C23×C32—21no—
154C22×C3⋊D3—31no—
155C22×C3⋊D3—34no—
156C22×C3⋊D3—64no—
157C22×C3⋊D3—64no—
158C22×C3⋊D3—616no—
159C22×C3⋊D3—68no—
160C22×C3⋊D3—632no—
161C22×C3⋊D3—64no—
162C22×C3⋊D3—61no—
163C22×C3⋊D3—124no—
164C22×C3⋊D3—1216no—
165C22×C3⋊D3—1232no—
166C32×Q8—32no—
167C32×Q8—61no—
168C32×D4—28no—
169C32×D4—24no—
170C32×D4—21no—
171C32×D4—28no—
172C32×D4C22×D318yes—
173C32×D4C22×D312yes—
174C2×C32×C4—24no—
175C2×C32×C4—61no—
176C2×C32×C4C22×D314yes—
177C2×C32×C4C22×D311yescharacteristic
178#638—64no—
179#638—616no—
180#638—632no—
181C2×C3⋊Dic3—21no—
182C2×C3⋊Dic3—31no—
183C2×C3⋊Dic3—34no—
184C2×C3⋊Dic3—64no—
185C4⋊(C3⋊D3)—32no—
186C4⋊(C3⋊D3)—38no—
187C4⋊(C3⋊D3)—68no—
188C4⋊(C3⋊D3)—68no—
189C4⋊(C3⋊D3)—64no—
190C4⋊(C3⋊D3)—61no—
191C4×C3⋊D3—32no—
192C4×C3⋊D3—32no—
193C4×C3⋊D3—38no—
194C4×C3⋊D3—38no—
195C4×C3⋊D3—68no—
196C4×C3⋊D3—68no—
197C4×C3⋊D3—64no—
198#634—21no—
199#634—64no—
200#634C22×D312yescomplemented
201C2×C3×Dic3—24no—
202C2×C3×Dic3—64no—
203C3×Dic6—24no—
204C3×Dic6—38no—
205C3×Dic6—64no—
206C3×Dic6C22×D318yescomplemented
207#627—616no—
208#624—616no—
209C32×C8—32no—
210C3⋊(C3⋊C8)C22×D312yescomplemented
211C3×C3⋊C8—38no—
212C3×C3⋊C8C22×D318yescomplemented
213C22×SD16—271noSylow
214C2×C33—42no—
215C2×C33C2×SD1612yescomplemented
216C2×C33C22×D411yescharacteristic
217C3⋊(C3⋊D3)—42no—
218C3⋊(C3⋊D3)C2×SD1614yescomplemented
219C23×D3—184no—
220C23×D3—181no—
221C23×D3—188no—
222C2×C3×Q8—34no—
223C2×C3×Q8—91no—
224C2×C3×D4—24no—
225C2×C3×D4D3214yes—
226C2×C3×D4C2×C3⋊D311yescharacteristic
227C2×#67—184no—
228C2×#67—181no—
229C2×#67—188no—
230D3×Q8—916no—
231D3×Q8—94no—
232D3×Q8—182no—
233D3×Q8—188no—
234D3×D4—916no—
235D3×D4—94no—
236D3×D4—1816no—
237D3×D4—188no—
238D3×D4—182no—
239D3×D4—1816no—
240C2×D12—94no—
241C2×D12—91no—
242C2×D12—184no—
243C2×C4×D3—94no—
244C2×C4×D3—91no—
245C2×C4×D3—184no—
246C2×C4×D3—181no—
247C2×C4×D3—184no—
248C2×Dic6—34no—
249C2×Dic6—181no—
250C2×Dic6—184no—
251C2×Dic6C2×C3⋊D311yescharacteristic, complemented
252C3×SD16—316no—
253C3×SD16—94no—
254C2×C3×C8—34no—
255C2×C3×C8—91no—
256#215—916no—
257#215—94no—
258#214—316no—
259#214C2×C3⋊D314yescomplemented
260C2×C3⋊C8—34no—
261C2×C3⋊C8C2×C3⋊D311yescharacteristic, complemented
262#204—916no—
263#204—94no—
264D3×C8—916no—
265D3×C8—94no—
266C22×C32—216no—
267C22×C32—28no—
268C22×C32—22no—
269C22×C32—24no—
270C22×C32—416no—
271C22×C32—48no—
272C22×C32—42no—
273C22×C32D3×D414yes—
274C22×C32D3×D411yescharacteristic
275C2×C3⋊D3—34no—
276C2×C3⋊D3—32no—
277C2×C3⋊D3—316no—
278C2×C3⋊D3—38no—
279C2×C3⋊D3—616no—
280C2×C3⋊D3—68no—
281C2×C3⋊D3—616no—
282C2×C3⋊D3—68no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—128no—
285C2×C3⋊D3—1232no—
286C2×C3⋊D3—1264no—
287C2×C3⋊D3—1216no—
288C2×C3⋊D3—128no—
289C2×C3⋊D3—122no—
290C32×C4—24no—
291C32×C4—24no—
292C32×C4—62no—
293C32×C4D3×D414yes—
294C32×C4D3×D411yescharacteristic
295C32×C4C23×D314yes—
296C32×C4C23×D311yescharacteristic
297C3⋊Dic3—22no—
298C3⋊Dic3—32no—
299C3⋊Dic3—38no—
300C3⋊Dic3—68no—
301C3×Dic3—28no—
302C3×Dic3—68no—
303C22×Q8—271no—
304C22×D4—271no—
305C2×SD16—31no—
306C2×SD16—91no—
307C2×SD16—271no—
308C2×SD16—271no—
309C2×SD16—278no—
310C22×C8—271no—
311C33C22×SD1611yescharacteristic, complemented, Sylow
312C23×C3—28no—
313C23×C3—24no—
314C23×C3—21no—
315C22×D3—94no—
316C22×D3—91no—
317C22×D3—184no—
318C22×D3—1816no—
319C22×D3—184no—
320C22×D3—188no—
321C22×D3—1832no—
322C22×D3—184no—
323C22×D3—181no—
324C22×D3—364no—
325C22×D3—3616no—
326C22×D3—3632no—
327C3×Q8—38no—
328C3×Q8—64no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×D4—28no—
332C3×D4—24no—
333C3×D4—21no—
334C3×D4—28no—
335C3×D4C2×D3218yes—
336C3×D4C22×C3⋊D312yes—
337C2×C3×C4—24no—
338C2×C3×C4—64no—
339C2×C3×C4—181no—
340C2×C3×C4C2×D3214yes—
341C2×C3×C4C22×C3⋊D311yescharacteristic
342#67—1816no—
343#67—184no—
344#67—1832no—
345C2×Dic3—21no—
346C2×Dic3—64no—
347C2×Dic3—94no—
348C2×Dic3—91no—
349C2×Dic3—184no—
350D12—98no—
351D12—92no—
352D12—188no—
353D12—188no—
354D12—184no—
355D12—181no—
356C4×D3—98no—
357C4×D3—92no—
358C4×D3—98no—
359C4×D3—92no—
360C4×D3—188no—
361C4×D3—188no—
362C4×D3—184no—
363C4×D3—1816no—
364Dic6—21no—
365Dic6—38no—
366Dic6—64no—
367Dic6—184no—
368Dic6—1816no—
369Dic6C22×C3⋊D312yescomplemented
370C3×C8—38no—
371C3×C8—92no—
372C3⋊C8—38no—
373C3⋊C8C22×C3⋊D312yescomplemented
374C2×C32—24no—
375C2×C32—28no—
376C2×C32—416no—
377C2×C32—48no—
378C2×C32—42no—
379C2×C32D3×SD1618yescomplemented
380C2×C32D3×SD1612yescomplemented
381C2×C32C2×D3×D414yes—
382C2×C32C2×D3×D411yescharacteristic
383C3⋊D3—34no—
384C3⋊D3—316no—
385C3⋊D3—616no—
386C3⋊D3—1216no—
387C3⋊D3—128no—
388C3⋊D3—122no—
389C24—541no—
390C2×Q8—31no—
391C2×Q8—91no—
392C2×Q8—274no—
393C2×Q8—541no—
394C2×Q8—542no—
395C2×D4—274no—
396C2×D4—271no—
397C2×D4—542no—
398C2×D4—541no—
399C2×D4D3×C3⋊D311yescharacteristic
400C22×C4—271no—
401C22×C4—541no—
402SD16—34no—
403SD16—94no—
404SD16—274no—
405SD16—274no—
406C2×C8—31no—
407C2×C8—91no—
408C2×C8—274no—
409C22×C3—216no—
410C22×C3—28no—
411C22×C3—22no—
412C22×C3—24no—
413C22×C3—416no—
414C22×C3—42no—
415C22×C3—48no—
416C22×C3#381114yes—
417C22×C3D4×C3⋊D311yescharacteristic
418C2×D3—98no—
419C2×D3—92no—
420C2×D3—94no—
421C2×D3—916no—
422C2×D3—188no—
423C2×D3—1816no—
424C2×D3—1816no—
425C2×D3—188no—
426C2×D3—182no—
427C2×D3—3632no—
428C2×D3—368no—
429C2×D3—3664no—
430C2×D3—3616no—
431C2×D3—362no—
432C2×D3—368no—
433C3×C4—24no—
434C3×C4—24no—
435C3×C4—68no—
436C3×C4—182no—
437C3×C4#381114yes—
438C3×C4D4×C3⋊D311yescharacteristic
439C3×C4C22×D3214yes—
440C3×C4C23×C3⋊D311yescharacteristic
441Dic3—22no—
442Dic3—68no—
443Dic3—98no—
444Dic3—92no—
445Dic3—188no—
446C32—24no—
447C32C2×D3×SD1614yescomplemented
448C32C2×D3×SD1611yescharacteristic, complemented
449C23—21no—
450C23—271no—
451C23—544no—
452C23—541no—
453C23—1084no—
454Q8—32no—
455Q8—61no—
456Q8—92no—
457Q8—181no—
458Q8—544no—
459D4—21no—
460D4—272no—
461D4—544no—
462D4—541no—
463D4C2×D3×C3⋊D312yes—
464C2×C4—61no—
465C2×C4—181no—
466C2×C4—271no—
467C2×C4—272no—
468C2×C4—272no—
469C2×C4—544no—
470C2×C4C2×D3×C3⋊D311yescharacteristic
471C8—32no—
472C8—92no—
473C2×C3—28no—
474C2×C3—24no—
475C2×C3—416no—
476C2×C3—48no—
477C2×C3—42no—
478C2×C3#6393018yescomplemented
479C2×C3SD16×C3⋊D312yescomplemented
480C2×C3C2×#381114yes—
481C2×C3C2×D4×C3⋊D311yescharacteristic
482D3—916no—
483D3—94no—
484D3—1816no—
485D3—3616no—
486D3—368no—
487D3—362no—
488C22—22no—
489C22—42no—
490C22—274no—
491C22—272no—
492C22—542no—
493C22—1088no—
494C22—1082no—
495C22#8903911yescharacteristic
496C4—62no—
497C4—182no—
498C4—272no—
499C4#8903911yescharacteristic
500C4C22×D3×C3⋊D311yescharacteristic
501C3—24no—
502C3C2×#6393014yescomplemented
503C3C2×SD16×C3⋊D311yescharacteristic, complemented
504C2—42no—
505C2—274no—
506C2—1082no—
507C2#1172355312yesdirect product factor
508C2C2×#8903911yescharacteristic
subgroupG/H
centerC22#89039
commutator, cosocleC33×C4C24
FrattiniC4C22×D3×C3⋊D3
FittingC2×C33×D4C22
socleC22×C33C2×D4
prime radicalC2×D4×C3⋊(C3⋊D3)C2
hypercenterC2×D4D3×C3⋊D3
hypocenterC33C22×SD16
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
2. C2×#89019C2semidirect
3. C2×C3×#3754C2semidirect
4. C2×C32×#214C2semidirect
5. #11723553C2direct (N × H)
6. C2×#88797C2semidirect
7. C2×#88795C2semidirect
8. C2×#88785C2semidirect
18. C2×C3×#634C22semidirect
19. C2×C32×Dic6C22semidirect
23. #89019C22semidirect
24. C3×#3754C22semidirect
25. C2×C3×C3⋊(C3⋊C8)C22semidirect
26. C32×#214C22semidirect
27. C2×C32×C3⋊C8C22semidirect
28. #88797C22semidirect
29. #88795C22semidirect
30. #88785C22semidirect
38. C2×#3754D3semidirect
43. C2×C3×#214D3semidirect
64. C3×#634C23semidirect
67. C32×Dic6C23semidirect
70. C3×C3⋊(C3⋊C8)C23semidirect
71. C32×C3⋊C8C23semidirect
100. C2×#634C2×D3semidirect
102. C2×C3×Dic6C2×D3semidirect
110. #3754C2×D3semidirect
111. C2×C3⋊(C3⋊C8)C2×D3semidirect
115. C3×#214C2×D3semidirect
117. C2×C3×C3⋊C8C2×D3semidirect
127. C2×C3⋊(C3⋊D3)SD16semidirect
144. C2×#214C3⋊D3semidirect
200. #634C22×D3semidirect
206. C3×Dic6C22×D3semidirect
210. C3⋊(C3⋊C8)C22×D3semidirect
212. C3×C3⋊C8C22×D3semidirect
215. C2×C33C2×SD16semidirect
218. C3⋊(C3⋊D3)C2×SD16semidirect
251. C2×Dic6C2×C3⋊D3semidirect
259. #214C2×C3⋊D3semidirect
261. C2×C3⋊C8C2×C3⋊D3semidirect
311. C33C22×SD16semidirect
369. Dic6C22×C3⋊D3semidirect
373. C3⋊C8C22×C3⋊D3semidirect
379. C2×C32D3×SD16semidirect
380. C2×C32D3×SD16semidirect
447. C32C2×D3×SD16semidirect
448. C32C2×D3×SD16semidirect
478. C2×C3#63930semidirect
479. C2×C3SD16×C3⋊D3semidirect
502. C3C2×#63930semidirect
503. C3C2×SD16×C3⋊D3semidirect
507. C2#11723553direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D4×C3⋊(C3⋊D3)2
C2×#890192
C2×C3×#37542
C2×C32×#2142
#117235532
C2×#887972
C2×#887952
C2×#887852
C2×SD16×C3⋊D33
C2×#639303