Finite Group Databasebeta

← #13140555

Subgroups — #13140555

Order 1728.

743 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 743 conjugacy classes of subgroups — too many to draw. Only its 87 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×D4×(C3⋊D3)⋊C3C211yesmaximal, characteristic, complemented
2C2×#3753⋊C3C211yesmaximal, characteristic, complemented
3#11721141C211yesmaximal, characteristic, complemented
4C2×#88432C211yesmaximal, characteristic, complemented
5#11720578C211yesmaximal, characteristic, complemented
6#11720317C211yesmaximal, characteristic, complemented
7#11720308C211yesmaximal, characteristic, complemented
8#10591456—31nomaximal
9#10591448—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×#638⋊C3—21no—
13D4×(C3⋊D3)⋊C3—22no—
14D4×(C3⋊D3)⋊C3—22no—
15C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic, complemented
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6198—21no—
19C4⋊#1084C2211yescharacteristic, complemented
20C2×#6180C2211yescharacteristic, complemented
21#3753⋊C3—22no—
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88450—21no—
24#88432—22no—
25C2×D4×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×D4—31no—
27C2×#3753D311yescharacteristic, complemented
28C2×C3×#213—31no—
29#63916—31no—
30#63909—31no—
31C2×#3714—31no—
32C2×#3714—31no—
33C3×#832—31no—
34C3×#832—31no—
35C3×#832—61no—
36#63564—31no—
37#63562—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×(C3⋊D3)⋊C3—24no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3—21no—
44C22×(C3⋊D3)⋊C3—42no—
45C22×(C3⋊D3)⋊C3—42no—
46C22×(C3⋊D3)⋊C3D411yescharacteristic
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)D412yescomplemented
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C2×#1084—21no—
51#638⋊C3—24no—
52C2×(C3⋊Dic3)⋊C3D411yescharacteristic
53C4⋊((C3⋊D3)⋊C3)—21no—
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D412yescomplemented
57C4×(C3⋊D3)⋊C3—21no—
58C4×(C3⋊D3)⋊C3—21no—
59#6198—42no—
60#6180—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62#5217—91no—
63#5156—91no—
64#4773—91no—
65C23×C3⋊D3—21no—
66C23×C3×D3—61no—
67C2×C32×D4—21no—
68C2×C32×D4C2×D311yescharacteristic
69C2×C32×D4C2×D311yescharacteristic
70C2×#638—21no—
71D4×C3⋊D3—22no—
72D4×C3⋊D3—22no—
73C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic, complemented
75C2×C4×C3⋊D3C2×D311yescharacteristic
76C2×C3×#67—61no—
77C3×D3×D4—62no—
78C3×D3×D4—62no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×C4×D3—31no—
82C2×#626—61no—
83C2×#626—61no—
84#3753—22no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
86C3×#213—62no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C2×C3×C3⋊C8—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—61no—
94#3722—61no—
95(C2×D3)⋊Dic3—61no—
96#3714—62no—
97#3714—62no—
98C22×H(3,𝔽3)—22no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)C2×D411yescharacteristic
102C2×(C3⋊D3)⋊C3—21no—
103C2×(C3⋊D3)⋊C3—21no—
104C2×(C3⋊D3)⋊C3—22no—
105C2×(C3⋊D3)⋊C3—48no—
106C2×(C3⋊D3)⋊C3—41no—
107C2×(C3⋊D3)⋊C3—41no—
108C2×(C3⋊D3)⋊C3—41no—
109C2×(C3⋊D3)⋊C3—41no—
110C2×(C3⋊D3)⋊C3—41no—
111C2×(C3⋊D3)⋊C3D812yescomplemented
112C4×H(3,𝔽3)C2×D411yescharacteristic
113C4×H(3,𝔽3)C2×D411yescharacteristic
114#1084—41no—
115(C3⋊Dic3)⋊C3—21no—
116C22×C3×D4—91no—
117C2×D3×D4—31no—
118C2×D3×D4—91no—
119C2×C3×D8—91no—
120#938—91no—
121C2×#213—31no—
122C2×#213—91no—
123C2×#213—91no—
124C2×#213—91no—
125#930—91no—
126C2×D24—91no—
127#895—91no—
128C3×#102—91no—
129#832—31no—
130#832—31no—
131#832—61no—
132#832—91no—
133#830—91no—
134#821—91no—
135(C2×D3)⋊C8—91no—
136(C2×D3)⋊C8—91no—
137#809—91no—
138C23×C32—21no—
139C23×C32—21no—
140C23×C32—41no—
141C22×C3⋊D3—24no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3—21no—
144C22×C3⋊D3—42no—
145C22×C3⋊D3—42no—
146C22×C3⋊D3#6711yescharacteristic
147C22×C3×D3—31no—
148C22×C3×D3—64no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—122no—
152C22×C3×D3—122no—
153C32×D4—21no—
154C32×D4—21no—
155C32×D4—22no—
156C32×D4—41no—
157C32×D4#6712yescomplemented
158C32×D4#6712yescomplemented
159C2×C32×C4—21no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162#638—24no—
163C2×C3⋊Dic3#6711yescharacteristic
164C4⋊(C3⋊D3)—21no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—21no—
167C4⋊(C3⋊D3)#6712yescomplemented
168C4×C3⋊D3—21no—
169C4×C3⋊D3—21no—
170C3×#67—64no—
171C2×C3×Dic3—31no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—121no—
175C3×D12—32no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—61no—
179C3×C4×D3—61no—
180C3×C4×D3—61no—
181#626—122no—
182#626—122no—
183C3⋊(C3⋊C8)—21no—
184C3×C3⋊C8—61no—
185C3×C3⋊C8—61no—
186C3×C3⋊C8—61no—
187C3×C3⋊C8—121no—
188#447—271noSylow
189C2×H(3,𝔽3)—42no—
190C2×H(3,𝔽3)#12011yescharacteristic
191C2×H(3,𝔽3)C2×D811yescharacteristic
192C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
193(C3⋊D3)⋊C3—22no—
194(C3⋊D3)⋊C3—42no—
195(C3⋊D3)⋊C3—81no—
196C24×C3—181no—
197C23×D3—61no—
198C23×D3—181no—
199C2×C3×D4—31no—
200C2×C3×D4—31no—
201C2×C3×D4—61no—
202C2×C3×D4—91no—
203C2×C3×D4—91no—
204C2×C3×D4—181no—
205C2×C3×D4—182no—
206C2×C3×D4—182no—
207C2×C3×D4D3211yescharacteristic
208C22×C3×C4—91no—
209C2×#67—61no—
210C2×#67—181no—
211C2×#67—181no—
212C2×#67—181no—
213D3×D4—62no—
214D3×D4—62no—
215D3×D4—182no—
216D3×D4—182no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—91no—
220C2×D12—91no—
221C2×D12—181no—
222C2×C4×D3—31no—
223C2×C4×D3—91no—
224C3×D8—182no—
225C2×C3×C8—91no—
226C2×C3×C8—91no—
227C3×C4⋊C4—91no—
228#217—181no—
229#213—62no—
230#213—182no—
231#213—182no—
232#213—182no—
233(C2×D3)⋊C4—181no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—61no—
238C2×C3⋊C8—31no—
239C2×C3⋊C8—31no—
240C2×C3⋊C8—31no—
241C2×C3⋊C8—61no—
242C2×C3⋊C8—91no—
243D24—182no—
244C22×C32—22no—
245C22×C32—22no—
246C22×C32—21no—
247C22×C32—42no—
248C22×C32—41no—
249C22×C32—41no—
250C22×C32—41no—
251C22×C32—41no—
252C22×C32—81no—
253C22×C32—81no—
254C22×C32D3×D411yescharacteristic
255C22×C32C2×#6711yescharacteristic
256C2×C3⋊D3—21no—
257C2×C3⋊D3—21no—
258C2×C3⋊D3—22no—
259C2×C3⋊D3—48no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—41no—
262C2×C3⋊D3—41no—
263C2×C3⋊D3—41no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3#21312yescomplemented
266C2×C3×D3—32no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—62no—
270C2×C3×D3—128no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4D3×D411yescharacteristic
279C32×C4C2×#6711yescharacteristic
280C32×C4C2×#6711yescharacteristic
281C32×C4C2×#6711yescharacteristic
282C3⋊Dic3—21no—
283C3×Dic3—61no—
284C3×Dic3—121no—
285C3×Dic3—121no—
286C3×Dic3—241no—
287C22×D4—91no—
288C2×D8—91no—
289C2×D8—271no—
290#121—271no—
291#102—91no—
292#102—271no—
293C22⋊C8—271no—
294H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
295C23×C3—21no—
296C23×C3—61no—
297C23×C3—61no—
298C23×C3—91no—
299C23×C3—121no—
300C23×C3—181no—
301C23×C3—181no—
302C23×C3—184no—
303C23×C3—362no—
304C23×C3—362no—
305C22×D3—31no—
306C22×D3—64no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—91no—
310C22×D3—122no—
311C22×D3—122no—
312C22×D3—181no—
313C22×D3—184no—
314C22×D3—181no—
315C22×D3—362no—
316C22×D3—362no—
317C3×D4—21no—
318C3×D4—32no—
319C3×D4—32no—
320C3×D4—61no—
321C3×D4—61no—
322C3×D4—62no—
323C3×D4—92no—
324C3×D4—121no—
325C3×D4—181no—
326C3×D4—181no—
327C3×D4—181no—
328C3×D4—184no—
329C3×D4#62512yes—
330C2×C3×C4—31no—
331C2×C3×C4—31no—
332C2×C3×C4—61no—
333C2×C3×C4—91no—
334C2×C3×C4—181no—
335C2×C3×C4—181no—
336C2×C3×C4—181no—
337C2×C3×C4C2×D3211yescharacteristic
338#67—64no—
339#67—184no—
340#67—362no—
341#67—362no—
342C2×Dic3—31no—
343C2×Dic3—61no—
344C2×Dic3—61no—
345C2×Dic3—91no—
346C2×Dic3—121no—
347D12—32no—
348D12—61no—
349D12—61no—
350D12—61no—
351D12—92no—
352D12—181no—
353D12—181no—
354D12—181no—
355D12—362no—
356C4×D3—61no—
357C4×D3—61no—
358C4×D3—181no—
359C4×D3—181no—
360C3×C8—181no—
361C3×C8—181no—
362C3⋊C8—61no—
363C3⋊C8—61no—
364C3⋊C8—61no—
365C3⋊C8—121no—
366C3⋊C8—181no—
367C2×C32—21no—
368C2×C32—21no—
369C2×C32—21no—
370C2×C32—42no—
371C2×C32—42no—
372C2×C32—82no—
373C2×C32D3×D811yescharacteristic
374C2×C32#91111yescharacteristic
375C2×C32C2×#21311yescharacteristic
376C2×C32#93211yescharacteristic
377C2×C32#93711yescharacteristic
378C2×C32#95311yescharacteristic
379C3⋊D3—22no—
380C3⋊D3—42no—
381C3⋊D3—81no—
382C3×D3—62no—
383C3×D3—122no—
384C3×D3—241no—
385C24—181no—
386C2×D4—91no—
387C2×D4—91no—
388C2×D4—181no—
389C2×D4—182no—
390C2×D4—182no—
391C2×D4—541no—
392C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
393C22×C4—91no—
394D8—182no—
395D8—542no—
396C2×C8—91no—
397C2×C8—91no—
398C4⋊C4—91no—
399C22⋊C4—541no—
400C22×C3—22no—
401C22×C3—31no—
402C22×C3—31no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—62no—
406C22×C3—62no—
407C22×C3—61no—
408C22×C3—92no—
409C22×C3—122no—
410C22×C3—121no—
411C22×C3—121no—
412C22×C3—121no—
413C22×C3—121no—
414C22×C3—181no—
415C22×C3—181no—
416C22×C3—182no—
417C22×C3—241no—
418C22×C3—241no—
419C22×C3—361no—
420C22×C3—361no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3—368no—
425C22×C3D3×#6711yescharacteristic
426C2×D3—32no—
427C2×D3—61no—
428C2×D3—61no—
429C2×D3—62no—
430C2×D3—92no—
431C2×D3—128no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—181no—
438C2×D3—181no—
439C2×D3—182no—
440C2×D3—368no—
441C2×D3—361no—
442C2×D3—361no—
443C2×D3—361no—
444C2×D3—361no—
445C2×D3—361no—
446C3×C4—31no—
447C3×C4—31no—
448C3×C4—31no—
449C3×C4—31no—
450C3×C4—61no—
451C3×C4—61no—
452C3×C4—181no—
453C3×C4—361no—
454C3×C4C2×#62511yescharacteristic
455C3×C4D3×#6711yescharacteristic
456Dic3—61no—
457Dic3—121no—
458Dic3—121no—
459Dic3—181no—
460Dic3—241no—
461C32—21no—
462C32#515611yescharacteristic, complemented
463C32#521711yescharacteristic, complemented
464C23—21no—
465C23—91no—
466C23—184no—
467C23—181no—
468C23—181no—
469C23—362no—
470C23—362no—
471D4—21no—
472D4—92no—
473D4—184no—
474D4—181no—
475D4—181no—
476D4—181no—
477D4—1082no—
478D4#619812yescomplemented
479C2×C4—91no—
480C2×C4—181no—
481C2×C4—181no—
482C2×C4—181no—
483C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
484C8—181no—
485C8—181no—
486C2×C3—31no—
487C2×C3—31no—
488C2×C3—31no—
489C2×C3—31no—
490C2×C3—31no—
491C2×C3—31no—
492C2×C3—42no—
493C2×C3—61no—
494C2×C3—61no—
495C2×C3—61no—
496C2×C3—122no—
497C2×C3—122no—
498C2×C3—182no—
499C2×C3—242no—
500C2×C3—362no—
501C2×C3—721no—
502C2×C3D3×#21311yescharacteristic
503C2×C3#6393211yescharacteristic
504C2×C3#6397611yescharacteristic
505D3—62no—
506D3—122no—
507D3—182no—
508D3—241no—
509D3—362no—
510D3—721no—
511C22—22no—
512C22—41no—
513C22—41no—
514C22—92no—
515C22—181no—
516C22—181no—
517C22—182no—
518C22—368no—
519C22—361no—
520C22—361no—
521C22—361no—
522C22—361no—
523C22—361no—
524C22#8867611yescharacteristic
525C4—181no—
526C4—361no—
527C4C2×#619811yescharacteristic
528C4#8867611yescharacteristic
529C3—31no—
530C3—31no—
531C3—61no—
532C3#1059144711yescharacteristic
533C2—42no—
534C2—182no—
535C2—362no—
536C2—721no—
537C2#1172119411yescharacteristic
538C2#1172120011yescharacteristic
539C2#1172131611yescharacteristic
subgroupG/H
centerC22#88676
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC22×C3D3×#67
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)#447
LCSG ▷ C2×C4×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. C2×D4×(C3⋊D3)⋊C3C2semidirect
2. C2×#3753⋊C3C2semidirect
3. #11721141C2semidirect
4. C2×#88432C2semidirect
5. #11720578C2semidirect
6. #11720317C2semidirect
7. #11720308C2semidirect
16. C2×C4⋊((C3⋊D3)⋊C3)C22semidirect
19. C4⋊#1084C22semidirect
20. C2×#6180C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
25. C2×D4×C3⋊D3D3semidirect
27. C2×#3753D3semidirect
48. D4×H(3,𝔽3)D4semidirect
56. C4⋊((C3⋊D3)⋊C3)D4semidirect
74. C2×C4⋊(C3⋊D3)C2×D3semidirect
85. C2×C3⋊(C3⋊C8)C2×D3semidirect
111. C2×(C3⋊D3)⋊C3D8semidirect
157. C32×D4#67semidirect
158. C32×D4#67semidirect
167. C4⋊(C3⋊D3)#67semidirect
265. C2×C3⋊D3#213semidirect
294. H(3,𝔽3)#447semidirect
462. C32#5156semidirect
463. C32#5217semidirect
478. D4#6198semidirect

Maximal subgroups

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

subgroupindex
C2×D4×(C3⋊D3)⋊C32
C2×#3753⋊C32
#117211412
C2×#884322
#117205782
#117203172
#117203082
#105914563
#105914483