Finite Group Databasebeta

← #13161990

Subgroups — #13161990

Order 1728.

580 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 580 conjugacy classes of subgroups — too many to draw. Only its 99 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
1C3×#64155C211yesmaximal, characteristic
2#11723667C211yesmaximal, characteristic
3C3×#63662C211yesmaximal, characteristic, complemented
4C3×#63662C211yesmaximal, characteristic, complemented
5#11722080C211yesmaximal, characteristic, complemented
6#11722079C211yesmaximal, characteristic, complemented
7#11722072C211yesmaximal, characteristic
8#10591561—31nomaximal
9#10591559—31nomaximal
10#10591558—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C3×#3752—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C3×#3750—21no—
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20#88814—21no—
21#88813—21no—
22#64155D311yescharacteristic
23C3×#946—31no—
24C3×#946—31no—
25C3×#946—61no—
26#63849—31no—
27#63841—31no—
28#63841—31no—
29#63662D311yescharacteristic, complemented
30#63662D311yescharacteristic, complemented
31C3×#835—31no—
32C3×#835—31no—
33C3×#835—31no—
34C3×#835—31no—
35C3×#835—61no—
36C3×#835—61no—
37#63576—31no—
38#63575—31no—
39#63575—31no—
40#63574—31no—
41#63573—31no—
42#63573—31no—
43#63557—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46C22×C3×C3⋊D3D411yescharacteristic
47C33×Q8—21no—
48C33×Q8—21no—
49C2×C33×C4—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D411yescharacteristic
55C3×C4×C3⋊D3—21no—
56C3×C4×C3⋊D3—21no—
57C3×C3⋊(C3⋊C8)—21no—
58C3×C3⋊(C3⋊C8)—21no—
59#5045—91no—
60#4815—91no—
61#4814—91no—
62C2×C32×Q8—21no—
63C2×C32×Q8—21no—
64C2×C32×Q8—21no—
65C2×C32×Q8—41no—
66C2×C32×Q8C2×D311yescharacteristic
67C2×C32×Q8C2×D311yescharacteristic
68C2×C32×Q8C2×D311yescharacteristic
69C2×C4×C3⋊D3C2×D311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73#3752—21no—
74C4⋊(C3⋊Dic3)C2×D311yescharacteristic
75C4⋊(C3⋊Dic3)C2×D311yescharacteristic
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77#3750—21no—
78C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
79C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
80C3×(C2×D3)⋊C4—61no—
81C3×(C2×D3)⋊C4—61no—
82C3×(C2×D3)⋊C4—62no—
83C3×C4⋊Dic3—31no—
84C3×C4⋊Dic3—31no—
85C3×C4⋊Dic3—31no—
86C3×C4⋊Dic3—31no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—61no—
90C3×C4⋊Dic3—61no—
91C3×C4⋊Dic3—61no—
92C3×Dic3⋊C4—61no—
93C3×Dic3⋊C4—61no—
94C3×Dic3⋊C4—62no—
95C2×C3×C3⋊C8—31no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—61no—
100C2×C3×C3⋊C8—61no—
101#3722—61no—
102(C2×D3)⋊Dic3—61no—
103(C2×D3)⋊Dic3—61no—
104Dic32—61no—
105Dic32—61no—
106Dic32—61no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C33×C4—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3×C3⋊Dic3—21no—
115C3×C3⋊Dic3—41no—
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C3×C22⋊Q8—91no—
119#946—31no—
120#946—31no—
121#946—61no—
122#926—91no—
123#892—91no—
124#892—91no—
125C3×#103—91no—
126C3×#103—91no—
127#835—31no—
128#835—31no—
129#835—31no—
130#835—31no—
131#835—61no—
132#835—61no—
133#830—91no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—91no—
136#818—91no—
137#817—91no—
138#808—91no—
139#808—91no—
140#807—91no—
141#807—91no—
142C22×C3⋊D3#6711yescharacteristic
143C22×C3×D3—31no—
144C22×C3×D3—31no—
145C22×C3×D3—61no—
146C32×Q8—21no—
147C32×Q8—21no—
148C32×Q8—22no—
149C32×Q8—22no—
150C32×Q8—21no—
151C32×Q8—21no—
152C32×Q8—21no—
153C32×Q8—21no—
154C32×Q8—41no—
155C32×Q8—41no—
156C32×Q8—41no—
157C32×Q8—41no—
158C32×Q8—42no—
159C32×Q8—42no—
160C2×C32×C4—21no—
161C2×C32×C4—22no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—41no—
169C2×C32×C4—41no—
170C2×C32×C4—42no—
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173C2×C32×C4C22×D311yescharacteristic
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—21no—
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3#6711yescharacteristic
178C4×C3⋊D3—21no—
179C4×C3⋊D3—21no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—61no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—61no—
187C2×C3×Dic3—61no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—121no—
191C2×C3×Dic3—121no—
192C3×C4×D3—61no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—121no—
197C3×C4×D3—121no—
198C3⋊(C3⋊C8)—21no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—121no—
205C3×C3⋊C8—121no—
206#485—271noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33#13711yescharacteristic
210C3×C3⋊D3—41no—
211C2×C3×Q8—21no—
212C2×C3×Q8—21no—
213C2×C3×Q8—21no—
214C2×C3×Q8—41no—
215C2×C3×Q8D3211yescharacteristic
216C2×C3×Q8D3211yescharacteristic
217C2×C3×Q8D3211yescharacteristic
218C22×C3×C4—91no—
219C2×C4×D3—31no—
220C2×C4×D3—31no—
221C2×C4×D3—61no—
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C3×C4⋊C4—91no—
225C3×C4⋊C4—91no—
226C3×C4⋊C4—91no—
227C3×C4⋊C4—181no—
228C3×C22⋊C4—181no—
229#217—181no—
230(C2×D3)⋊C4—61no—
231(C2×D3)⋊C4—61no—
232(C2×D3)⋊C4—62no—
233(C2×D3)⋊C4—181no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243C4⋊Dic3—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246Dic3⋊C4—62no—
247C4×Dic3—181no—
248C4×Dic3—181no—
249C4×Dic3—181no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—61no—
255C2×C3⋊C8—61no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—41no—
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C2×C3⋊D3—21no—
264C2×C3⋊D3—21no—
265C2×C3⋊D3—21no—
266C2×C3×D3—62no—
267C2×C3×D3—61no—
268C2×C3×D3—61no—
269C2×C3×D3—61no—
270C2×C3×D3—61no—
271C2×C3×D3—61no—
272C2×C3×D3—61no—
273C2×C3×D3—121no—
274C2×C3×D3—121no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4—41no—
283C32×C4—42no—
284C32×C4—42no—
285C32×C4—42no—
286C32×C4—44no—
287C32×C4—41no—
288C32×C4—41no—
289C32×C4—41no—
290C32×C4#23511yescharacteristic
291C32×C4#23511yescharacteristic
292C32×C4#23711yescharacteristic
293C32×C4#23711yescharacteristic
294C32×C4#23711yescharacteristic
295C32×C4#23711yescharacteristic
296C3⋊Dic3—21no—
297C3⋊Dic3—41no—
298C3⋊Dic3—41no—
299C3⋊Dic3—41no—
300C3×Dic3—61no—
301C3×Dic3—61no—
302C3×Dic3—121no—
303C3×Dic3—122no—
304C3×Dic3—121no—
305C3×Dic3—121no—
306C3×Dic3—121no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—241no—
311C3×Dic3—241no—
312C22⋊Q8—91no—
313#117—271no—
314#107—271no—
315#106—271no—
316#103—91no—
317#103—91no—
318C22⋊C8—271no—
319C33#48511yescharacteristic, complemented, Sylow
320C23×C3—91no—
321C22×D3—31no—
322C22×D3—31no—
323C22×D3—61no—
324C3×Q8—22no—
325C3×Q8—22no—
326C3×Q8—21no—
327C3×Q8—21no—
328C3×Q8—21no—
329C3×Q8—21no—
330C3×Q8—21no—
331C3×Q8—21no—
332C3×Q8—41no—
333C3×Q8—41no—
334C3×Q8—41no—
335C3×Q8—41no—
336C3×Q8—42no—
337C3×Q8—42no—
338C2×C3×C4—22no—
339C2×C3×C4—21no—
340C2×C3×C4—21no—
341C2×C3×C4—21no—
342C2×C3×C4—21no—
343C2×C3×C4—21no—
344C2×C3×C4—21no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—42no—
348C2×C3×C4—41no—
349C2×C3×C4—91no—
350C2×C3×C4—181no—
351C2×C3×C4—181no—
352C2×C3×C4—181no—
353C2×C3×C4—181no—
354C2×C3×C4—181no—
355C2×C3×C4C2×D3211yescharacteristic
356C2×C3×C4C2×D3211yescharacteristic
357C2×C3×C4C2×D3211yescharacteristic
358C2×Dic3—31no—
359C2×Dic3—31no—
360C2×Dic3—61no—
361C2×Dic3—62no—
362C2×Dic3—61no—
363C2×Dic3—61no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—61no—
367C2×Dic3—61no—
368C2×Dic3—121no—
369C2×Dic3—121no—
370C4×D3—61no—
371C4×D3—61no—
372C4×D3—61no—
373C4×D3—61no—
374C4×D3—121no—
375C4×D3—121no—
376C3×C8—181no—
377C3×C8—181no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—61no—
381C3⋊C8—61no—
382C3⋊C8—121no—
383C3⋊C8—121no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—41no—
396C2×C32#88311yescharacteristic
397C2×C32#88311yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91611yescharacteristic
400C2×C32#91811yescharacteristic
401C2×C32#91911yescharacteristic
402C2×C32#93511yescharacteristic
403C2×C32#94211yescharacteristic
404C2×C32#95011yescharacteristic
405C3⋊D3—41no—
406C3×D3—122no—
407C3×D3—121no—
408C3×D3—121no—
409C2×Q8#111311yescharacteristic
410C22×C4—91no—
411C2×C8—91no—
412C2×C8—91no—
413C4⋊C4—91no—
414C4⋊C4—91no—
415C4⋊C4—91no—
416C4⋊C4—181no—
417C22⋊C4—181no—
418C22⋊C4—541no—
419C42—541no—
420C22×C3—21no—
421C22×C3—21no—
422C22×C3—21no—
423C22×C3—41no—
424C22×C3—181no—
425C22×C3—181no—
426C22×C3—181no—
427C22×C3D3×#6711yescharacteristic
428C22×C3D3×#6711yescharacteristic
429C22×C3#381111yescharacteristic
430C2×D3—62no—
431C2×D3—61no—
432C2×D3—61no—
433C2×D3—61no—
434C2×D3—61no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—121no—
438C2×D3—121no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—42no—
446C3×C4—42no—
447C3×C4—42no—
448C3×C4—44no—
449C3×C4—41no—
450C3×C4—41no—
451C3×C4—41no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—181no—
455C3×C4—361no—
456C3×C4—361no—
457C3×C4—361no—
458C3×C4#380411yescharacteristic
459C3×C4#380411yescharacteristic
460C3×C4#380411yescharacteristic
461C3×C4#380411yescharacteristic
462C3×C4#380511yescharacteristic
463C3×C4#380511yescharacteristic
464Dic3—61no—
465Dic3—61no—
466Dic3—121no—
467Dic3—122no—
468Dic3—121no—
469Dic3—121no—
470Dic3—121no—
471Dic3—121no—
472Dic3—121no—
473Dic3—121no—
474Dic3—241no—
475Dic3—241no—
476C32—21no—
477C32—21no—
478C32—21no—
479C32—41no—
480C32#481411yescharacteristic, complemented
481C32#481511yescharacteristic, complemented
482C32#504511yescharacteristic, complemented
483C23—91no—
484Q8—21no—
485Q8—21no—
486C2×C4—21no—
487C2×C4—91no—
488C2×C4—181no—
489C2×C4—181no—
490C2×C4—181no—
491C2×C4—181no—
492C2×C4—181no—
493C2×C4C2×#111311yescharacteristic
494C8—181no—
495C8—181no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—21no—
504C2×C3—21no—
505C2×C3—41no—
506C2×C3—41no—
507C2×C3—41no—
508C2×C3—361no—
509C2×C3#6394611yescharacteristic
510C2×C3#6394711yescharacteristic
511C2×C3#6394811yescharacteristic
512C2×C3#6394911yescharacteristic
513C2×C3#6395011yescharacteristic
514C2×C3#6395111yescharacteristic
515C2×C3#6396911yescharacteristic
516C2×C3#6396911yescharacteristic
517C2×C3#6397511yescharacteristic
518D3—122no—
519D3—121no—
520D3—121no—
521C22—181no—
522C22—181no—
523C22—181no—
524C22#8904911yescharacteristic
525C4—41no—
526C4—181no—
527C4—361no—
528C4—361no—
529C4—361no—
530C4#8904611yescharacteristic
531C4#8904611yescharacteristic
532C3—21no—
533C3—21no—
534C3—21no—
535C3—41no—
536C3#1059155811yescharacteristic, complemented
537C3#1059155911yescharacteristic, complemented
538C3#1059156111yescharacteristic, complemented
539C2—361no—
540C2#1172371211yescharacteristic
541C2#1172371311yescharacteristic
542C2#1172372411yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC22×C3×C3⋊D3D4
hypercenterC2×Q8#1113
hypocenterC33#485
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
3. C3×#63662C2semidirect
4. C3×#63662C2semidirect
5. #11722080C2semidirect
6. #11722079C2semidirect
29. #63662D3semidirect
30. #63662D3semidirect
319. C33#485semidirect
480. C32#4814semidirect
481. C32#4815semidirect
482. C32#5045semidirect
536. C3#10591558semidirect
537. C3#10591559semidirect
538. C3#10591561semidirect

Maximal subgroups

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

subgroupindex
C3×#641552
#117236672
C3×#636622
#117220802
#117220792
#117220722
#105915613
#105915593
#105915583