Finite Group Databasebeta

← C2×#11720854 #13149840

Subgroups — C2×#11720854 #13149840

Order 1728.

999 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 999 conjugacy classes of subgroups — too many to draw. Only its 155 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×#88652C211yesmaximal, characteristic, complemented
2C2×#88648C211yesmaximal, characteristic, complemented
3C2×#88528C211yesmaximal, characteristic, complemented
4C2×#3744⋊C3C211yesmaximal, characteristic, complemented
5C2×C8×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
6C2×#88439C211yesmaximal, characteristic, complemented
7C2×#88436C211yesmaximal, characteristic, complemented
8#11720854C218yesmaximal, direct product factor
9C2×#63809—31nomaximal
10C2×#63806—31nomaximal
11C2×C4⋊#272C2211yescharacteristic
12C2×#6229C2211yescharacteristic
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#634⋊C3C2211yescharacteristic
16C2×#6200—21no—
17C2×#6199—21no—
18C2×#6198—21no—
19C2×((C3⋊D3)⋊C3)⋊C4—21no—
20#88652—22no—
21#88652C2214yescomplemented
22#88648—22no—
23#88648C2214yescomplemented
24C2×C8×H(3,𝔽3)C2211yescharacteristic
25#88528C2214yescomplemented
26C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic
27#3744⋊C3C2214yescomplemented
28C8×(C3⋊D3)⋊C3C2214yescomplemented
29#88439C2214yescomplemented
30#88436C2214yescomplemented
31C2×#3799—31no—
32C2×#3796—31no—
33C2×#3796—31no—
34C2×#3795—31no—
35C2×#3744D311yescharacteristic, complemented
36C2×C8×C3⋊D3D311yescharacteristic, complemented
37C2×C3×#204—31no—
38C2×C3×#204—31no—
39C2×C3×#204—31no—
40C2×C3×#204—61no—
41C2×C3×D3×C8—31no—
42C2×#3719—31no—
43C2×#3718—31no—
44C2×#3714—31no—
45C2×#3713—31no—
46#63809—38no—
47#63806—38no—
48C22×#272—21no—
49C22×(C3⋊D3)⋊C3—21no—
50C22×(C3⋊D3)⋊C3D411yescharacteristic
51C2×C4×H(3,𝔽3)C2311yescharacteristic
52C2×#1084—21no—
53C4⋊#272—21no—
54C4⋊#272C2312yes—
55#6229—21no—
56#6229C2312yes—
57C2×(C3⋊Dic3)⋊C3—21no—
58C2×(C3⋊Dic3)⋊C3D411yescharacteristic
59C4⋊((C3⋊D3)⋊C3)—21no—
60C4⋊((C3⋊D3)⋊C3)C2312yes—
61C4×(C3⋊D3)⋊C3D412yes—
62C4×(C3⋊D3)⋊C3C2312yes—
63#634⋊C3—21no—
64#634⋊C3C2312yes—
65#6200—24no—
66#6199—24no—
67#6198—24no—
68((C3⋊D3)⋊C3)⋊C4—24no—
69C8×H(3,𝔽3)C2312yes—
70(C3⋊(C3⋊C8))⋊C3C2312yes—
71C2×#916—91no—
72C2×#916—91no—
73C2×#904—91no—
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—61no—
81C2×C3×C4×D3—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86C2×#626—61no—
87C2×#626—61no—
88C2×#626—61no—
89C2×#625—61no—
90C2×#624—61no—
91C2×D3×Dic3—61no—
92C2×D3×Dic3—61no—
93C2×D3×Dic3—61no—
94#3799—34no—
95#3799—62no—
96#3796—34no—
97#3796—34no—
98#3796—62no—
99#3796—62no—
100#3795—34no—
101#3795—62no—
102C2×C32×C8—21no—
103C2×C32×C8C2×D311yescharacteristic
104C2×C32×C8C2×D311yescharacteristic
105C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
106#3744C2×D314yescomplemented
107C8×C3⋊D3C2×D314yescomplemented
108C2×C3×C3⋊C8—31no—
109C3×#204—34no—
110C3×#204—34no—
111C3×#204—34no—
112C3×#204—64no—
113C3×D3×C8—34no—
114#3719—34no—
115#3718—34no—
116#3714—34no—
117#3713—34no—
118C22×H(3,𝔽3)C2×D411yescharacteristic
119C2×#272—22no—
120C2×#272—42no—
121C2×(C3⋊D3)⋊C3—22no—
122C2×(C3⋊D3)⋊C3—22no—
123C2×(C3⋊D3)⋊C3—42no—
124C2×(C3⋊D3)⋊C3C2×D412yes—
125C4×H(3,𝔽3)C2×D411yescharacteristic
126C4×H(3,𝔽3)C2411yescharacteristic
127#1084—22no—
128(C3⋊Dic3)⋊C3—22no—
129(C3⋊Dic3)⋊C3C2×D412yes—
130C2×#239—91no—
131C2×#239—91no—
132C2×#237—91no—
133C2×#237—91no—
134C2×#235—91no—
135C2×#235—91no—
136C2×C3×SD16—91no—
137C2×C3×SD16—91no—
138C22×C3×C8—91no—
139C2×#216—91no—
140C2×#216—91no—
141C2×#213—91no—
142C2×#213—91no—
143#916—98no—
144#916—98no—
145C2×Dic12—91no—
146#904—98no—
147C2×D24—91no—
148C2×#204—31no—
149C2×#204—31no—
150C2×#204—31no—
151C2×#204—61no—
152C2×#204—91no—
153C2×D3×C8—31no—
154C2×D3×C8—91no—
155C22×C3⋊D3—21no—
156C22×C3⋊D3D1211yescharacteristic
157C22×C3×D3—31no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C22×C3×D3—121no—
162C2×C32×C4—21no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3D1211yescharacteristic
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)C22×D312yes—
169C4×C3⋊D3D1212yes—
170C4×C3⋊D3C22×D312yes—
171#634—21no—
172#634C22×D312yes—
173C2×C3×Dic3—31no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C3×D12—32no—
179C3×D12—32no—
180C3×D12—32no—
181C3×D12—62no—
182C3×D12—61no—
183C3×D12—61no—
184C3×D12—61no—
185C3×D12—121no—
186C3×C4×D3—32no—
187C3×C4×D3—32no—
188C3×Dic6—32no—
189C3×Dic6—32no—
190C3×Dic6—32no—
191C3×Dic6—62no—
192C3×Dic6—61no—
193C3×Dic6—61no—
194C3×Dic6—61no—
195C3×Dic6—121no—
196#626—64no—
197#626—64no—
198#626—64no—
199#625—64no—
200#624—64no—
201D3×Dic3—64no—
202D3×Dic3—64no—
203D3×Dic3—64no—
204C32×C8—22no—
205C32×C8C22×D312yes—
206C32×C8C22×D312yes—
207C3⋊(C3⋊C8)C22×D312yes—
208C3×C3⋊C8—32no—
209C2×#135—271noSylow
210C2×H(3,𝔽3)#13512yescomplemented
211C2×H(3,𝔽3)C22×D411yescharacteristic
212#272—42no—
213(C3⋊D3)⋊C3—22no—
214(C3⋊D3)⋊C3—42no—
215C2×C3×Q8—91no—
216C2×C3×Q8—91no—
217C2×C3×D4—91no—
218C2×C3×D4—91no—
219C22×C3×C4—91no—
220C2×#67—181no—
221C2×#67—181no—
222C2×#67—181no—
223C2×#67—181no—
224C22×Dic3—181no—
225C22×Dic3—181no—
226#239—94no—
227#239—94no—
228#239—182no—
229#239—182no—
230#237—94no—
231#237—94no—
232#237—182no—
233#237—182no—
234#235—94no—
235#235—94no—
236#235—182no—
237#235—182no—
238C2×D12—31no—
239C2×D12—31no—
240C2×D12—31no—
241C2×D12—61no—
242C2×D12—91no—
243C2×D12—181no—
244C2×D12—181no—
245C2×C4×D3—31no—
246C2×C4×D3—91no—
247C2×C4×D3—181no—
248C2×C4×D3—181no—
249C2×C4×D3—181no—
250C2×C4×D3—181no—
251C2×Dic6—31no—
252C2×Dic6—31no—
253C2×Dic6—31no—
254C2×Dic6—61no—
255C2×Dic6—91no—
256C3×SD16—94no—
257C3×SD16—94no—
258C2×C3×C8—31no—
259C2×C3×C8—31no—
260C2×C3×C8—61no—
261C2×C3×C8—94no—
262C2×C3×C8—91no—
263C2×C3×C8D3211yescharacteristic
264#216—94no—
265#216—94no—
266#213—94no—
267#213—94no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—91no—
270Dic12—94no—
271D24—94no—
272#204—34no—
273#204—34no—
274#204—34no—
275#204—64no—
276#204—94no—
277D3×C8—34no—
278D3×C8—94no—
279C22×C32—21no—
280C22×C32C2×D1211yescharacteristic
281C22×C32D3×D411yescharacteristic
282C2×C3⋊D3—22no—
283C2×C3⋊D3—22no—
284C2×C3⋊D3—42no—
285C2×C3⋊D3C2×D1212yes—
286C2×C3×D3—32no—
287C2×C3×D3—62no—
288C2×C3×D3—62no—
289C2×C3×D3—62no—
290C2×C3×D3—62no—
291C2×C3×D3—122no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—242no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4C2×D1211yescharacteristic
299C32×C4D3×D411yescharacteristic
300C32×C4C23×D311yescharacteristic
301C32×C4C23×D311yescharacteristic
302C3⋊Dic3—22no—
303C3⋊Dic3C2×D1212yes—
304C3×Dic3—32no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—62no—
308C3×Dic3—122no—
309C2×#41—271no—
310C2×#41—271no—
311#135—278no—
312C2×Q16—271no—
313C2×SD16—91no—
314C2×SD16—91no—
315C2×D8—271no—
316C22×C8—91no—
317H(3,𝔽3)C2×#13511yescharacteristic, complemented, Sylow
318C23×C3—91no—
319C23×C3—181no—
320C23×C3—181no—
321C22×D3—31no—
322C22×D3—61no—
323C22×D3—61no—
324C22×D3—61no—
325C22×D3—91no—
326C22×D3—121no—
327C22×D3—181no—
328C3×Q8—92no—
329C3×Q8—92no—
330C3×Q8—181no—
331C3×Q8—181no—
332C3×D4—92no—
333C3×D4—92no—
334C3×D4—181no—
335C3×D4—181no—
336C2×C3×C4—31no—
337C2×C3×C4—31no—
338C2×C3×C4—61no—
339C2×C3×C4—92no—
340C2×C3×C4—92no—
341C2×C3×C4—91no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4C2×D3211yescharacteristic
345#67—184no—
346#67—184no—
347#67—184no—
348#67—184no—
349C2×Dic3—31no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—61no—
353C2×Dic3—91no—
354C2×Dic3—121no—
355C2×Dic3—181no—
356C2×Dic3—184no—
357C2×Dic3—184no—
358D12—32no—
359D12—32no—
360D12—32no—
361D12—62no—
362D12—61no—
363D12—61no—
364D12—61no—
365D12—92no—
366D12—121no—
367D12—181no—
368D12—184no—
369D12—184no—
370C4×D3—32no—
371C4×D3—32no—
372C4×D3—92no—
373C4×D3—92no—
374C4×D3—184no—
375C4×D3—184no—
376C4×D3—184no—
377C4×D3—184no—
378Dic6—32no—
379Dic6—32no—
380Dic6—32no—
381Dic6—62no—
382Dic6—61no—
383Dic6—61no—
384Dic6—61no—
385Dic6—92no—
386Dic6—121no—
387Dic6—181no—
388C3×C8—32no—
389C3×C8—32no—
390C3×C8—62no—
391C3×C8—92no—
392C3×C8C2×D3212yes—
393C3⋊C8—32no—
394C3⋊C8—92no—
395C2×C32—21no—
396C2×C32—22no—
397C2×C32#90412yescomplemented
398C2×C32#91612yescomplemented
399C2×C32C22×D1211yescharacteristic
400C2×C32C2×D3×D411yescharacteristic
401C3⋊D3—22no—
402C3⋊D3—42no—
403C3×D3—62no—
404C3×D3—122no—
405C3×D3—122no—
406C3×D3—122no—
407C3×D3—242no—
408#41—274no—
409#41—274no—
410#41—542no—
411#41—542no—
412C2×Q8—91no—
413C2×Q8—91no—
414C2×D4—91no—
415C2×D4—91no—
416C2×D4—541no—
417C2×D4—541no—
418C22×C4—91no—
419C22×C4—541no—
420C22×C4—541no—
421Q16—274no—
422SD16—94no—
423SD16—94no—
424D8—274no—
425C2×C8—91no—
426C2×C8—94no—
427C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic
428C22×C3—31no—
429C22×C3—31no—
430C22×C3—61no—
431C22×C3—92no—
432C22×C3—182no—
433C22×C3—182no—
434C22×C3—182no—
435C22×C3—362no—
436C22×C3—362no—
437C22×C3D3×D1211yescharacteristic
438C2×D3—32no—
439C2×D3—62no—
440C2×D3—62no—
441C2×D3—62no—
442C2×D3—62no—
443C2×D3—92no—
444C2×D3—122no—
445C2×D3—122no—
446C2×D3—122no—
447C2×D3—122no—
448C2×D3—182no—
449C2×D3—182no—
450C2×D3—242no—
451C2×D3—362no—
452C3×C4—31no—
453C3×C4—31no—
454C3×C4—31no—
455C3×C4—31no—
456C3×C4—61no—
457C3×C4—61no—
458C3×C4—92no—
459C3×C4—182no—
460C3×C4—182no—
461C3×C4D3×D1211yescharacteristic
462C3×C4C22×D3211yescharacteristic
463Dic3—32no—
464Dic3—62no—
465Dic3—62no—
466Dic3—62no—
467Dic3—92no—
468Dic3—122no—
469Dic3—182no—
470C32—21no—
471C32C2×#90411yescharacteristic, complemented
472C32C2×#91611yescharacteristic, complemented
473C23—91no—
474C23—181no—
475C23—181no—
476Q8—92no—
477Q8—92no—
478Q8—181no—
479Q8—181no—
480D4—92no—
481D4—92no—
482D4—181no—
483D4—181no—
484D4—544no—
485D4—544no—
486C2×C4—91no—
487C2×C4—92no—
488C2×C4—92no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—544no—
492C2×C4—544no—
493C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
494C8—92no—
495C8C2×((C3⋊D3)⋊C3)⋊C212yes—
496C2×C3—32no—
497C2×C3—32no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—62no—
501C2×C3—61no—
502C2×C3—182no—
503C2×C3—362no—
504C2×C3—362no—
505C2×C3#6380612yes—
506C2×C3C2×D3×D1211yescharacteristic
507D3—62no—
508D3—122no—
509D3—122no—
510D3—122no—
511D3—182no—
512D3—242no—
513D3—362no—
514C22—92no—
515C22—182no—
516C22—182no—
517C22—182no—
518C22—362no—
519C22—362no—
520C22#8865411yescharacteristic
521C4—92no—
522C4—182no—
523C4—182no—
524C4#8865411yescharacteristic
525C4C22×((C3⋊D3)⋊C3)⋊C211yescharacteristic
526C3—31no—
527C3—31no—
528C3—61no—
529C3C2×#6380611yescharacteristic
530C2—182no—
531C2—362no—
532C2—362no—
533C2#1172085412yesdirect product factor
534C2C2×#8865411yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC4×H(3,𝔽3)C24
FrattiniC3×C4C22×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)C2×#135
LCSG ▷ C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88652C2semidirect
2. C2×#88648C2semidirect
3. C2×#88528C2semidirect
4. C2×#3744⋊C3C2semidirect
5. C2×C8×(C3⋊D3)⋊C3C2semidirect
6. C2×#88439C2semidirect
7. C2×#88436C2semidirect
8. #11720854C2direct (N × H)
21. #88652C22semidirect
23. #88648C22semidirect
25. #88528C22semidirect
27. #3744⋊C3C22semidirect
28. C8×(C3⋊D3)⋊C3C22semidirect
29. #88439C22semidirect
30. #88436C22semidirect
35. C2×#3744D3semidirect
36. C2×C8×C3⋊D3D3semidirect
106. #3744C2×D3semidirect
107. C8×C3⋊D3C2×D3semidirect
210. C2×H(3,𝔽3)#135semidirect
317. H(3,𝔽3)C2×#135semidirect
397. C2×C32#904semidirect
398. C2×C32#916semidirect
471. C32C2×#904semidirect
472. C32C2×#916semidirect
533. C2#11720854direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#886522
C2×#886482
C2×#885282
C2×#3744⋊C32
C2×C8×(C3⋊D3)⋊C32
C2×#884392
C2×#884362
#117208542
C2×#638093
C2×#638063