Finite Group Databasebeta

← #13156558

Subgroups — #13156558

Order 1728.

602 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#63911C211yesmaximal, characteristic
2C3×#63869C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722033C211yesmaximal, characteristic, complemented
5#11722031C211yesmaximal, characteristic, complemented
6#11722019C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591477—31nomaximal
9#10590550—31nomaximal
10#10590279—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×C3×#626—21no—
14C2×C3×#626—21no—
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C32×C4×Dic3C2211yescharacteristic
20C3×#3724—21no—
21C3×Dic3⋊Dic3—21no—
22C3×#940—31no—
23C3×#890—31no—
24C3×#874—31no—
25C3×#870—31no—
26#63911D311yescharacteristic
27#63869D311yescharacteristic, complemented
28#63659—31no—
29#63636—31no—
30#63630—31no—
31C3×#832—31no—
32C3×#809—31no—
33#63564—31no—
34#63563—31no—
35#63562—31no—
36#63562—31no—
37#63562—31no—
38#63562—31no—
39#63562—31no—
40#63562—61no—
41#63562—61no—
42#63562D311yescharacteristic, complemented
43#63554—31no—
44#63554—31no—
45#63554—61no—
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3×C3⋊Dic3—21no—
50C3×C4⋊(C3⋊D3)—21no—
51C3×C4⋊(C3⋊D3)—21no—
52C2×C32×Dic3—21no—
53C2×C32×Dic3D411yescharacteristic
54C2×C32×Dic3D411yescharacteristic
55C32×D12—21no—
56C32×D12—21no—
57C3×#626—42no—
58C3×#626—42no—
59C3⋊(C3⋊(C3⋊C8))—21no—
60C3×C3⋊(C3⋊C8)—21no—
61#5071—91no—
62#5067—91no—
63#4764—91no—
64C2×C32×D4—31no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C3×#67—61no—
67C2×C3×#67—61no—
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12—61no—
73C2×C3×D12—61no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×#626—21no—
77C2×#626—21no—
78C32×C4⋊C4—31no—
79C32×C42—31no—
80C4⋊(C3⋊Dic3)C2×D311yescharacteristic
81C2×C3⋊(C3⋊C8)—31no—
82C2×C3⋊(C3⋊C8)—31no—
83C2×C3⋊(C3⋊C8)—31no—
84C2×C3⋊(C3⋊C8)—61no—
85C2×C3⋊(C3⋊C8)—61no—
86C2×C3⋊(C3⋊C8)—61no—
87C2×C3⋊(C3⋊C8)—121no—
88C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—31no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3—61no—
94C3×C4⋊Dic3C2×D311yescharacteristic
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×Dic3⋊C4—61no—
97C3×Dic3⋊C4—61no—
98C3×Dic3⋊C4—61no—
99C3×C4×Dic3—21no—
100C3×C4×Dic3C2×D311yescharacteristic
101C3×C4×Dic3C2×D311yescharacteristic
102C2×C3×C3⋊C8—31no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—61no—
105#3724—21no—
106Dic3⋊Dic3—21no—
107C22×C33C2×D411yescharacteristic
108C2×C3×C3⋊D3—41no—
109C2×C3×C3⋊D3—41no—
110C2×C3×C3⋊D3—41no—
111C2×C32×D3—41no—
112C2×C32×D3—41no—
113C2×C32×D3—41no—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C3×C3⋊Dic3—41no—
117C32×Dic3—21no—
118C32×Dic3—21no—
119C32×Dic3—41no—
120C3×#127—91no—
121C3×#125—91no—
122#940—31no—
123#890—31no—
124#874—31no—
125#870—31no—
126C3×#102—91no—
127#832—31no—
128#832—91no—
129#832—91no—
130#832—91no—
131#832—91no—
132#832—181no—
133#821—91no—
134#815—91no—
135#809—31no—
136#809—91no—
137#809—91no—
138#809—91no—
139#809—91no—
140#809—181no—
141#803—91no—
142#803—91no—
143#803—181no—
144C23×C32—61no—
145C22×C3⋊D3—21no—
146C22×C3×D3—21no—
147C22×C3×D3—21no—
148C22×C3×D3—41no—
149C22×C3×D3—61no—
150C22×C3×D3—61no—
151C22×C3×D3—121no—
152C32×D4—61no—
153C32×D4—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—31no—
158C2×C32×C4—31no—
159C2×C32×C4—41no—
160C2×C32×C4—61no—
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C32×C4C22×D311yescharacteristic
164C2×C3⋊Dic3—21no—
165C4⋊(C3⋊D3)—21no—
166C4⋊(C3⋊D3)—21no—
167C3×#67—122no—
168C3×#67—122no—
169C2×C3×Dic3—21no—
170C2×C3×Dic3—21no—
171C2×C3×Dic3—21no—
172C2×C3×Dic3—21no—
173C2×C3×Dic3—41no—
174C2×C3×Dic3—61no—
175C2×C3×Dic3—61no—
176C2×C3×Dic3—121no—
177C2×C3×Dic3#6711yescharacteristic
178C2×C3×Dic3#6711yescharacteristic
179C2×C3×Dic3#6711yescharacteristic
180C2×C3×Dic3#6711yescharacteristic
181C3×D12—22no—
182C3×D12—21no—
183C3×D12—21no—
184C3×D12—21no—
185C3×D12—21no—
186C3×D12—41no—
187C3×D12—62no—
188C3×D12—61no—
189C3×D12—61no—
190C3×D12—61no—
191C3×D12—61no—
192C3×D12—121no—
193C3×D12—122no—
194C3×D12—122no—
195#626—42no—
196#626—42no—
197C3⋊(C3⋊C8)—21no—
198C3⋊(C3⋊C8)—61no—
199C3⋊(C3⋊C8)—62no—
200C3⋊(C3⋊C8)—61no—
201C3⋊(C3⋊C8)—61no—
202C3⋊(C3⋊C8)—121no—
203C3⋊(C3⋊C8)—121no—
204C3⋊(C3⋊C8)—122no—
205C3×C3⋊C8—62no—
206C3×C3⋊C8—61no—
207C3×C3⋊C8—61no—
208#490—271noSylow
209C2×C33#12411yescharacteristic
210C2×C33AGL(1,ℤ/8)11yescharacteristic
211C2×C33AGL(1,ℤ/8)11yescharacteristic
212C3×C3⋊D3—81no—
213C32×D3—81no—
214C2×C3×D4—31no—
215C2×C3×D4—31no—
216C2×C3×D4—61no—
217C2×C3×D4—91no—
218C2×C3×D4—181no—
219C2×C3×D4—181no—
220C2×#67—61no—
221C2×#67—61no—
222C2×D12—31no—
223C2×D12—31no—
224C2×D12—61no—
225C2×D12—61no—
226C2×D12—61no—
227C2×D12D3211yescharacteristic
228C2×C3×C8—91no—
229C3×C4⋊C4—31no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—61no—
232C3×C4⋊C4—91no—
233C3×C4⋊C4—181no—
234C3×C4⋊C4—181no—
235C3×C42—31no—
236C3×C42—31no—
237C3×C42—61no—
238C4⋊Dic3—31no—
239C4⋊Dic3—31no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3D3211yescharacteristic
243Dic3⋊C4—61no—
244Dic3⋊C4—61no—
245Dic3⋊C4—61no—
246C4×Dic3D3211yescharacteristic
247C2×C3⋊C8—31no—
248C2×C3⋊C8—31no—
249C2×C3⋊C8—61no—
250C2×C3⋊C8—91no—
251C2×C3⋊C8—91no—
252C2×C3⋊C8—91no—
253C2×C3⋊C8—181no—
254C2×C3⋊C8—181no—
255C2×C3⋊C8—181no—
256C2×C3⋊C8—361no—
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×C32C2×#6711yescharacteristic
266C22×C32C2×#6711yescharacteristic
267C2×C3⋊D3—41no—
268C2×C3⋊D3—41no—
269C2×C3⋊D3—41no—
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—81no—
278C2×C3×D3—81no—
279C2×C3×D3—122no—
280C2×C3×D3—121no—
281C2×C3×D3—121no—
282C2×C3×D3—121no—
283C2×C3×D3—121no—
284C2×C3×D3—121no—
285C2×C3×D3—121no—
286C2×C3×D3—241no—
287C2×C3×D3—241no—
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—61no—
297C32×C4—61no—
298C32×C4—121no—
299C32×C4#23511yescharacteristic
300C32×C4#23711yescharacteristic
301C32×C4#23711yescharacteristic
302C32×C4#23711yescharacteristic
303C32×C4#23911yescharacteristic
304C32×C4#23911yescharacteristic
305C3⋊Dic3—41no—
306C3×Dic3—21no—
307C3×Dic3—21no—
308C3×Dic3—21no—
309C3×Dic3—21no—
310C3×Dic3—41no—
311C3×Dic3—41no—
312C3×Dic3—41no—
313C3×Dic3—41no—
314C3×Dic3—81no—
315C3×Dic3—121no—
316C3×Dic3—121no—
317C3×Dic3—241no—
318#127—91no—
319#125—91no—
320#102—91no—
321#102—271no—
322#102—271no—
323#102—271no—
324#97—271no—
325C33#49011yescharacteristic, complemented, Sylow
326C23×C3—61no—
327C23×C3—61no—
328C23×C3—121no—
329C23×C3—181no—
330C22×D3—21no—
331C22×D3—61no—
332C22×D3—61no—
333C22×D3—121no—
334C3×D4—62no—
335C3×D4—61no—
336C3×D4—61no—
337C3×D4—61no—
338C3×D4—61no—
339C3×D4—121no—
340C3×D4—181no—
341C3×D4—181no—
342C3×D4—362no—
343C3×D4—362no—
344C2×C3×C4—21no—
345C2×C3×C4—21no—
346C2×C3×C4—21no—
347C2×C3×C4—31no—
348C2×C3×C4—31no—
349C2×C3×C4—31no—
350C2×C3×C4—31no—
351C2×C3×C4—41no—
352C2×C3×C4—61no—
353C2×C3×C4—61no—
354C2×C3×C4—61no—
355C2×C3×C4—61no—
356C2×C3×C4—121no—
357C2×C3×C4—181no—
358C2×C3×C4C2×D3211yescharacteristic
359C2×C3×C4C2×D3211yescharacteristic
360C2×C3×C4C2×D3211yescharacteristic
361#67—122no—
362#67—122no—
363C2×Dic3—21no—
364C2×Dic3—61no—
365C2×Dic3—61no—
366C2×Dic3—121no—
367C2×Dic3#62511yescharacteristic
368C2×Dic3#62511yescharacteristic
369D12—21no—
370D12—21no—
371D12—62no—
372D12—61no—
373D12—61no—
374D12—61no—
375D12—61no—
376D12—121no—
377D12—122no—
378D12—122no—
379C3×C8—181no—
380C3⋊C8—62no—
381C3⋊C8—61no—
382C3⋊C8—61no—
383C3⋊C8—182no—
384C3⋊C8—181no—
385C3⋊C8—181no—
386C3⋊C8—181no—
387C3⋊C8—361no—
388C3⋊C8—361no—
389C3⋊C8—362no—
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#88511yescharacteristic
404C2×C32#91111yescharacteristic
405C2×C32#91411yescharacteristic
406C2×C32#93211yescharacteristic
407C2×C32#93211yescharacteristic
408C2×C32#93611yescharacteristic
409C2×C32#94711yescharacteristic
410C2×C32#94911yescharacteristic
411C2×C32#94911yescharacteristic
412C3⋊D3—81no—
413C3×D3—82no—
414C3×D3—81no—
415C3×D3—81no—
416C3×D3—242no—
417C3×D3—241no—
418C3×D3—241no—
419C2×D4—31no—
420C2×D4—91no—
421C2×D4—181no—
422C2×D4—181no—
423C2×C8—91no—
424C2×C8—271no—
425C4⋊C4—31no—
426C4⋊C4—91no—
427C4⋊C4—181no—
428C4⋊C4—181no—
429C42—31no—
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—241no—
442C22×C3—241no—
443C22×C3—361no—
444C22×C3—361no—
445C22×C3—361no—
446C22×C3C2×#62511yescharacteristic
447C22×C3D3×#6711yescharacteristic
448C22×C3D3×#6711yescharacteristic
449C2×D3—41no—
450C2×D3—41no—
451C2×D3—41no—
452C2×D3—122no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
457C2×D3—121no—
458C2×D3—121no—
459C2×D3—241no—
460C2×D3—241no—
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—61no—
470C3×C4—61no—
471C3×C4—61no—
472C3×C4—61no—
473C3×C4—121no—
474C3×C4—121no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—241no—
478C3×C4—361no—
479C3×C4#379611yescharacteristic
480C3×C4#379611yescharacteristic
481C3×C4#379611yescharacteristic
482C3×C4#379911yescharacteristic
483C3×C4#380411yescharacteristic
484C3×C4#380511yescharacteristic
485Dic3—21no—
486Dic3—21no—
487Dic3—41no—
488Dic3—121no—
489Dic3—121no—
490Dic3—241no—
491C32—21no—
492C32—21no—
493C32—21no—
494C32—41no—
495C32#476411yescharacteristic, complemented
496C32#506711yescharacteristic, complemented
497C32#507111yescharacteristic, complemented
498C23—61no—
499C23—181no—
500D4—61no—
501D4—61no—
502D4—181no—
503D4—181no—
504D4—362no—
505D4—362no—
506C2×C4—31no—
507C2×C4—31no—
508C2×C4—61no—
509C2×C4—181no—
510C2×C4D3311yescharacteristic
511C8—181no—
512C8—541no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—21no—
516C2×C3—21no—
517C2×C3—21no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—21no—
522C2×C3—41no—
523C2×C3—41no—
524C2×C3—41no—
525C2×C3—242no—
526C2×C3—241no—
527C2×C3—241no—
528C2×C3—721no—
529C2×C3#6382311yescharacteristic
530C2×C3#6382311yescharacteristic
531C2×C3#6386311yescharacteristic
532C2×C3#6390711yescharacteristic
533C2×C3#6392511yescharacteristic
534C2×C3#6393211yescharacteristic
535C2×C3#6393411yescharacteristic
536C2×C3#6393911yescharacteristic
537C2×C3#6395911yescharacteristic
538D3—81no—
539D3—242no—
540D3—241no—
541D3—241no—
542C22—121no—
543C22—121no—
544C22—121no—
545C22—361no—
546C22—361no—
547C22—361no—
548C22D3×#62511yescharacteristic
549C4—61no—
550C4—61no—
551C4—121no—
552C4—361no—
553C4#8896211yescharacteristic
554C4#8896411yescharacteristic
555C3—21no—
556C3—21no—
557C3—21no—
558C3—41no—
559C3#1059027911yescharacteristic, complemented
560C3#1059055011yescharacteristic, complemented
561C3#1059147711yescharacteristic, complemented
562C2—241no—
563C2—721no—
564C2#1172286611yescharacteristic
565C2#1172286711yescharacteristic
566C2#1172300111yescharacteristic
subgroupG/H
centerC22D3×#625
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC3×#63911C2
hypocenterC33#490
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. C3×#63869C2semidirect
3. #11722075C2semidirect
4. #11722033C2semidirect
5. #11722031C2semidirect
6. #11722019C2semidirect
7. C3×#63562C2semidirect
27. #63869D3semidirect
42. #63562D3semidirect
325. C33#490semidirect
495. C32#4764semidirect
496. C32#5067semidirect
497. C32#5071semidirect
559. C3#10590279semidirect
560. C3#10590550semidirect
561. C3#10591477semidirect

Maximal subgroups

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

subgroupindex
C3×#639112
C3×#638692
#117220752
#117220332
#117220312
#117220192
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