Finite Group Databasebeta

← #13156434

Subgroups — #13156434

Order 1728.

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

normal subgroups only

This group has 582 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#11723667C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3#11722053C211yesmaximal, characteristic, complemented
4#11722031C211yesmaximal, characteristic, complemented
5#11722023C211yesmaximal, characteristic, complemented
6C3×#63575C211yesmaximal, characteristic, complemented
7C3×#63563C211yesmaximal, characteristic, complemented
8#10591423—31nomaximal
9#10590651—31nomaximal
10#10590446—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88814—21no—
20#88813—21no—
21C3×(C2×D3)⋊Dic3—21no—
22C3×#938—31no—
23C3×#895—31no—
24#63916D311yescharacteristic
25#63849—31no—
26#63841—31no—
27#63841—31no—
28#63659—31no—
29#63647—31no—
30#63635—31no—
31C3×#832—31no—
32C3×#821—31no—
33C3×#818—31no—
34C3×#807—31no—
35#63575D311yescharacteristic, complemented
36#63574—31no—
37#63574—31no—
38#63574—61no—
39#63563D311yescharacteristic, complemented
40#63562—31no—
41#63562—31no—
42#63562—61no—
43(C2×D3)⋊(C3⋊C8)—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—61no—
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C2×C3×C3⋊Dic3—21no—
51C2×C3×C3⋊Dic3D411yescharacteristic
52C3×C4×C3⋊D3—21no—
53C3×C4×C3⋊D3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C3⋊(C3⋊(C3⋊C8))—21no—
59C32×C3⋊C8—21no—
60#5035—91no—
61#4885—91no—
62#4784—91no—
63C2×C32×D4—31no—
64C2×C4×C3⋊D3C2×D311yescharacteristic
65C2×C3×#67—61no—
66C2×C3×D12—21no—
67C2×C3×D12—61no—
68C2×C3×D12C2×D311yescharacteristic
69C2×C3×D12C2×D311yescharacteristic
70C2×C3×C4×D3—31no—
71C2×C3×C4×D3—31no—
72C2×C3×C4×D3—61no—
73C2×#626—21no—
74C2×C32×C8—31no—
75C32×C4⋊C4—31no—
76C4⋊(C3⋊Dic3)C2×D311yescharacteristic
77C4⋊(C3⋊Dic3)C2×D311yescharacteristic
78C2×C3⋊(C3⋊C8)—31no—
79C2×C3⋊(C3⋊C8)—31no—
80C2×C3⋊(C3⋊C8)—31no—
81C2×C3⋊(C3⋊C8)—61no—
82C2×C3⋊(C3⋊C8)—61no—
83C2×C3⋊(C3⋊C8)—61no—
84C2×C3⋊(C3⋊C8)—121no—
85C3×#217—61no—
86C3×(C2×D3)⋊C4—61no—
87C3×C4⋊Dic3—21no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C2×C3×C3⋊C8—21no—
97C2×C3×C3⋊C8C2×D311yescharacteristic
98C2×C3×C3⋊C8C2×D311yescharacteristic
99#3722—61no—
100(C2×D3)⋊Dic3—21no—
101(C2×D3)⋊Dic3—61no—
102(C2×D3)⋊Dic3—61no—
103Dic32—61no—
104Dic32—61no—
105Dic32—61no—
106C22×C33C2×D411yescharacteristic
107C2×C3×C3⋊D3—21no—
108C2×C3×C3⋊D3—21no—
109C2×C3×C3⋊D3—21no—
110C2×C32×D3—41no—
111C2×C32×D3—41no—
112C2×C32×D3—41no—
113C33×C4#4111yescharacteristic
114C33×C4#4111yescharacteristic
115C3×C3⋊Dic3—21no—
116C3×C3⋊Dic3—41no—
117C3×C3⋊Dic3—41no—
118C32×Dic3—41no—
119C3×#121—91no—
120#938—31no—
121#926—91no—
122#895—31no—
123#892—91no—
124#892—91no—
125C3×#107—91no—
126C3×#102—91no—
127#832—31no—
128#832—91no—
129#832—91no—
130#832—181no—
131#830—91no—
132#821—31no—
133(C2×D3)⋊C8—91no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—181no—
136#818—31no—
137#809—91no—
138#808—91no—
139#808—91no—
140#808—91no—
141#808—181no—
142#807—31no—
143C23×C32—61no—
144C22×C3⋊D3#6711yescharacteristic
145C22×C3×D3—21no—
146C22×C3×D3—21no—
147C22×C3×D3—31no—
148C22×C3×D3—31no—
149C22×C3×D3—41no—
150C22×C3×D3—61no—
151C32×D4—61no—
152C32×D4—61no—
153C2×C32×C4—21no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—41no—
157C2×C32×C4—61no—
158C2×C32×C4C22×D311yescharacteristic
159C2×C32×C4C22×D311yescharacteristic
160C2×C32×C4C22×D311yescharacteristic
161C2×C3⋊Dic3—21no—
162C2×C3⋊Dic3—21no—
163C2×C3⋊Dic3#6711yescharacteristic
164C4×C3⋊D3—21no—
165C4×C3⋊D3—21no—
166C3×#67—122no—
167C2×C3×Dic3—21no—
168C2×C3×Dic3—21no—
169C2×C3×Dic3—31no—
170C2×C3×Dic3—31no—
171C2×C3×Dic3—41no—
172C2×C3×Dic3—61no—
173C2×C3×Dic3—61no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—61no—
177C2×C3×Dic3—121no—
178C2×C3×Dic3—121no—
179C3×D12—22no—
180C3×D12—21no—
181C3×D12—21no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—41no—
185C3×D12—122no—
186C3×C4×D3—61no—
187C3×C4×D3—61no—
188C3×C4×D3—61no—
189C3×C4×D3—61no—
190C3×C4×D3—121no—
191C3×C4×D3—121no—
192#626—42no—
193C32×C8—61no—
194C3⋊(C3⋊C8)—61no—
195C3⋊(C3⋊C8)—62no—
196C3⋊(C3⋊C8)—61no—
197C3⋊(C3⋊C8)—61no—
198C3⋊(C3⋊C8)—121no—
199C3⋊(C3⋊C8)—121no—
200C3⋊(C3⋊C8)—122no—
201C3×C3⋊C8—21no—
202C3×C3⋊C8—21no—
203C3×C3⋊C8—41no—
204#482—271noSylow
205C2×C33#12311yescharacteristic
206C2×C33#13511yescharacteristic
207C2×C33AGL(1,ℤ/8)11yescharacteristic
208C3×C3⋊D3—41no—
209C32×D3—81no—
210C2×C3×D4—31no—
211C2×C3×D4—31no—
212C2×C3×D4—61no—
213C2×C3×D4—181no—
214C22×C3×C4—91no—
215C2×#67—61no—
216C2×D12—61no—
217C2×D12D3211yescharacteristic
218C2×C4×D3—31no—
219C2×C4×D3—31no—
220C2×C4×D3—61no—
221C2×C3×C8—31no—
222C2×C3×C8—31no—
223C2×C3×C8—61no—
224C3×C4⋊C4—31no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—61no—
227C3×C4⋊C4—91no—
228C3×C4⋊C4—91no—
229C3×C22⋊C4—181no—
230#217—61no—
231#217—181no—
232(C2×D3)⋊C4—61no—
233(C2×D3)⋊C4—181no—
234(C2×D3)⋊C4—181no—
235C4⋊Dic3—31no—
236C4⋊Dic3—31no—
237C4⋊Dic3—31no—
238C4⋊Dic3—31no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3D3211yescharacteristic
242C4×Dic3—181no—
243C4×Dic3—181no—
244C4×Dic3—181no—
245C2×C3⋊C8—91no—
246C2×C3⋊C8—91no—
247C2×C3⋊C8—91no—
248C2×C3⋊C8—181no—
249C2×C3⋊C8—181no—
250C2×C3⋊C8—181no—
251C2×C3⋊C8—361no—
252C2×C3⋊C8D3211yescharacteristic
253C22×C32—21no—
254C22×C32—21no—
255C22×C32—21no—
256C22×C32—41no—
257C22×C32—121no—
258C22×C32—121no—
259C22×C32—121no—
260C22×C32D3×D411yescharacteristic
261C22×C32D3×D411yescharacteristic
262C22×C32C2×#6711yescharacteristic
263C2×C3⋊D3—21no—
264C2×C3⋊D3—21no—
265C2×C3⋊D3—21no—
266C2×C3×D3—42no—
267C2×C3×D3—41no—
268C2×C3×D3—41no—
269C2×C3×D3—41no—
270C2×C3×D3—41no—
271C2×C3×D3—41no—
272C2×C3×D3—41no—
273C2×C3×D3—62no—
274C2×C3×D3—61no—
275C2×C3×D3—61no—
276C2×C3×D3—61no—
277C2×C3×D3—61no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—81no—
281C2×C3×D3—81no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C32×C4—21no—
285C32×C4—21no—
286C32×C4—21no—
287C32×C4—21no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—41no—
291C32×C4—41no—
292C32×C4—121no—
293C32×C4#23511yescharacteristic
294C32×C4#23511yescharacteristic
295C32×C4#23711yescharacteristic
296C32×C4#23711yescharacteristic
297C32×C4#23711yescharacteristic
298C32×C4#23911yescharacteristic
299C3⋊Dic3—21no—
300C3⋊Dic3—41no—
301C3⋊Dic3—41no—
302C3×Dic3—41no—
303C3×Dic3—41no—
304C3×Dic3—61no—
305C3×Dic3—61no—
306C3×Dic3—81no—
307C3×Dic3—121no—
308C3×Dic3—121no—
309C3×Dic3—121no—
310C3×Dic3—121no—
311C3×Dic3—121no—
312C3×Dic3—241no—
313C3×Dic3—241no—
314#121—91no—
315#117—271no—
316#107—91no—
317#106—271no—
318#102—91no—
319#102—271no—
320C22⋊C8—271no—
321C33#48211yescharacteristic, complemented, Sylow
322C23×C3—61no—
323C23×C3—61no—
324C23×C3—91no—
325C23×C3—121no—
326C22×D3—21no—
327C22×D3—31no—
328C22×D3—31no—
329C22×D3—61no—
330C3×D4—62no—
331C3×D4—61no—
332C3×D4—61no—
333C3×D4—61no—
334C3×D4—61no—
335C3×D4—121no—
336C3×D4—362no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—41no—
341C2×C3×C4—61no—
342C2×C3×C4—61no—
343C2×C3×C4—91no—
344C2×C3×C4—121no—
345C2×C3×C4—181no—
346C2×C3×C4—181no—
347C2×C3×C4—181no—
348C2×C3×C4—181no—
349C2×C3×C4C2×D3211yescharacteristic
350C2×C3×C4C2×D3211yescharacteristic
351C2×C3×C4C2×D3211yescharacteristic
352#67—122no—
353C2×Dic3—21no—
354C2×Dic3—31no—
355C2×Dic3—31no—
356C2×Dic3—61no—
357C2×Dic3—61no—
358C2×Dic3—61no—
359C2×Dic3—61no—
360C2×Dic3—61no—
361C2×Dic3—121no—
362C2×Dic3—121no—
363D12—21no—
364D12—21no—
365D12—122no—
366C4×D3—61no—
367C4×D3—61no—
368C4×D3—61no—
369C4×D3—61no—
370C4×D3—121no—
371C4×D3—121no—
372C3×C8—61no—
373C3×C8—61no—
374C3×C8—121no—
375C3⋊C8—21no—
376C3⋊C8—182no—
377C3⋊C8—181no—
378C3⋊C8—181no—
379C3⋊C8—181no—
380C3⋊C8—361no—
381C3⋊C8—361no—
382C3⋊C8—362no—
383C2×C32—21no—
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—41no—
393C2×C32—41no—
394C2×C32—41no—
395C2×C32—241no—
396C2×C32#88311yescharacteristic
397C2×C32#89411yescharacteristic
398C2×C32#91111yescharacteristic
399C2×C32#91211yescharacteristic
400C2×C32#91411yescharacteristic
401C2×C32#91911yescharacteristic
402C2×C32#93211yescharacteristic
403C2×C32#93511yescharacteristic
404C2×C32#95011yescharacteristic
405C3⋊D3—41no—
406C3×D3—82no—
407C3×D3—81no—
408C3×D3—81no—
409C3×D3—122no—
410C3×D3—121no—
411C3×D3—121no—
412C2×D4—31no—
413C2×D4—181no—
414C22×C4—91no—
415C2×C8—31no—
416C2×C8—271no—
417C4⋊C4—31no—
418C4⋊C4—91no—
419C4⋊C4—91no—
420C22⋊C4—181no—
421C22⋊C4—541no—
422C42—541no—
423C22×C3—21no—
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—41no—
427C22×C3—122no—
428C22×C3—121no—
429C22×C3—121no—
430C22×C3—121no—
431C22×C3—121no—
432C22×C3—121no—
433C22×C3—121no—
434C22×C3—181no—
435C22×C3—181no—
436C22×C3—181no—
437C22×C3—241no—
438C22×C3—241no—
439C22×C3#380211yescharacteristic
440C22×C3D3×#6711yescharacteristic
441C22×C3D3×#6711yescharacteristic
442C2×D3—41no—
443C2×D3—41no—
444C2×D3—41no—
445C2×D3—62no—
446C2×D3—61no—
447C2×D3—61no—
448C2×D3—61no—
449C2×D3—61no—
450C2×D3—61no—
451C2×D3—61no—
452C2×D3—121no—
453C2×D3—121no—
454C3×C4—21no—
455C3×C4—21no—
456C3×C4—21no—
457C3×C4—21no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—41no—
461C3×C4—41no—
462C3×C4—121no—
463C3×C4—121no—
464C3×C4—181no—
465C3×C4—241no—
466C3×C4—361no—
467C3×C4—361no—
468C3×C4#379511yescharacteristic
469C3×C4#379611yescharacteristic
470C3×C4#379911yescharacteristic
471C3×C4#380411yescharacteristic
472C3×C4#380411yescharacteristic
473C3×C4#380511yescharacteristic
474Dic3—41no—
475Dic3—61no—
476Dic3—61no—
477Dic3—121no—
478Dic3—121no—
479Dic3—121no—
480Dic3—121no—
481Dic3—121no—
482Dic3—241no—
483Dic3—241no—
484C32—21no—
485C32—21no—
486C32—21no—
487C32—41no—
488C32#478411yescharacteristic, complemented
489C32#488511yescharacteristic, complemented
490C32#503511yescharacteristic, complemented
491C23—61no—
492C23—91no—
493D4—61no—
494D4—61no—
495D4—362no—
496C2×C4—61no—
497C2×C4—91no—
498C2×C4—181no—
499C2×C4—181no—
500C2×C4—181no—
501C2×C4—181no—
502C2×C4D3311yescharacteristic
503C8—61no—
504C8—541no—
505C2×C3—21no—
506C2×C3—21no—
507C2×C3—21no—
508C2×C3—21no—
509C2×C3—21no—
510C2×C3—21no—
511C2×C3—21no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—41no—
515C2×C3—41no—
516C2×C3—41no—
517C2×C3—242no—
518C2×C3—241no—
519C2×C3—241no—
520C2×C3—361no—
521C2×C3#6379811yescharacteristic
522C2×C3#6381011yescharacteristic
523C2×C3#6385411yescharacteristic
524C2×C3#6388411yescharacteristic
525C2×C3#6393211yescharacteristic
526C2×C3#6393311yescharacteristic
527C2×C3#6393411yescharacteristic
528C2×C3#6394911yescharacteristic
529C2×C3#6396911yescharacteristic
530D3—81no—
531D3—122no—
532D3—121no—
533D3—121no—
534C22—121no—
535C22—121no—
536C22—121no—
537C22—181no—
538C22—181no—
539C22—181no—
540C22#8895311yescharacteristic
541C4—121no—
542C4—181no—
543C4—361no—
544C4—361no—
545C4#8896111yescharacteristic
546C4#8896211yescharacteristic
547C3—21no—
548C3—21no—
549C3—21no—
550C3—41no—
551C3#1059044611yescharacteristic, complemented
552C3#1059065111yescharacteristic, complemented
553C3#1059142311yescharacteristic, complemented
554C2—241no—
555C2—361no—
556C2#1172286011yescharacteristic
557C2#1172287511yescharacteristic
558C2#1172299411yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63916C2
hypocenterC33#482
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. #11723667C2semidirect
3. #11722053C2semidirect
4. #11722031C2semidirect
5. #11722023C2semidirect
6. C3×#63575C2semidirect
7. C3×#63563C2semidirect
35. #63575D3semidirect
39. #63563D3semidirect
321. C33#482semidirect
488. C32#4784semidirect
489. C32#4885semidirect
490. C32#5035semidirect
551. C3#10590446semidirect
552. C3#10590651semidirect
553. C3#10591423semidirect

Maximal subgroups

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

subgroupindex
#117236672
C3×#639162
#117220532
#117220312
#117220232
C3×#635752
C3×#635632
#105914233
#105906513
#105904463