Finite Group Databasebeta

← #13156566

Subgroups — #13156566

Order 1728.

674 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 96 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#11723529C211yesmaximal, characteristic
2#11723450C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722021C211yesmaximal, characteristic, complemented
5C3×#63564C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591480—31nomaximal
9#10590888—31nomaximal
10#10590554—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6290—21no—
14C2×#6290—21no—
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88807—21no—
21Dic3⋊(C3⋊Dic3)—21no—
22#64149—31no—
23#64099—31no—
24#63914—31no—
25#63914—31no—
26#63914—61no—
27#63872—31no—
28#63872—31no—
29#63872—61no—
30#63642—31no—
31C3×#832—31no—
32C3×#832—31no—
33C3×#821—31no—
34C3×#821—31no—
35C3×#809—31no—
36C3×#809—31no—
37#63564—31no—
38#63564D311yescharacteristic, complemented
39#63563—31no—
40#63563D311yescharacteristic, complemented
41#63562—31no—
42#63562D311yescharacteristic, complemented
43#63555—31no—
44#63555—31no—
45#63555—61no—
46C22×C3×C3⋊D3—21no—
47C22×C32×D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C2×C3⋊(C3⋊Dic3)D411yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C3×C4⋊(C3⋊D3)—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C2×C32×Dic3—21no—
55C32×D12—21no—
56C32×D12—21no—
57#6290—42no—
58#6290—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#4764—91no—
62#4764—91no—
63#4764—91no—
64C2×C32×D4—31no—
65C2×#638—61no—
66C2×#638—61no—
67C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
68C2×C3×D12—21no—
69C2×C3×D12—31no—
70C2×C3×D12—31no—
71C2×C3×D12—61no—
72C2×C3×D12C2×D311yescharacteristic
73C2×C3×D12C2×D311yescharacteristic
74C2×#625—61no—
75C2×#625—61no—
76C2×#625—61no—
77C2×#625—61no—
78C2×#625—62no—
79C2×#625—62no—
80C2×C32×C8—31no—
81C32×C4⋊C4—31no—
82C4⋊(C3⋊Dic3)—61no—
83C4⋊(C3⋊Dic3)C2×D311yescharacteristic
84#3750—61no—
85C4×C3⋊Dic3—31no—
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—61no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—61no—
91C4×C3⋊Dic3—121no—
92C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
93C3×C4⋊Dic3—21no—
94C3×C4⋊Dic3—31no—
95C3×C4⋊Dic3—31no—
96C3×C4⋊Dic3—61no—
97C3×C4⋊Dic3C2×D311yescharacteristic
98C3×C4⋊Dic3C2×D311yescharacteristic
99C2×C3×C3⋊C8—21no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—31no—
102C2×C3×C3⋊C8—61no—
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C2×C3×C3⋊C8C2×D311yescharacteristic
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—61no—
107Dic3⋊Dic3—61no—
108Dic3⋊Dic3—61no—
109Dic3⋊Dic3—62no—
110Dic3⋊Dic3—62no—
111C22×C33C2×D411yescharacteristic
112C2×C3×C3⋊D3—41no—
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C32×D3—41no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C33×C4#4111yescharacteristic
119C33×C4#4111yescharacteristic
120C3⋊(C3⋊Dic3)—21no—
121C3⋊(C3⋊Dic3)—21no—
122C3×C3⋊Dic3—41no—
123C32×Dic3—41no—
124#940—91no—
125#940—91no—
126#940—91no—
127#940—181no—
128#890—91no—
129#890—91no—
130#890—91no—
131#890—181no—
132C3×#102—91no—
133C3×#102—91no—
134C3×#102—91no—
135#832—31no—
136#832—31no—
137#832—91no—
138#821—31no—
139#821—31no—
140#821—91no—
141#815—91no—
142#815—91no—
143#815—91no—
144#815—181no—
145#809—31no—
146#809—31no—
147#809—91no—
148C23×C32—61no—
149C22×C3⋊D3—21no—
150C22×C3×D3—22no—
151C22×C3×D3—21no—
152C22×C3×D3—21no—
153C22×C3×D3—62no—
154C22×C3×D3—61no—
155C22×C3×D3—61no—
156C32×D4—61no—
157C32×D4—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166#638—122no—
167#638—122no—
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—61no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—121no—
182C2×C3⋊Dic3—121no—
183C4⋊(C3⋊D3)—21no—
184C4⋊(C3⋊D3)—21no—
185C2×C3×Dic3—22no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—62no—
189C2×C3×Dic3—61no—
190C2×C3×Dic3—61no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—41no—
196C3×D12—41no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—121no—
202C3×D12—121no—
203#625—124no—
204#625—124no—
205#625—122no—
206#625—122no—
207#625—122no—
208#625—122no—
209C32×C8—61no—
210C3⋊(C3⋊C8)—21no—
211C3×C3⋊C8—22no—
212C3×C3⋊C8—21no—
213C3×C3⋊C8—21no—
214C3×C3⋊C8—62no—
215C3×C3⋊C8—61no—
216C3×C3⋊C8—61no—
217#490—271noSylow
218C2×C33#12411yescharacteristic
219C2×C33AGL(1,ℤ/8)11yescharacteristic
220C2×C33AGL(1,ℤ/8)11yescharacteristic
221C3×C3⋊D3—81no—
222C32×D3—81no—
223C2×C3×D4—31no—
224C2×C3×D4—31no—
225C2×C3×D4—61no—
226C2×C3×D4—91no—
227C2×#67—181no—
228C2×#67—181no—
229C2×#67—181no—
230C2×#67—181no—
231C2×#67—181no—
232C2×#67—181no—
233C2×#67—182no—
234C2×#67—182no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12D3211yescharacteristic
239C2×C3×C8—31no—
240C2×C3×C8—31no—
241C2×C3×C8—61no—
242C2×C3×C8—91no—
243C3×C4⋊C4—31no—
244C3×C4⋊C4—31no—
245C3×C4⋊C4—61no—
246C3×C4⋊C4—91no—
247C4⋊Dic3—31no—
248C4⋊Dic3—31no—
249C4⋊Dic3—61no—
250C4⋊Dic3—181no—
251C4⋊Dic3—181no—
252C4⋊Dic3—181no—
253C4⋊Dic3—182no—
254C4⋊Dic3D3211yescharacteristic
255Dic3⋊C4—181no—
256Dic3⋊C4—181no—
257Dic3⋊C4—181no—
258Dic3⋊C4—182no—
259C4×Dic3—91no—
260C4×Dic3—91no—
261C4×Dic3—91no—
262C4×Dic3—181no—
263C4×Dic3—181no—
264C4×Dic3—181no—
265C4×Dic3—361no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—61no—
269C2×C3⋊C8D3211yescharacteristic
270C22×C32—21no—
271C22×C32—21no—
272C22×C32—21no—
273C22×C32—41no—
274C22×C32—121no—
275C22×C32—121no—
276C22×C32—121no—
277C22×C32D3×D411yescharacteristic
278C22×C32D3×D411yescharacteristic
279C22×C32D3×D411yescharacteristic
280C2×C3⋊D3—41no—
281C2×C3⋊D3—41no—
282C2×C3⋊D3—41no—
283C2×C3×D3—42no—
284C2×C3×D3—42no—
285C2×C3×D3—42no—
286C2×C3×D3—41no—
287C2×C3×D3—41no—
288C2×C3×D3—41no—
289C2×C3×D3—41no—
290C2×C3×D3—41no—
291C2×C3×D3—41no—
292C2×C3×D3—122no—
293C2×C3×D3—122no—
294C2×C3×D3—122no—
295C2×C3×D3—121no—
296C2×C3×D3—121no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—121no—
310C32×C4#23511yescharacteristic
311C32×C4#23511yescharacteristic
312C32×C4#23511yescharacteristic
313C32×C4#23711yescharacteristic
314C32×C4#23711yescharacteristic
315C32×C4#23711yescharacteristic
316C3⋊Dic3—41no—
317C3⋊Dic3—61no—
318C3⋊Dic3—61no—
319C3⋊Dic3—62no—
320C3⋊Dic3—62no—
321C3⋊Dic3—61no—
322C3⋊Dic3—61no—
323C3⋊Dic3—61no—
324C3⋊Dic3—61no—
325C3⋊Dic3—121no—
326C3⋊Dic3—121no—
327C3⋊Dic3—121no—
328C3⋊Dic3—121no—
329C3⋊Dic3—122no—
330C3⋊Dic3—122no—
331C3×Dic3—42no—
332C3×Dic3—41no—
333C3×Dic3—41no—
334C3×Dic3—122no—
335C3×Dic3—121no—
336C3×Dic3—121no—
337#127—271no—
338#125—271no—
339#102—91no—
340#102—91no—
341#102—91no—
342#102—271no—
343#97—271no—
344C33#49011yescharacteristic, complemented, Sylow
345C23×C3—62no—
346C23×C3—61no—
347C23×C3—61no—
348C23×C3—181no—
349C22×D3—21no—
350C22×D3—62no—
351C22×D3—61no—
352C22×D3—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—61no—
357C3×D4—121no—
358C3×D4—121no—
359C3×D4—181no—
360C3×D4—181no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—41no—
365C2×C3×C4—62no—
366C2×C3×C4—61no—
367C2×C3×C4—61no—
368C2×C3×C4—181no—
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372#67—362no—
373#67—362no—
374#67—364no—
375#67—364no—
376#67—362no—
377#67—362no—
378#67—362no—
379#67—362no—
380C2×Dic3—21no—
381C2×Dic3—62no—
382C2×Dic3—61no—
383C2×Dic3—61no—
384C2×Dic3—91no—
385C2×Dic3—91no—
386C2×Dic3—91no—
387C2×Dic3—91no—
388C2×Dic3—91no—
389C2×Dic3—91no—
390C2×Dic3—181no—
391C2×Dic3—181no—
392C2×Dic3—181no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—361no—
397C2×Dic3—361no—
398D12—21no—
399D12—21no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—121no—
405D12—121no—
406C3×C8—62no—
407C3×C8—61no—
408C3×C8—61no—
409C3×C8—181no—
410C3⋊C8—21no—
411C3⋊C8—62no—
412C3⋊C8—61no—
413C3⋊C8—61no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—41no—
424C2×C32—41no—
425C2×C32—41no—
426C2×C32—241no—
427C2×C32#88511yescharacteristic
428C2×C32#88511yescharacteristic
429C2×C32#88511yescharacteristic
430C2×C32#91111yescharacteristic
431C2×C32#91111yescharacteristic
432C2×C32#91111yescharacteristic
433C2×C32#91411yescharacteristic
434C2×C32#91411yescharacteristic
435C2×C32#91411yescharacteristic
436C3⋊D3—81no—
437C3×D3—82no—
438C3×D3—81no—
439C3×D3—81no—
440C3×D3—242no—
441C3×D3—241no—
442C3×D3—241no—
443C2×D4—31no—
444C2×D4—91no—
445C2×D4—541no—
446C2×D4—541no—
447C2×C8—31no—
448C2×C8—91no—
449C4⋊C4—31no—
450C4⋊C4—91no—
451C4⋊C4—541no—
452C4⋊C4—541no—
453C42—271no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—21no—
457C22×C3—41no—
458C22×C3—122no—
459C22×C3—122no—
460C22×C3—122no—
461C22×C3—121no—
462C22×C3—121no—
463C22×C3—121no—
464C22×C3—121no—
465C22×C3—121no—
466C22×C3—121no—
467C22×C3—361no—
468C22×C3—361no—
469C22×C3—361no—
470C22×C3#380211yescharacteristic
471C22×C3#381111yescharacteristic
472C22×C3#381111yescharacteristic
473C2×D3—41no—
474C2×D3—41no—
475C2×D3—41no—
476C2×D3—122no—
477C2×D3—122no—
478C2×D3—122no—
479C2×D3—121no—
480C2×D3—121no—
481C2×D3—121no—
482C2×D3—121no—
483C2×D3—121no—
484C2×D3—121no—
485C3×C4—21no—
486C3×C4—21no—
487C3×C4—21no—
488C3×C4—21no—
489C3×C4—21no—
490C3×C4—21no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—122no—
494C3×C4—121no—
495C3×C4—121no—
496C3×C4—361no—
497C3×C4#379511yescharacteristic
498C3×C4#379511yescharacteristic
499C3×C4#379811yescharacteristic
500C3×C4#380411yescharacteristic
501C3×C4#380411yescharacteristic
502C3×C4#380511yescharacteristic
503Dic3—41no—
504Dic3—122no—
505Dic3—121no—
506Dic3—121no—
507Dic3—182no—
508Dic3—182no—
509Dic3—181no—
510Dic3—181no—
511Dic3—181no—
512Dic3—181no—
513Dic3—181no—
514Dic3—181no—
515Dic3—361no—
516Dic3—361no—
517Dic3—361no—
518Dic3—361no—
519Dic3—362no—
520Dic3—362no—
521C32—21no—
522C32—21no—
523C32—21no—
524C32—41no—
525C32#476411yescharacteristic, complemented
526C32#476411yescharacteristic, complemented
527C32#476411yescharacteristic, complemented
528C23—61no—
529C23—181no—
530D4—61no—
531D4—61no—
532D4—181no—
533D4—181no—
534D4—1082no—
535D4—1082no—
536C2×C4—61no—
537C2×C4—181no—
538C2×C4—271no—
539C2×C4—271no—
540C2×C4D3311yescharacteristic
541C8—61no—
542C8—181no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—21no—
547C2×C3—21no—
548C2×C3—21no—
549C2×C3—21no—
550C2×C3—21no—
551C2×C3—21no—
552C2×C3—41no—
553C2×C3—41no—
554C2×C3—41no—
555C2×C3—242no—
556C2×C3—241no—
557C2×C3—241no—
558C2×C3—721no—
559C2×C3#6379811yescharacteristic
560C2×C3#6379811yescharacteristic
561C2×C3#6391311yescharacteristic
562C2×C3#6391511yescharacteristic
563C2×C3#6392711yescharacteristic
564C2×C3#6393711yescharacteristic
565C2×C3#6393711yescharacteristic
566C2×C3#6394111yescharacteristic
567C2×C3#6395311yescharacteristic
568D3—81no—
569D3—242no—
570D3—241no—
571D3—241no—
572C22—121no—
573C22—121no—
574C22—121no—
575C22—361no—
576C22—361no—
577C22—361no—
578C22#8895411yescharacteristic
579C4—121no—
580C4—361no—
581C4—541no—
582C4—541no—
583C4#8896011yescharacteristic
584C4#8896111yescharacteristic
585C3—21no—
586C3—21no—
587C3—21no—
588C3—41no—
589C3#1059055411yescharacteristic, complemented
590C3#1059088811yescharacteristic, complemented
591C3#1059148011yescharacteristic, complemented
592C2—241no—
593C2—721no—
594C2#1172287711yescharacteristic
595C2#1172287811yescharacteristic
596C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723529C2
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. #11723450C2semidirect
3. #11722075C2semidirect
4. #11722021C2semidirect
5. C3×#63564C2semidirect
6. C3×#63563C2semidirect
7. C3×#63562C2semidirect
38. #63564D3semidirect
40. #63563D3semidirect
42. #63562D3semidirect
344. C33#490semidirect
525. C32#4764semidirect
526. C32#4764semidirect
527. C32#4764semidirect
589. C3#10590554semidirect
590. C3#10590888semidirect
591. C3#10591480semidirect

Maximal subgroups

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

subgroupindex
#117235292
#117234502
#117220752
#117220212
C3×#635642
C3×#635632
C3×#635622
#105914803
#105908883
#105905543