Finite Group Databasebeta

← #13140643

Subgroups — #13140643

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×#88545C211yesmaximal, characteristic, complemented
3#64147⋊C3C211yesmaximal, characteristic, complemented
4C2×#3753⋊C3C211yesmaximal, characteristic, complemented
5#11720324C211yesmaximal, characteristic, complemented
6#11720318C211yesmaximal, characteristic, complemented
7#11720317C211yesmaximal, characteristic, complemented
8#10591516—31nomaximal
9#10591515—31nomaximal
10C22×((C3⋊D3)⋊C3)⋊C2—21no—
11C2×D4×H(3,𝔽3)C2211yescharacteristic, complemented
12C2×C4⋊#272C2211yescharacteristic
13C2×#638⋊C3—21no—
14C2×C4⋊((C3⋊D3)⋊C3)C2211yescharacteristic
15C2×C4×(C3⋊D3)⋊C3C2211yescharacteristic
16C2×#6200—21no—
17#88654—22no—
18#88654—22no—
19#88545—22no—
20C2×#6180C2211yescharacteristic, complemented
21#3757⋊C3—21no—
22#3753⋊C3—22no—
23C4⋊((C3⋊Dic3)⋊C3)C2211yescharacteristic, complemented
24C2×(C3⋊(C3⋊C8))⋊C3C2211yescharacteristic, complemented
25C2×#3802—31no—
26C2×D3×D12—31no—
27#64147D311yescharacteristic, complemented
28C2×#3753D311yescharacteristic, complemented
29C3×#938—31no—
30C2×C3×#213—31no—
31C2×C3×#213—31no—
32C2×C3×#213—31no—
33C2×C3×#213—61no—
34#63564—31no—
35#63563—31no—
36#63563—31no—
37#63562—31no—
38#63557—31no—
39(C2×D3)⋊(C3⋊C8)—31no—
40C23×H(3,𝔽3)—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—
47D4×H(3,𝔽3)—21no—
48D4×H(3,𝔽3)—21no—
49C2×C4×H(3,𝔽3)C2311yescharacteristic
50C4⋊#272—21no—
51C4⋊#272D412yescomplemented
52#638⋊C3—42no—
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—24no—
60#6180—21no—
61(C3⋊(C3⋊C8))⋊C3—21no—
62#5156—91no—
63#5156—91no—
64#5036—91no—
65C22×D32—61no—
66C22×D32—61no—
67C2×C32×D4—21no—
68C2×C32×D4C2×D311yescharacteristic, complemented
69C2×C32×D4C2×D311yescharacteristic, complemented
70C2×#638—21no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×C4×C3⋊D3C2×D311yescharacteristic
73C2×C3×#67—61no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—31no—
77C2×C3×D12—61no—
78C2×C3×C4×D3—31no—
79C2×#626—61no—
80C2×#625—61no—
81#3802—62no—
82#3802—62no—
83D3×D12—62no—
84D3×D12—62no—
85#3757—21no—
86#3753—22no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic, complemented
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic, complemented
89C3×#217—61no—
90C3×#213—62no—
91C3×#213—62no—
92C3×#213—62no—
93C3×#213—64no—
94C3×C4⋊Dic3—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C22×H(3,𝔽3)—41no—
100C22×H(3,𝔽3)—41no—
101C22×H(3,𝔽3)—41no—
102C22×H(3,𝔽3)C2×D411yescharacteristic
103C2×#272—22no—
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—
121C2×C3×D8—91no—
122C3×#121—91no—
123#938—31no—
124#938—91no—
125C2×#213—31no—
126C2×#213—31no—
127C2×#213—31no—
128C2×#213—61no—
129C2×#213—91no—
130#832—91no—
131#832—91no—
132#830—91no—
133#821—91no—
134#821—91no—
135(C2×D3)⋊C8—91no—
136(C2×D3)⋊C8—91no—
137#809—91no—
138#809—91no—
139C23×C32—21no—
140C23×C32—21no—
141C23×C32—41no—
142C22×C3⋊D3—21no—
143C22×C3⋊D3#6711yescharacteristic
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C22×C3×D3—61no—
148C22×C3×D3—121no—
149C2×D32—64no—
150C2×D32—64no—
151C2×D32—122no—
152C2×D32—122no—
153C2×D32—122no—
154C2×D32—122no—
155C32×D4—22no—
156C32×D4—21no—
157C32×D4—21no—
158C32×D4—21no—
159C32×D4—21no—
160C32×D4—41no—
161C2×C32×C4—21no—
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164#638—42no—
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—122no—
172C2×C3×Dic3—31no—
173C2×C3×Dic3—61no—
174C3×D12—32no—
175C3×D12—32no—
176C3×D12—32no—
177C3×D12—62no—
178C3×D12—61no—
179C3×D12—61no—
180C3×D12—61no—
181C3×D12—121no—
182C3×C4×D3—61no—
183C3×C4×D3—61no—
184#626—64no—
185#625—64no—
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)—81no—
193C2×H(3,𝔽3)#12011yescharacteristic
194C2×H(3,𝔽3)C2×D811yescharacteristic
195C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
196#272—42no—
197(C3⋊D3)⋊C3—22no—
198(C3⋊D3)⋊C3—42no—
199C23×D3—181no—
200C23×D3—181no—
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×D4D3211yescharacteristic
209C22×C3×C4—91no—
210C2×#67—61no—
211C2×#67—181no—
212C2×#67—181no—
213C2×#67—181no—
214D3×D4—182no—
215D3×D4—182no—
216D3×D4—182no—
217D3×D4—182no—
218C2×D12—31no—
219C2×D12—31no—
220C2×D12—31no—
221C2×D12—61no—
222C2×D12—91no—
223C2×D12—182no—
224C2×D12—182no—
225C2×D12—181no—
226C2×C4×D3—31no—
227C2×C4×D3—91no—
228C3×D8—182no—
229C3×D8—182no—
230C2×C3×C8—91no—
231C2×C3×C8—91no—
232C3×C4⋊C4—91no—
233C3×C22⋊C4—181no—
234#217—61no—
235#217—181no—
236#213—62no—
237#213—62no—
238#213—62no—
239#213—64no—
240#213—182no—
241C4⋊Dic3—31no—
242C4⋊Dic3—91no—
243C2×C3⋊C8—31no—
244C2×C3⋊C8—31no—
245C2×C3⋊C8—31no—
246C2×C3⋊C8—61no—
247C2×C3⋊C8—91no—
248C22×C32—21no—
249C22×C32—42no—
250C22×C32—41no—
251C22×C32—41no—
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—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—62no—
271C2×C3×D3—62no—
272C2×C3×D3—122no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C2×C3×D3—121no—
276C2×C3×D3—121no—
277C2×C3×D3—121no—
278C2×C3×D3—121no—
279C2×C3×D3—241no—
280C2×C3×D3—241no—
281D32—128no—
282D32—128no—
283C32×C4—21no—
284C32×C4—21no—
285C32×C4D3×D411yescharacteristic
286C32×C4D3×D411yescharacteristic
287C32×C4D3×D411yescharacteristic
288C32×C4C2×#6711yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—41no—
291C3×Dic3—61no—
292C3×Dic3—121no—
293C22×D4—271no—
294C2×D8—91no—
295C2×D8—91no—
296#121—91no—
297#102—271no—
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—181no—
308C22×D3—31no—
309C22×D3—61no—
310C22×D3—61no—
311C22×D3—61no—
312C22×D3—91no—
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—21no—
325C3×D4—21no—
326C3×D4—61no—
327C3×D4—61no—
328C3×D4—61no—
329C3×D4—61no—
330C3×D4—62no—
331C3×D4—92no—
332C3×D4—92no—
333C3×D4—121no—
334C3×D4—181no—
335C3×D4—181no—
336C3×D4—362no—
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—122no—
346#67—184no—
347#67—184no—
348#67—362no—
349C2×Dic3—31no—
350C2×Dic3—61no—
351C2×Dic3—91no—
352C2×Dic3—181no—
353D12—32no—
354D12—32no—
355D12—32no—
356D12—62no—
357D12—61no—
358D12—61no—
359D12—61no—
360D12—92no—
361D12—121no—
362D12—181no—
363D12—184no—
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—82no—
379C2×C32—81no—
380C2×C32—81no—
381C2×C32D3×D811yescharacteristic
382C2×C32#91111yescharacteristic
383C2×C32C2×#21311yescharacteristic
384C2×C32#93711yescharacteristic
385C2×C32#93711yescharacteristic
386C2×C32#94911yescharacteristic
387C3⋊D3—22no—
388C3⋊D3—42no—
389C3×D3—62no—
390C3×D3—122no—
391C3×D3—122no—
392C3×D3—122no—
393C3×D3—242no—
394C24—541no—
395C2×D4—91no—
396C2×D4—91no—
397C2×D4—181no—
398C2×D4—541no—
399C2×D4—542no—
400C2×D4—542no—
401C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic, complemented
402C22×C4—91no—
403D8—182no—
404D8—182no—
405C2×C8—91no—
406C2×C8—91no—
407C4⋊C4—91no—
408C22⋊C4—181no—
409C22×C3—31no—
410C22×C3—31no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—61no—
415C22×C3—92no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3—121no—
420C22×C3—121no—
421C22×C3—121no—
422C22×C3—122no—
423C22×C3—181no—
424C22×C3—181no—
425C22×C3—182no—
426C22×C3—182no—
427C22×C3—241no—
428C22×C3—241no—
429C22×C3—361no—
430C22×C3—361no—
431C22×C3—361no—
432C22×C3—361no—
433C22×C3D3×#6711yescharacteristic
434C2×D3—32no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—62no—
438C2×D3—62no—
439C2×D3—62no—
440C2×D3—92no—
441C2×D3—122no—
442C2×D3—121no—
443C2×D3—121no—
444C2×D3—121no—
445C2×D3—121no—
446C2×D3—121no—
447C2×D3—121no—
448C2×D3—181no—
449C2×D3—181no—
450C2×D3—182no—
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×#6711yescharacteristic
467C3×C4#381111yescharacteristic
468Dic3—61no—
469Dic3—121no—
470Dic3—181no—
471Dic3—361no—
472C32—21no—
473C32#503611yescharacteristic, complemented
474C32#515611yescharacteristic, complemented
475C23—21no—
476C23—91no—
477C23—181no—
478C23—181no—
479C23—544no—
480C23—1082no—
481C23—1082no—
482D4—21no—
483D4—21no—
484D4—92no—
485D4—92no—
486D4—181no—
487D4—181no—
488D4—362no—
489D4—544no—
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—61no—
504C2×C3—61no—
505C2×C3—61no—
506C2×C3—81no—
507C2×C3—182no—
508C2×C3—241no—
509C2×C3—241no—
510C2×C3—242no—
511C2×C3—362no—
512C2×C3—362no—
513C2×C3D3×#21311yescharacteristic
514C2×C3#6392511yescharacteristic
515C2×C3#6398311yescharacteristic
516D3—62no—
517D3—122no—
518D3—122no—
519D3—122no—
520D3—182no—
521D3—242no—
522D3—362no—
523C22—41no—
524C22—41no—
525C22—41no—
526C22—92no—
527C22—181no—
528C22—181no—
529C22—182no—
530C22—182no—
531C22—361no—
532C22—361no—
533C22—361no—
534C22—361no—
535C22—1088no—
536C22#8867611yescharacteristic
537C4—181no—
538C4—361no—
539C4#8867611yescharacteristic
540C4#8867711yescharacteristic
541C3—31no—
542C3—31no—
543C3—61no—
544C3#1059151511yescharacteristic
545C2—81no—
546C2—182no—
547C2—362no—
548C2—362no—
549C2#1172119411yescharacteristic
550C2#1172119511yescharacteristic
551C2#1172132411yescharacteristic
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×#88654C2semidirect
2. C2×#88545C2semidirect
3. #64147⋊C3C2semidirect
4. C2×#3753⋊C3C2semidirect
5. #11720324C2semidirect
6. #11720318C2semidirect
7. #11720317C2semidirect
11. C2×D4×H(3,𝔽3)C22semidirect
20. C2×#6180C22semidirect
23. C4⋊((C3⋊Dic3)⋊C3)C22semidirect
24. C2×(C3⋊(C3⋊C8))⋊C3C22semidirect
27. #64147D3semidirect
28. C2×#3753D3semidirect
51. C4⋊#272D4semidirect
56. C4⋊((C3⋊D3)⋊C3)D4semidirect
68. C2×C32×D4C2×D3semidirect
69. C2×C32×D4C2×D3semidirect
87. C4⋊(C3⋊Dic3)C2×D3semidirect
88. C2×C3⋊(C3⋊C8)C2×D3semidirect
111. C2×(C3⋊D3)⋊C3D8semidirect
168. C4⋊(C3⋊D3)#67semidirect
265. C2×C3⋊D3#213semidirect
300. H(3,𝔽3)#447semidirect
401. C2×D4((C3⋊D3)⋊C3)⋊C2semidirect
473. C32#5036semidirect
474. C32#5156semidirect

Maximal subgroups

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

subgroupindex
C2×#886542
C2×#885452
#64147⋊C32
C2×#3753⋊C32
#117203242
#117203182
#117203172
#105915163
#105915153