Finite Group Databasebeta

← #13156514

Subgroups — #13156514

Order 1728.

722 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 100 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#11723536C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3#11722075C211yesmaximal, characteristic, complemented
4#11722018C211yesmaximal, characteristic, complemented
5C3×#63564C211yesmaximal, characteristic, complemented
6C3×#63563C211yesmaximal, characteristic, complemented
7C3×#63562C211yesmaximal, characteristic, complemented
8#10591402—31nomaximal
9#10590885—31nomaximal
10#10590552—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×D12C2211yescharacteristic
13C2×#6293—21no—
14C2×#6290—21no—
15C2×#6290—21no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
18C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
19C32×C4⋊Dic3C2211yescharacteristic
20C2×C32×C3⋊C8C2211yescharacteristic
21#88806—21no—
22#64149—31no—
23#64097—31no—
24#63919—31no—
25#63919—31no—
26#63919—61no—
27#63914—31no—
28#63914—31no—
29#63914—61no—
30C8×C3⋊Dic3—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#63553—31no—
44#63553—31no—
45#63553—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#6293—42no—
58#6290—42no—
59#6290—42no—
60C3×C3⋊(C3⋊C8)—21no—
61C32×C3⋊C8—21no—
62#4760—91no—
63#4760—91no—
64#4760—91no—
65C2×C32×D4—31no—
66C2×#638—61no—
67C2×#638—61no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634—61no—
70C2×C3×D12—21no—
71C2×C3×D12—31no—
72C2×C3×D12—31no—
73C2×C3×D12—61no—
74C2×C3×D12C2×D311yescharacteristic
75C2×C3×D12C2×D311yescharacteristic
76C2×#627—61no—
77C2×#627—61no—
78C2×#627—62no—
79C2×#625—61no—
80C2×#625—61no—
81C2×#625—61no—
82C2×#625—61no—
83C2×#625—62no—
84C2×#625—62no—
85C2×C32×C8—31no—
86C32×C4⋊C4—31no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3750—61no—
89C4×C3⋊Dic3—31no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—31no—
92C4×C3⋊Dic3—61no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—61no—
95C4×C3⋊Dic3—121no—
96C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
97C3×C4⋊Dic3—21no—
98C3×C4⋊Dic3—31no—
99C3×C4⋊Dic3—31no—
100C3×C4⋊Dic3—61no—
101C3×C4⋊Dic3C2×D311yescharacteristic
102C3×C4⋊Dic3C2×D311yescharacteristic
103C2×C3×C3⋊C8—21no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—61no—
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C2×C3×C3⋊C8C2×D311yescharacteristic
109#3724—61no—
110#3724—61no—
111#3724—62no—
112C22×C33C2×D411yescharacteristic
113C2×C3×C3⋊D3—41no—
114C2×C3×C3⋊D3—41no—
115C2×C3×C3⋊D3—41no—
116C2×C32×D3—41no—
117C2×C32×D3—41no—
118C2×C32×D3—41no—
119C33×C4#4111yescharacteristic
120C33×C4#4111yescharacteristic
121C3⋊(C3⋊Dic3)D812yes—
122C3⋊(C3⋊Dic3)SD1612yes—
123C3×C3⋊Dic3—41no—
124C32×Dic3—41no—
125#940—91no—
126#940—91no—
127#940—91no—
128#940—181no—
129#888—91no—
130#888—91no—
131#888—91no—
132#888—181no—
133C3×#102—91no—
134C3×#102—91no—
135C3×#102—91no—
136#832—31no—
137#832—31no—
138#832—91no—
139#821—31no—
140#821—31no—
141#821—91no—
142C8×Dic3—91no—
143C8×Dic3—91no—
144C8×Dic3—91no—
145C8×Dic3—181no—
146#809—31no—
147#809—31no—
148#809—91no—
149C23×C32—61no—
150C22×C3⋊D3—21no—
151C22×C3×D3—22no—
152C22×C3×D3—21no—
153C22×C3×D3—21no—
154C22×C3×D3—62no—
155C22×C3×D3—61no—
156C22×C3×D3—61no—
157C32×D4—61no—
158C32×D4—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167#638—122no—
168#638—122no—
169C2×C3⋊Dic3—21no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—31no—
173C2×C3⋊Dic3—31no—
174C2×C3⋊Dic3—31no—
175C2×C3⋊Dic3—31no—
176C2×C3⋊Dic3—61no—
177C2×C3⋊Dic3—61no—
178C2×C3⋊Dic3—61no—
179C2×C3⋊Dic3—61no—
180C2×C3⋊Dic3—61no—
181C2×C3⋊Dic3—61no—
182C2×C3⋊Dic3—121no—
183C2×C3⋊Dic3—121no—
184C4⋊(C3⋊D3)—21no—
185C4⋊(C3⋊D3)—21no—
186#634—122no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—62no—
191C2×C3×Dic3—61no—
192C2×C3×Dic3—61no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—21no—
197C3×D12—41no—
198C3×D12—41no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—61no—
203C3×D12—121no—
204C3×D12—121no—
205#627—122no—
206#627—122no—
207#627—124no—
208#625—122no—
209#625—122no—
210#625—122no—
211#625—122no—
212#625—124no—
213#625—124no—
214C32×C8—61no—
215C3⋊(C3⋊C8)—21no—
216C3×C3⋊C8—22no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—62no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222#486—271noSylow
223C2×C33#12411yescharacteristic
224C2×C33C2×D811yescharacteristic
225C2×C33C2×SD1611yescharacteristic
226C3×C3⋊D3—81no—
227C32×D3—81no—
228C2×C3×D4—31no—
229C2×C3×D4—31no—
230C2×C3×D4—61no—
231C2×C3×D4—91no—
232C2×#67—181no—
233C2×#67—181no—
234C2×#67—181no—
235C2×#67—181no—
236C2×#67—181no—
237C2×#67—181no—
238C2×#67—182no—
239C2×#67—182no—
240C2×D12—31no—
241C2×D12—31no—
242C2×D12—61no—
243C2×D12D3211yescharacteristic
244C2×Dic6—181no—
245C2×Dic6—181no—
246C2×Dic6—181no—
247C2×Dic6—182no—
248C2×C3×C8—31no—
249C2×C3×C8—31no—
250C2×C3×C8—61no—
251C2×C3×C8—91no—
252C3×C4⋊C4—31no—
253C3×C4⋊C4—31no—
254C3×C4⋊C4—61no—
255C3×C4⋊C4—91no—
256C4⋊Dic3—31no—
257C4⋊Dic3—31no—
258C4⋊Dic3—61no—
259C4⋊Dic3D3211yescharacteristic
260Dic3⋊C4—181no—
261Dic3⋊C4—181no—
262Dic3⋊C4—181no—
263Dic3⋊C4—182no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—361no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8D3211yescharacteristic
275C22×C32—21no—
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—41no—
279C22×C32—121no—
280C22×C32—121no—
281C22×C32—121no—
282C22×C32D3×D411yescharacteristic
283C22×C32D3×D411yescharacteristic
284C22×C32D3×D411yescharacteristic
285C2×C3⋊D3—41no—
286C2×C3⋊D3—41no—
287C2×C3⋊D3—41no—
288C2×C3×D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—42no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—122no—
300C2×C3×D3—121no—
301C2×C3×D3—121no—
302C2×C3×D3—121no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—41no—
313C32×C4—41no—
314C32×C4—121no—
315C32×C4#23511yescharacteristic
316C32×C4#23511yescharacteristic
317C32×C4#23511yescharacteristic
318C32×C4#23711yescharacteristic
319C32×C4#23711yescharacteristic
320C32×C4#23711yescharacteristic
321C3⋊Dic3—32no—
322C3⋊Dic3—32no—
323C3⋊Dic3—32no—
324C3⋊Dic3—32no—
325C3⋊Dic3—32no—
326C3⋊Dic3—32no—
327C3⋊Dic3—41no—
328C3⋊Dic3—62no—
329C3⋊Dic3—62no—
330C3⋊Dic3—62no—
331C3⋊Dic3—62no—
332C3⋊Dic3—62no—
333C3⋊Dic3—62no—
334C3⋊Dic3—122no—
335C3⋊Dic3—122no—
336C3×Dic3—42no—
337C3×Dic3—41no—
338C3×Dic3—41no—
339C3×Dic3—122no—
340C3×Dic3—121no—
341C3×Dic3—121no—
342C4⋊Q8—271no—
343#127—271no—
344#102—91no—
345#102—91no—
346#102—91no—
347#102—271no—
348C4×C8—271no—
349C33#48611yescharacteristic, complemented, Sylow
350C23×C3—62no—
351C23×C3—61no—
352C23×C3—61no—
353C23×C3—181no—
354C22×D3—21no—
355C22×D3—62no—
356C22×D3—61no—
357C22×D3—61no—
358C3×D4—61no—
359C3×D4—61no—
360C3×D4—61no—
361C3×D4—61no—
362C3×D4—121no—
363C3×D4—121no—
364C3×D4—181no—
365C3×D4—181no—
366C2×C3×C4—21no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—41no—
370C2×C3×C4—62no—
371C2×C3×C4—61no—
372C2×C3×C4—61no—
373C2×C3×C4—181no—
374C2×C3×C4C2×D3211yescharacteristic
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377#67—362no—
378#67—362no—
379#67—362no—
380#67—362no—
381#67—362no—
382#67—362no—
383#67—364no—
384#67—364no—
385C2×Dic3—21no—
386C2×Dic3—62no—
387C2×Dic3—61no—
388C2×Dic3—61no—
389C2×Dic3—91no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—91no—
394C2×Dic3—91no—
395C2×Dic3—181no—
396C2×Dic3—181no—
397C2×Dic3—181no—
398C2×Dic3—181no—
399C2×Dic3—181no—
400C2×Dic3—181no—
401C2×Dic3—361no—
402C2×Dic3—361no—
403D12—21no—
404D12—21no—
405D12—61no—
406D12—61no—
407D12—61no—
408D12—61no—
409D12—121no—
410D12—121no—
411Dic6—362no—
412Dic6—362no—
413Dic6—362no—
414Dic6—364no—
415C3×C8—62no—
416C3×C8—61no—
417C3×C8—61no—
418C3×C8—181no—
419C3⋊C8—21no—
420C3⋊C8—62no—
421C3⋊C8—61no—
422C3⋊C8—61no—
423C2×C32—21no—
424C2×C32—21no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—21no—
429C2×C32—21no—
430C2×C32—21no—
431C2×C32—21no—
432C2×C32—41no—
433C2×C32—41no—
434C2×C32—41no—
435C2×C32—241no—
436C2×C32#88511yescharacteristic
437C2×C32#88511yescharacteristic
438C2×C32#88511yescharacteristic
439C2×C32D3×D811yescharacteristic
440C2×C32D3×D811yescharacteristic
441C2×C32D3×D811yescharacteristic
442C2×C32D3×SD1611yescharacteristic
443C2×C32D3×SD1611yescharacteristic
444C2×C32D3×SD1611yescharacteristic
445C3⋊D3—81no—
446C3×D3—82no—
447C3×D3—81no—
448C3×D3—81no—
449C3×D3—242no—
450C3×D3—241no—
451C3×D3—241no—
452C2×Q8—541no—
453C2×D4—31no—
454C2×D4—91no—
455C2×D4—541no—
456C2×D4—541no—
457C2×C8—31no—
458C2×C8—91no—
459C4⋊C4—31no—
460C4⋊C4—91no—
461C4⋊C4—541no—
462C42—271no—
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—21no—
466C22×C3—41no—
467C22×C3—122no—
468C22×C3—122no—
469C22×C3—122no—
470C22×C3—121no—
471C22×C3—121no—
472C22×C3—121no—
473C22×C3—121no—
474C22×C3—121no—
475C22×C3—121no—
476C22×C3—361no—
477C22×C3—361no—
478C22×C3—361no—
479C22×C3#380211yescharacteristic
480C22×C3#381111yescharacteristic
481C22×C3#381111yescharacteristic
482C2×D3—41no—
483C2×D3—41no—
484C2×D3—41no—
485C2×D3—122no—
486C2×D3—122no—
487C2×D3—122no—
488C2×D3—121no—
489C2×D3—121no—
490C2×D3—121no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C3×C4—21no—
495C3×C4—21no—
496C3×C4—21no—
497C3×C4—21no—
498C3×C4—21no—
499C3×C4—21no—
500C3×C4—41no—
501C3×C4—41no—
502C3×C4—122no—
503C3×C4—121no—
504C3×C4—121no—
505C3×C4—361no—
506C3×C4#379511yescharacteristic
507C3×C4#379511yescharacteristic
508C3×C4#379811yescharacteristic
509C3×C4#380411yescharacteristic
510C3×C4#380411yescharacteristic
511C3×C4#380511yescharacteristic
512Dic3—41no—
513Dic3—92no—
514Dic3—92no—
515Dic3—92no—
516Dic3—92no—
517Dic3—92no—
518Dic3—92no—
519Dic3—122no—
520Dic3—121no—
521Dic3—121no—
522Dic3—182no—
523Dic3—182no—
524Dic3—182no—
525Dic3—182no—
526Dic3—182no—
527Dic3—182no—
528Dic3—362no—
529Dic3—362no—
530C32—21no—
531C32—21no—
532C32—21no—
533C32—41no—
534C32#476011yescharacteristic, complemented
535C32#476011yescharacteristic, complemented
536C32#476011yescharacteristic, complemented
537C23—61no—
538C23—181no—
539Q8—1082no—
540D4—61no—
541D4—61no—
542D4—181no—
543D4—181no—
544D4—1082no—
545D4—1082no—
546C2×C4—61no—
547C2×C4—181no—
548C2×C4—271no—
549C2×C4—271no—
550C2×C4D3311yescharacteristic
551C8—61no—
552C8—181no—
553C2×C3—21no—
554C2×C3—21no—
555C2×C3—21no—
556C2×C3—21no—
557C2×C3—21no—
558C2×C3—21no—
559C2×C3—21no—
560C2×C3—21no—
561C2×C3—21no—
562C2×C3—41no—
563C2×C3—41no—
564C2×C3—41no—
565C2×C3—242no—
566C2×C3—241no—
567C2×C3—241no—
568C2×C3—721no—
569C2×C3#6379611yescharacteristic
570C2×C3#6379711yescharacteristic
571C2×C3#6391311yescharacteristic
572C2×C3#6391511yescharacteristic
573C2×C3#6392611yescharacteristic
574C2×C3#6392611yescharacteristic
575C2×C3#6393011yescharacteristic
576C2×C3#6394011yescharacteristic
577C2×C3#6395311yescharacteristic
578D3—81no—
579D3—242no—
580D3—241no—
581D3—241no—
582C22—121no—
583C22—121no—
584C22—121no—
585C22—361no—
586C22—361no—
587C22—361no—
588C22#8895411yescharacteristic
589C4—121no—
590C4—272no—
591C4—272no—
592C4—361no—
593C4#8896011yescharacteristic
594C4#8896111yescharacteristic
595C3—21no—
596C3—21no—
597C3—21no—
598C3—41no—
599C3#1059055211yescharacteristic, complemented
600C3#1059088511yescharacteristic, complemented
601C3#1059140211yescharacteristic, complemented
602C2—241no—
603C2—721no—
604C2#1172283211yescharacteristic
605C2#1172284111yescharacteristic
606C2#1172300911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723529C2
hypocenterC33#486
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. #11723536C2semidirect
3. #11722075C2semidirect
4. #11722018C2semidirect
5. C3×#63564C2semidirect
6. C3×#63563C2semidirect
7. C3×#63562C2semidirect
38. #63564D3semidirect
40. #63563D3semidirect
42. #63562D3semidirect
349. C33#486semidirect
534. C32#4760semidirect
535. C32#4760semidirect
536. C32#4760semidirect
599. C3#10590552semidirect
600. C3#10590885semidirect
601. C3#10591402semidirect

Maximal subgroups

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

subgroupindex
#117235362
#117235292
#117220752
#117220182
C3×#635642
C3×#635632
C3×#635622
#105914023
#105908853
#105905523