Finite Group Databasebeta

← #13156354

Subgroups — #13156354

Order 1728.

626 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 626 conjugacy classes of subgroups — too many to draw. Only its 100 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#11723501C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic
4#11722050C211yesmaximal, characteristic, complemented
5C3×#63575C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic
8#10590780—31nomaximal
9#10590474—31nomaximal
10#10590435—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×Dic3×C3⋊D3—21no—
13C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4⋊Dic3C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20Dic3⋊(C3⋊Dic3)—21no—
21#88803—21no—
22#64102—31no—
23#64099—31no—
24#63898—31no—
25#63898—31no—
26#63898—61no—
27#63872—31no—
28#63872—31no—
29#63872—61no—
30#63641—31no—
31C3×#821—31no—
32C3×#818—31no—
33C3×#809—31no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#807—31no—
37#63575D311yescharacteristic, complemented
38#63574D311yescharacteristic, complemented
39#63572—31no—
40#63572—31no—
41#63572—61no—
42#63564—31no—
43#63564D311yescharacteristic
44#63563—31no—
45#63562—31no—
46C22×C3×C3⋊D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3×C3⋊Dic3—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C3×C4⋊(C3⋊D3)Q812yes—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55Dic3×C3⋊D3—42no—
56C3×C3⋊(C3⋊C8)—21no—
57C32×C3⋊C8—21no—
58#4891—91no—
59#4871—91no—
60#4766—91no—
61C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
62C2×C4×C3⋊D3—61no—
63C2×C3×D12—31no—
64C2×C3×D12—31no—
65C2×C3×D12—61no—
66C2×D3×Dic3—61no—
67C2×D3×Dic3—61no—
68C2×D3×Dic3—62no—
69C2×C32×C8—31no—
70C32×C4⋊C4—31no—
71C32×C4⋊C4—31no—
72#3752—61no—
73C4⋊(C3⋊Dic3)—61no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75#3750—61no—
76C4×C3⋊Dic3—31no—
77C4×C3⋊Dic3—31no—
78C4×C3⋊Dic3—31no—
79C4×C3⋊Dic3—61no—
80C4×C3⋊Dic3—61no—
81C4×C3⋊Dic3—61no—
82C4×C3⋊Dic3—121no—
83C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—61no—
89C3×C4⋊Dic3C2×D311yescharacteristic
90C3×C4⋊Dic3C2×D311yescharacteristic
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C2×C3×C3⋊C8—21no—
94C2×C3×C3⋊C8—31no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—61no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99Dic3⋊Dic3—61no—
100Dic3⋊Dic3—61no—
101Dic3⋊Dic3—61no—
102Dic3⋊Dic3—61no—
103Dic3⋊Dic3—62no—
104Dic3⋊Dic3—62no—
105(C2×D3)⋊Dic3—61no—
106(C2×D3)⋊Dic3—61no—
107(C2×D3)⋊Dic3—62no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—22no—
110C2×C3×C3⋊D3—41no—
111C2×C3×C3⋊D3—41no—
112C33×C4C2×Q811yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—21no—
115C3⋊(C3⋊Dic3)—21no—
116C3×C3⋊Dic3—41no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119D4×Dic3—91no—
120#893—91no—
121#893—91no—
122#893—181no—
123#890—91no—
124#890—91no—
125#890—91no—
126#890—181no—
127C3×#107—91no—
128C3×#106—91no—
129C3×#102—91no—
130#832—91no—
131#821—31no—
132#821—91no—
133#818—31no—
134Dic3⋊C8—91no—
135Dic3⋊C8—91no—
136Dic3⋊C8—91no—
137Dic3⋊C8—181no—
138#809—31no—
139#809—91no—
140#808—31no—
141#808—31no—
142#807—31no—
143C22×C3⋊D3—21no—
144C22×C3×D3—62no—
145C22×C3×D3—61no—
146C22×C3×D3—61no—
147C2×C32×C4—21no—
148C2×C32×C4—21no—
149C2×C32×C4—21no—
150C2×C32×C4—41no—
151C2×C32×C4—61no—
152C2×C32×C4—61no—
153C2×C32×C4C22×D311yescharacteristic
154C2×C32×C4C22×D311yescharacteristic
155C2×C32×C4C22×D311yescharacteristic
156C2×C3⋊Dic3—21no—
157C2×C3⋊Dic3—31no—
158C2×C3⋊Dic3—31no—
159C2×C3⋊Dic3—31no—
160C2×C3⋊Dic3—31no—
161C2×C3⋊Dic3—31no—
162C2×C3⋊Dic3—31no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—61no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—121no—
171C4⋊(C3⋊D3)—21no—
172C4⋊(C3⋊D3)Dic612yes—
173C4×C3⋊D3—122no—
174C2×C3×Dic3—22no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—62no—
181C2×C3×Dic3—61no—
182C2×C3×Dic3—61no—
183C3×D12—32no—
184C3×D12—32no—
185C3×D12—62no—
186C3×D12—62no—
187C3×D12—61no—
188C3×D12—61no—
189D3×Dic3—124no—
190D3×Dic3—122no—
191D3×Dic3—122no—
192C32×C8—61no—
193C3⋊(C3⋊C8)—21no—
194C3×C3⋊C8—21no—
195C3×C3⋊C8—21no—
196C3×C3⋊C8—41no—
197C3×C3⋊C8—61no—
198C3×C3⋊C8—61no—
199C3×C3⋊C8—121no—
200#478—271noSylow
201C2×C33C22⋊Q811yescharacteristic
202C2×C33#13511yescharacteristic
203C2×C33AGL(1,ℤ/8)11yescharacteristic
204C3×C3⋊D3—42no—
205C2×C3×D4—91no—
206C22×Dic3—181no—
207C2×D12—31no—
208C2×D12—31no—
209C2×D12—61no—
210C2×C4×D3—181no—
211C2×C4×D3—181no—
212C2×C4×D3—182no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—91no—
217C3×C4⋊C4—31no—
218C3×C4⋊C4—31no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—31no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—91no—
224#217—181no—
225(C2×D3)⋊C4—181no—
226(C2×D3)⋊C4—181no—
227(C2×D3)⋊C4—182no—
228C4⋊Dic3—31no—
229C4⋊Dic3—31no—
230C4⋊Dic3—61no—
231C4⋊Dic3—181no—
232C4⋊Dic3—181no—
233C4⋊Dic3—181no—
234C4⋊Dic3—182no—
235C4⋊Dic3D3211yescharacteristic
236C4⋊Dic3D3211yescharacteristic
237Dic3⋊C4—181no—
238Dic3⋊C4—181no—
239Dic3⋊C4—181no—
240Dic3⋊C4—182no—
241C4×Dic3—91no—
242C4×Dic3—91no—
243C4×Dic3—91no—
244C4×Dic3—181no—
245C4×Dic3—181no—
246C4×Dic3—181no—
247C4×Dic3—361no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—61no—
251C2×C3⋊C8D3211yescharacteristic
252C22×C32—21no—
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—41no—
256C22×C32D3×D411yescharacteristic
257C22×C32D3×D411yescharacteristic
258C22×C32D3×D411yescharacteristic
259C2×C3⋊D3—22no—
260C2×C3⋊D3—41no—
261C2×C3⋊D3—41no—
262C2×C3×D3—64no—
263C2×C3×D3—62no—
264C2×C3×D3—62no—
265C2×C3×D3—122no—
266C2×C3×D3—122no—
267C2×C3×D3—121no—
268C2×C3×D3—121no—
269C2×C3×D3—121no—
270C2×C3×D3—121no—
271C32×C4—21no—
272C32×C4—21no—
273C32×C4—21no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—41no—
278C32×C4—41no—
279C32×C4—121no—
280C32×C4—121no—
281C32×C4C2×Dic611yescharacteristic
282C32×C4#23511yescharacteristic
283C32×C4#23511yescharacteristic
284C32×C4#23711yescharacteristic
285C32×C4D3×Q811yescharacteristic
286C32×C4D3×Q811yescharacteristic
287C3⋊Dic3—41no—
288C3⋊Dic3—61no—
289C3⋊Dic3—61no—
290C3⋊Dic3—62no—
291C3⋊Dic3—62no—
292C3⋊Dic3—61no—
293C3⋊Dic3—61no—
294C3⋊Dic3—61no—
295C3⋊Dic3—61no—
296C3⋊Dic3—121no—
297C3⋊Dic3—121no—
298C3⋊Dic3—121no—
299C3⋊Dic3—121no—
300C3⋊Dic3—122no—
301C3⋊Dic3—122no—
302C3×Dic3—42no—
303C3×Dic3—42no—
304C3×Dic3—41no—
305C3×Dic3—41no—
306C3×Dic3—41no—
307C3×Dic3—41no—
308C3×Dic3—122no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311#125—271no—
312C4×D4—271no—
313#107—91no—
314#106—91no—
315C4⋊C8—271no—
316#102—91no—
317#102—271no—
318C33#47811yescharacteristic, complemented, Sylow
319C23×C3—181no—
320C22×D3—62no—
321C22×D3—61no—
322C22×D3—61no—
323C3×D4—92no—
324C3×D4—181no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—41no—
329C2×C3×C4—62no—
330C2×C3×C4—62no—
331C2×C3×C4—61no—
332C2×C3×C4—61no—
333C2×C3×C4—61no—
334C2×C3×C4—61no—
335C2×C3×C4—181no—
336C2×C3×C4C2×D3211yescharacteristic
337C2×C3×C4C2×D3211yescharacteristic
338C2×C3×C4C2×D3211yescharacteristic
339C2×Dic3—21no—
340C2×Dic3—21no—
341C2×Dic3—62no—
342C2×Dic3—61no—
343C2×Dic3—61no—
344C2×Dic3—91no—
345C2×Dic3—91no—
346C2×Dic3—91no—
347C2×Dic3—91no—
348C2×Dic3—91no—
349C2×Dic3—91no—
350C2×Dic3—181no—
351C2×Dic3—181no—
352C2×Dic3—181no—
353C2×Dic3—181no—
354C2×Dic3—181no—
355C2×Dic3—181no—
356C2×Dic3—361no—
357C2×Dic3—361no—
358C2×Dic3—362no—
359D12—32no—
360D12—32no—
361D12—62no—
362D12—62no—
363D12—61no—
364D12—61no—
365C4×D3—364no—
366C4×D3—362no—
367C4×D3—362no—
368C3×C8—61no—
369C3×C8—61no—
370C3×C8—121no—
371C3×C8—181no—
372C3⋊C8—21no—
373C3⋊C8—61no—
374C3⋊C8—61no—
375C3⋊C8—121no—
376C2×C32—21no—
377C2×C32—21no—
378C2×C32—21no—
379C2×C32—21no—
380C2×C32—21no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—41no—
386C2×C32—41no—
387C2×C32—41no—
388C2×C32#87811yescharacteristic
389C2×C32#89611yescharacteristic
390C2×C32#89611yescharacteristic
391C2×C32#91111yescharacteristic
392C2×C32#91211yescharacteristic
393C2×C32#91411yescharacteristic
394C2×C32#91411yescharacteristic
395C2×C32#91611yescharacteristic
396C2×C32#91911yescharacteristic
397C3⋊D3—42no—
398C3×D3—124no—
399C3×D3—122no—
400C3×D3—122no—
401C2×D4—91no—
402C22×C4—541no—
403C2×C8—31no—
404C2×C8—91no—
405C4⋊C4—31no—
406C4⋊C4—31no—
407C4⋊C4—91no—
408C4⋊C4—541no—
409C4⋊C4—541no—
410C22⋊C4—541no—
411C42—271no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—182no—
417C22×C3—361no—
418C22×C3—361no—
419C22×C3#380211yescharacteristic
420C22×C3#381111yescharacteristic
421C22×C3#381111yescharacteristic
422C2×D3—64no—
423C2×D3—62no—
424C2×D3—62no—
425C2×D3—122no—
426C2×D3—122no—
427C2×D3—121no—
428C2×D3—121no—
429C2×D3—121no—
430C2×D3—121no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—122no—
440C3×C4—122no—
441C3×C4—121no—
442C3×C4—121no—
443C3×C4—121no—
444C3×C4—121no—
445C3×C4—361no—
446C3×C4D3×Dic611yescharacteristic
447C3×C4D3×Dic611yescharacteristic
448C3×C4#379511yescharacteristic
449C3×C4#379711yescharacteristic
450C3×C4#379811yescharacteristic
451C3×C4#380411yescharacteristic
452Dic3—41no—
453Dic3—41no—
454Dic3—122no—
455Dic3—121no—
456Dic3—121no—
457Dic3—182no—
458Dic3—182no—
459Dic3—181no—
460Dic3—181no—
461Dic3—181no—
462Dic3—181no—
463Dic3—181no—
464Dic3—181no—
465Dic3—361no—
466Dic3—361no—
467Dic3—361no—
468Dic3—361no—
469Dic3—362no—
470Dic3—362no—
471C32—21no—
472C32—21no—
473C32—21no—
474C32—41no—
475C32#476611yescharacteristic, complemented
476C32#487111yescharacteristic, complemented
477C32#489111yescharacteristic, complemented
478C23—181no—
479D4—92no—
480D4—181no—
481C2×C4—61no—
482C2×C4—61no—
483C2×C4—181no—
484C2×C4—271no—
485C2×C4—271no—
486C2×C4—1082no—
487C2×C4D3311yescharacteristic
488C8—61no—
489C8—181no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—41no—
500C2×C3—41no—
501C2×C3—41no—
502C2×C3—362no—
503C2×C3#6379811yescharacteristic
504C2×C3#6381011yescharacteristic
505C2×C3#6386511yescharacteristic
506C2×C3#6389611yescharacteristic
507C2×C3#6389911yescharacteristic
508C2×C3#6393611yescharacteristic
509C2×C3#6393711yescharacteristic
510C2×C3#6394111yescharacteristic
511C2×C3#6395011yescharacteristic
512D3—124no—
513D3—122no—
514D3—122no—
515C22—182no—
516C22—361no—
517C22—361no—
518C22#8895411yescharacteristic
519C4—121no—
520C4—121no—
521C4—361no—
522C4—541no—
523C4—541no—
524C4#8895711yescharacteristic
525C4#8896011yescharacteristic
526C3—21no—
527C3—21no—
528C3—21no—
529C3—41no—
530C3#1059043511yescharacteristic, complemented
531C3#1059047411yescharacteristic, complemented
532C3#1059078011yescharacteristic, complemented
533C2—362no—
534C2#1172286211yescharacteristic
535C2#1172287811yescharacteristic
536C2#1172298311yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(C3⋊D3)C22
hypocenterC33#478
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11723450C2semidirect
4. #11722050C2semidirect
5. C3×#63575C2semidirect
6. C3×#63574C2semidirect
37. #63575D3semidirect
38. #63574D3semidirect
318. C33#478semidirect
475. C32#4766semidirect
476. C32#4871semidirect
477. C32#4891semidirect
530. C3#10590435semidirect
531. C3#10590474semidirect
532. C3#10590780semidirect

Maximal subgroups

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

subgroupindex
#117235012
#117234502
#117220752
#117220502
C3×#635752
C3×#635742
C3×#635642
#105907803
#105904743
#105904353