Finite Group Databasebeta

← #13156580

Subgroups — #13156580

Order 1728.

818 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 818 conjugacy classes of subgroups — too many to draw. Only its 112 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
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723529C211yesmaximal, characteristic
3#11723451C211yesmaximal, characteristic, complemented
4C2×C3×#3717C211yesmaximal, characteristic, complemented
5C2×C3×#3716C211yesmaximal, characteristic, complemented
6C2×C3×#3715C211yesmaximal, characteristic, complemented
7#11722018C211yesmaximal, characteristic, complemented
8#10591620—31nomaximal
9#10591492—31nomaximal
10#10590884—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×#6290—22no—
16C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
17C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
18C2×C32×C3⋊C8C2211yescharacteristic
19#88811—22no—
20#88806—22no—
21C3×#3717—22no—
22C3×#3716—22no—
23C3×#3715—22no—
24#64153—31no—
25#64149—31no—
26C2×C3×#215—31no—
27C2×C3×#215—31no—
28C2×C3×#214—31no—
29C2×C3×#214—31no—
30C2×C3×#204—31no—
31C2×C3×#204—31no—
32#63914—31no—
33#63914—31no—
34#63914—61no—
35#63873—31no—
36#63873—31no—
37#63873—61no—
38C2×#3717—31no—
39C2×#3717D311yescharacteristic, complemented
40C2×#3716—31no—
41C2×#3716D311yescharacteristic, complemented
42C2×#3715—31no—
43C2×#3715D311yescharacteristic, complemented
44C8×C3⋊Dic3—31no—
45#63553—31no—
46#63553—31no—
47#63553—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3⋊(C3⋊Dic3)D412yes—
52C2×C3×C3⋊Dic3—21no—
53C3×C4⋊(C3⋊D3)—21no—
54C3×C4⋊(C3⋊D3)—21no—
55C3×#634—21no—
56C3×#634—21no—
57C2×C32×Dic3—21no—
58C32×D12—21no—
59C32×D12—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62#6290—44no—
63C3×C3⋊(C3⋊C8)D412yescomplemented
64C32×C3⋊C8D412yescomplemented
65#5175—91no—
66#5175—91no—
67#5175—91no—
68C2×C32×Q8—31no—
69C2×C32×D4—31no—
70C2×#638—62no—
71C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
72C2×#634C2×D311yescharacteristic
73C2×C3×D12—21no—
74C2×C3×D12—31no—
75C2×C3×D12—31no—
76C2×C3×D12—61no—
77C2×C3×D12C2×D311yescharacteristic
78C2×C3×D12C2×D311yescharacteristic
79C2×C3×Dic6—21no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6C2×D311yescharacteristic
84C2×C3×Dic6C2×D311yescharacteristic
85C2×#625—62no—
86C2×#625—62no—
87C2×#625—64no—
88C2×C32×C8—31no—
89#3750—62no—
90C4×C3⋊Dic3—31no—
91C4×C3⋊Dic3—31no—
92C4×C3⋊Dic3—31no—
93C4×C3⋊Dic3—61no—
94C4×C3⋊Dic3—61no—
95C4×C3⋊Dic3—61no—
96C4×C3⋊Dic3—121no—
97C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
98C3×#215—62no—
99C3×#215—62no—
100C3×#214—62no—
101C3×#214—62no—
102C2×C3×C3⋊C8—21no—
103C2×C3×C3⋊C8—31no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—61no—
106C2×C3×C3⋊C8C2×D311yescharacteristic
107C2×C3×C3⋊C8C2×D311yescharacteristic
108C3×#204—62no—
109C3×#204—62no—
110#3724—62no—
111#3724—62no—
112#3724—64no—
113#3717—22no—
114#3717—62no—
115#3716—22no—
116#3716—62no—
117#3715—22no—
118#3715—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—42no—
122C2×C32×D3—41no—
123C2×C32×D3—42no—
124C33×C4C2×D411yescharacteristic
125C33×C4C2×D411yescharacteristic
126C3⋊(C3⋊Dic3)SD1614yescomplemented
127C3×C3⋊Dic3—41no—
128C32×Dic3—41no—
129C2×C3×SD16—91no—
130C2×C3×SD16—91no—
131C2×C3×SD16—91no—
132#944—91no—
133#944—91no—
134#944—91no—
135#944—181no—
136C2×#215—31no—
137C2×#215—31no—
138C2×#215—91no—
139#940—91no—
140#940—91no—
141#940—91no—
142#940—181no—
143C2×#214—31no—
144C2×#214—31no—
145C2×#214—91no—
146C2×#204—31no—
147C2×#204—31no—
148C2×#204—91no—
149C8×Dic3—91no—
150C8×Dic3—91no—
151C8×Dic3—91no—
152C8×Dic3—181no—
153C23×C32—61no—
154C22×C3⋊D3—21no—
155C22×C3×D3—22no—
156C22×C3×D3—21no—
157C22×C3×D3—21no—
158C22×C3×D3—62no—
159C22×C3×D3—61no—
160C22×C3×D3—61no—
161C32×Q8—61no—
162C32×Q8—61no—
163C32×D4—61no—
164C32×D4—61no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—41no—
169C2×C32×C4—61no—
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C32×C4C22×D311yescharacteristic
173#638—124no—
174C2×C3⋊Dic3—21no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—32no—
177C2×C3⋊Dic3—32no—
178C2×C3⋊Dic3—62no—
179C2×C3⋊Dic3—62no—
180C2×C3⋊Dic3—62no—
181C2×C3⋊Dic3—122no—
182C4⋊(C3⋊D3)—21no—
183C4⋊(C3⋊D3)—21no—
184#634—21no—
185#634—21no—
186C2×C3×Dic3—22no—
187C2×C3×Dic3—21no—
188C2×C3×Dic3—21no—
189C2×C3×Dic3—62no—
190C2×C3×Dic3—61no—
191C2×C3×Dic3—61no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—21no—
195C3×D12—21no—
196C3×D12—41no—
197C3×D12—41no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—61no—
201C3×D12—61no—
202C3×D12—121no—
203C3×D12—121no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—21no—
208C3×Dic6—41no—
209C3×Dic6—41no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—121no—
215C3×Dic6—121no—
216#625—124no—
217#625—124no—
218#625—128no—
219C32×C8—32no—
220C3⋊(C3⋊C8)#6712yescomplemented
221C3×C3⋊C8—22no—
222C3×C3⋊C8—32no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—62no—
225C3×C3⋊C8#6712yescomplemented
226C3×C3⋊C8#6712yescomplemented
227#492—271noSylow
228C2×C33#12711yescharacteristic
229C2×C33C2×SD1612yes—
230C3×C3⋊D3—81no—
231C32×D3—81no—
232C2×C3×Q8—31no—
233C2×C3×Q8—31no—
234C2×C3×Q8—61no—
235C2×C3×Q8—91no—
236C2×C3×D4—31no—
237C2×C3×D4—31no—
238C2×C3×D4—61no—
239C2×C3×D4—91no—
240C2×#67—182no—
241C2×#67—182no—
242C2×#67—182no—
243C2×#67—184no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—61no—
247C2×D12D3211yescharacteristic
248C2×Dic6—31no—
249C2×Dic6—31no—
250C2×Dic6—61no—
251C2×Dic6D3211yescharacteristic
252C3×SD16—182no—
253C3×SD16—182no—
254C3×SD16—182no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—61no—
258C2×C3×C8—91no—
259#215—62no—
260#215—62no—
261#215—182no—
262#214—62no—
263#214—62no—
264#214—182no—
265Dic3⋊C4—182no—
266Dic3⋊C4—182no—
267Dic3⋊C4—182no—
268Dic3⋊C4—184no—
269C4×Dic3—91no—
270C4×Dic3—91no—
271C4×Dic3—91no—
272C4×Dic3—181no—
273C4×Dic3—181no—
274C4×Dic3—181no—
275C4×Dic3—361no—
276C2×C3⋊C8—31no—
277C2×C3⋊C8—31no—
278C2×C3⋊C8—61no—
279C2×C3⋊C8D3211yescharacteristic
280#204—62no—
281#204—62no—
282#204—182no—
283C22×C32—21no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—41no—
287C22×C32—122no—
288C22×C32—121no—
289C22×C32D3×D411yescharacteristic
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C2×C3⋊D3—42no—
293C2×C3⋊D3—41no—
294C2×C3×D3—44no—
295C2×C3×D3—42no—
296C2×C3×D3—42no—
297C2×C3×D3—42no—
298C2×C3×D3—41no—
299C2×C3×D3—41no—
300C2×C3×D3—124no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—122no—
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×C4D3×D411yescharacteristic
316C32×C4D3×D411yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4C2×#6711yescharacteristic
319C32×C4C2×#6711yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C3⋊Dic3—34no—
322C3⋊Dic3—34no—
323C3⋊Dic3—34no—
324C3⋊Dic3—41no—
325C3⋊Dic3—64no—
326C3⋊Dic3—64no—
327C3⋊Dic3—64no—
328C3⋊Dic3—124no—
329C3×Dic3—42no—
330C3×Dic3—41no—
331C3×Dic3—41no—
332C3×Dic3—122no—
333C3×Dic3—121no—
334C3×Dic3—121no—
335C2×SD16—91no—
336C2×SD16—91no—
337C2×SD16—91no—
338C2×SD16—271no—
339C4⋊Q8—271no—
340#127—271no—
341C4×C8—271no—
342C33#49211yescharacteristic, complemented, Sylow
343C23×C3—62no—
344C23×C3—61no—
345C23×C3—61no—
346C23×C3—181no—
347C22×D3—21no—
348C22×D3—62no—
349C22×D3—61no—
350C22×D3—61no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×Q8—61no—
355C3×Q8—121no—
356C3×Q8—121no—
357C3×Q8—181no—
358C3×Q8—181no—
359C3×D4—61no—
360C3×D4—61no—
361C3×D4—61no—
362C3×D4—61no—
363C3×D4—121no—
364C3×D4—121no—
365C3×D4—181no—
366C3×D4—181no—
367C2×C3×C4—21no—
368C2×C3×C4—21no—
369C2×C3×C4—21no—
370C2×C3×C4—41no—
371C2×C3×C4—62no—
372C2×C3×C4—61no—
373C2×C3×C4—61no—
374C2×C3×C4—181no—
375C2×C3×C4C2×D3211yescharacteristic
376C2×C3×C4C2×D3211yescharacteristic
377C2×C3×C4C2×D3211yescharacteristic
378#67—364no—
379#67—364no—
380#67—364no—
381#67—368no—
382C2×Dic3—21no—
383C2×Dic3—62no—
384C2×Dic3—61no—
385C2×Dic3—61no—
386C2×Dic3—92no—
387C2×Dic3—92no—
388C2×Dic3—92no—
389C2×Dic3—182no—
390C2×Dic3—182no—
391C2×Dic3—182no—
392C2×Dic3—362no—
393D12—21no—
394D12—21no—
395D12—61no—
396D12—61no—
397D12—61no—
398D12—61no—
399D12—121no—
400D12—121no—
401Dic6—21no—
402Dic6—21no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—61no—
407Dic6—121no—
408Dic6—121no—
409C3×C8—32no—
410C3×C8—32no—
411C3×C8—62no—
412C3×C8—92no—
413C3⋊C8—32no—
414C3⋊C8—32no—
415C3⋊C8—62no—
416C3⋊C8#62512yescomplemented
417C2×C32—22no—
418C2×C32—21no—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—41no—
424C2×C32—42no—
425C2×C32—241no—
426C2×C32D3×SD1612yes—
427C2×C32D3×SD1612yes—
428C2×C32D3×SD1612yes—
429C2×C32#94011yescharacteristic
430C2×C32#94011yescharacteristic
431C2×C32#94011yescharacteristic
432C3⋊D3—81no—
433C3×D3—82no—
434C3×D3—81no—
435C3×D3—81no—
436C3×D3—242no—
437C3×D3—241no—
438C3×D3—241no—
439C2×Q8—31no—
440C2×Q8—91no—
441C2×D4—31no—
442C2×D4—91no—
443C2×D4—542no—
444SD16—182no—
445SD16—182no—
446SD16—182no—
447SD16—542no—
448C2×C8—31no—
449C2×C8—91no—
450C4⋊C4—542no—
451C42—271no—
452C22×C3—21no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—41no—
456C22×C3—124no—
457C22×C3—122no—
458C22×C3—122no—
459C22×C3—121no—
460C22×C3—122no—
461C22×C3—121no—
462C22×C3—362no—
463C22×C3—361no—
464C22×C3#380211yescharacteristic
465C22×C3#381111yescharacteristic
466C22×C3#381111yescharacteristic
467C2×D3—42no—
468C2×D3—41no—
469C2×D3—122no—
470C2×D3—124no—
471C2×D3—122no—
472C2×D3—121no—
473C2×D3—122no—
474C2×D3—121no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—21no—
481C3×C4—41no—
482C3×C4—41no—
483C3×C4—122no—
484C3×C4—121no—
485C3×C4—121no—
486C3×C4—361no—
487C3×C4#380211yescharacteristic
488C3×C4C2×#62511yescharacteristic
489C3×C4D3×#6711yescharacteristic
490C3×C4D3×#6711yescharacteristic
491C3×C4D3×#6711yescharacteristic
492C3×C4D3×#6711yescharacteristic
493Dic3—41no—
494Dic3—94no—
495Dic3—94no—
496Dic3—94no—
497Dic3—122no—
498Dic3—121no—
499Dic3—121no—
500Dic3—184no—
501Dic3—184no—
502Dic3—184no—
503Dic3—364no—
504C32—21no—
505C32—21no—
506C32—21no—
507C32—41no—
508C32#517511yescharacteristic, complemented
509C32#517511yescharacteristic, complemented
510C32#517511yescharacteristic, complemented
511C23—61no—
512C23—181no—
513Q8—61no—
514Q8—61no—
515Q8—181no—
516Q8—181no—
517D4—61no—
518D4—61no—
519D4—181no—
520D4—181no—
521D4—1084no—
522C2×C4—61no—
523C2×C4—181no—
524C2×C4—272no—
525C2×C4D3311yescharacteristic
526C8—32no—
527C8—92no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—21no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—41no—
535C2×C3—42no—
536C2×C3—242no—
537C2×C3—241no—
538C2×C3—241no—
539C2×C3—721no—
540C2×C3#6379612yes—
541C2×C3#6391411yescharacteristic
542C2×C3#6393012yes—
543C2×C3#6394012yes—
544C2×C3#6397911yescharacteristic
545C2×C3#6397911yescharacteristic
546D3—81no—
547D3—242no—
548D3—241no—
549D3—241no—
550C22—121no—
551C22—122no—
552C22—361no—
553C22—362no—
554C22#8895411yescharacteristic
555C4—121no—
556C4—274no—
557C4—361no—
558C4D3×#62511yescharacteristic
559C4#8895311yescharacteristic
560C3—21no—
561C3—21no—
562C3—21no—
563C3—41no—
564C3#1059088411yescharacteristic, complemented
565C3#1059149211yescharacteristic, complemented
566C3#1059162011yescharacteristic, complemented
567C2—241no—
568C2—721no—
569C2#1172284112yes—
570C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radical#11723529C2
hypocenterC33#492
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. C2×#88811C2semidirect
3. #11723451C2semidirect
4. C2×C3×#3717C2semidirect
5. C2×C3×#3716C2semidirect
6. C2×C3×#3715C2semidirect
7. #11722018C2semidirect
39. C2×#3717D3semidirect
41. C2×#3716D3semidirect
43. C2×#3715D3semidirect
63. C3×C3⋊(C3⋊C8)D4semidirect
64. C32×C3⋊C8D4semidirect
126. C3⋊(C3⋊Dic3)SD16semidirect
220. C3⋊(C3⋊C8)#67semidirect
225. C3×C3⋊C8#67semidirect
226. C3×C3⋊C8#67semidirect
342. C33#492semidirect
416. C3⋊C8#625semidirect
508. C32#5175semidirect
509. C32#5175semidirect
510. C32#5175semidirect
564. C3#10590884semidirect
565. C3#10591492semidirect
566. C3#10591620semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117235292
#117234512
C2×C3×#37172
C2×C3×#37162
C2×C3×#37152
#117220182
#105916203
#105914923
#105908843