Finite Group Databasebeta

← #13156431

Subgroups — #13156431

Order 1728.

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

normal subgroups only

This group has 612 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#11723404C211yesmaximal, characteristic, complemented
2C3×#63916C211yesmaximal, characteristic
3#11722072C211yesmaximal, characteristic, complemented
4#11722057C211yesmaximal, characteristic, complemented
5#11722032C211yesmaximal, characteristic, complemented
6C3×#63574C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591423—31nomaximal
9#10590513—31nomaximal
10#10590449—31nomaximal
11C2×C3×C4×C3⋊D3C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C4⋊(C3⋊(C3⋊Dic3))C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19#88803—21no—
20Dic3×C3⋊Dic3—21no—
21C3×(C2×D3)⋊Dic3—21no—
22#64101—31no—
23C3×#938—31no—
24C3×#895—31no—
25#63916D311yescharacteristic
26#63841—31no—
27#63841—31no—
28#63841—61no—
29#63659—31no—
30#63634—31no—
31C3×#832—31no—
32C3×#809—31no—
33C3×#808—31no—
34C3×#808—31no—
35#63575—31no—
36#63575—31no—
37#63575—61no—
38#63574D311yescharacteristic, complemented
39#63563—31no—
40#63563—31no—
41#63563—61no—
42#63562D311yescharacteristic, complemented
43#63557—31no—
44(C2×D3)⋊(C3⋊C8)—31no—
45(C2×D3)⋊(C3⋊C8)—31no—
46C22×C3×C3⋊D3D411yescharacteristic
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)—21no—
50C2×C3×C3⋊Dic3D411yescharacteristic
51C3×C4×C3⋊D3—21no—
52C3×C4×C3⋊D3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C3×C3⋊(C3⋊C8)—21no—
60#5035—91no—
61#4866—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—
74C32×C4⋊C4—31no—
75C32×C4⋊C4—31no—
76#3752—61no—
77C4⋊(C3⋊Dic3)—31no—
78C4⋊(C3⋊Dic3)—31no—
79C4⋊(C3⋊Dic3)—31no—
80C4⋊(C3⋊Dic3)—61no—
81C4⋊(C3⋊Dic3)—61no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)—121no—
84C4×C3⋊Dic3—61no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
87C3×#217—61no—
88C3×(C2×D3)⋊C4—61no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2×D311yescharacteristic
92C3×C4⋊Dic3C2×D311yescharacteristic
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
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(C2×D3)⋊Dic3—21no—
102(C2×D3)⋊Dic3—61no—
103(C2×D3)⋊Dic3—61no—
104(C2×D3)⋊Dic3—62no—
105Dic32—61no—
106Dic32—61no—
107Dic32—62no—
108C22×C33C2×D411yescharacteristic
109C2×C3×C3⋊D3—21no—
110C2×C3×C3⋊D3—21no—
111C2×C3×C3⋊D3—21no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C2×C32×D3—41no—
115C33×C4#4111yescharacteristic
116C33×C4#4111yescharacteristic
117C3⋊(C3⋊Dic3)—41no—
118C3×C3⋊Dic3—21no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121C3×#121—91no—
122#938—31no—
123#926—91no—
124#895—31no—
125#892—91no—
126#892—91no—
127#892—181no—
128C3×#106—91no—
129C3×#102—91no—
130#832—31no—
131#832—91no—
132#832—91no—
133#832—181no—
134#830—91no—
135#821—91no—
136(C2×D3)⋊C8—91no—
137(C2×D3)⋊C8—91no—
138#818—91no—
139#809—31no—
140#808—31no—
141#808—31no—
142#807—91no—
143#807—91no—
144#807—181no—
145C23×C32—61no—
146C22×C3⋊D3#6711yescharacteristic
147C22×C3×D3—21no—
148C22×C3×D3—21no—
149C22×C3×D3—31no—
150C22×C3×D3—31no—
151C22×C3×D3—41no—
152C22×C3×D3—61no—
153C32×D4—61no—
154C32×D4—61no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—21no—
158C2×C32×C4—41no—
159C2×C32×C4—61no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—61no—
165C2×C3⋊Dic3—62no—
166C2×C3⋊Dic3—61no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—121no—
169C2×C3⋊Dic3—121no—
170C2×C3⋊Dic3—122no—
171C2×C3⋊Dic3#6711yescharacteristic
172C4×C3⋊D3—21no—
173C4×C3⋊D3—21no—
174C3×#67—122no—
175C2×C3×Dic3—22no—
176C2×C3×Dic3—21no—
177C2×C3×Dic3—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—31no—
181C2×C3×Dic3—31no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—61no—
184C3×D12—22no—
185C3×D12—21no—
186C3×D12—21no—
187C3×D12—21no—
188C3×D12—21no—
189C3×D12—41no—
190C3×D12—122no—
191C3×C4×D3—62no—
192C3×C4×D3—62no—
193C3×C4×D3—61no—
194C3×C4×D3—61no—
195C3×C4×D3—61no—
196C3×C4×D3—61no—
197#626—42no—
198C3⋊(C3⋊C8)—21no—
199C3⋊(C3⋊C8)—21no—
200C3×C3⋊C8—62no—
201C3×C3⋊C8—61no—
202C3×C3⋊C8—61no—
203C3×C3⋊C8—61no—
204C3×C3⋊C8—61no—
205C3×C3⋊C8—121no—
206#482—271noSylow
207C2×C33#12311yescharacteristic
208C2×C33#13511yescharacteristic
209C2×C33AGL(1,ℤ/8)11yescharacteristic
210C3×C3⋊D3—41no—
211C32×D3—81no—
212C2×C3×D4—31no—
213C2×C3×D4—31no—
214C2×C3×D4—61no—
215C2×C3×D4—181no—
216C22×C3×C4—91no—
217C2×#67—61no—
218C2×D12—61no—
219C2×D12D3211yescharacteristic
220C2×C4×D3—31no—
221C2×C4×D3—31no—
222C2×C4×D3—61no—
223C2×C3×C8—91no—
224C2×C3×C8—91no—
225C3×C4⋊C4—31no—
226C3×C4⋊C4—31no—
227C3×C4⋊C4—31no—
228C3×C4⋊C4—31no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4—61no—
231C3×C22⋊C4—181no—
232#217—61no—
233#217—181no—
234(C2×D3)⋊C4—61no—
235(C2×D3)⋊C4—181no—
236(C2×D3)⋊C4—181no—
237(C2×D3)⋊C4—182no—
238C4⋊Dic3—91no—
239C4⋊Dic3—91no—
240C4⋊Dic3—91no—
241C4⋊Dic3—181no—
242C4⋊Dic3—181no—
243C4⋊Dic3—181no—
244C4⋊Dic3—361no—
245C4⋊Dic3D3211yescharacteristic
246C4⋊Dic3D3211yescharacteristic
247C4×Dic3—181no—
248C4×Dic3—181no—
249C4×Dic3—181no—
250C4×Dic3—182no—
251C2×C3⋊C8—31no—
252C2×C3⋊C8—31no—
253C2×C3⋊C8—31no—
254C2×C3⋊C8—31no—
255C2×C3⋊C8—61no—
256C2×C3⋊C8—61no—
257C22×C32—21no—
258C22×C32—21no—
259C22×C32—21no—
260C22×C32—41no—
261C22×C32—121no—
262C22×C32—121no—
263C22×C32—121no—
264C22×C32D3×D411yescharacteristic
265C22×C32D3×D411yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—21no—
268C2×C3⋊D3—21no—
269C2×C3⋊D3—21no—
270C2×C3×D3—42no—
271C2×C3×D3—41no—
272C2×C3×D3—41no—
273C2×C3×D3—41no—
274C2×C3×D3—41no—
275C2×C3×D3—41no—
276C2×C3×D3—41no—
277C2×C3×D3—62no—
278C2×C3×D3—61no—
279C2×C3×D3—61no—
280C2×C3×D3—61no—
281C2×C3×D3—61no—
282C2×C3×D3—61no—
283C2×C3×D3—61no—
284C2×C3×D3—81no—
285C2×C3×D3—81no—
286C2×C3×D3—121no—
287C2×C3×D3—121no—
288C32×C4—21no—
289C32×C4—21no—
290C32×C4—21no—
291C32×C4—21no—
292C32×C4—21no—
293C32×C4—21no—
294C32×C4—41no—
295C32×C4—41no—
296C32×C4—121no—
297C32×C4—121no—
298C32×C4#23511yescharacteristic
299C32×C4#23511yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23711yescharacteristic
303C32×C4#23911yescharacteristic
304C3⋊Dic3—21no—
305C3⋊Dic3—121no—
306C3⋊Dic3—122no—
307C3⋊Dic3—121no—
308C3⋊Dic3—121no—
309C3⋊Dic3—241no—
310C3⋊Dic3—241no—
311C3⋊Dic3—242no—
312C3×Dic3—42no—
313C3×Dic3—41no—
314C3×Dic3—41no—
315C3×Dic3—41no—
316C3×Dic3—41no—
317C3×Dic3—62no—
318C3×Dic3—61no—
319C3×Dic3—61no—
320C3×Dic3—81no—
321#121—91no—
322#117—271no—
323#107—271no—
324#106—91no—
325#102—91no—
326#102—271no—
327C22⋊C8—271no—
328C33#48211yescharacteristic, complemented, Sylow
329C23×C3—61no—
330C23×C3—61no—
331C23×C3—91no—
332C23×C3—121no—
333C22×D3—21no—
334C22×D3—31no—
335C22×D3—31no—
336C22×D3—61no—
337C3×D4—62no—
338C3×D4—61no—
339C3×D4—61no—
340C3×D4—61no—
341C3×D4—61no—
342C3×D4—121no—
343C3×D4—362no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—41no—
348C2×C3×C4—62no—
349C2×C3×C4—61no—
350C2×C3×C4—61no—
351C2×C3×C4—61no—
352C2×C3×C4—61no—
353C2×C3×C4—91no—
354C2×C3×C4—121no—
355C2×C3×C4—181no—
356C2×C3×C4—181no—
357C2×C3×C4C2×D3211yescharacteristic
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360#67—122no—
361C2×Dic3—21no—
362C2×Dic3—21no—
363C2×Dic3—31no—
364C2×Dic3—31no—
365C2×Dic3—61no—
366C2×Dic3—182no—
367C2×Dic3—181no—
368C2×Dic3—181no—
369C2×Dic3—181no—
370C2×Dic3—361no—
371C2×Dic3—361no—
372C2×Dic3—362no—
373D12—21no—
374D12—21no—
375D12—122no—
376C4×D3—62no—
377C4×D3—62no—
378C4×D3—61no—
379C4×D3—61no—
380C4×D3—61no—
381C4×D3—61no—
382C3×C8—181no—
383C3×C8—181no—
384C3⋊C8—62no—
385C3⋊C8—61no—
386C3⋊C8—61no—
387C3⋊C8—61no—
388C3⋊C8—61no—
389C3⋊C8—121no—
390C2×C32—21no—
391C2×C32—21no—
392C2×C32—21no—
393C2×C32—21no—
394C2×C32—21no—
395C2×C32—21no—
396C2×C32—21no—
397C2×C32—21no—
398C2×C32—21no—
399C2×C32—41no—
400C2×C32—41no—
401C2×C32—41no—
402C2×C32—241no—
403C2×C32#88311yescharacteristic
404C2×C32#89411yescharacteristic
405C2×C32#91211yescharacteristic
406C2×C32#91411yescharacteristic
407C2×C32#91411yescharacteristic
408C2×C32#91611yescharacteristic
409C2×C32#93211yescharacteristic
410C2×C32#93511yescharacteristic
411C2×C32#95011yescharacteristic
412C3⋊D3—41no—
413C3×D3—82no—
414C3×D3—81no—
415C3×D3—81no—
416C3×D3—122no—
417C3×D3—121no—
418C3×D3—121no—
419C2×D4—31no—
420C2×D4—181no—
421C22×C4—91no—
422C2×C8—91no—
423C2×C8—91no—
424C4⋊C4—31no—
425C4⋊C4—31no—
426C4⋊C4—271no—
427C22⋊C4—181no—
428C22⋊C4—541no—
429C42—541no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3—21no—
433C22×C3—41no—
434C22×C3—122no—
435C22×C3—121no—
436C22×C3—121no—
437C22×C3—121no—
438C22×C3—121no—
439C22×C3—121no—
440C22×C3—121no—
441C22×C3—181no—
442C22×C3—181no—
443C22×C3—181no—
444C22×C3—241no—
445C22×C3—241no—
446C22×C3D3×#6711yescharacteristic
447C22×C3D3×#6711yescharacteristic
448C22×C3#381111yescharacteristic
449C2×D3—41no—
450C2×D3—41no—
451C2×D3—41no—
452C2×D3—62no—
453C2×D3—61no—
454C2×D3—61no—
455C2×D3—61no—
456C2×D3—61no—
457C2×D3—61no—
458C2×D3—61no—
459C2×D3—121no—
460C2×D3—121no—
461C3×C4—21no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—41no—
468C3×C4—41no—
469C3×C4—122no—
470C3×C4—121no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—181no—
475C3×C4—241no—
476C3×C4#379511yescharacteristic
477C3×C4#379611yescharacteristic
478C3×C4#379611yescharacteristic
479C3×C4#379811yescharacteristic
480C3×C4#380411yescharacteristic
481C3×C4#380511yescharacteristic
482Dic3—41no—
483Dic3—41no—
484Dic3—62no—
485Dic3—61no—
486Dic3—61no—
487Dic3—362no—
488Dic3—361no—
489Dic3—361no—
490Dic3—361no—
491Dic3—721no—
492Dic3—721no—
493Dic3—722no—
494C32—21no—
495C32—21no—
496C32—21no—
497C32—41no—
498C32#478411yescharacteristic, complemented
499C32#486611yescharacteristic, complemented
500C32#503511yescharacteristic, complemented
501C23—61no—
502C23—91no—
503D4—61no—
504D4—61no—
505D4—362no—
506C2×C4—61no—
507C2×C4—61no—
508C2×C4—91no—
509C2×C4—181no—
510C2×C4—181no—
511C2×C4—541no—
512C2×C4D3311yescharacteristic
513C8—181no—
514C8—181no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—21no—
523C2×C3—21no—
524C2×C3—41no—
525C2×C3—41no—
526C2×C3—41no—
527C2×C3—242no—
528C2×C3—241no—
529C2×C3—241no—
530C2×C3—361no—
531C2×C3#6385311yescharacteristic
532C2×C3#6388411yescharacteristic
533C2×C3#6393311yescharacteristic
534C2×C3#6393411yescharacteristic
535C2×C3#6393411yescharacteristic
536C2×C3#6393611yescharacteristic
537C2×C3#6394111yescharacteristic
538C2×C3#6394711yescharacteristic
539C2×C3#6396911yescharacteristic
540D3—81no—
541D3—122no—
542D3—121no—
543D3—121no—
544C22—121no—
545C22—121no—
546C22—121no—
547C22—181no—
548C22—181no—
549C22—181no—
550C22#8895211yescharacteristic
551C4—121no—
552C4—121no—
553C4—181no—
554C4—1081no—
555C4#8896011yescharacteristic
556C4#8896411yescharacteristic
557C3—21no—
558C3—21no—
559C3—21no—
560C3—41no—
561C3#1059044911yescharacteristic, complemented
562C3#1059051311yescharacteristic, complemented
563C3#1059142311yescharacteristic, complemented
564C2—241no—
565C2—361no—
566C2#1172285711yescharacteristic
567C2#1172287311yescharacteristic
568C2#1172299111yescharacteristic
subgroupG/H
centerC22#88952
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. #11723404C2semidirect
3. #11722072C2semidirect
4. #11722057C2semidirect
5. #11722032C2semidirect
6. C3×#63574C2semidirect
7. C3×#63562C2semidirect
38. #63574D3semidirect
42. #63562D3semidirect
328. C33#482semidirect
498. C32#4784semidirect
499. C32#4866semidirect
500. C32#5035semidirect
561. C3#10590449semidirect
562. C3#10590513semidirect
563. C3#10591423semidirect

Maximal subgroups

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

subgroupindex
#117234042
C3×#639162
#117220722
#117220572
#117220322
C3×#635742
C3×#635622
#105914233
#105905133
#105904493