Finite Group Databasebeta

← #13156639

Subgroups — #13156639

Order 1728.

762 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 114 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×#88812C211yesmaximal, characteristic, complemented
2#11723441C211yesmaximal, characteristic
3C2×#88797C211yesmaximal, characteristic
4C2×#88795C211yesmaximal, characteristic
5C3×#63866C211yesmaximal, characteristic, complemented
6C2×C3×#3719C211yesmaximal, characteristic, complemented
7C3×#63554C211yesmaximal, characteristic, complemented
8#10591608—31nomaximal
9#10590899—31nomaximal
10#10590374—31nomaximal
11C2×C4⋊(C3⋊(C3⋊D3))C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
16C32×C4×Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88812—22no—
19#88804—22no—
20#88797—22no—
21#88795—22no—
22C3×Dic3⋊Dic3—22no—
23C3×#3719—22no—
24C2×#3755—31no—
25C2×#3744—31no—
26#64084—31no—
27C3×#944—31no—
28C2×C3×#216—31no—
29C2×C3×Dic12—31no—
30C3×#869—31no—
31#63866D311yescharacteristic, complemented
32#63864—31no—
33#63864—31no—
34#63864—61no—
35C2×#3719—31no—
36C2×#3719—31no—
37C2×#3719D311yescharacteristic, complemented
38C2×#3718—31no—
39C2×#3717—31no—
40C2×#3717—31no—
41C2×#3717—31no—
42C2×#3717—31no—
43C2×#3717—61no—
44C2×#3717—61no—
45C3×#815—31no—
46C3×#803—31no—
47#63554D311yescharacteristic, complemented
48C22×C3⋊(C3⋊D3)—21no—
49C2×C33×C4C2311yescharacteristic
50C4⋊(C3⋊(C3⋊D3))—21no—
51C4⋊(C3⋊(C3⋊D3))—21no—
52C2×C3×C3⋊Dic3—21no—
53C2×C3×C3⋊Dic3—21no—
54C3×#634—21no—
55C3×#634—21no—
56C3×#634—21no—
57C3×#634—21no—
58C2×C32×Dic3—21no—
59C2×C32×Dic3D412yes—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64#5190—91no—
65#5086—91no—
66#4715—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)—31no—
69C2×C4⋊(C3⋊D3)—31no—
70C2×C4⋊(C3⋊D3)—31no—
71C2×C4⋊(C3⋊D3)—61no—
72C2×C4⋊(C3⋊D3)—61no—
73C2×C4⋊(C3⋊D3)—61no—
74C2×C4⋊(C3⋊D3)—121no—
75C2×#634C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—31no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—61no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86C2×C32×C8—31no—
87C32×C42—31no—
88#3755—62no—
89#3752—62no—
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91#3744—62no—
92C3×#216—62no—
93C3×C4⋊Dic3—62no—
94C3×Dic3⋊C4—62no—
95C3×C4×Dic3—21no—
96C3×C4×Dic3C2×D311yescharacteristic
97C3×C4×Dic3C2×D311yescharacteristic
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8—31no—
100C2×C3×C3⋊C8—31no—
101C2×C3×C3⋊C8—61no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×Dic12—62no—
105Dic3⋊Dic3—22no—
106#3722—64no—
107#3722—62no—
108#3722—62no—
109#3719—22no—
110#3719—62no—
111#3719—62no—
112#3718—62no—
113#3717—64no—
114#3717—62no—
115#3717—62no—
116#3717—64no—
117#3717—62no—
118#3717—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3⋊(C3⋊D3)—41no—
121C2×C3⋊(C3⋊D3)—42no—
122C33×C4C2×D411yescharacteristic
123C33×C4C2×D411yescharacteristic
124C3×C3⋊Dic3—41no—
125C3×C3⋊Dic3—41no—
126C32×Dic3—22no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C3×C4⋊Q8—91no—
130#947—91no—
131#944—31no—
132C2×#216—31no—
133C2×#216—91no—
134C2×#216—91no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—91no—
138C2×#215—181no—
139C2×Dic12—31no—
140C2×Dic12—91no—
141C2×#204—91no—
142C2×#204—91no—
143C2×#204—91no—
144C2×#204—181no—
145#875—91no—
146#875—91no—
147#875—181no—
148#869—31no—
149C3×#97—91no—
150#815—31no—
151#803—31no—
152C22×C3⋊D3—61no—
153C22×C3⋊D3—62no—
154C22×C3⋊D3—61no—
155C22×C3⋊D3—61no—
156C22×C3⋊D3—121no—
157C22×C3⋊D3—121no—
158C22×C3⋊D3—122no—
159C32×Q8—61no—
160C32×Q8—61no—
161C2×C32×C4—21no—
162C2×C32×C4—21no—
163C2×C32×C4—21no—
164C2×C32×C4—32no—
165C2×C32×C4—41no—
166C2×C32×C4—61no—
167C2×C32×C4C22×D311yescharacteristic
168C2×C32×C4C22×D311yescharacteristic
169C2×C32×C4C22×D311yescharacteristic
170C2×C3⋊Dic3—21no—
171C2×C3⋊Dic3—21no—
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—
186#634—21no—
187#634—21no—
188#634—21no—
189#634—21no—
190C2×C3×Dic3—21no—
191C2×C3×Dic3—21no—
192C2×C3×Dic3—22no—
193C2×C3×Dic3—41no—
194C2×C3×Dic3—62no—
195C2×C3×Dic3—61no—
196C2×C3×Dic3—61no—
197C2×C3×Dic3—61no—
198C2×C3×Dic3—61no—
199C2×C3×Dic3—121no—
200C2×C3×Dic3D1212yes—
201C2×C3×Dic3#6712yes—
202C3×Dic6—22no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—21no—
207C3×Dic6—41no—
208C3×Dic6—62no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—61no—
213C3×Dic6—61no—
214C3×Dic6—61no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—121no—
218C3×Dic6—121no—
219C3×Dic6—121no—
220C32×C8—32no—
221C3⋊(C3⋊C8)D1212yes—
222C3×C3⋊C8—22no—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—62no—
226C3×C3⋊C8D1212yes—
227C3×C3⋊C8#6712yes—
228#497—271noSylow
229C2×C33#12711yescharacteristic
230C2×C33#13712yes—
231C3⋊(C3⋊D3)—81no—
232C2×C3×Q8—31no—
233C2×C3×Q8—31no—
234C2×C3×Q8—61no—
235C2×C3×Q8—91no—
236C2×C3×Q8—91no—
237C2×D12—91no—
238C2×D12—91no—
239C2×D12—91no—
240C2×D12—181no—
241C2×D12—181no—
242C2×D12—181no—
243C2×D12—361no—
244C2×Dic6—31no—
245C2×Dic6—31no—
246C2×Dic6—31no—
247C2×Dic6—31no—
248C2×Dic6—61no—
249C2×Dic6—61no—
250C2×Dic6D3211yescharacteristic
251C3×Q16—182no—
252C2×C3×C8—31no—
253C2×C3×C8—31no—
254C2×C3×C8—61no—
255C2×C3×C8—91no—
256C3×C4⋊C4—182no—
257C3×C42—31no—
258C3×C42—31no—
259C3×C42—61no—
260#216—62no—
261#216—182no—
262#216—182no—
263#215—182no—
264#215—182no—
265#215—184no—
266#215—182no—
267(C2×D3)⋊C4—182no—
268(C2×D3)⋊C4—182no—
269(C2×D3)⋊C4—184no—
270(C2×D3)⋊C4—182no—
271C4⋊Dic3—62no—
272Dic3⋊C4—62no—
273C4×Dic3D3211yescharacteristic
274C2×C3⋊C8—31no—
275C2×C3⋊C8—31no—
276C2×C3⋊C8—61no—
277C2×C3⋊C8D3211yescharacteristic
278Dic12—62no—
279Dic12—182no—
280#204—182no—
281#204—182no—
282#204—184no—
283#204—182no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32C2×D1211yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32C2×#6711yescharacteristic
291C2×C3⋊D3—122no—
292C2×C3⋊D3—122no—
293C2×C3⋊D3—122no—
294C2×C3⋊D3—124no—
295C2×C3⋊D3—121no—
296C2×C3⋊D3—122no—
297C2×C3⋊D3—121no—
298C2×C3⋊D3—121no—
299C2×C3⋊D3—124no—
300C2×C3⋊D3—122no—
301C2×C3⋊D3—122no—
302C2×C3⋊D3—242no—
303C2×C3⋊D3—242no—
304C2×C3⋊D3—244no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—21no—
308C32×C4—21no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—41no—
312C32×C4—41no—
313C32×C4—62no—
314C32×C4—121no—
315C32×C4C2×D1211yescharacteristic
316C32×C4C2×D1211yescharacteristic
317C32×C4D3×D411yescharacteristic
318C32×C4D3×D411yescharacteristic
319C32×C4D3×D411yescharacteristic
320C32×C4C2×#6711yescharacteristic
321C3⋊Dic3—41no—
322C3⋊Dic3—41no—
323C3×Dic3—22no—
324C3×Dic3—22no—
325C3×Dic3—42no—
326C3×Dic3—42no—
327C3×Dic3—41no—
328C3×Dic3—41no—
329C3×Dic3—122no—
330C3×Dic3—122no—
331C3×Dic3—121no—
332C3×Dic3—121no—
333C3×Dic3—121no—
334C3×Dic3—121no—
335C2×Q16—91no—
336C2×Q16—271no—
337C2×SD16—271no—
338C2×SD16—271no—
339C4⋊Q8—91no—
340#124—271no—
341#97—91no—
342C33#49711yescharacteristic, complemented, Sylow
343C22×D3—182no—
344C22×D3—181no—
345C22×D3—181no—
346C22×D3—181no—
347C22×D3—361no—
348C22×D3—361no—
349C22×D3—362no—
350C3×Q8—62no—
351C3×Q8—61no—
352C3×Q8—61no—
353C3×Q8—61no—
354C3×Q8—61no—
355C3×Q8—121no—
356C3×Q8—181no—
357C3×Q8—181no—
358C3×Q8—181no—
359C3×Q8—181no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—32no—
364C2×C3×C4—32no—
365C2×C3×C4—41no—
366C2×C3×C4—62no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—121no—
370C2×C3×C4—181no—
371C2×C3×C4—181no—
372C2×C3×C4C2×D3211yescharacteristic
373C2×C3×C4C2×D3211yescharacteristic
374C2×C3×C4C2×D3211yescharacteristic
375C2×Dic3—21no—
376C2×Dic3—62no—
377C2×Dic3—61no—
378C2×Dic3—61no—
379C2×Dic3—61no—
380C2×Dic3—61no—
381C2×Dic3—121no—
382C2×Dic3#62612yes—
383D12—182no—
384D12—182no—
385D12—181no—
386D12—181no—
387D12—181no—
388D12—181no—
389D12—181no—
390D12—181no—
391D12—361no—
392D12—361no—
393D12—361no—
394D12—361no—
395D12—362no—
396D12—362no—
397Dic6—21no—
398Dic6—21no—
399Dic6—62no—
400Dic6—61no—
401Dic6—61no—
402Dic6—61no—
403Dic6—61no—
404Dic6—61no—
405Dic6—61no—
406Dic6—61no—
407Dic6—61no—
408Dic6—121no—
409Dic6—121no—
410Dic6—121no—
411C3×C8—32no—
412C3×C8—32no—
413C3×C8—62no—
414C3×C8—92no—
415C3⋊C8—32no—
416C3⋊C8—32no—
417C3⋊C8—62no—
418C3⋊C8#62612yes—
419C2×C32—22no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—21no—
423C2×C32—22no—
424C2×C32—21no—
425C2×C32—41no—
426C2×C32—42no—
427C2×C32#87411yescharacteristic
428C2×C32#90912yes—
429C2×C32#91812yes—
430C2×C32#94011yescharacteristic
431C2×C32#94011yescharacteristic
432C2×C32#94212yes—
433C3⋊D3—242no—
434C3⋊D3—242no—
435C3⋊D3—242no—
436C3⋊D3—244no—
437C3⋊D3—241no—
438C3⋊D3—241no—
439C3⋊D3—241no—
440C2×Q8—31no—
441C2×Q8—91no—
442C2×Q8—91no—
443C2×D4—271no—
444Q16—182no—
445Q16—542no—
446SD16—542no—
447SD16—542no—
448C2×C8—31no—
449C2×C8—91no—
450C4⋊C4—182no—
451C22⋊C4—542no—
452C42—31no—
453C22×C3—21no—
454C22×C3—21no—
455C22×C3—21no—
456C22×C3—41no—
457C22×C3D3×D1211yescharacteristic
458C22×C3C2×#62611yescharacteristic
459C22×C3D3×#6711yescharacteristic
460C2×D3—362no—
461C2×D3—362no—
462C2×D3—362no—
463C2×D3—364no—
464C2×D3—364no—
465C2×D3—362no—
466C2×D3—362no—
467C2×D3—361no—
468C2×D3—361no—
469C2×D3—362no—
470C2×D3—361no—
471C2×D3—722no—
472C2×D3—722no—
473C2×D3—724no—
474C3×C4—21no—
475C3×C4—21no—
476C3×C4—21no—
477C3×C4—21no—
478C3×C4—21no—
479C3×C4—21no—
480C3×C4—41no—
481C3×C4—41no—
482C3×C4—62no—
483C3×C4—62no—
484C3×C4—122no—
485C3×C4—122no—
486C3×C4—121no—
487C3×C4—121no—
488C3×C4—361no—
489C3×C4—361no—
490C3×C4D3×D1211yescharacteristic
491C3×C4D3×D1211yescharacteristic
492C3×C4D3×D1211yescharacteristic
493C3×C4C2×#62611yescharacteristic
494C3×C4D3×#6711yescharacteristic
495C3×C4#381111yescharacteristic
496Dic3—22no—
497Dic3—41no—
498Dic3—122no—
499Dic3—122no—
500Dic3—121no—
501Dic3—121no—
502Dic3—121no—
503Dic3—121no—
504C32—21no—
505C32—21no—
506C32—21no—
507C32—41no—
508C32#471511yescharacteristic, complemented
509C32#508611yescharacteristic, complemented
510C32#519011yescharacteristic, complemented
511C23—541no—
512Q8—61no—
513Q8—61no—
514Q8—181no—
515Q8—181no—
516Q8—181no—
517Q8—181no—
518D4—541no—
519D4—541no—
520C2×C4—32no—
521C2×C4—61no—
522C2×C4—181no—
523C2×C4—181no—
524C2×C4D3311yescharacteristic
525C8—32no—
526C8—92no—
527C2×C3—21no—
528C2×C3—21no—
529C2×C3—21no—
530C2×C3—22no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—41no—
534C2×C3—42no—
535C2×C3#6380112yes—
536C2×C3#6383312yes—
537C2×C3#6391111yescharacteristic
538C2×C3#6391111yescharacteristic
539C2×C3#6394812yes—
540C2×C3#6397911yescharacteristic
541D3—722no—
542D3—722no—
543D3—722no—
544D3—724no—
545D3—721no—
546D3—721no—
547D3—721no—
548C22—1081no—
549C22—1082no—
550C22D3×#62611yescharacteristic
551C4—62no—
552C4—121no—
553C4—361no—
554C4—361no—
555C4D3×#62611yescharacteristic
556C4#8895511yescharacteristic
557C3—21no—
558C3—21no—
559C3—21no—
560C3—41no—
561C3#1059037411yescharacteristic, complemented
562C3#1059089911yescharacteristic, complemented
563C3#1059160811yescharacteristic, complemented
564C2—2161no—
565C2#1172288512yes—
566C2#1172303011yescharacteristic
subgroupG/H
centerC22D3×#626
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C4⋊(C3⋊(C3⋊D3))C22
hypocenterC33#497
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×#88812C2semidirect
5. C3×#63866C2semidirect
6. C2×C3×#3719C2semidirect
7. C3×#63554C2semidirect
31. #63866D3semidirect
37. C2×#3719D3semidirect
47. #63554D3semidirect
342. C33#497semidirect
508. C32#4715semidirect
509. C32#5086semidirect
510. C32#5190semidirect
561. C3#10590374semidirect
562. C3#10590899semidirect
563. C3#10591608semidirect

Maximal subgroups

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

subgroupindex
C2×#888122
#117234412
C2×#887972
C2×#887952
C3×#638662
C2×C3×#37192
C3×#635542
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#105908993
#105903743