Finite Group Databasebeta

← #13156416

Subgroups — #13156416

Order 1728.

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

normal subgroups only

This group has 746 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#11723532C211yesmaximal, characteristic
2#11723503C211yesmaximal, characteristic, complemented
3#11722075C211yesmaximal, characteristic, complemented
4#11722024C211yesmaximal, characteristic, complemented
5C3×#63574C211yesmaximal, characteristic, complemented
6C3×#63573C211yesmaximal, characteristic, complemented
7C3×#63564C211yesmaximal, characteristic, complemented
8#10590649—31nomaximal
9#10590511—31nomaximal
10#10590445—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×#6292—21no—
14C2×#6289—21no—
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C32×C4⋊Dic3C2211yescharacteristic
19C2×C32×C3⋊C8C2211yescharacteristic
20#88806—21no—
21#88803—21no—
22#64104—31no—
23C4⋊C4×C3⋊D3—31no—
24#63916—31no—
25#63916—31no—
26#63916—61no—
27#63900—31no—
28#63900—31no—
29#63900—61no—
30#63647—31no—
31C3×#821—31no—
32C3×#817—31no—
33C3×#809—31no—
34C3×#808—31no—
35C3×#808—31no—
36C3×#808—31no—
37#63574D311yescharacteristic, complemented
38#63573D311yescharacteristic, complemented
39#63564—31no—
40#63564D311yescharacteristic, complemented
41#63563—31no—
42#63562—31no—
43#63557—31no—
44#63557—31no—
45#63557—61no—
46C22×C3⋊(C3⋊D3)D411yescharacteristic
47C22×C3×C3⋊D3—21no—
48C2×C33×C4C2311yescharacteristic
49C2×C3⋊(C3⋊Dic3)D411yescharacteristic
50C4×C3⋊(C3⋊D3)—21no—
51C4×C3⋊(C3⋊D3)—21no—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C2×C32×Dic3—21no—
56C2×C32×Dic3—21no—
57#6292—42no—
58#6289—42no—
59C3×C3⋊(C3⋊C8)—21no—
60C32×C3⋊C8—21no—
61#4863—91no—
62#4863—91no—
63#4777—91no—
64C2×C4⋊(C3⋊D3)—61no—
65C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—31no—
68C2×C4×C3⋊D3—31no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3—61no—
71C2×C4×C3⋊D3—61no—
72C2×C4×C3⋊D3—61no—
73C2×C4×C3⋊D3—121no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×#626—61no—
78C2×#626—61no—
79C2×#626—62no—
80C2×#624—61no—
81C2×#624—61no—
82C2×#624—62no—
83C2×C32×C8—31no—
84C32×C4⋊C4—31no—
85C32×C4⋊C4—31no—
86#3752—61no—
87C4⋊(C3⋊Dic3)C2×D311yescharacteristic
88#3750—61no—
89C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3—21no—
92C3×C4⋊Dic3—31no—
93C3×C4⋊Dic3—31no—
94C3×C4⋊Dic3—61no—
95C3×C4⋊Dic3C2×D311yescharacteristic
96C3×C4⋊Dic3C2×D311yescharacteristic
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
105#3724—61no—
106#3724—61no—
107#3724—62no—
108(C2×D3)⋊Dic3—61no—
109(C2×D3)⋊Dic3—61no—
110(C2×D3)⋊Dic3—62no—
111C22×C33C2×D411yescharacteristic
112C2×C3⋊(C3⋊D3)—21no—
113C2×C3⋊(C3⋊D3)—21no—
114C2×C3⋊(C3⋊D3)SD1612yes—
115C2×C3×C3⋊D3—41no—
116C2×C3×C3⋊D3—41no—
117C2×C3×C3⋊D3—41no—
118C33×C4#4111yescharacteristic
119C33×C4#4111yescharacteristic
120C3⋊(C3⋊Dic3)—21no—
121C3×C3⋊Dic3—41no—
122C32×Dic3—41no—
123C32×Dic3—41no—
124#938—91no—
125#895—91no—
126#895—91no—
127#895—181no—
128D3×C4⋊C4—91no—
129D3×C4⋊C4—91no—
130D3×C4⋊C4—91no—
131D3×C4⋊C4—181no—
132C3×#106—91no—
133C3×#106—91no—
134C3×#102—91no—
135#832—91no—
136#821—31no—
137#821—91no—
138(C2×D3)⋊C8—91no—
139(C2×D3)⋊C8—91no—
140(C2×D3)⋊C8—91no—
141(C2×D3)⋊C8—181no—
142#817—31no—
143#809—31no—
144#809—91no—
145#808—31no—
146#808—31no—
147#808—31no—
148C22×C3⋊D3—21no—
149C22×C3⋊D3—31no—
150C22×C3⋊D3—31no—
151C22×C3⋊D3—31no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—61no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—121no—
156C22×C3×D3—62no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—21no—
162C2×C32×C4—41no—
163C2×C32×C4—61no—
164C2×C32×C4—61no—
165C2×C32×C4C22×D311yescharacteristic
166C2×C32×C4C22×D311yescharacteristic
167C2×C32×C4C22×D311yescharacteristic
168C2×C3⋊Dic3—21no—
169C2×C3⋊Dic3—31no—
170C2×C3⋊Dic3—31no—
171C2×C3⋊Dic3—31no—
172C2×C3⋊Dic3—61no—
173C2×C3⋊Dic3—61no—
174C2×C3⋊Dic3—61no—
175C2×C3⋊Dic3—121no—
176C4⋊(C3⋊D3)—21no—
177C4⋊(C3⋊D3)—21no—
178C4⋊(C3⋊D3)—122no—
179C4×C3⋊D3—61no—
180C4×C3⋊D3—61no—
181C4×C3⋊D3—62no—
182C4×C3⋊D3—62no—
183C4×C3⋊D3—61no—
184C4×C3⋊D3—61no—
185C4×C3⋊D3—61no—
186C4×C3⋊D3—61no—
187C4×C3⋊D3—121no—
188C4×C3⋊D3—121no—
189C4×C3⋊D3—121no—
190C4×C3⋊D3—121no—
191C4×C3⋊D3—122no—
192C4×C3⋊D3—122no—
193C4×C3⋊D3—122no—
194C2×C3×Dic3—22no—
195C2×C3×Dic3—22no—
196C2×C3×Dic3—21no—
197C2×C3×Dic3—21no—
198C2×C3×Dic3—21no—
199C2×C3×Dic3—21no—
200C2×C3×Dic3—62no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—61no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×D12—121no—
209#626—122no—
210#626—122no—
211#626—124no—
212#624—124no—
213#624—122no—
214#624—122no—
215C32×C8—61no—
216C3⋊(C3⋊C8)—21no—
217C3×C3⋊C8—21no—
218C3×C3⋊C8—21no—
219C3×C3⋊C8—41no—
220C3×C3⋊C8—61no—
221C3×C3⋊C8—61no—
222C3×C3⋊C8—121no—
223#481—271noSylow
224C2×C33#12311yescharacteristic
225C2×C33C2×SD1611yescharacteristic
226C2×C33AGL(1,ℤ/8)11yescharacteristic
227C3⋊(C3⋊D3)—22no—
228C3×C3⋊D3—81no—
229C2×C3×D4—91no—
230C2×#67—181no—
231C2×D12—31no—
232C2×D12—31no—
233C2×D12—61no—
234C2×D12—181no—
235C2×D12—181no—
236C2×D12—182no—
237C2×C4×D3—91no—
238C2×C4×D3—91no—
239C2×C4×D3—91no—
240C2×C4×D3—181no—
241C2×C4×D3—181no—
242C2×C4×D3—181no—
243C2×C4×D3—181no—
244C2×C4×D3—181no—
245C2×C4×D3—181no—
246C2×C4×D3—182no—
247C2×C4×D3—361no—
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—31no—
255C3×C4⋊C4—31no—
256C3×C4⋊C4—61no—
257C3×C4⋊C4—61no—
258C3×C4⋊C4—91no—
259#217—181no—
260(C2×D3)⋊C4—181no—
261(C2×D3)⋊C4—181no—
262(C2×D3)⋊C4—182no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—61no—
266C4⋊Dic3D3211yescharacteristic
267C4⋊Dic3D3211yescharacteristic
268Dic3⋊C4—181no—
269Dic3⋊C4—181no—
270Dic3⋊C4—181no—
271Dic3⋊C4—182no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—31no—
274C2×C3⋊C8—61no—
275C2×C3⋊C8D3211yescharacteristic
276C22×C32—21no—
277C22×C32—21no—
278C22×C32—21no—
279C22×C32—41no—
280C22×C32D3×D411yescharacteristic
281C22×C32D3×D411yescharacteristic
282C22×C32D3×D411yescharacteristic
283C2×C3⋊D3—32no—
284C2×C3⋊D3—32no—
285C2×C3⋊D3—32no—
286C2×C3⋊D3—41no—
287C2×C3⋊D3—41no—
288C2×C3⋊D3—41no—
289C2×C3⋊D3—62no—
290C2×C3⋊D3—62no—
291C2×C3⋊D3—62no—
292C2×C3⋊D3—61no—
293C2×C3⋊D3—61no—
294C2×C3⋊D3—62no—
295C2×C3⋊D3—62no—
296C2×C3⋊D3—61no—
297C2×C3⋊D3—61no—
298C2×C3⋊D3—61no—
299C2×C3⋊D3—61no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—121no—
302C2×C3⋊D3—121no—
303C2×C3⋊D3—121no—
304C2×C3⋊D3—121no—
305C2×C3⋊D3—122no—
306C2×C3⋊D3—122no—
307C2×C3×D3—122no—
308C2×C3×D3—122no—
309C2×C3×D3—122no—
310C2×C3×D3—121no—
311C2×C3×D3—121no—
312C2×C3×D3—121no—
313C2×C3×D3—121no—
314C2×C3×D3—121no—
315C2×C3×D3—121no—
316C32×C4—21no—
317C32×C4—21no—
318C32×C4—21no—
319C32×C4—21no—
320C32×C4—21no—
321C32×C4—21no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4—121no—
325C32×C4—121no—
326C32×C4#23511yescharacteristic
327C32×C4#23511yescharacteristic
328C32×C4#23511yescharacteristic
329C32×C4#23711yescharacteristic
330C32×C4#23911yescharacteristic
331C32×C4#23911yescharacteristic
332C3⋊Dic3—41no—
333C3⋊Dic3—61no—
334C3⋊Dic3—62no—
335C3⋊Dic3—61no—
336C3⋊Dic3—61no—
337C3⋊Dic3—121no—
338C3⋊Dic3—121no—
339C3⋊Dic3—122no—
340C3×Dic3—42no—
341C3×Dic3—42no—
342C3×Dic3—41no—
343C3×Dic3—41no—
344C3×Dic3—41no—
345C3×Dic3—41no—
346C3×Dic3—122no—
347C3×Dic3—121no—
348C3×Dic3—121no—
349#121—271no—
350C2×C4⋊C4—271no—
351#106—91no—
352#106—91no—
353#102—91no—
354#102—271no—
355C22⋊C8—271no—
356C33#48111yescharacteristic, complemented, Sylow
357C23×C3—181no—
358C22×D3—62no—
359C22×D3—61no—
360C22×D3—61no—
361C22×D3—91no—
362C22×D3—91no—
363C22×D3—91no—
364C22×D3—181no—
365C22×D3—181no—
366C22×D3—181no—
367C22×D3—361no—
368C3×D4—181no—
369C3×D4—181no—
370C2×C3×C4—21no—
371C2×C3×C4—21no—
372C2×C3×C4—21no—
373C2×C3×C4—41no—
374C2×C3×C4—62no—
375C2×C3×C4—62no—
376C2×C3×C4—61no—
377C2×C3×C4—61no—
378C2×C3×C4—61no—
379C2×C3×C4—61no—
380C2×C3×C4—181no—
381C2×C3×C4C2×D3211yescharacteristic
382C2×C3×C4C2×D3211yescharacteristic
383C2×C3×C4C2×D3211yescharacteristic
384#67—362no—
385C2×Dic3—21no—
386C2×Dic3—21no—
387C2×Dic3—62no—
388C2×Dic3—61no—
389C2×Dic3—61no—
390C2×Dic3—91no—
391C2×Dic3—91no—
392C2×Dic3—91no—
393C2×Dic3—181no—
394C2×Dic3—181no—
395C2×Dic3—181no—
396C2×Dic3—361no—
397D12—61no—
398D12—61no—
399D12—61no—
400D12—61no—
401D12—121no—
402D12—121no—
403D12—362no—
404D12—362no—
405D12—364no—
406C4×D3—182no—
407C4×D3—182no—
408C4×D3—181no—
409C4×D3—181no—
410C4×D3—181no—
411C4×D3—181no—
412C4×D3—181no—
413C4×D3—181no—
414C4×D3—361no—
415C4×D3—361no—
416C4×D3—361no—
417C4×D3—361no—
418C4×D3—362no—
419C4×D3—362no—
420C4×D3—362no—
421C4×D3—364no—
422C4×D3—362no—
423C4×D3—362no—
424C3×C8—61no—
425C3×C8—61no—
426C3×C8—121no—
427C3×C8—181no—
428C3⋊C8—21no—
429C3⋊C8—61no—
430C3⋊C8—61no—
431C3⋊C8—121no—
432C2×C32—21no—
433C2×C32—21no—
434C2×C32—21no—
435C2×C32—21no—
436C2×C32—21no—
437C2×C32—21no—
438C2×C32—21no—
439C2×C32—21no—
440C2×C32—21no—
441C2×C32—41no—
442C2×C32—41no—
443C2×C32—41no—
444C2×C32#88311yescharacteristic
445C2×C32#89411yescharacteristic
446C2×C32#89411yescharacteristic
447C2×C32#91111yescharacteristic
448C2×C32D3×SD1611yescharacteristic
449C2×C32D3×SD1611yescharacteristic
450C2×C32D3×SD1611yescharacteristic
451C2×C32#91411yescharacteristic
452C2×C32#91411yescharacteristic
453C3⋊D3—62no—
454C3⋊D3—64no—
455C3⋊D3—62no—
456C3⋊D3—62no—
457C3⋊D3—81no—
458C3⋊D3—122no—
459C3⋊D3—122no—
460C3⋊D3—124no—
461C3×D3—242no—
462C3×D3—241no—
463C3×D3—241no—
464C2×D4—91no—
465C2×D4—541no—
466C22×C4—271no—
467C22×C4—541no—
468C2×C8—31no—
469C2×C8—91no—
470C4⋊C4—31no—
471C4⋊C4—31no—
472C4⋊C4—91no—
473C4⋊C4—541no—
474C22⋊C4—541no—
475C22×C3—21no—
476C22×C3—21no—
477C22×C3—21no—
478C22×C3—41no—
479C22×C3—361no—
480C22×C3—361no—
481C22×C3—361no—
482C22×C3#380211yescharacteristic
483C22×C3#381111yescharacteristic
484C22×C3#381111yescharacteristic
485C2×D3—92no—
486C2×D3—92no—
487C2×D3—92no—
488C2×D3—122no—
489C2×D3—122no—
490C2×D3—122no—
491C2×D3—121no—
492C2×D3—121no—
493C2×D3—121no—
494C2×D3—121no—
495C2×D3—121no—
496C2×D3—121no—
497C2×D3—182no—
498C2×D3—182no—
499C2×D3—182no—
500C2×D3—182no—
501C2×D3—182no—
502C2×D3—181no—
503C2×D3—181no—
504C2×D3—181no—
505C2×D3—181no—
506C2×D3—181no—
507C2×D3—181no—
508C2×D3—362no—
509C2×D3—361no—
510C2×D3—361no—
511C2×D3—361no—
512C2×D3—361no—
513C2×D3—362no—
514C2×D3—362no—
515C3×C4—21no—
516C3×C4—21no—
517C3×C4—21no—
518C3×C4—21no—
519C3×C4—21no—
520C3×C4—21no—
521C3×C4—41no—
522C3×C4—41no—
523C3×C4—122no—
524C3×C4—122no—
525C3×C4—121no—
526C3×C4—121no—
527C3×C4—121no—
528C3×C4—121no—
529C3×C4—361no—
530C3×C4#379611yescharacteristic
531C3×C4#379811yescharacteristic
532C3×C4#379911yescharacteristic
533C3×C4#379911yescharacteristic
534C3×C4#379911yescharacteristic
535C3×C4#380411yescharacteristic
536Dic3—41no—
537Dic3—41no—
538Dic3—122no—
539Dic3—121no—
540Dic3—121no—
541Dic3—182no—
542Dic3—181no—
543Dic3—181no—
544Dic3—181no—
545Dic3—361no—
546Dic3—361no—
547Dic3—362no—
548C32—21no—
549C32—21no—
550C32—21no—
551C32—41no—
552C32#477711yescharacteristic, complemented
553C32#486311yescharacteristic, complemented
554C32#486311yescharacteristic, complemented
555C23—181no—
556C23—271no—
557D4—181no—
558D4—181no—
559D4—1082no—
560C2×C4—61no—
561C2×C4—61no—
562C2×C4—181no—
563C2×C4—271no—
564C2×C4—541no—
565C2×C4—541no—
566C2×C4—1082no—
567C2×C4D3311yescharacteristic
568C8—61no—
569C8—181no—
570C2×C3—21no—
571C2×C3—21no—
572C2×C3—21no—
573C2×C3—21no—
574C2×C3—21no—
575C2×C3—21no—
576C2×C3—21no—
577C2×C3—21no—
578C2×C3—21no—
579C2×C3—41no—
580C2×C3—41no—
581C2×C3—41no—
582C2×C3—721no—
583C2×C3#6379611yescharacteristic
584C2×C3#6379811yescharacteristic
585C2×C3#6385311yescharacteristic
586C2×C3#6385411yescharacteristic
587C2×C3#6389711yescharacteristic
588C2×C3#6393011yescharacteristic
589C2×C3#6393711yescharacteristic
590C2×C3#6394011yescharacteristic
591C2×C3#6394111yescharacteristic
592D3—184no—
593D3—182no—
594D3—182no—
595D3—182no—
596D3—242no—
597D3—241no—
598D3—241no—
599D3—362no—
600D3—362no—
601D3—364no—
602C22—272no—
603C22—361no—
604C22—361no—
605C22—361no—
606C22—541no—
607C22—541no—
608C22#8895411yescharacteristic
609C4—121no—
610C4—121no—
611C4—361no—
612C4—541no—
613C4#8896211yescharacteristic
614C4#8896511yescharacteristic
615C3—21no—
616C3—21no—
617C3—21no—
618C3—41no—
619C3#1059044511yescharacteristic, complemented
620C3#1059051111yescharacteristic, complemented
621C3#1059064911yescharacteristic, complemented
622C2—542no—
623C2—721no—
624C2#1172284111yescharacteristic
625C2#1172287811yescharacteristic
626C2#1172300011yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723532C2
hypocenterC33#481
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. #11723503C2semidirect
3. #11722075C2semidirect
4. #11722024C2semidirect
5. C3×#63574C2semidirect
6. C3×#63573C2semidirect
7. C3×#63564C2semidirect
37. #63574D3semidirect
38. #63573D3semidirect
40. #63564D3semidirect
356. C33#481semidirect
552. C32#4777semidirect
553. C32#4863semidirect
554. C32#4863semidirect
619. C3#10590445semidirect
620. C3#10590511semidirect
621. C3#10590649semidirect

Maximal subgroups

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

subgroupindex
#117235322
#117235032
#117220752
#117220242
C3×#635742
C3×#635732
C3×#635642
#105906493
#105905113
#105904453