Finite Group Databasebeta

← #13156550

Subgroups — #13156550

Order 1728.

586 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 586 conjugacy classes of subgroups — too many to draw. Only its 98 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#11723535C211yesmaximal, characteristic, complemented
2C3×#63863C211yesmaximal, characteristic
3#11722071C211yesmaximal, characteristic, complemented
4#11722038C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic
6C3×#63565C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic
8#10591624—31nomaximal
9#10591488—31nomaximal
10#10590541—31nomaximal
11C2×C32×D12C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6293—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88807—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64097—31no—
23C3×#947—31no—
24C3×#936—31no—
25#63918—31no—
26#63918—31no—
27#63918—61no—
28#63863D311yescharacteristic
29#63662—31no—
30#63659—31no—
31C3×#835—31no—
32C3×#832—31no—
33C3×#810—31no—
34C3×#809—31no—
35#63566—31no—
36#63566—31no—
37#63566—61no—
38#63565D311yescharacteristic, complemented
39#63563—31no—
40#63563—31no—
41#63563—61no—
42#63562D311yescharacteristic
43#63555—31no—
44#63554—31no—
45#63554—31no—
46C22×C32×D3—21no—
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)—21no—
49C2×C3×C3⋊Dic3D411yescharacteristic
50C2×C3×C3⋊Dic3D411yescharacteristic
51C2×C32×Dic3—21no—
52C2×C32×Dic3—21no—
53C32×D12—21no—
54C32×D12—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57#6293—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5055—91no—
61#4794—91no—
62#4765—91no—
63C2×C32×Q8—31no—
64C2×C32×D4—31no—
65C2×#634—61no—
66C2×C3×D12—21no—
67C2×C3×D12C2×D311yescharacteristic
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6C2×D311yescharacteristic
71C2×C3×Dic6C2×D311yescharacteristic
72C2×#627—61no—
73C2×#627—61no—
74C2×#627—62no—
75C32×C4⋊C4—31no—
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—
83#3750—61no—
84C4×C3⋊Dic3C2×D311yescharacteristic
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×#217—61no—
88C3×#217—61no—
89C3×(C2×D3)⋊C4—61no—
90C3×(C2×D3)⋊C4—61no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—31no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8—61no—
103Dic3⋊Dic3—61no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—62no—
106(C2×D3)⋊Dic3—21no—
107(C2×D3)⋊Dic3—21no—
108C22×C33C2×D411yescharacteristic
109C2×C32×D3—41no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C33×C4#4111yescharacteristic
113C33×C4#4111yescharacteristic
114C3⋊(C3⋊Dic3)—41no—
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—21no—
117C32×Dic3—41no—
118C32×Dic3—41no—
119C3×#124—91no—
120#947—31no—
121#936—31no—
122#888—91no—
123#888—91no—
124#888—181no—
125#869—91no—
126C3×#103—91no—
127C3×#102—91no—
128#835—31no—
129#835—91no—
130#835—91no—
131#835—181no—
132#832—31no—
133#832—91no—
134#832—91no—
135#832—181no—
136#821—91no—
137#816—91no—
138#815—91no—
139#815—91no—
140#810—31no—
141#809—31no—
142#803—91no—
143C23×C32—61no—
144C22×C3×D3—21no—
145C22×C3×D3—21no—
146C22×C3×D3—41no—
147C32×Q8—61no—
148C32×Q8—61no—
149C32×D4—61no—
150C32×D4—61no—
151C2×C32×C4—21no—
152C2×C32×C4—21no—
153C2×C32×C4—21no—
154C2×C32×C4—41no—
155C2×C32×C4—61no—
156C2×C32×C4—61no—
157C2×C32×C4C22×D311yescharacteristic
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C3⋊Dic3—61no—
161C2×C3⋊Dic3—62no—
162C2×C3⋊Dic3—61no—
163C2×C3⋊Dic3—61no—
164C2×C3⋊Dic3—121no—
165C2×C3⋊Dic3—121no—
166C2×C3⋊Dic3—122no—
167C2×C3⋊Dic3#6711yescharacteristic
168C2×C3⋊Dic3#6711yescharacteristic
169#634—122no—
170C2×C3×Dic3—22no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—21no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3—31no—
176C2×C3×Dic3—31no—
177C2×C3×Dic3—31no—
178C2×C3×Dic3—31no—
179C2×C3×Dic3—41no—
180C2×C3×Dic3—61no—
181C2×C3×Dic3—61no—
182C3×D12—22no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—41no—
188C3×Dic6—22no—
189C3×Dic6—21no—
190C3×Dic6—21no—
191C3×Dic6—21no—
192C3×Dic6—21no—
193C3×Dic6—41no—
194#627—124no—
195#627—122no—
196#627—122no—
197C3⋊(C3⋊C8)—21no—
198C3⋊(C3⋊C8)—21no—
199C3×C3⋊C8—61no—
200C3×C3⋊C8—61no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—121no—
204C3×C3⋊C8—121no—
205#489—271noSylow
206C2×C33#12411yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C2×C33#13711yescharacteristic
209C32×D3—81no—
210C2×C3×Q8—31no—
211C2×C3×Q8—31no—
212C2×C3×Q8—61no—
213C2×C3×D4—31no—
214C2×C3×D4—31no—
215C2×C3×D4—61no—
216C2×D12D3211yescharacteristic
217C2×Dic6—181no—
218C2×Dic6—181no—
219C2×Dic6—181no—
220C2×Dic6—182no—
221C2×Dic6D3211yescharacteristic
222C2×C3×C8—91no—
223C2×C3×C8—91no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C22⋊C4—181no—
228C3×C22⋊C4—181no—
229C3×C42—91no—
230#217—61no—
231#217—61no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—61no—
234C4⋊Dic3—91no—
235C4⋊Dic3—91no—
236C4⋊Dic3—91no—
237C4⋊Dic3—181no—
238C4⋊Dic3—181no—
239C4⋊Dic3—181no—
240C4⋊Dic3—181no—
241C4⋊Dic3—361no—
242C4⋊Dic3D3211yescharacteristic
243Dic3⋊C4—181no—
244Dic3⋊C4—181no—
245Dic3⋊C4—182no—
246C4×Dic3—31no—
247C4×Dic3—31no—
248C4×Dic3—61no—
249C2×C3⋊C8—31no—
250C2×C3⋊C8—31no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—61no—
254C2×C3⋊C8—61no—
255C22×C32—21no—
256C22×C32—21no—
257C22×C32—21no—
258C22×C32—41no—
259C22×C32—121no—
260C22×C32—121no—
261C22×C32—121no—
262C22×C32D3×D411yescharacteristic
263C22×C32D3×D411yescharacteristic
264C22×C32C2×#6711yescharacteristic
265C2×C3×D3—42no—
266C2×C3×D3—41no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C2×C3×D3—41no—
272C2×C3×D3—81no—
273C2×C3×D3—81no—
274C32×C4—21no—
275C32×C4—21no—
276C32×C4—21no—
277C32×C4—21no—
278C32×C4—21no—
279C32×C4—21no—
280C32×C4—41no—
281C32×C4—41no—
282C32×C4—121no—
283C32×C4—121no—
284C32×C4#23511yescharacteristic
285C32×C4#23511yescharacteristic
286C32×C4#23711yescharacteristic
287C32×C4#23711yescharacteristic
288C32×C4#23711yescharacteristic
289C32×C4#23711yescharacteristic
290C3⋊Dic3—21no—
291C3⋊Dic3—21no—
292C3⋊Dic3—121no—
293C3⋊Dic3—122no—
294C3⋊Dic3—121no—
295C3⋊Dic3—121no—
296C3⋊Dic3—241no—
297C3⋊Dic3—241no—
298C3⋊Dic3—242no—
299C3×Dic3—42no—
300C3×Dic3—42no—
301C3×Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—41no—
305C3×Dic3—62no—
306C3×Dic3—62no—
307C3×Dic3—61no—
308C3×Dic3—61no—
309C3×Dic3—61no—
310C3×Dic3—61no—
311C4⋊Q8—271no—
312#124—91no—
313#103—91no—
314#103—271no—
315#102—91no—
316#102—271no—
317#97—271no—
318C33#48911yescharacteristic, complemented, Sylow
319C23×C3—61no—
320C23×C3—61no—
321C23×C3—121no—
322C22×D3—21no—
323C3×Q8—62no—
324C3×Q8—61no—
325C3×Q8—61no—
326C3×Q8—61no—
327C3×Q8—61no—
328C3×Q8—121no—
329C3×D4—62no—
330C3×D4—61no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—61no—
334C3×D4—121no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—41no—
339C2×C3×C4—62no—
340C2×C3×C4—61no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—61no—
344C2×C3×C4—91no—
345C2×C3×C4—91no—
346C2×C3×C4—121no—
347C2×C3×C4C2×D3211yescharacteristic
348C2×C3×C4C2×D3211yescharacteristic
349C2×C3×C4C2×D3211yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—31no—
353C2×Dic3—31no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—182no—
359C2×Dic3—181no—
360C2×Dic3—181no—
361C2×Dic3—181no—
362C2×Dic3—361no—
363C2×Dic3—361no—
364C2×Dic3—362no—
365D12—21no—
366D12—21no—
367Dic6—21no—
368Dic6—21no—
369Dic6—362no—
370Dic6—364no—
371Dic6—362no—
372Dic6—362no—
373C3×C8—181no—
374C3×C8—181no—
375C3⋊C8—61no—
376C3⋊C8—61no—
377C3⋊C8—61no—
378C3⋊C8—61no—
379C3⋊C8—121no—
380C3⋊C8—121no—
381C2×C32—21no—
382C2×C32—21no—
383C2×C32—21no—
384C2×C32—21no—
385C2×C32—21no—
386C2×C32—21no—
387C2×C32—21no—
388C2×C32—21no—
389C2×C32—21no—
390C2×C32—41no—
391C2×C32—41no—
392C2×C32—41no—
393C2×C32—241no—
394C2×C32#88511yescharacteristic
395C2×C32#88511yescharacteristic
396C2×C32#91111yescharacteristic
397C2×C32#91411yescharacteristic
398C2×C32#91511yescharacteristic
399C2×C32#91811yescharacteristic
400C2×C32#93611yescharacteristic
401C2×C32#94911yescharacteristic
402C2×C32#95111yescharacteristic
403C3×D3—82no—
404C3×D3—81no—
405C3×D3—81no—
406C2×Q8—31no—
407C2×Q8—541no—
408C2×D4—31no—
409C2×C8—91no—
410C2×C8—91no—
411C4⋊C4—31no—
412C4⋊C4—271no—
413C4⋊C4—541no—
414C22⋊C4—181no—
415C22⋊C4—181no—
416C42—91no—
417C22×C3—21no—
418C22×C3—21no—
419C22×C3—21no—
420C22×C3—41no—
421C22×C3—122no—
422C22×C3—121no—
423C22×C3—121no—
424C22×C3—121no—
425C22×C3—121no—
426C22×C3—121no—
427C22×C3—121no—
428C22×C3—241no—
429C22×C3—241no—
430C22×C3D3×#6711yescharacteristic
431C22×C3D3×#6711yescharacteristic
432C22×C3#381111yescharacteristic
433C2×D3—41no—
434C2×D3—41no—
435C2×D3—41no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—21no—
441C3×C4—21no—
442C3×C4—41no—
443C3×C4—41no—
444C3×C4—122no—
445C3×C4—122no—
446C3×C4—121no—
447C3×C4—121no—
448C3×C4—121no—
449C3×C4—121no—
450C3×C4—181no—
451C3×C4—181no—
452C3×C4#379511yescharacteristic
453C3×C4#379511yescharacteristic
454C3×C4#380411yescharacteristic
455C3×C4#380411yescharacteristic
456C3×C4#380511yescharacteristic
457C3×C4#380511yescharacteristic
458Dic3—41no—
459Dic3—41no—
460Dic3—62no—
461Dic3—62no—
462Dic3—61no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466Dic3—362no—
467Dic3—361no—
468Dic3—361no—
469Dic3—361no—
470Dic3—721no—
471Dic3—721no—
472Dic3—722no—
473C32—21no—
474C32—21no—
475C32—21no—
476C32—41no—
477C32#476511yescharacteristic, complemented
478C32#479411yescharacteristic, complemented
479C32#505511yescharacteristic, complemented
480C23—61no—
481Q8—61no—
482Q8—61no—
483Q8—1082no—
484D4—61no—
485D4—61no—
486C2×C4—61no—
487C2×C4—61no—
488C2×C4—91no—
489C2×C4—91no—
490C2×C4—541no—
491C2×C4D3311yescharacteristic
492C8—181no—
493C8—181no—
494C2×C3—21no—
495C2×C3—21no—
496C2×C3—21no—
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—21no—
501C2×C3—21no—
502C2×C3—21no—
503C2×C3—41no—
504C2×C3—41no—
505C2×C3—41no—
506C2×C3—242no—
507C2×C3—241no—
508C2×C3—241no—
509C2×C3#6391311yescharacteristic
510C2×C3#6392511yescharacteristic
511C2×C3#6392911yescharacteristic
512C2×C3#6393711yescharacteristic
513C2×C3#6393911yescharacteristic
514C2×C3#6394311yescharacteristic
515C2×C3#6395111yescharacteristic
516C2×C3#6395911yescharacteristic
517C2×C3#6395911yescharacteristic
518D3—81no—
519C22—121no—
520C22—121no—
521C22—121no—
522C22#8895211yescharacteristic
523C4—121no—
524C4—121no—
525C4—181no—
526C4—181no—
527C4—1081no—
528C4#8896111yescharacteristic
529C4#8896111yescharacteristic
530C3—21no—
531C3—21no—
532C3—21no—
533C3—41no—
534C3#1059054111yescharacteristic, complemented
535C3#1059148811yescharacteristic, complemented
536C3#1059162411yescharacteristic, complemented
537C2—241no—
538C2#1172287411yescharacteristic
539C2#1172288811yescharacteristic
540C2#1172300411yescharacteristic
subgroupG/H
centerC22#88952
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C32×D12C22
hypocenterC33#489
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
1. #11723535C2semidirect
3. #11722071C2semidirect
4. #11722038C2semidirect
6. C3×#63565C2semidirect
38. #63565D3semidirect
318. C33#489semidirect
477. C32#4765semidirect
478. C32#4794semidirect
479. C32#5055semidirect
534. C3#10590541semidirect
535. C3#10591488semidirect
536. C3#10591624semidirect

Maximal subgroups

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

subgroupindex
#117235352
C3×#638632
#117220712
#117220382
#117220322
C3×#635652
C3×#635622
#105916243
#105914883
#105905413