Finite Group Databasebeta

← #13156457

Subgroups — #13156457

Order 1728.

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

normal subgroups only

This group has 686 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#11723503C211yesmaximal, characteristic
2#11723456C211yesmaximal, characteristic
3#11722079C211yesmaximal, characteristic, complemented
4#11722024C211yesmaximal, characteristic
5C3×#63573C211yesmaximal, characteristic, complemented
6C3×#63566C211yesmaximal, characteristic, complemented
7C3×#63565C211yesmaximal, characteristic, complemented
8#10591553—31nomaximal
9#10590686—31nomaximal
10#10590529—31nomaximal
11C2×C4×C3⋊(C3⋊D3)C2211yescharacteristic
12C2×C32×Dic6C2211yescharacteristic
13C2×#6289—21no—
14C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
15C3×C4⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C32×C4⋊Dic3C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88807—21no—
20#88806—21no—
21#88804—21no—
22#64155—31no—
23C4⋊C4×C3⋊D3—31no—
24#63900—31no—
25#63900—31no—
26#63900—61no—
27#63871—31no—
28#63871—31no—
29#63871—61no—
30#63647—31no—
31C3×#835—31no—
32C3×#835—31no—
33C3×#817—31no—
34C3×#816—31no—
35C3×#810—31no—
36C3×#808—31no—
37#63574—31no—
38#63573—31no—
39#63573—31no—
40#63573D311yescharacteristic, complemented
41#63566D311yescharacteristic, complemented
42#63565D311yescharacteristic, complemented
43#63557—31no—
44#63557—31no—
45#63557—61no—
46C22×C3⋊(C3⋊D3)D411yescharacteristic
47C2×C33×C4C2311yescharacteristic
48C2×C3⋊(C3⋊Dic3)D411yescharacteristic
49C4×C3⋊(C3⋊D3)—21no—
50C4×C3⋊(C3⋊D3)—21no—
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C32×Dic3—21no—
54C2×C32×Dic3—21no—
55C32×Dic6—21no—
56C32×Dic6—21no—
57#6289—42no—
58C3×C3⋊(C3⋊C8)—21no—
59C32×C3⋊C8—21no—
60#4862—91no—
61#4805—91no—
62#4805—91no—
63C2×C32×Q8—31no—
64C2×C4×C3⋊D3—31no—
65C2×C4×C3⋊D3—31no—
66C2×C4×C3⋊D3—31no—
67C2×C4×C3⋊D3—61no—
68C2×C4×C3⋊D3—61no—
69C2×C4×C3⋊D3—61no—
70C2×C4×C3⋊D3—61no—
71C2×C4×C3⋊D3—121no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6C2×D311yescharacteristic
74C2×C3×Dic6C2×D311yescharacteristic
75C2×#624—61no—
76C2×#624—61no—
77C2×#624—62no—
78C2×C32×C8—31no—
79C32×C4⋊C4—31no—
80#3752—61no—
81C4⋊(C3⋊Dic3)C2×D311yescharacteristic
82C4⋊(C3⋊Dic3)C2×D311yescharacteristic
83#3750—61no—
84#3750—61no—
85C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3—31no—
88C3×C4⋊Dic3—31no—
89C3×C4⋊Dic3—31no—
90C3×C4⋊Dic3—31no—
91C3×C4⋊Dic3—61no—
92C3×C4⋊Dic3—61no—
93C3×C4⋊Dic3C2×D311yescharacteristic
94C3×C4⋊Dic3C2×D311yescharacteristic
95C2×C3×C3⋊C8—21no—
96C2×C3×C3⋊C8—31no—
97C2×C3×C3⋊C8—31no—
98C2×C3×C3⋊C8—61no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101#3724—61no—
102#3724—61no—
103#3724—62no—
104Dic3⋊Dic3—61no—
105Dic3⋊Dic3—61no—
106Dic3⋊Dic3—62no—
107#3722—61no—
108#3722—61no—
109#3722—62no—
110C22×C33C2×D411yescharacteristic
111C2×C3⋊(C3⋊D3)—21no—
112C2×C3⋊(C3⋊D3)—21no—
113C2×C3⋊(C3⋊D3)SD1612yes—
114C33×C4#4111yescharacteristic
115C33×C4#4111yescharacteristic
116C3⋊(C3⋊Dic3)—21no—
117C3×C3⋊Dic3—41no—
118C3×C3⋊Dic3—41no—
119C32×Dic3—41no—
120C32×Dic3—41no—
121#946—91no—
122#946—91no—
123#946—181no—
124#897—91no—
125D3×C4⋊C4—91no—
126D3×C4⋊C4—91no—
127D3×C4⋊C4—91no—
128D3×C4⋊C4—181no—
129C3×#106—91no—
130C3×#103—91no—
131C3×#103—91no—
132#835—31no—
133#835—31no—
134(C2×D3)⋊C8—91no—
135(C2×D3)⋊C8—91no—
136(C2×D3)⋊C8—91no—
137(C2×D3)⋊C8—181no—
138#817—31no—
139#817—91no—
140#816—31no—
141#810—31no—
142#808—31no—
143#808—91no—
144#808—91no—
145C22×C3⋊D3—31no—
146C22×C3⋊D3—31no—
147C22×C3⋊D3—31no—
148C22×C3⋊D3—61no—
149C22×C3⋊D3—61no—
150C22×C3⋊D3—61no—
151C22×C3⋊D3—121no—
152C32×Q8—61no—
153C32×Q8—61no—
154C2×C32×C4—21no—
155C2×C32×C4—21no—
156C2×C32×C4—21no—
157C2×C32×C4—41no—
158C2×C32×C4—61no—
159C2×C32×C4—61no—
160C2×C32×C4C22×D311yescharacteristic
161C2×C32×C4C22×D311yescharacteristic
162C2×C32×C4C22×D311yescharacteristic
163C2×C3⋊Dic3—21no—
164C2×C3⋊Dic3—21no—
165C2×C3⋊Dic3—31no—
166C2×C3⋊Dic3—31no—
167C2×C3⋊Dic3—31no—
168C2×C3⋊Dic3—61no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C4×C3⋊D3—61no—
173C4×C3⋊D3—61no—
174C4×C3⋊D3—62no—
175C4×C3⋊D3—62no—
176C4×C3⋊D3—61no—
177C4×C3⋊D3—61no—
178C4×C3⋊D3—61no—
179C4×C3⋊D3—61no—
180C4×C3⋊D3—121no—
181C4×C3⋊D3—121no—
182C4×C3⋊D3—121no—
183C4×C3⋊D3—121no—
184C4×C3⋊D3—122no—
185C4×C3⋊D3—122no—
186C4×C3⋊D3—122no—
187C2×C3×Dic3—22no—
188C2×C3×Dic3—22no—
189C2×C3×Dic3—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—21no—
193C2×C3×Dic3—62no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C3×Dic6—21no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—41no—
204C3×Dic6—41no—
205#624—122no—
206#624—122no—
207#624—124no—
208C32×C8—61no—
209C3⋊(C3⋊C8)—21no—
210C3×C3⋊C8—21no—
211C3×C3⋊C8—21no—
212C3×C3⋊C8—41no—
213C3×C3⋊C8—61no—
214C3×C3⋊C8—61no—
215C3×C3⋊C8—121no—
216#483—271noSylow
217C2×C33#12311yescharacteristic
218C2×C33C2×SD1611yescharacteristic
219C2×C33#13711yescharacteristic
220C3⋊(C3⋊D3)—22no—
221C2×C3×Q8—31no—
222C2×C3×Q8—31no—
223C2×C3×Q8—61no—
224C2×C4×D3—91no—
225C2×C4×D3—91no—
226C2×C4×D3—91no—
227C2×C4×D3—181no—
228C2×C4×D3—181no—
229C2×C4×D3—181no—
230C2×C4×D3—181no—
231C2×C4×D3—181no—
232C2×C4×D3—181no—
233C2×C4×D3—182no—
234C2×C4×D3—361no—
235C2×Dic6D3211yescharacteristic
236C2×C3×C8—31no—
237C2×C3×C8—31no—
238C2×C3×C8—61no—
239C2×C3×C8—91no—
240C3×C4⋊C4—31no—
241C3×C4⋊C4—31no—
242C3×C4⋊C4—61no—
243C3×C4⋊C4—91no—
244C3×C4⋊C4—91no—
245(C2×D3)⋊C4—181no—
246(C2×D3)⋊C4—181no—
247(C2×D3)⋊C4—181no—
248(C2×D3)⋊C4—182no—
249C4⋊Dic3—31no—
250C4⋊Dic3—31no—
251C4⋊Dic3—31no—
252C4⋊Dic3—31no—
253C4⋊Dic3—61no—
254C4⋊Dic3—61no—
255C4⋊Dic3—181no—
256C4⋊Dic3D3211yescharacteristic
257Dic3⋊C4—181no—
258Dic3⋊C4—181no—
259Dic3⋊C4—181no—
260Dic3⋊C4—181no—
261Dic3⋊C4—181no—
262Dic3⋊C4—182no—
263Dic3⋊C4—182no—
264C2×C3⋊C8—31no—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—61no—
267C2×C3⋊C8D3211yescharacteristic
268C22×C32—21no—
269C22×C32—21no—
270C22×C32—21no—
271C22×C32—41no—
272C22×C32D3×D411yescharacteristic
273C22×C32D3×D411yescharacteristic
274C22×C32D3×D411yescharacteristic
275C2×C3⋊D3—32no—
276C2×C3⋊D3—32no—
277C2×C3⋊D3—32no—
278C2×C3⋊D3—62no—
279C2×C3⋊D3—62no—
280C2×C3⋊D3—62no—
281C2×C3⋊D3—61no—
282C2×C3⋊D3—61no—
283C2×C3⋊D3—62no—
284C2×C3⋊D3—62no—
285C2×C3⋊D3—61no—
286C2×C3⋊D3—61no—
287C2×C3⋊D3—61no—
288C2×C3⋊D3—61no—
289C2×C3⋊D3—122no—
290C2×C3⋊D3—121no—
291C2×C3⋊D3—121no—
292C2×C3⋊D3—121no—
293C2×C3⋊D3—121no—
294C2×C3⋊D3—122no—
295C2×C3⋊D3—122no—
296C32×C4—21no—
297C32×C4—21no—
298C32×C4—21no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—41no—
303C32×C4—41no—
304C32×C4—121no—
305C32×C4—121no—
306C32×C4#23511yescharacteristic
307C32×C4#23511yescharacteristic
308C32×C4#23511yescharacteristic
309C32×C4#23711yescharacteristic
310C32×C4#23711yescharacteristic
311C32×C4#23911yescharacteristic
312C3⋊Dic3—41no—
313C3⋊Dic3—41no—
314C3⋊Dic3—61no—
315C3⋊Dic3—62no—
316C3⋊Dic3—61no—
317C3⋊Dic3—61no—
318C3⋊Dic3—121no—
319C3⋊Dic3—121no—
320C3⋊Dic3—122no—
321C3×Dic3—42no—
322C3×Dic3—42no—
323C3×Dic3—41no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—122no—
328C3×Dic3—122no—
329C3×Dic3—121no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C3×Dic3—121no—
333C22⋊Q8—271no—
334C2×C4⋊C4—271no—
335#106—91no—
336#106—271no—
337#103—91no—
338#103—91no—
339C22⋊C8—271no—
340C33#48311yescharacteristic, complemented, Sylow
341C22×D3—91no—
342C22×D3—91no—
343C22×D3—91no—
344C22×D3—181no—
345C22×D3—181no—
346C22×D3—181no—
347C22×D3—361no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—61no—
351C3×Q8—61no—
352C3×Q8—121no—
353C3×Q8—121no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—62no—
359C2×C3×C4—62no—
360C2×C3×C4—61no—
361C2×C3×C4—61no—
362C2×C3×C4—61no—
363C2×C3×C4—61no—
364C2×C3×C4—181no—
365C2×C3×C4—181no—
366C2×C3×C4C2×D3211yescharacteristic
367C2×C3×C4C2×D3211yescharacteristic
368C2×C3×C4C2×D3211yescharacteristic
369C2×Dic3—21no—
370C2×Dic3—21no—
371C2×Dic3—62no—
372C2×Dic3—62no—
373C2×Dic3—61no—
374C2×Dic3—61no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—91no—
378C2×Dic3—91no—
379C2×Dic3—91no—
380C2×Dic3—181no—
381C2×Dic3—181no—
382C2×Dic3—181no—
383C2×Dic3—361no—
384C4×D3—182no—
385C4×D3—182no—
386C4×D3—181no—
387C4×D3—181no—
388C4×D3—181no—
389C4×D3—181no—
390C4×D3—181no—
391C4×D3—181no—
392C4×D3—361no—
393C4×D3—361no—
394C4×D3—361no—
395C4×D3—361no—
396C4×D3—362no—
397C4×D3—362no—
398C4×D3—362no—
399C4×D3—362no—
400C4×D3—362no—
401C4×D3—364no—
402Dic6—21no—
403Dic6—21no—
404C3×C8—61no—
405C3×C8—61no—
406C3×C8—121no—
407C3×C8—181no—
408C3⋊C8—21no—
409C3⋊C8—61no—
410C3⋊C8—61no—
411C3⋊C8—121no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—21no—
415C2×C32—21no—
416C2×C32—21no—
417C2×C32—21no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—21no—
421C2×C32—41no—
422C2×C32—41no—
423C2×C32—41no—
424C2×C32#88311yescharacteristic
425C2×C32#88311yescharacteristic
426C2×C32#89411yescharacteristic
427C2×C32D3×SD1611yescharacteristic
428C2×C32D3×SD1611yescharacteristic
429C2×C32D3×SD1611yescharacteristic
430C2×C32#91511yescharacteristic
431C2×C32#91811yescharacteristic
432C2×C32#91811yescharacteristic
433C3⋊D3—62no—
434C3⋊D3—64no—
435C3⋊D3—62no—
436C3⋊D3—62no—
437C3⋊D3—122no—
438C3⋊D3—122no—
439C3⋊D3—124no—
440C2×Q8—31no—
441C22×C4—271no—
442C22×C4—541no—
443C2×C8—31no—
444C2×C8—91no—
445C4⋊C4—31no—
446C4⋊C4—91no—
447C4⋊C4—91no—
448C4⋊C4—541no—
449C4⋊C4—541no—
450C22⋊C4—541no—
451C22×C3—21no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—41no—
455C22×C3#380211yescharacteristic
456C22×C3#381111yescharacteristic
457C22×C3#381111yescharacteristic
458C2×D3—92no—
459C2×D3—92no—
460C2×D3—92no—
461C2×D3—182no—
462C2×D3—182no—
463C2×D3—182no—
464C2×D3—182no—
465C2×D3—182no—
466C2×D3—181no—
467C2×D3—181no—
468C2×D3—181no—
469C2×D3—181no—
470C2×D3—181no—
471C2×D3—181no—
472C2×D3—362no—
473C2×D3—361no—
474C2×D3—361no—
475C2×D3—361no—
476C2×D3—361no—
477C2×D3—362no—
478C2×D3—362no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—41no—
486C3×C4—41no—
487C3×C4—122no—
488C3×C4—122no—
489C3×C4—121no—
490C3×C4—121no—
491C3×C4—121no—
492C3×C4—121no—
493C3×C4—361no—
494C3×C4—361no—
495C3×C4#379611yescharacteristic
496C3×C4#379811yescharacteristic
497C3×C4#379911yescharacteristic
498C3×C4#380411yescharacteristic
499C3×C4#380411yescharacteristic
500C3×C4#380411yescharacteristic
501Dic3—41no—
502Dic3—41no—
503Dic3—122no—
504Dic3—122no—
505Dic3—121no—
506Dic3—121no—
507Dic3—121no—
508Dic3—121no—
509Dic3—182no—
510Dic3—181no—
511Dic3—181no—
512Dic3—181no—
513Dic3—361no—
514Dic3—361no—
515Dic3—362no—
516C32—21no—
517C32—21no—
518C32—21no—
519C32—41no—
520C32#480511yescharacteristic, complemented
521C32#480511yescharacteristic, complemented
522C32#486211yescharacteristic, complemented
523C23—271no—
524Q8—61no—
525Q8—61no—
526C2×C4—61no—
527C2×C4—61no—
528C2×C4—181no—
529C2×C4—181no—
530C2×C4—271no—
531C2×C4—541no—
532C2×C4—541no—
533C2×C4—1082no—
534C2×C4D3311yescharacteristic
535C8—61no—
536C8—181no—
537C2×C3—21no—
538C2×C3—21no—
539C2×C3—21no—
540C2×C3—21no—
541C2×C3—21no—
542C2×C3—21no—
543C2×C3—21no—
544C2×C3—21no—
545C2×C3—21no—
546C2×C3—41no—
547C2×C3—41no—
548C2×C3—41no—
549C2×C3#6379611yescharacteristic
550C2×C3#6381111yescharacteristic
551C2×C3#6385311yescharacteristic
552C2×C3#6385411yescharacteristic
553C2×C3#6393011yescharacteristic
554C2×C3#6394011yescharacteristic
555C2×C3#6394511yescharacteristic
556C2×C3#6395111yescharacteristic
557C2×C3#6397511yescharacteristic
558D3—184no—
559D3—182no—
560D3—182no—
561D3—182no—
562D3—362no—
563D3—362no—
564D3—364no—
565C22—272no—
566C22—541no—
567C22—541no—
568C22#8895411yescharacteristic
569C4—121no—
570C4—121no—
571C4—361no—
572C4—361no—
573C4—541no—
574C4#8895911yescharacteristic
575C4#8896211yescharacteristic
576C3—21no—
577C3—21no—
578C3—21no—
579C3—41no—
580C3#1059052911yescharacteristic, complemented
581C3#1059068611yescharacteristic, complemented
582C3#1059155311yescharacteristic, complemented
583C2—542no—
584C2#1172284111yescharacteristic
585C2#1172289411yescharacteristic
586C2#1172299611yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC22×C3⋊(C3⋊D3)D4
hypocenterC33#483
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
3. #11722079C2semidirect
5. C3×#63573C2semidirect
6. C3×#63566C2semidirect
7. C3×#63565C2semidirect
40. #63573D3semidirect
41. #63566D3semidirect
42. #63565D3semidirect
340. C33#483semidirect
520. C32#4805semidirect
521. C32#4805semidirect
522. C32#4862semidirect
580. C3#10590529semidirect
581. C3#10590686semidirect
582. C3#10591553semidirect

Maximal subgroups

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

subgroupindex
#117235032
#117234562
#117220792
#117220242
C3×#635732
C3×#635662
C3×#635652
#105915533
#105906863
#105905293