Finite Group Databasebeta

← #13139050

Subgroups — #13139050

Order 1728.

769 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 769 conjugacy classes of subgroups — too many to draw. Only its 79 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×#88654C211yesmaximal, characteristic, complemented
2C2×#88529C211yesmaximal, characteristic, complemented
3#11721156C211yesmaximal, characteristic, complemented
4C2×#88436C211yesmaximal, characteristic, complemented
5#63648⋊C3C211yesmaximal, characteristic, complemented
6#63647⋊C3C211yesmaximal, characteristic, complemented
7#11720324C211yesmaximal, characteristic, complemented
8#10590945—31nomaximal
9#10590944—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×C4⋊#272C2211yescharacteristic
12C2×C4⋊#272C2211yescharacteristic, complemented
13C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
14C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
15C2×#6200—21no—
16C2×#6200—21no—
17#88654—22no—
18#88654—22no—
19C2×C8×H(3,𝔽3)C2211yescharacteristic, complemented
20#88529—22no—
21C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
22C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
23#88451—21no—
24#88436—22no—
25C2×#3802—31no—
26C2×D3×D12—31no—
27C2×C3×D24—31no—
28C2×C3×D24—31no—
29C2×C3×D24—61no—
30#63916—31no—
31#63908—31no—
32C2×#3714—31no—
33C2×#3713—31no—
34#63648D311yescharacteristic, complemented
35#63647D311yescharacteristic, complemented
36C3×#821—31no—
37C3×(C2×D3)⋊C8—31no—
38#63563—31no—
39#63562—31no—
40C22×#272—21no—
41C22×#272—21no—
42C22×(C3⋊D3)⋊C3—21no—
43C22×(C3⋊D3)⋊C3D411yescharacteristic
44C2×((C3⋊D3)⋊C3)⋊C2—24no—
45C2×((C3⋊D3)⋊C3)⋊C2—42no—
46C2×((C3⋊D3)⋊C3)⋊C2—42no—
47C2×C4×H(3,𝔽3)C2311yescharacteristic
48C4⋊#272—21no—
49C4⋊#272—21no—
50C4⋊#272—21no—
51C4⋊#272D412yescomplemented
52C2×(C3⋊Dic3)⋊C3—21no—
53C2×(C3⋊Dic3)⋊C3D411yescharacteristic
54C4⋊((C3⋊D3)⋊C3)—21no—
55C4⋊((C3⋊D3)⋊C3)D412yescomplemented
56C4×(C3⋊D3)⋊C3—21no—
57C4×(C3⋊D3)⋊C3—21no—
58#6200—24no—
59#6200—42no—
60C8×H(3,𝔽3)—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62#5156—91no—
63#4773—91no—
64#4732—91no—
65C22×D32—61no—
66C22×D32—61no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C4×C3⋊D3C2×D311yescharacteristic
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12—61no—
76C2×C3×C4×D3—31no—
77C2×#626—61no—
78C2×#626—61no—
79C2×#625—61no—
80C2×#625—61no—
81#3802—62no—
82#3802—62no—
83D3×D12—62no—
84D3×D12—62no—
85C2×C32×C8—21no—
86C2×C32×C8C2×D311yescharacteristic, complemented
87C2×C32×C8C2×D311yescharacteristic, complemented
88C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
90C3×C4⋊Dic3—31no—
91C2×C3×C3⋊C8—31no—
92C3×D24—62no—
93C3×D24—62no—
94C3×D24—64no—
95(C2×D3)⋊Dic3—61no—
96(C2×D3)⋊Dic3—61no—
97#3714—62no—
98#3713—62no—
99C22×H(3,𝔽3)C2×D411yescharacteristic
100C2×#272—22no—
101C2×#272—41no—
102C2×#272—41no—
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—41no—
110C2×(C3⋊D3)⋊C3—41no—
111C2×(C3⋊D3)⋊C3D812yescomplemented
112((C3⋊D3)⋊C3)⋊C2—48no—
113C4×H(3,𝔽3)C2×D411yescharacteristic
114C4×H(3,𝔽3)C2×D411yescharacteristic
115(C3⋊Dic3)⋊C3—21no—
116(C3⋊Dic3)⋊C3—41no—
117C2×D3×D4—91no—
118C2×D3×D4—91no—
119C22×D12—91no—
120C2×C3×D8—91no—
121#938—91no—
122C2×#213—91no—
123C2×#213—91no—
124#930—91no—
125C2×D24—31no—
126C2×D24—31no—
127C2×D24—61no—
128C2×D24—91no—
129#895—91no—
130C3×#102—91no—
131C3×C22⋊C8—91no—
132#832—91no—
133#821—31no—
134#821—91no—
135#821—91no—
136(C2×D3)⋊C8—31no—
137(C2×D3)⋊C8—91no—
138#809—91no—
139C22×C3⋊D3—21no—
140C22×C3⋊D3D1211yescharacteristic
141C22×C3×D3—31no—
142C22×C3×D3—61no—
143C22×C3×D3—61no—
144C22×C3×D3—61no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C22×C3×D3—121no—
148C22×C3×D3—121no—
149C2×D32—64no—
150C2×D32—64no—
151C2×D32—122no—
152C2×D32—122no—
153C2×D32—122no—
154C2×D32—122no—
155C2×C32×C4—21no—
156C2×C32×C4C22×D311yescharacteristic
157C2×C32×C4C22×D311yescharacteristic
158C2×C3⋊Dic3—21no—
159C2×C3⋊Dic3D1211yescharacteristic
160C4⋊(C3⋊D3)—21no—
161C4⋊(C3⋊D3)D1212yescomplemented
162C4×C3⋊D3—21no—
163C4×C3⋊D3—21no—
164C2×C3×Dic3—31no—
165C2×C3×Dic3—61no—
166C3×D12—32no—
167C3×D12—32no—
168C3×D12—32no—
169C3×D12—62no—
170C3×D12—62no—
171C3×D12—61no—
172C3×D12—61no—
173C3×D12—61no—
174C3×D12—61no—
175C3×D12—61no—
176C3×D12—61no—
177C3×D12—61no—
178C3×D12—121no—
179C3×D12—121no—
180C3×C4×D3—61no—
181C3×C4×D3—61no—
182#626—64no—
183#626—122no—
184#625—64no—
185#625—122no—
186C32×C8—22no—
187C32×C8—21no—
188C32×C8—21no—
189C3⋊(C3⋊C8)—21no—
190C3×C3⋊C8—61no—
191#447—271noSylow
192C2×H(3,𝔽3)#12011yescharacteristic
193C2×H(3,𝔽3)C2×D811yescharacteristic
194C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
195#272—42no—
196#272—81no—
197(C3⋊D3)⋊C3—22no—
198(C3⋊D3)⋊C3—42no—
199C23×D3—181no—
200C23×D3—181no—
201C23×D3—181no—
202C2×C3×D4—91no—
203C2×C3×D4—91no—
204C2×C3×D4—91no—
205C22×C3×C4—91no—
206C2×#67—181no—
207C2×#67—181no—
208C2×#67—181no—
209C2×#67—181no—
210D3×D4—182no—
211D3×D4—182no—
212D3×D4—182no—
213D3×D4—182no—
214C2×D12—31no—
215C2×D12—31no—
216C2×D12—31no—
217C2×D12—31no—
218C2×D12—31no—
219C2×D12—61no—
220C2×D12—61no—
221C2×D12—91no—
222C2×D12—182no—
223C2×D12—182no—
224C2×D12—181no—
225C2×D12—181no—
226C2×C4×D3—31no—
227C2×C4×D3—91no—
228C3×D8—182no—
229C2×C3×C8—31no—
230C2×C3×C8—31no—
231C2×C3×C8—61no—
232C2×C3×C8—91no—
233C2×C3×C8D3211yescharacteristic
234C3×C4⋊C4—91no—
235#217—181no—
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—91no—
244D24—62no—
245D24—62no—
246D24—64no—
247D24—182no—
248C22×C32—21no—
249C22×C32C2×D1211yescharacteristic
250C22×C32D3×D411yescharacteristic
251C2×C3⋊D3—21no—
252C2×C3⋊D3—21no—
253C2×C3⋊D3—22no—
254C2×C3⋊D3—41no—
255C2×C3⋊D3—41no—
256C2×C3⋊D3D2412yescomplemented
257C2×C3×D3—32no—
258C2×C3×D3—61no—
259C2×C3×D3—61no—
260C2×C3×D3—62no—
261C2×C3×D3—62no—
262C2×C3×D3—62no—
263C2×C3×D3—122no—
264C2×C3×D3—121no—
265C2×C3×D3—121no—
266C2×C3×D3—121no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—122no—
271C2×C3×D3—121no—
272C2×C3×D3—121no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—241no—
278C2×C3×D3—241no—
279C2×C3×D3—241no—
280C2×C3×D3—241no—
281D32—128no—
282D32—128no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4C2×D1211yescharacteristic
286C32×C4D3×D411yescharacteristic
287C32×C4D3×D411yescharacteristic
288C32×C4D3×D411yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—41no—
291C3×Dic3—61no—
292C3×Dic3—121no—
293C22×D4—271no—
294C2×D8—91no—
295C2×D8—271no—
296#121—271no—
297#102—91no—
298#102—271no—
299C22⋊C8—91no—
300H(3,𝔽3)#44711yescharacteristic, complemented, Sylow
301C23×C3—91no—
302C23×C3—181no—
303C23×C3—181no—
304C23×C3—181no—
305C22×D3—31no—
306C22×D3—61no—
307C22×D3—61no—
308C22×D3—61no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—91no—
312C22×D3—121no—
313C22×D3—121no—
314C22×D3—181no—
315C22×D3—184no—
316C22×D3—184no—
317C22×D3—184no—
318C22×D3—362no—
319C22×D3—362no—
320C22×D3—362no—
321C22×D3—362no—
322C22×D3—362no—
323C22×D3—362no—
324C3×D4—92no—
325C3×D4—92no—
326C3×D4—181no—
327C3×D4—181no—
328C3×D4—181no—
329C3×D4—181no—
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—184no—
339#67—184no—
340#67—362no—
341#67—362no—
342C2×Dic3—31no—
343C2×Dic3—61no—
344C2×Dic3—91no—
345C2×Dic3—181no—
346D12—32no—
347D12—32no—
348D12—32no—
349D12—62no—
350D12—61no—
351D12—61no—
352D12—61no—
353D12—61no—
354D12—61no—
355D12—61no—
356D12—62no—
357D12—61no—
358D12—92no—
359D12—121no—
360D12—121no—
361D12—181no—
362D12—184no—
363D12—362no—
364C4×D3—61no—
365C4×D3—61no—
366C4×D3—181no—
367C4×D3—181no—
368C3×C8—21no—
369C3×C8—61no—
370C3×C8—61no—
371C3×C8—62no—
372C3×C8—181no—
373C3⋊C8—61no—
374C3⋊C8—181no—
375C2×C32—21no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32#88211yescharacteristic
379C2×C32#88211yescharacteristic
380C2×C32C2×D2411yescharacteristic
381C2×C32#90811yescharacteristic
382C2×C32D3×D811yescharacteristic
383C2×C32#91411yescharacteristic
384C3⋊D3—22no—
385C3⋊D3—42no—
386C3×D3—62no—
387C3×D3—122no—
388C3×D3—122no—
389C3×D3—122no—
390C3×D3—242no—
391C3×D3—242no—
392C3×D3—241no—
393C3×D3—241no—
394C24—541no—
395C2×D4—91no—
396C2×D4—91no—
397C2×D4—91no—
398C2×D4—541no—
399C2×D4—542no—
400C2×D4—542no—
401C2×D4—541no—
402C22×C4—91no—
403D8—182no—
404D8—542no—
405C2×C8—91no—
406C2×C8((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
407C4⋊C4—91no—
408C22⋊C4—541no—
409C22×C3—31no—
410C22×C3—31no—
411C22×C3—61no—
412C22×C3—92no—
413C22×C3—181no—
414C22×C3—181no—
415C22×C3—182no—
416C22×C3—182no—
417C22×C3—361no—
418C22×C3—361no—
419C22×C3—361no—
420C22×C3—361no—
421C22×C3—361no—
422C22×C3—361no—
423C22×C3—361no—
424C22×C3D3×D1211yescharacteristic
425C2×D3—32no—
426C2×D3—61no—
427C2×D3—61no—
428C2×D3—62no—
429C2×D3—62no—
430C2×D3—62no—
431C2×D3—92no—
432C2×D3—122no—
433C2×D3—121no—
434C2×D3—121no—
435C2×D3—121no—
436C2×D3—121no—
437C2×D3—121no—
438C2×D3—121no—
439C2×D3—121no—
440C2×D3—121no—
441C2×D3—121no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—122no—
446C2×D3—181no—
447C2×D3—181no—
448C2×D3—182no—
449C2×D3—241no—
450C2×D3—241no—
451C2×D3—241no—
452C2×D3—241no—
453C2×D3—361no—
454C2×D3—361no—
455C2×D3—368no—
456C2×D3—368no—
457C2×D3—368no—
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×C4D3×D1211yescharacteristic
467C3×C4#381111yescharacteristic
468Dic3—61no—
469Dic3—121no—
470Dic3—181no—
471Dic3—361no—
472C32—21no—
473C32#473211yescharacteristic, complemented
474C32#477311yescharacteristic, complemented
475C23—91no—
476C23—181no—
477C23—181no—
478C23—181no—
479C23—544no—
480C23—1082no—
481C23—1082no—
482D4—92no—
483D4—92no—
484D4—181no—
485D4—181no—
486D4—181no—
487D4—181no—
488D4—544no—
489D4—1082no—
490C2×C4—91no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4—181no—
494C2×C4C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
495C8—21no—
496C8—181no—
497C2×C3—31no—
498C2×C3—31no—
499C2×C3—31no—
500C2×C3—31no—
501C2×C3—31no—
502C2×C3—31no—
503C2×C3—61no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—182no—
507C2×C3—362no—
508C2×C3—362no—
509C2×C3—721no—
510C2×C3D3×D2411yescharacteristic
511C2×C3#6379411yescharacteristic
512C2×C3#6392311yescharacteristic
513D3—62no—
514D3—122no—
515D3—122no—
516D3—122no—
517D3—182no—
518D3—242no—
519D3—241no—
520D3—241no—
521D3—242no—
522D3—362no—
523C22—92no—
524C22—181no—
525C22—181no—
526C22—182no—
527C22—182no—
528C22—361no—
529C22—361no—
530C22—361no—
531C22—361no—
532C22—361no—
533C22—361no—
534C22—361no—
535C22—1088no—
536C22#8865411yescharacteristic
537C4—181no—
538C4—361no—
539C4#8865411yescharacteristic
540C4#8867711yescharacteristic
541C3—31no—
542C3—31no—
543C3—61no—
544C3#1059094711yescharacteristic
545C2—182no—
546C2—362no—
547C2—362no—
548C2—721no—
549C2#1172084711yescharacteristic
550C2#1172084811yescharacteristic
551C2#1172116311yescharacteristic
subgroupG/H
centerC22#88654
commutator, cosocleC2×C4×H(3,𝔽3)C23
FrattiniC2×C3×C4C2×D32
FittingC2×C8×H(3,𝔽3)C22
socleC22×C3D3×D12
hypercenterC2×C8((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×C8

Complements

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

NH ≅ G/Ntype
1. C2×#88654C2semidirect
2. C2×#88529C2semidirect
3. #11721156C2semidirect
4. C2×#88436C2semidirect
5. #63648⋊C3C2semidirect
6. #63647⋊C3C2semidirect
7. #11720324C2semidirect
12. C2×C4⋊#272C22semidirect
19. C2×C8×H(3,𝔽3)C22semidirect
21. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
22. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
34. #63648D3semidirect
35. #63647D3semidirect
51. C4⋊#272D4semidirect
55. C4⋊((C3⋊D3)⋊C3)D4semidirect
86. C2×C32×C8C2×D3semidirect
87. C2×C32×C8C2×D3semidirect
88. C4⋊(C3⋊Dic3)C2×D3semidirect
89. C2×C3⋊(C3⋊C8)C2×D3semidirect
111. C2×(C3⋊D3)⋊C3D8semidirect
161. C4⋊(C3⋊D3)D12semidirect
256. C2×C3⋊D3D24semidirect
300. H(3,𝔽3)#447semidirect
406. C2×C8((C3⋊D3)⋊C3)⋊C2semidirect
473. C32#4732semidirect
474. C32#4773semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#885292
#117211562
C2×#884362
#63648⋊C32
#63647⋊C32
#117203242
#105909453
#105909443