Finite Group Databasebeta

← C2×D3×Ω+(4,𝔽3) #72242403

Subgroups — C2×D3×Ω+(4,𝔽3) #72242403

Order 3456.

1568 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 1568 conjugacy classes of subgroups — too many to draw. Only its 91 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×Ω+(4,𝔽3)C211yesmaximal, characteristic, complemented
2D3×Ω+(4,𝔽3)C212yesmaximal, direct product factor
3C2×D3×#997C312yesmaximal, complemented
4C2×D3×#994C312yesmaximal, complemented
5C22×Ω+(4,𝔽3)—31nomaximal
6C2×C3×D3×SL(2,𝔽3)—42nomaximal
7C3×Ω+(4,𝔽3)C2211yescharacteristic, complemented
8C2×C3×#997C2×C312yescomplemented
9C2×C3×#994C2×C312yescomplemented
10D3×#997C2×C314yescomplemented
11D3×#994—34no—
12D3×#994C2×C314yescomplemented
13C2×D3×#41⋊C3—32no—
14C2×C3⋊#994—32no—
15C2×Ω+(4,𝔽3)—22no—
16C2×Ω+(4,𝔽3)—22no—
17C2×Ω+(4,𝔽3)—32no—
18C2×Ω+(4,𝔽3)D311yescharacteristic, direct product factor
19C2×C32×SL(2,𝔽3)—42no—
20C3×D3×SL(2,𝔽3)—44no—
21C2×D3×H(5,𝔽2)C3211yescharacteristic, complemented
22C22×#997—32no—
23C22×#994—32no—
24C22×D3×A4—122no—
25C2×C3×D3×Q8—42no—
26C3×#997C22×C312yescomplemented
27C3×#994—32no—
28C3×#994C22×C312yescomplemented
29C2×C3×#41⋊C3—32no—
30C22×C3×SL(2,𝔽3)—122no—
31D3×#41⋊C3—34no—
32D3×#41⋊C3—34no—
33C3⋊#994—34no—
34C3⋊#994—34no—
35C2×D3×SL(2,𝔽3)—44no—
36C2×D3×SL(2,𝔽3)A412yescomplemented
37C2×C3⋊(#41⋊C3)—32no—
38Ω+(4,𝔽3)—22no—
39Ω+(4,𝔽3)—22no—
40Ω+(4,𝔽3)C2×D311yescharacteristic, direct product factor
41C22×C32×D3—161no—
42C32×SL(2,𝔽3)—42no—
43C2×C3×H(5,𝔽2)C2×C3211yescharacteristic, complemented
44D3×H(5,𝔽2)—34no—
45D3×H(5,𝔽2)—92no—
46D3×H(5,𝔽2)C2×C3212yescomplemented
47C2×#1009—32no—
48C2×D3×#41—32no—
49C2×#1004—91no—
50C22×D3×D4—91no—
51C2×#997—22no—
52C2×#997—34no—
53C2×#997C3×D312yescomplemented
54C2×#994—22no—
55C2×#994—34no—
56C2×#994—92no—
57C2×#994—94no—
58C2×#994C3×D312yescomplemented
59C22×#41⋊C3—92no—
60C22×C3×A4—122no—
61C2×D3×A4—124no—
62C2×D3×A4—128no—
63C2×C32×Q8—42no—
64C3×D3×Q8—44no—
65C3×D3×Q8—124no—
66C2×C3×C4×D3—122no—
67C2×C3×Dic6—122no—
68C3×#41⋊C3—32no—
69C3×#41⋊C3—32no—
70C2×C3×SL(2,𝔽3)—42no—
71C2×C3×SL(2,𝔽3)—44no—
72C2×C3×SL(2,𝔽3)—84no—
73C2×C3×SL(2,𝔽3)—82no—
74C2×C3×SL(2,𝔽3)—82no—
75C2×C3×SL(2,𝔽3)—124no—
76C2×C3×SL(2,𝔽3)C2×A412yescomplemented
77D3×SL(2,𝔽3)—48no—
78D3×SL(2,𝔽3)C2×A414yescomplemented
79C3⋊(#41⋊C3)—34no—
80C22×H(5,𝔽2)—31noSylow
81C22×C33—161no—
82C2×C32×D3—162no—
83C2×C32×D3—164no—
84C24×D3—32no—
85C23×A4—362no—
86C3×H(5,𝔽2)—32no—
87C3×H(5,𝔽2)—91no—
88C3×H(5,𝔽2)C22×C3211yescharacteristic, complemented
89C2×C3×#41—32no—
90C22×C3×Q8—122no—
91C22×C3×D4—91no—
92C22×#67—92no—
93#1009—34no—
94#1009—34no—
95#1009—94no—
96#1009—94no—
97D3×#41—34no—
98D3×#41—34no—
99D3×#41—94no—
100D3×#41—94no—
101C2×#239—32no—
102C2×D3×Q8C3×A412yes—
103#1004—94no—
104#1004—98no—
105#1004—92no—
106#1004—92no—
107C2×#237—92no—
108C2×D3×D4—92no—
109C2×D3×D4—94no—
110C2×D3×D4—94no—
111C2×D3×D4—98no—
112C2×D3×D4—92no—
113C2×D3×D4—92no—
114C2×#235—92no—
115C22×D12—91no—
116C22×C4×D3—91no—
117#997—22no—
118#997C2×C3×D312yescomplemented
119#994—22no—
120#994—32no—
121#994—62no—
122#994—94no—
123#994—94no—
124#994C2×C3×D312yescomplemented
125C2×#41⋊C3—32no—
126C2×#41⋊C3—62no—
127C2×#41⋊C3—94no—
128C2×#41⋊C3—94no—
129C2×#41⋊C3—92no—
130C22×SL(2,𝔽3)—32no—
131C22×SL(2,𝔽3)—124no—
132C23×C32—481no—
133C22×C3×D3—42no—
134C22×C3×D3—162no—
135C2×C3×A4—122no—
136C2×C3×A4—124no—
137D3×A4—128no—
138C32×Q8—42no—
139C32×Q8—122no—
140C2×C32×C4—122no—
141C2×C3×Dic3—122no—
142C3×C4×D3—124no—
143C3×C4×D3—124no—
144C3×Dic6—124no—
145C3×Dic6—124no—
146C3×SL(2,𝔽3)—42no—
147C3×SL(2,𝔽3)—44no—
148C3×SL(2,𝔽3)—84no—
149C3×SL(2,𝔽3)—82no—
150C3×SL(2,𝔽3)—82no—
151C3×SL(2,𝔽3)C22×A412yescomplemented
152C2×H(5,𝔽2)—32no—
153C2×H(5,𝔽2)—92no—
154C2×H(5,𝔽2)—94no—
155C2×H(5,𝔽2)—271no—
156C2×H(5,𝔽2)—272no—
157C2×H(5,𝔽2)C32×D311yescharacteristic, complemented
158C22×#41—92no—
159C23×D4—271no—
160C2×C33—162no—
161C2×C33—161no—
162C32×D3—164no—
163C24×C3—32no—
164C23×D3—34no—
165C23×D3—91no—
166C23×D3—92no—
167C23×D3—94no—
168C23×D3—128no—
169C22×A4—122no—
170C22×A4—242no—
171C22×A4—364no—
172C22×A4—368no—
173C3×#41—32no—
174C3×#41—32no—
175C3×#41—92no—
176C3×#41—92no—
177C2×C3×Q8—42no—
178C2×C3×Q8—82no—
179C2×C3×Q8—124no—
180C2×C3×Q8—362no—
181C2×C3×Q8—364no—
182C2×C3×Q8C2×C3×A412yes—
183C2×C3×D4—92no—
184C2×C3×D4—92no—
185C2×C3×D4—94no—
186C2×C3×D4—91no—
187C2×C3×D4—91no—
188C22×C3×C4—91no—
189C22×C3×C4—362no—
190C2×#67—94no—
191C2×#67—94no—
192C2×#67—98no—
193C2×#67—94no—
194C22×Dic3—91no—
195#239—34no—
196#239—94no—
197D3×Q8—34no—
198D3×Q8C2×C3×A414yes—
199#237—98no—
200#237—94no—
201#237—94no—
202D3×D4—94no—
203D3×D4—94no—
204D3×D4—98no—
205D3×D4—916no—
206D3×D4—98no—
207D3×D4—98no—
208#235—94no—
209#235—94no—
210#235—94no—
211#235—94no—
212C2×D12—92no—
213C2×D12—92no—
214C2×D12—94no—
215C2×D12—92no—
216C2×C4×D3—32no—
217C2×C4×D3—92no—
218C2×C4×D3—94no—
219C2×C4×D3—92no—
220C2×C4×D3—92no—
221C2×Dic6—32no—
222#41⋊C3—32no—
223#41⋊C3—32no—
224#41⋊C3—62no—
225#41⋊C3—62no—
226#41⋊C3—94no—
227C2×SL(2,𝔽3)—22no—
228C2×SL(2,𝔽3)—34no—
229C2×SL(2,𝔽3)—44no—
230C2×SL(2,𝔽3)—84no—
231C2×SL(2,𝔽3)—128no—
232C2×SL(2,𝔽3)D3×A412yescomplemented
233C22×C32—42no—
234C22×C32—161no—
235C22×C32—162no—
236C22×C32—322no—
237C22×C32—322no—
238C22×C32—484no—
239C22×C32—482no—
240C2×C3×D3—44no—
241C2×C3×D3—48no—
242C2×C3×D3—164no—
243C2×C3×D3—168no—
244C3×A4—122no—
245C32×C4—122no—
246C32×C4—122no—
247C3×Dic3—124no—
248C25—92no—
249H(5,𝔽2)—32no—
250H(5,𝔽2)—91no—
251H(5,𝔽2)—94no—
252H(5,𝔽2)—94no—
253H(5,𝔽2)—272no—
254H(5,𝔽2)—278no—
255H(5,𝔽2)—274no—
256H(5,𝔽2)—272no—
257H(5,𝔽2)—274no—
258H(5,𝔽2)—274no—
259H(5,𝔽2)C2×C32×D311yescharacteristic, complemented
260C2×#41—32no—
261C2×#41—94no—
262C2×#41—94no—
263C2×#41—92no—
264C2×#41—272no—
265C2×#41—272no—
266C2×#41—274no—
267C2×#41—274no—
268C22×Q8—32no—
269C22×D4—91no—
270C22×D4—272no—
271C22×D4—271no—
272C22×D4—274no—
273C22×D4—274no—
274C22×D4—278no—
275C22×D4—272no—
276C22×D4—272no—
277C22×D4—272no—
278C23×C4—271no—
279C33—161noSylow
280C23×C3—32no—
281C23×C3—91no—
282C23×C3—92no—
283C23×C3—122no—
284C23×C3—124no—
285C23×C3—482no—
286C22×D3—91no—
287C22×D3—94no—
288C22×D3—92no—
289C22×D3—92no—
290C22×D3—94no—
291C22×D3—128no—
292C22×D3—184no—
293C22×D3—368no—
294C22×D3—364no—
295C22×D3A4211yescharacteristic
296C2×A4—122no—
297C2×A4—124no—
298C2×A4—242no—
299C2×A4—244no—
300C2×A4—368no—
301C3×Q8—32no—
302C3×Q8—42no—
303C3×Q8—82no—
304C3×Q8—122no—
305C3×Q8—242no—
306C3×Q8—364no—
307C3×Q8—364no—
308C3×Q8C22×C3×A412yes—
309C3×D4—92no—
310C3×D4—92no—
311C3×D4—94no—
312C2×C3×C4—32no—
313C2×C3×C4—91no—
314C2×C3×C4—92no—
315C2×C3×C4—91no—
316C2×C3×C4—122no—
317C2×C3×C4—242no—
318C2×C3×C4—364no—
319C2×C3×C4—364no—
320C2×C3×C4—362no—
321#67—98no—
322#67—98no—
323C2×Dic3—32no—
324C2×Dic3—92no—
325C2×Dic3—92no—
326D12—94no—
327D12—94no—
328C4×D3—34no—
329C4×D3—34no—
330C4×D3—94no—
331C4×D3—94no—
332Dic6—34no—
333Dic6—34no—
334SL(2,𝔽3)—22no—
335SL(2,𝔽3)—44no—
336SL(2,𝔽3)—84no—
337SL(2,𝔽3)C2×D3×A412yescomplemented
338C2×C32—44no—
339C2×C32—42no—
340C2×C32—161no—
341C2×C32—162no—
342C2×C32—164no—
343C2×C32—162no—
344C2×C32—322no—
345C2×C32—324no—
346C2×C32—322no—
347C2×C32—324no—
348C2×C32—484no—
349C3×D3—48no—
350C3×D3—168no—
351C24—32no—
352C24—94no—
353C24—271no—
354C24—272no—
355C24—274no—
356C24—368no—
357#41—32no—
358#41—32no—
359#41—92no—
360#41—92no—
361#41—94no—
362#41—274no—
363#41—274no—
364#41—274no—
365#41—274no—
366#41—278no—
367#41—274no—
368#41—274no—
369#41—274no—
370C2×Q8—34no—
371C2×Q8—92no—
372C2×Q8—94no—
373C2×Q8C3×D3×A412yes—
374C2×D4—92no—
375C2×D4—92no—
376C2×D4—94no—
377C2×D4—91no—
378C2×D4—91no—
379C2×D4—272no—
380C2×D4—274no—
381C2×D4—274no—
382C2×D4—278no—
383C2×D4—274no—
384C2×D4—272no—
385C2×D4—274no—
386C2×D4—272no—
387C2×D4—2716no—
388C2×D4—278no—
389C2×D4—278no—
390C2×D4—274no—
391C2×D4—278no—
392C2×D4—274no—
393C22×C4—91no—
394C22×C4—92no—
395C22×C4—272no—
396C22×C4—272no—
397C22×C4—274no—
398C22×C4—272no—
399C22×C4—271no—
400C22×C3—42no—
401C22×C3—82no—
402C22×C3—91no—
403C22×C3—91no—
404C22×C3—124no—
405C22×C3—128no—
406C22×C3—122no—
407C22×C3—162no—
408C22×C3—182no—
409C22×C3—322no—
410C22×C3—362no—
411C22×C3—484no—
412C22×C3—488no—
413C22×C3C2×A4211yescharacteristic
414C2×D3—92no—
415C2×D3—182no—
416C2×D3—184no—
417C2×D3—184no—
418C2×D3—364no—
419C2×D3—364no—
420C2×D3Ω+(4,𝔽3)14yesdirect product factor
421C2×D3C2×A4212yes—
422A4—122no—
423A4—242no—
424C3×C4—32no—
425C3×C4—32no—
426C3×C4—122no—
427C3×C4—122no—
428C3×C4—242no—
429C3×C4—242no—
430C3×C4—364no—
431Dic3—34no—
432C32—42no—
433C32—161no—
434C32—162no—
435C32—322no—
436C32—322no—
437C23—31no—
438C23—32no—
439C23—91no—
440C23—92no—
441C23—124no—
442C23—272no—
443C23—272no—
444C23—274no—
445C23—274no—
446C23—271no—
447C23—368no—
448C23—544no—
449C23—1084no—
450C23—1088no—
451Q8—32no—
452Q8—94no—
453Q8—94no—
454Q8C2×C3×D3×A412yes—
455D4—92no—
456D4—92no—
457D4—94no—
458D4—274no—
459D4—274no—
460D4—278no—
461D4—278no—
462C2×C4—32no—
463C2×C4—91no—
464C2×C4—92no—
465C2×C4—91no—
466C2×C4—94no—
467C2×C4—94no—
468C2×C4—92no—
469C2×C4—274no—
470C2×C4—274no—
471C2×C4—272no—
472C2×C4—272no—
473C2×C3—42no—
474C2×C3—44no—
475C2×C3—82no—
476C2×C3—84no—
477C2×C3—128no—
478C2×C3—162no—
479C2×C3—164no—
480C2×C3—181no—
481C2×C3—181no—
482C2×C3—322no—
483C2×C3—324no—
484C2×C3—488no—
485C2×C3C2×Ω+(4,𝔽3)12yescomplemented
486C2×C3C22×A4211yescharacteristic
487D3—182no—
488D3C2×Ω+(4,𝔽3)14yesdirect product factor
489C22—32no—
490C22—34no—
491C22—91no—
492C22—91no—
493C22—122no—
494C22—182no—
495C22—272no—
496C22—362no—
497C22—544no—
498C22—544no—
499C22—542no—
500C22—1084no—
501C22—1084no—
502C22D3×A4211yescharacteristic
503C4—32no—
504C4—32no—
505C4—94no—
506C3—42no—
507C3—82no—
508C3—162no—
509C3—322no—
510C3C22×Ω+(4,𝔽3)11yescharacteristic, complemented
511C2—34no—
512C2—181no—
513C2—181no—
514C2—542no—
515C2D3×Ω+(4,𝔽3)12yesdirect product factor
516C2C2×D3×A4211yescharacteristic
subgroupG/H
center, hypercenterC22D3×A42
commutator, cosocle, hypocenterC3×H(5,𝔽2)C22×C32
FrattiniC2C2×D3×A42
FittingC2×C3×H(5,𝔽2)C2×C32
socleC22×C3C2×A42
LCSG ▷ C3×H(5,𝔽2)
UCSC1 ◁ C22

Complements

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

NH ≅ G/Ntype
1. C2×C3×Ω+(4,𝔽3)C2semidirect
2. D3×Ω+(4,𝔽3)C2direct (N × H)
3. C2×D3×#997C3semidirect
4. C2×D3×#994C3semidirect
7. C3×Ω+(4,𝔽3)C22semidirect
8. C2×C3×#997C2×C3semidirect
9. C2×C3×#994C2×C3semidirect
10. D3×#997C2×C3semidirect
12. D3×#994C2×C3semidirect
18. C2×Ω+(4,𝔽3)D3direct (N × H)
21. C2×D3×H(5,𝔽2)C32semidirect
26. C3×#997C22×C3semidirect
28. C3×#994C22×C3semidirect
36. C2×D3×SL(2,𝔽3)A4semidirect
40. Ω+(4,𝔽3)C2×D3direct (N × H)
43. C2×C3×H(5,𝔽2)C2×C32semidirect
46. D3×H(5,𝔽2)C2×C32semidirect
53. C2×#997C3×D3semidirect
58. C2×#994C3×D3semidirect
76. C2×C3×SL(2,𝔽3)C2×A4semidirect
78. D3×SL(2,𝔽3)C2×A4semidirect
88. C3×H(5,𝔽2)C22×C32semidirect
118. #997C2×C3×D3semidirect
124. #994C2×C3×D3semidirect
151. C3×SL(2,𝔽3)C22×A4semidirect
157. C2×H(5,𝔽2)C32×D3semidirect
232. C2×SL(2,𝔽3)D3×A4semidirect
259. H(5,𝔽2)C2×C32×D3semidirect
337. SL(2,𝔽3)C2×D3×A4semidirect
420. C2×D3Ω+(4,𝔽3)direct (N × H)
485. C2×C3C2×Ω+(4,𝔽3)semidirect
488. D3C2×Ω+(4,𝔽3)direct (N × H)
510. C3C22×Ω+(4,𝔽3)semidirect
515. C2D3×Ω+(4,𝔽3)direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×Ω+(4,𝔽3)2
D3×Ω+(4,𝔽3)2
C2×D3×#9973
C2×D3×#9943
C22×Ω+(4,𝔽3)3
C2×C3×D3×SL(2,𝔽3)4