Finite Group Databasebeta

← #13140605

Subgroups — #13140605

Order 1728.

859 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 859 conjugacy classes of subgroups — too many to draw. Only its 69 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
1#11722664C211yesmaximal, characteristic, complemented
2#64146⋊C3C211yesmaximal, characteristic, complemented
3#11721321C212yesmaximal, complemented
4#63661⋊C3C211yesmaximal, characteristic, complemented
5#11720335C212yesmaximal, complemented
6#10591531—31nomaximal
7#10591530—31nomaximal
8C23×(C3⋊D3)⋊C3C2211yescharacteristic, complemented
9C2×D4×H(3,𝔽3)C2211yescharacteristic
10C2×#6235—21no—
11C2×#6235C2212yes—
12#88744—22no—
13D4×#272—22no—
14#88740—21no—
15C2×#638⋊C3—21no—
16C2×#6200—22no—
17#3757⋊C3C2212yescomplemented
18#3757⋊C3C2211yescharacteristic, complemented
19#3752⋊C3—21no—
20#88450—22no—
21C3×#1004—31no—
22C3×#1004—31no—
23C3×#1004—61no—
24#64146D311yescharacteristic, complemented
25C3×#937—31no—
26#63981—32no—
27#63976—32no—
28#63661D311yescharacteristic, complemented
29C3×#834—31no—
30#63580—32no—
31#63580—32no—
32C23×H(3,𝔽3)D411yescharacteristic, complemented
33C23×H(3,𝔽3)C2311yescharacteristic
34C22×#272—21no—
35C22×#272D412yescomplemented
36C22×(C3⋊D3)⋊C3—22no—
37C22×(C3⋊D3)⋊C3—21no—
38C22×(C3⋊D3)⋊C3—42no—
39D4×H(3,𝔽3)—22no—
40C2×C4×H(3,𝔽3)D411yescharacteristic, complemented
41#6235—22no—
42#6235—22no—
43#6235—22no—
44C2×#1084—21no—
45C2×#1084D412yescomplemented
46C4⋊#272—21no—
47C4×#272—22no—
48#6229—21no—
49#638⋊C3—42no—
50C2×(C3⋊Dic3)⋊C3—22no—
51C2×(C3⋊Dic3)⋊C3—21no—
52#6200—44no—
53#5244—91no—
54#5032—91no—
55#4741—91no—
56C23×C3⋊D3C2×D311yescharacteristic, complemented
57C23×C3×D3—31no—
58C2×C32×D4—21no—
59C2×C32×D4C2×D311yescharacteristic
60C2×C32×D4C2×D311yescharacteristic
61C2×#638—21no—
62C2×C3×#67—32no—
63C2×C3×#67—32no—
64C2×C3×#67—62no—
65C2×C3×#67—62no—
66C2×C3×#67—61no—
67C2×C3×#67—61no—
68C2×C3×#67—61no—
69C3×#237—62no—
70C3×#237—62no—
71C3×#237—64no—
72C3×D3×D4—62no—
73C3×D3×D4—62no—
74C3×D3×D4—64no—
75C3×#235—62no—
76C3×#235—61no—
77C3×#235—61no—
78C2×#626—62no—
79C2×#625—62no—
80#3757C2×D311yescharacteristic, complemented
81#3757C2×D312yescomplemented
82#3752—21no—
83C3×#217—31no—
84C3×#217—32no—
85C3×(C2×D3)⋊C4—61no—
86#3722—62no—
87(C2×D3)⋊Dic3—62no—
88C22×H(3,𝔽3)—21no—
89C22×H(3,𝔽3)—41no—
90C22×H(3,𝔽3)—41no—
91C22×H(3,𝔽3)C2×D411yescharacteristic
92C22×H(3,𝔽3)C2×D412yes—
93C2×#272—22no—
94C2×#272—42no—
95C2×#272—81no—
96C2×(C3⋊D3)⋊C3—24no—
97C2×(C3⋊D3)⋊C3—22no—
98C2×(C3⋊D3)⋊C3—42no—
99C2×(C3⋊D3)⋊C3—41no—
100C2×(C3⋊D3)⋊C3—41no—
101C4×H(3,𝔽3)—21no—
102#1084—22no—
103(C3⋊Dic3)⋊C3—42no—
104(C3⋊Dic3)⋊C3—41no—
105C3×H(5,𝔽2)—91no—
106#1004—31no—
107#1004—31no—
108#1004—61no—
109C3×#120—91no—
110#953—92no—
111#937—31no—
112#937—91no—
113#937—92no—
114#882—92no—
115C3×#99—91no—
116#834—31no—
117#834—91no—
118#834—92no—
119#806—92no—
120#806—92no—
121C23×C32—21no—
122C23×C32—21no—
123C23×C32D1211yescharacteristic, complemented
124C23×C32#6711yescharacteristic, complemented
125C23×C32C22×D311yescharacteristic
126C23×C32C22×D311yescharacteristic
127C22×C3⋊D3—22no—
128C22×C3⋊D3—21no—
129C22×C3⋊D3—42no—
130C22×C3×D3—32no—
131C22×C3×D3—32no—
132C22×C3×D3—62no—
133C22×C3×D3—61no—
134C22×C3×D3—62no—
135C22×C3×D3—62no—
136C22×C3×D3—61no—
137C22×C3×D3—61no—
138C22×C3×D3—122no—
139C32×D4—24no—
140C32×D4—22no—
141C32×D4—22no—
142C2×C32×C4—21no—
143C2×C32×C4D1211yescharacteristic, complemented
144C2×C32×C4#6711yescharacteristic, complemented
145#638—42no—
146C2×C3⋊Dic3—22no—
147C2×C3⋊Dic3—21no—
148C3×#67—64no—
149C3×#67—64no—
150C3×#67—62no—
151C3×#67—62no—
152C3×#67—64no—
153C3×#67—62no—
154C3×#67—62no—
155C3×#67—62no—
156C3×#67—62no—
157C3×#67—122no—
158C2×C3×Dic3—32no—
159C2×C3×Dic3—32no—
160C2×C3×Dic3—62no—
161C2×C3×Dic3—62no—
162C2×C3×Dic3—61no—
163C2×C3×Dic3—61no—
164C2×C3×Dic3—61no—
165C2×C3×Dic3—62no—
166C3×D12—62no—
167C3×D12—61no—
168C3×D12—61no—
169C3×C4×D3—64no—
170C3×C4×D3—62no—
171C3×C4×D3—62no—
172C3×Dic6—62no—
173C3×Dic6—61no—
174C3×Dic6—61no—
175#626—124no—
176#625—124no—
177B(4,𝔽2)—271noSylow
178C2×H(3,𝔽3)—22no—
179C2×H(3,𝔽3)—21no—
180C2×H(3,𝔽3)—41no—
181C2×H(3,𝔽3)#12011yescharacteristic
182#272—42no—
183(C3⋊D3)⋊C3—42no—
184C24×C3—91no—
185C23×D3—31no—
186C23×D3—91no—
187C3×#41—182no—
188C3×#41—181no—
189C2×C3×D4—31no—
190C2×C3×D4—31no—
191C2×C3×D4—61no—
192C2×C3×D4—92no—
193C2×C3×D4—182no—
194C2×C3×D4—181no—
195C2×C3×D4—181no—
196C2×C3×D4D3211yescharacteristic
197C2×#67—32no—
198C2×#67—32no—
199C2×#67—62no—
200C2×#67—61no—
201C2×#67—61no—
202C2×#67—62no—
203C2×#67—61no—
204C2×#67—181no—
205C2×#67—182no—
206C2×#67—182no—
207#237—62no—
208#237—62no—
209#237—64no—
210D3×D4—62no—
211D3×D4—62no—
212D3×D4—64no—
213#235—61no—
214#235—61no—
215#235—62no—
216C2×D12—182no—
217C3×C22⋊C4—91no—
218C3×C22⋊C4—92no—
219C3×C22⋊C4—181no—
220#217—32no—
221#217—31no—
222#217—91no—
223#217—92no—
224#217—182no—
225(C2×D3)⋊C4—61no—
226(C2×D3)⋊C4—181no—
227(C2×D3)⋊C4—182no—
228(C2×D3)⋊C4—182no—
229C22×C32—22no—
230C22×C32—21no—
231C22×C32—21no—
232C22×C32—21no—
233C22×C32—22no—
234C22×C32—41no—
235C22×C32—41no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—81no—
239C22×C32—81no—
240C22×C32C2×D1211yescharacteristic
241C22×C32D3×D412yes—
242C22×C32D3×D412yes—
243C22×C32C2×#6711yescharacteristic
244C2×C3⋊D3—24no—
245C2×C3⋊D3—22no—
246C2×C3⋊D3—42no—
247C2×C3⋊D3—41no—
248C2×C3⋊D3—41no—
249C2×C3×D3—64no—
250C2×C3×D3—62no—
251C2×C3×D3—64no—
252C2×C3×D3—62no—
253C2×C3×D3—62no—
254C2×C3×D3—121no—
255C2×C3×D3—122no—
256C2×C3×D3—121no—
257C2×C3×D3—122no—
258C2×C3×D3—122no—
259C2×C3×D3—242no—
260C2×C3×D3—242no—
261C2×C3×D3—241no—
262C2×C3×D3—241no—
263C32×C4—22no—
264C32×C4—21no—
265C32×C4—21no—
266C3⋊Dic3—42no—
267C3⋊Dic3—41no—
268C3×Dic3—64no—
269C3×Dic3—62no—
270C3×Dic3—62no—
271C3×Dic3—122no—
272C3×Dic3—121no—
273H(5,𝔽2)—91no—
274#120—91no—
275#120—272no—
276#99—91no—
277#99—272no—
278H(3,𝔽3)B(4,𝔽2)11yescharacteristic, complemented, Sylow
279C23×C3—31no—
280C23×C3—31no—
281C23×C3—31no—
282C23×C3—31no—
283C23×C3—61no—
284C23×C3—61no—
285C23×C3—92no—
286C23×C3—181no—
287C23×C3—182no—
288C23×C3—181no—
289C23×C3—362no—
290C23×C3#62611yescharacteristic
291C23×C3C2×D3211yescharacteristic
292C22×D3—32no—
293C22×D3—32no—
294C22×D3—62no—
295C22×D3—61no—
296C22×D3—62no—
297C22×D3—61no—
298C22×D3—61no—
299C22×D3—62no—
300C22×D3—122no—
301C22×D3—182no—
302C22×D3—181no—
303C22×D3—362no—
304C3×Q8—181no—
305C3×D4—22no—
306C3×D4—62no—
307C3×D4—62no—
308C3×D4—64no—
309C3×D4—181no—
310C3×D4—182no—
311C3×D4—182no—
312C3×D4—182no—
313C3×D4—362no—
314C2×C3×C4—31no—
315C2×C3×C4—31no—
316C2×C3×C4—61no—
317C2×C3×C4—92no—
318C2×C3×C4—182no—
319C2×C3×C4—181no—
320C2×C3×C4—182no—
321C2×C3×C4—181no—
322C2×C3×C4#62611yescharacteristic
323#67—62no—
324#67—62no—
325#67—62no—
326#67—62no—
327#67—64no—
328#67—62no—
329#67—62no—
330#67—64no—
331#67—64no—
332#67—122no—
333#67—362no—
334#67—364no—
335#67—364no—
336C2×Dic3—32no—
337C2×Dic3—32no—
338C2×Dic3—62no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—62no—
342C2×Dic3—61no—
343C2×Dic3—62no—
344C2×Dic3—181no—
345C2×Dic3—182no—
346D12—61no—
347D12—61no—
348D12—62no—
349D12—364no—
350C4×D3—62no—
351C4×D3—62no—
352C4×D3—64no—
353Dic6—61no—
354Dic6—61no—
355Dic6—62no—
356C2×C32—21no—
357C2×C32—22no—
358C2×C32—22no—
359C2×C32—21no—
360C2×C32—21no—
361C2×C32—42no—
362C2×C32—41no—
363C2×C32—42no—
364C2×C32—41no—
365C2×C32—41no—
366C2×C32#88211yescharacteristic
367C2×C32#93711yescharacteristic
368C3⋊D3—42no—
369C3×D3—124no—
370C3×D3—122no—
371C3×D3—122no—
372C3×D3—122no—
373C24—91no—
374#41—182no—
375#41—181no—
376C2×D4—92no—
377C2×D4—182no—
378C2×D4—181no—
379C2×D4—181no—
380C2×D4—542no—
381C2×D4((C3⋊D3)⋊C3)⋊C211yescharacteristic
382C22⋊C4—92no—
383C22⋊C4—91no—
384C22⋊C4—181no—
385C22⋊C4—542no—
386C22×C3—21no—
387C22×C3—31no—
388C22×C3—31no—
389C22×C3—32no—
390C22×C3—32no—
391C22×C3—41no—
392C22×C3—41no—
393C22×C3—61no—
394C22×C3—61no—
395C22×C3—62no—
396C22×C3—61no—
397C22×C3—62no—
398C22×C3—121no—
399C22×C3—121no—
400C22×C3—121no—
401C22×C3—121no—
402C22×C3—184no—
403C22×C3—182no—
404C22×C3—182no—
405C22×C3—241no—
406C22×C3—241no—
407C22×C3—362no—
408C22×C3—361no—
409C22×C3—362no—
410C22×C3—361no—
411C22×C3—721no—
412C22×C3C2×#62611yescharacteristic
413C22×C3#381112yes—
414C2×D3—62no—
415C2×D3—64no—
416C2×D3—62no—
417C2×D3—62no—
418C2×D3—64no—
419C2×D3—121no—
420C2×D3—122no—
421C2×D3—121no—
422C2×D3—122no—
423C2×D3—122no—
424C2×D3—184no—
425C2×D3—182no—
426C2×D3—242no—
427C2×D3—241no—
428C2×D3—241no—
429C2×D3—242no—
430C2×D3—361no—
431C2×D3—361no—
432C2×D3—362no—
433C3×C4—21no—
434C3×C4—61no—
435C3×C4—61no—
436C3×C4—62no—
437C3×C4—182no—
438C3×C4—362no—
439C3×C4—361no—
440Dic3—62no—
441Dic3—62no—
442Dic3—64no—
443Dic3—121no—
444Dic3—122no—
445Dic3—361no—
446Dic3—362no—
447C32—21no—
448C32#474111yescharacteristic, complemented
449C32#503211yescharacteristic, complemented
450C23—92no—
451C23—181no—
452C23—181no—
453C23—182no—
454C23—362no—
455C23#620011yescharacteristic, complemented
456C23C2×((C3⋊D3)⋊C3)⋊C211yescharacteristic
457Q8—181no—
458D4—22no—
459D4—181no—
460D4—182no—
461D4—182no—
462D4—182no—
463D4—362no—
464D4—1084no—
465C2×C4—92no—
466C2×C4—182no—
467C2×C4—181no—
468C2×C4—181no—
469C2×C4—182no—
470C2×C4#620011yescharacteristic, complemented
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—31no—
474C2×C3—31no—
475C2×C3—41no—
476C2×C3—62no—
477C2×C3—61no—
478C2×C3—62no—
479C2×C3—61no—
480C2×C3—61no—
481C2×C3—122no—
482C2×C3—121no—
483C2×C3—121no—
484C2×C3—121no—
485C2×C3—122no—
486C2×C3—362no—
487C2×C3—362no—
488C2×C3#6398111yescharacteristic
489D3—122no—
490D3—122no—
491D3—124no—
492D3—122no—
493D3—362no—
494C22—21no—
495C22—41no—
496C22—41no—
497C22—182no—
498C22—184no—
499C22—182no—
500C22—362no—
501C22—361no—
502C22—362no—
503C22—361no—
504C22—721no—
505C22C2×#620011yescharacteristic
506C22#8867712yes—
507C4—21no—
508C4—182no—
509C4—361no—
510C4—362no—
511C3—31no—
512C3—31no—
513C3—61no—
514C3#1059153211yescharacteristic
515C2—21no—
516C2—22no—
517C2—41no—
518C2—362no—
519C2—362no—
520C2#1172132311yescharacteristic
subgroupG/H
centerC2#11721323
commutator, cosocleC23×H(3,𝔽3)C23
FrattiniC23×C3C2×D32
FittingC2×D4×H(3,𝔽3)C22
socleC2×C3#63981
hypercenterC2×D4((C3⋊D3)⋊C3)⋊C2
hypocenterH(3,𝔽3)B(4,𝔽2)
LCSG ▷ C23×H(3,𝔽3) ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C2 ◁ C23 ◁ C2×D4

Complements

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

NH ≅ G/Ntype
1. #11722664C2semidirect
2. #64146⋊C3C2semidirect
3. #11721321C2semidirect
4. #63661⋊C3C2semidirect
5. #11720335C2semidirect
8. C23×(C3⋊D3)⋊C3C22semidirect
17. #3757⋊C3C22semidirect
18. #3757⋊C3C22semidirect
24. #64146D3semidirect
28. #63661D3semidirect
32. C23×H(3,𝔽3)D4semidirect
35. C22×#272D4semidirect
40. C2×C4×H(3,𝔽3)D4semidirect
45. C2×#1084D4semidirect
56. C23×C3⋊D3C2×D3semidirect
80. #3757C2×D3semidirect
81. #3757C2×D3semidirect
123. C23×C32D12semidirect
124. C23×C32#67semidirect
143. C2×C32×C4D12semidirect
144. C2×C32×C4#67semidirect
278. H(3,𝔽3)B(4,𝔽2)semidirect
448. C32#4741semidirect
449. C32#5032semidirect
455. C23#6200semidirect
470. C2×C4#6200semidirect

Maximal subgroups

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

subgroupindex
#117226642
#64146⋊C32
#117213212
#63661⋊C32
#117203352
#105915313
#105915303