Finite Group Databasebeta

← #13140606

Subgroups — #13140606

Order 1728.

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

normal subgroups only

This group has 769 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×#88545C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63659⋊C3C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591456—31nomaximal
9#10591447—31nomaximal
10C23×(C3⋊D3)⋊C3—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic
12C2×C4⋊#272C2211yescharacteristic, complemented
13C2×#638⋊C3—21no—
14D4×(C3⋊D3)⋊C3—22no—
15D4×(C3⋊D3)⋊C3—22no—
16C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
17C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
18C2×#6200—21no—
19#88545—22no—
20C2×#6180C2211yescharacteristic, complemented
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88451—21no—
24#88436—22no—
25C2×D4×C3⋊D3D311yescharacteristic, complemented
26C2×C3×D3×D4—31no—
27C2×C3×#213—31no—
28C2×C3×#213—31no—
29C2×C3×#213—61no—
30#63916—31no—
31#63908—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63659D311yescharacteristic, complemented
35C3×#832—31no—
36#63564—31no—
37#63563—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—21no—
41C22×#272—21no—
42C22×(C3⋊D3)⋊C3—24no—
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
50C4⋊#272—21no—
51C4⋊#272—21no—
52#638⋊C3—24no—
53C2×(C3⋊Dic3)⋊C3—21no—
54C2×(C3⋊Dic3)⋊C3D411yescharacteristic
55C4⋊((C3⋊D3)⋊C3)—21no—
56C4⋊((C3⋊D3)⋊C3)D412yescomplemented
57C4×(C3⋊D3)⋊C3—21no—
58C4×(C3⋊D3)⋊C3—21no—
59#6200—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⋊D3C2×D311yescharacteristic
75C2×C3×#67—61no—
76C3×D3×D4—62no—
77C3×D3×D4—62no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×C4×D3—31no—
83C2×#626—61no—
84C2×#625—61no—
85C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
87C3×#213—64no—
88C3×#213—62no—
89C3×#213—62no—
90C3×C4⋊Dic3—31no—
91C2×C3×C3⋊C8—31no—
92C2×C3×C3⋊C8—31no—
93C2×C3×C3⋊C8—31no—
94C2×C3×C3⋊C8—61no—
95(C2×D3)⋊Dic3—61no—
96(C2×D3)⋊Dic3—61no—
97#3714—62no—
98#3713—62no—
99C22×H(3,𝔽3)—22no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)—41no—
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C2×#272—41no—
104C2×#272—41no—
105C2×#272—41no—
106C2×(C3⋊D3)⋊C3—21no—
107C2×(C3⋊D3)⋊C3—21no—
108C2×(C3⋊D3)⋊C3—22no—
109C2×(C3⋊D3)⋊C3—48no—
110C2×(C3⋊D3)⋊C3—41no—
111C2×(C3⋊D3)⋊C3—41no—
112C2×(C3⋊D3)⋊C3D812yescomplemented
113C4×H(3,𝔽3)C2×D411yescharacteristic
114C4×H(3,𝔽3)C2×D411yescharacteristic
115(C3⋊Dic3)⋊C3—21no—
116(C3⋊Dic3)⋊C3—41no—
117C22×C3×D4—91no—
118C2×D3×D4—31no—
119C2×D3×D4—91no—
120C2×C3×D8—91no—
121#938—91no—
122C2×#213—31no—
123C2×#213—31no—
124C2×#213—61no—
125C2×#213—91no—
126C2×#213—91no—
127#930—91no—
128C2×D24—91no—
129#895—91no—
130C3×#102—91no—
131#832—31no—
132#832—91no—
133#832—91no—
134#830—91no—
135#821—91no—
136(C2×D3)⋊C8—91no—
137(C2×D3)⋊C8—91no—
138#809—91no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—41no—
142C22×C3⋊D3—24no—
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—61no—
152C22×C3×D3—121no—
153C22×C3×D3—122no—
154C22×C3×D3—122no—
155C32×D4—21no—
156C32×D4—21no—
157C32×D4—22no—
158C32×D4—41no—
159C32×D4D1212yescomplemented
160C32×D4#6712yescomplemented
161C2×C32×C4—21no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164#638—24no—
165C2×C3⋊Dic3—21no—
166C2×C3⋊Dic3#6711yescharacteristic
167C4⋊(C3⋊D3)—21no—
168C4⋊(C3⋊D3)#6712yescomplemented
169C4×C3⋊D3—21no—
170C4×C3⋊D3—21no—
171C3×#67—64no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—61no—
174C3×D12—32no—
175C3×D12—62no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—61no—
181C3×D12—121no—
182C3×C4×D3—61no—
183C3×C4×D3—61no—
184#626—122no—
185#625—122no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—62no—
188C3×C3⋊C8—61no—
189C3×C3⋊C8—61no—
190C3×C3⋊C8—61no—
191#447—271noSylow
192C2×H(3,𝔽3)—42no—
193C2×H(3,𝔽3)#12011yescharacteristic
194C2×H(3,𝔽3)C2×D811yescharacteristic
195C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
196#272—81no—
197(C3⋊D3)⋊C3—22no—
198(C3⋊D3)⋊C3—42no—
199C24×C3—181no—
200C23×D3—61no—
201C23×D3—181no—
202C2×C3×D4—31no—
203C2×C3×D4—31no—
204C2×C3×D4—61no—
205C2×C3×D4—91no—
206C2×C3×D4—91no—
207C2×C3×D4—181no—
208C2×C3×D4—182no—
209C2×C3×D4—182no—
210C2×C3×D4D3211yescharacteristic
211C22×C3×C4—91no—
212C2×#67—61no—
213C2×#67—181no—
214C2×#67—181no—
215C2×#67—181no—
216D3×D4—62no—
217D3×D4—62no—
218D3×D4—182no—
219D3×D4—182no—
220C2×D12—31no—
221C2×D12—31no—
222C2×D12—31no—
223C2×D12—61no—
224C2×D12—91no—
225C2×D12—181no—
226C2×C4×D3—31no—
227C2×C4×D3—91no—
228C3×D8—182no—
229C2×C3×C8—91no—
230C2×C3×C8—91no—
231C3×C4⋊C4—91no—
232#217—181no—
233#213—62no—
234#213—62no—
235#213—64no—
236#213—182no—
237#213—182no—
238(C2×D3)⋊C4—181no—
239(C2×D3)⋊C4—181no—
240C4⋊Dic3—31no—
241C4⋊Dic3—91no—
242C2×C3⋊C8—31no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—61no—
246C2×C3⋊C8—91no—
247D24—182no—
248C22×C32—22no—
249C22×C32—22no—
250C22×C32—21no—
251C22×C32—42no—
252C22×C32—41no—
253C22×C32—41no—
254C22×C32—41no—
255C22×C32—41no—
256C22×C32—81no—
257C22×C32—81no—
258C22×C32D3×D411yescharacteristic
259C22×C32C2×#6711yescharacteristic
260C2×C3⋊D3—21no—
261C2×C3⋊D3—21no—
262C2×C3⋊D3—22no—
263C2×C3⋊D3—48no—
264C2×C3⋊D3—41no—
265C2×C3⋊D3—41no—
266C2×C3⋊D3#21312yescomplemented
267C2×C3×D3—32no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—62no—
271C2×C3×D3—128no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—122no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—121no—
280C2×C3×D3—121no—
281C2×C3×D3—241no—
282C2×C3×D3—241no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4C2×D1211yescharacteristic
286C32×C4D3×D411yescharacteristic
287C32×C4C2×#6711yescharacteristic
288C32×C4C2×#6711yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—41no—
291C3×Dic3—61no—
292C3×Dic3—121no—
293C22×D4—91no—
294C2×D8—91no—
295C2×D8—271no—
296#121—271no—
297#102—91no—
298#102—271no—
299C22⋊C8—271no—
300H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
301C23×C3—21no—
302C23×C3—61no—
303C23×C3—61no—
304C23×C3—91no—
305C23×C3—121no—
306C23×C3—181no—
307C23×C3—184no—
308C23×C3—181no—
309C23×C3—362no—
310C23×C3—362no—
311C22×D3—31no—
312C22×D3—64no—
313C22×D3—61no—
314C22×D3—61no—
315C22×D3—61no—
316C22×D3—91no—
317C22×D3—121no—
318C22×D3—122no—
319C22×D3—122no—
320C22×D3—181no—
321C22×D3—184no—
322C22×D3—362no—
323C22×D3—362no—
324C3×D4—21no—
325C3×D4—32no—
326C3×D4—32no—
327C3×D4—61no—
328C3×D4—61no—
329C3×D4—62no—
330C3×D4—92no—
331C3×D4—121no—
332C3×D4—181no—
333C3×D4—184no—
334C3×D4—181no—
335C3×D4—181no—
336C3×D4#62612yes—
337C2×C3×C4—31no—
338C2×C3×C4—31no—
339C2×C3×C4—61no—
340C2×C3×C4—91no—
341C2×C3×C4—181no—
342C2×C3×C4—181no—
343C2×C3×C4—181no—
344C2×C3×C4C2×D3211yescharacteristic
345#67—64no—
346#67—184no—
347#67—362no—
348#67—362no—
349C2×Dic3—31no—
350C2×Dic3—61no—
351C2×Dic3—91no—
352C2×Dic3—181no—
353D12—32no—
354D12—61no—
355D12—61no—
356D12—61no—
357D12—61no—
358D12—62no—
359D12—61no—
360D12—92no—
361D12—121no—
362D12—181no—
363D12—362no—
364C4×D3—61no—
365C4×D3—61no—
366C4×D3—181no—
367C4×D3—181no—
368C3×C8—181no—
369C3×C8—181no—
370C3⋊C8—61no—
371C3⋊C8—61no—
372C3⋊C8—62no—
373C3⋊C8—61no—
374C3⋊C8—181no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—42no—
379C2×C32—42no—
380C2×C32—82no—
381C2×C32#88211yescharacteristic
382C2×C32D3×D811yescharacteristic
383C2×C32#91411yescharacteristic
384C2×C32C2×#21311yescharacteristic
385C2×C32#93211yescharacteristic
386C2×C32#95311yescharacteristic
387C3⋊D3—22no—
388C3⋊D3—42no—
389C3×D3—62no—
390C3×D3—122no—
391C3×D3—242no—
392C3×D3—241no—
393C3×D3—241no—
394C24—181no—
395C2×D4—91no—
396C2×D4—91no—
397C2×D4—181no—
398C2×D4—182no—
399C2×D4—182no—
400C2×D4—541no—
401C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
402C22×C4—91no—
403D8—182no—
404D8—542no—
405C2×C8—91no—
406C2×C8—91no—
407C4⋊C4—91no—
408C22⋊C4—541no—
409C22×C3—22no—
410C22×C3—31no—
411C22×C3—31no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—62no—
415C22×C3—62no—
416C22×C3—61no—
417C22×C3—92no—
418C22×C3—122no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3—121no—
422C22×C3—121no—
423C22×C3—181no—
424C22×C3—181no—
425C22×C3—182no—
426C22×C3—241no—
427C22×C3—241no—
428C22×C3—361no—
429C22×C3—361no—
430C22×C3—368no—
431C22×C3—361no—
432C22×C3—361no—
433C22×C3—361no—
434C22×C3D3×#6711yescharacteristic
435C2×D3—32no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—62no—
439C2×D3—92no—
440C2×D3—128no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—121no—
449C2×D3—122no—
450C2×D3—181no—
451C2×D3—181no—
452C2×D3—182no—
453C2×D3—241no—
454C2×D3—241no—
455C2×D3—368no—
456C2×D3—361no—
457C2×D3—361no—
458C3×C4—31no—
459C3×C4—31no—
460C3×C4—31no—
461C3×C4—31no—
462C3×C4—61no—
463C3×C4—61no—
464C3×C4—181no—
465C3×C4—361no—
466C3×C4C2×#62611yescharacteristic
467C3×C4D3×#6711yescharacteristic
468Dic3—61no—
469Dic3—121no—
470Dic3—181no—
471Dic3—361no—
472C32—21no—
473C32#477311yescharacteristic, complemented
474C32#521711yescharacteristic, complemented
475C23—21no—
476C23—91no—
477C23—184no—
478C23—181no—
479C23—181no—
480C23—362no—
481C23—362no—
482D4—21no—
483D4—92no—
484D4—184no—
485D4—181no—
486D4—181no—
487D4—181no—
488D4—1082no—
489D4#620012yescomplemented
490C2×C4—91no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4—181no—
494C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
495C8—181no—
496C8—181no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—42no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—61no—
507C2×C3—122no—
508C2×C3—122no—
509C2×C3—182no—
510C2×C3—242no—
511C2×C3—362no—
512C2×C3—721no—
513C2×C3D3×#21311yescharacteristic
514C2×C3#6393411yescharacteristic
515C2×C3#6397711yescharacteristic
516D3—62no—
517D3—122no—
518D3—182no—
519D3—241no—
520D3—241no—
521D3—242no—
522D3—362no—
523C22—22no—
524C22—41no—
525C22—41no—
526C22—92no—
527C22—181no—
528C22—181no—
529C22—182no—
530C22—368no—
531C22—361no—
532C22—361no—
533C22—361no—
534C22—361no—
535C22—361no—
536C22#8867611yescharacteristic
537C4—181no—
538C4—361no—
539C4C2×#620011yescharacteristic
540C4#8867611yescharacteristic
541C3—31no—
542C3—31no—
543C3—61no—
544C3#1059144811yescharacteristic
545C2—42no—
546C2—182no—
547C2—362no—
548C2—721no—
549C2#1172119411yescharacteristic
550C2#1172119611yescharacteristic
551C2#1172132111yescharacteristic
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×#88545C2semidirect
3. #11721156C2semidirect
4. C2×#88436C2semidirect
5. #63659⋊C3C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
12. C2×C4⋊#272C22semidirect
20. C2×#6180C22semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
25. C2×D4×C3⋊D3D3semidirect
34. #63659D3semidirect
48. D4×H(3,𝔽3)D4semidirect
56. C4⋊((C3⋊D3)⋊C3)D4semidirect
85. C4⋊(C3⋊Dic3)C2×D3semidirect
86. C2×C3⋊(C3⋊C8)C2×D3semidirect
112. C2×(C3⋊D3)⋊C3D8semidirect
159. C32×D4D12semidirect
160. C32×D4#67semidirect
168. C4⋊(C3⋊D3)#67semidirect
266. C2×C3⋊D3#213semidirect
300. H(3,𝔽3)#447semidirect
473. C32#4773semidirect
474. C32#5217semidirect
489. D4#6200semidirect

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×#885452
#117211562
C2×#884362
#63659⋊C32
#117203182
#117203172
#105914563
#105914473