Finite Group Databasebeta

← #13156642

Subgroups — #13156642

Order 1728.

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

normal subgroups only

This group has 730 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
1#11723677C211yesmaximal, characteristic
2#11723455C211yesmaximal, characteristic, complemented
3C2×#88800C211yesmaximal, characteristic, complemented
4C2×#88796C211yesmaximal, characteristic
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3717C211yesmaximal, characteristic
7C3×#63555C211yesmaximal, characteristic, complemented
8#10591611—31nomaximal
9#10590899—31nomaximal
10#10590897—31nomaximal
11C2×#6308C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C2×C32×C3⋊C8C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—22no—
19#88807—22no—
20#88800—22no—
21#88796—22no—
22C3×#3719—22no—
23C3×#3717—22no—
24#64153—31no—
25C2×#3746—31no—
26C2×#3744—31no—
27C2×C3×#216—31no—
28C2×C3×#215—31no—
29C2×C3×Dic12—31no—
30C2×C3×#204—31no—
31#63866—31no—
32#63866—31no—
33#63866—61no—
34#63864—31no—
35#63863—31no—
36#63860—31no—
37C2×#3719—31no—
38C2×#3719—31no—
39C2×#3719—61no—
40C2×#3719D311yescharacteristic, complemented
41C2×#3717D311yescharacteristic
42C2×#3716—31no—
43C2×#3716—31no—
44C2×#3716—61no—
45C3×#815—31no—
46C3×#815—31no—
47#63555D311yescharacteristic, complemented
48C22×C3×C3⋊D3—21no—
49C2×C33×C4C2311yescharacteristic
50C2×C3⋊(C3⋊Dic3)—21no—
51#6308—21no—
52#6308—21no—
53C2×C3×C3⋊Dic3—21no—
54C2×C3×C3⋊Dic3D412yes—
55C3×C4⋊(C3⋊D3)—21no—
56C3×C4⋊(C3⋊D3)—21no—
57C3×#634—21no—
58C3×#634—21no—
59C2×C32×Dic3—21no—
60C32×Dic6—21no—
61C32×Dic6—21no—
62C32×C3⋊C8D412yes—
63C32×C3⋊C8D412yes—
64#5190—91no—
65#5167—91no—
66#4715—91no—
67C2×C32×Q8—31no—
68C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
69C2×#634—31no—
70C2×#634—31no—
71C2×#634—31no—
72C2×#634—61no—
73C2×#634—61no—
74C2×#634—61no—
75C2×#634—121no—
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—61no—
80C2×C3×Dic6—21no—
81C2×C3×Dic6—31no—
82C2×C3×Dic6—31no—
83C2×C3×Dic6—61no—
84C2×C3×Dic6C2×D311yescharacteristic
85C2×C3×Dic6C2×D311yescharacteristic
86C2×C32×C8—31no—
87C2×C32×C8—31no—
88#3750—62no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90#3746—62no—
91#3744—62no—
92C3×#216—62no—
93C3×#215—62no—
94C3×C4×Dic3—31no—
95C3×C4×Dic3—31no—
96C3×C4×Dic3—61no—
97C2×C3×C3⋊C8—21no—
98C2×C3×C3⋊C8—21no—
99C2×C3×C3⋊C8C2×D311yescharacteristic
100C2×C3×C3⋊C8C2×D311yescharacteristic
101C2×C3×C3⋊C8C2×D311yescharacteristic
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C3×Dic12—62no—
104C3×#204—62no—
105Dic3⋊Dic3—62no—
106Dic3⋊Dic3—62no—
107Dic3⋊Dic3—64no—
108#3722—62no—
109(C2×D3)⋊Dic3—62no—
110(C2×D3)⋊Dic3—62no—
111#3719—22no—
112#3719—64no—
113#3719—62no—
114#3719—62no—
115#3717—22no—
116#3716—64no—
117#3716—62no—
118#3716—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—22no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×SD16—91no—
130#947—91no—
131#944—91no—
132#944—91no—
133#944—181no—
134C2×#216—31no—
135C2×#216—91no—
136C2×#215—31no—
137#936—91no—
138C2×#214—91no—
139C2×Dic12—31no—
140C2×Dic12—91no—
141C2×Dic12—91no—
142C2×Dic12—181no—
143C2×#204—31no—
144C2×#204—91no—
145C2×#204—91no—
146C2×#204—181no—
147#875—91no—
148#869—91no—
149C3×#97—91no—
150#815—31no—
151#815—31no—
152C22×C3⋊D3—21no—
153C22×C3×D3—61no—
154C22×C3×D3—61no—
155C22×C3×D3—121no—
156C32×Q8—61no—
157C32×Q8—61no—
158C2×C32×C4—21no—
159C2×C32×C4—21no—
160C2×C32×C4—21no—
161C2×C32×C4—41no—
162C2×C32×C4—61no—
163C2×C32×C4C22×D311yescharacteristic
164C2×C32×C4C22×D311yescharacteristic
165C2×C32×C4C22×D311yescharacteristic
166C2×C3⋊Dic3—21no—
167C2×C3⋊Dic3—61no—
168C2×C3⋊Dic3—62no—
169C2×C3⋊Dic3—61no—
170C2×C3⋊Dic3—61no—
171C2×C3⋊Dic3—121no—
172C2×C3⋊Dic3—121no—
173C2×C3⋊Dic3—122no—
174C2×C3⋊Dic3D1212yes—
175C4⋊(C3⋊D3)—21no—
176C4⋊(C3⋊D3)—21no—
177#634—21no—
178#634—21no—
179#634—61no—
180#634—61no—
181#634—62no—
182#634—62no—
183#634—61no—
184#634—61no—
185#634—61no—
186#634—61no—
187#634—121no—
188#634—121no—
189#634—121no—
190#634—121no—
191#634—122no—
192#634—122no—
193C2×C3×Dic3—22no—
194C2×C3×Dic3—21no—
195C2×C3×Dic3—21no—
196C2×C3×Dic3—32no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—62no—
199C2×C3×Dic3—61no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—121no—
202C3×D12—62no—
203C3×D12—61no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—121no—
208C3×Dic6—21no—
209C3×Dic6—21no—
210C3×Dic6—21no—
211C3×Dic6—21no—
212C3×Dic6—41no—
213C3×Dic6—41no—
214C3×Dic6—62no—
215C3×Dic6—61no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—121no—
220C32×C8—32no—
221C32×C8—32no—
222C3×C3⋊C8—22no—
223C3×C3⋊C8—22no—
224C3×C3⋊C8D1212yes—
225C3×C3⋊C8D1212yes—
226C3×C3⋊C8#6712yes—
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×D4—91no—
237C2×D12—31no—
238C2×D12—31no—
239C2×D12—61no—
240C2×Dic6—31no—
241C2×Dic6—31no—
242C2×Dic6—61no—
243C2×Dic6—91no—
244C2×Dic6—91no—
245C2×Dic6—91no—
246C2×Dic6—181no—
247C2×Dic6—181no—
248C2×Dic6—181no—
249C2×Dic6—361no—
250C2×Dic6D3211yescharacteristic
251C3×Q16—182no—
252C3×SD16—182no—
253C2×C3×C8—31no—
254C2×C3×C8—31no—
255C2×C3×C8—31no—
256C2×C3×C8—31no—
257C2×C3×C8—61no—
258C2×C3×C8—61no—
259C3×C42—91no—
260#217—182no—
261#216—62no—
262#216—182no—
263#215—62no—
264#214—182no—
265(C2×D3)⋊C4—182no—
266(C2×D3)⋊C4—182no—
267C4⋊Dic3—182no—
268Dic3⋊C4—182no—
269Dic3⋊C4—182no—
270Dic3⋊C4—184no—
271C4×Dic3—31no—
272C4×Dic3—31no—
273C4×Dic3—61no—
274C2×C3⋊C8D3211yescharacteristic
275C2×C3⋊C8D3211yescharacteristic
276Dic12—62no—
277Dic12—182no—
278Dic12—182no—
279Dic12—184no—
280#204—62no—
281#204—182no—
282#204—182no—
283#204—184no—
284C22×C32—21no—
285C22×C32—21no—
286C22×C32—21no—
287C22×C32—41no—
288C22×C32C2×D1211yescharacteristic
289C22×C32D3×D411yescharacteristic
290C22×C32D3×D411yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—42no—
293C2×C3×D3—122no—
294C2×C3×D3—121no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—121no—
298C2×C3×D3—242no—
299C32×C4—21no—
300C32×C4—21no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—41no—
306C32×C4—41no—
307C32×C4—121no—
308C32×C4C2×D1211yescharacteristic
309C32×C4C2×D1211yescharacteristic
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4C2×#6711yescharacteristic
313C32×C4C2×#6711yescharacteristic
314C3⋊Dic3—22no—
315C3⋊Dic3—41no—
316C3⋊Dic3—122no—
317C3⋊Dic3—122no—
318C3⋊Dic3—122no—
319C3⋊Dic3—124no—
320C3⋊Dic3—121no—
321C3⋊Dic3—121no—
322C3⋊Dic3—121no—
323C3×Dic3—42no—
324C3×Dic3—41no—
325C3×Dic3—41no—
326C3×Dic3—64no—
327C3×Dic3—62no—
328C3×Dic3—62no—
329C3×Dic3—122no—
330C3×Dic3—121no—
331C3×Dic3—121no—
332C2×Q16—91no—
333C2×Q16—271no—
334C2×SD16—91no—
335C2×SD16—271no—
336C4⋊Q8—271no—
337#124—271no—
338#97—91no—
339C33#49711yescharacteristic, complemented, Sylow
340C23×C3—181no—
341C22×D3—61no—
342C22×D3—61no—
343C22×D3—121no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—121no—
349C3×Q8—121no—
350C3×Q8—181no—
351C3×Q8—181no—
352C3×D4—181no—
353C3×D4—181no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—41no—
358C2×C3×C4—62no—
359C2×C3×C4—61no—
360C2×C3×C4—61no—
361C2×C3×C4—92no—
362C2×C3×C4—181no—
363C2×C3×C4C2×D3211yescharacteristic
364C2×C3×C4C2×D3211yescharacteristic
365C2×C3×C4C2×D3211yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—32no—
368C2×Dic3—32no—
369C2×Dic3—62no—
370C2×Dic3—61no—
371C2×Dic3—61no—
372C2×Dic3—121no—
373C2×Dic3—182no—
374C2×Dic3—181no—
375C2×Dic3—181no—
376C2×Dic3—181no—
377C2×Dic3—361no—
378C2×Dic3—361no—
379C2×Dic3—362no—
380D12—62no—
381D12—61no—
382D12—61no—
383D12—61no—
384D12—61no—
385D12—121no—
386Dic6—21no—
387Dic6—21no—
388Dic6—62no—
389Dic6—61no—
390Dic6—61no—
391Dic6—61no—
392Dic6—61no—
393Dic6—121no—
394Dic6—182no—
395Dic6—182no—
396Dic6—181no—
397Dic6—181no—
398Dic6—181no—
399Dic6—181no—
400Dic6—181no—
401Dic6—181no—
402Dic6—361no—
403Dic6—361no—
404Dic6—361no—
405Dic6—361no—
406Dic6—362no—
407Dic6—362no—
408C3×C8—32no—
409C3×C8—32no—
410C3×C8—32no—
411C3×C8—32no—
412C3×C8—62no—
413C3×C8—62no—
414C3⋊C8#62612yes—
415C3⋊C8#62612yes—
416C2×C32—22no—
417C2×C32—22no—
418C2×C32—21no—
419C2×C32—21no—
420C2×C32—22no—
421C2×C32—21no—
422C2×C32—41no—
423C2×C32—42no—
424C2×C32#87411yescharacteristic
425C2×C32#90912yes—
426C2×C32#91512yes—
427C2×C32#91812yes—
428C2×C32#94011yescharacteristic
429C2×C32#94011yescharacteristic
430C3⋊D3—81no—
431C3×D3—242no—
432C3×D3—241no—
433C3×D3—241no—
434C2×Q8—31no—
435C2×Q8—91no—
436C2×Q8—271no—
437C2×D4—91no—
438Q16—182no—
439Q16—542no—
440SD16—182no—
441SD16—542no—
442C2×C8—31no—
443C2×C8—31no—
444C4⋊C4—542no—
445C22⋊C4—542no—
446C42—91no—
447C22×C3—21no—
448C22×C3—21no—
449C22×C3—21no—
450C22×C3—41no—
451C22×C3—362no—
452C22×C3—361no—
453C22×C3D3×D1211yescharacteristic
454C22×C3D3×D1211yescharacteristic
455C22×C3#381111yescharacteristic
456C2×D3—122no—
457C2×D3—122no—
458C2×D3—121no—
459C2×D3—122no—
460C2×D3—121no—
461C2×D3—242no—
462C3×C4—21no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—41no—
469C3×C4—41no—
470C3×C4—122no—
471C3×C4—121no—
472C3×C4—121no—
473C3×C4—182no—
474C3×C4—361no—
475C3×C4D3×D1211yescharacteristic
476C3×C4D3×D1211yescharacteristic
477C3×C4C2×#62611yescharacteristic
478C3×C4C2×#62611yescharacteristic
479C3×C4D3×#6711yescharacteristic
480C3×C4D3×#6711yescharacteristic
481Dic3—41no—
482Dic3—64no—
483Dic3—62no—
484Dic3—62no—
485Dic3—122no—
486Dic3—121no—
487Dic3—121no—
488Dic3—362no—
489Dic3—362no—
490Dic3—362no—
491Dic3—364no—
492Dic3—361no—
493Dic3—361no—
494Dic3—361no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#471511yescharacteristic, complemented
500C32#516711yescharacteristic, complemented
501C32#519011yescharacteristic, complemented
502C23—181no—
503Q8—61no—
504Q8—61no—
505Q8—181no—
506Q8—181no—
507Q8—541no—
508Q8—541no—
509D4—181no—
510D4—181no—
511C2×C4—61no—
512C2×C4—92no—
513C2×C4—181no—
514C2×C4—541no—
515C2×C4D3311yescharacteristic
516C8—32no—
517C8—32no—
518C2×C3—21no—
519C2×C3—21no—
520C2×C3—21no—
521C2×C3—22no—
522C2×C3—22no—
523C2×C3—22no—
524C2×C3—41no—
525C2×C3—42no—
526C2×C3—721no—
527C2×C3#6380012yes—
528C2×C3#6380112yes—
529C2×C3#6391111yescharacteristic
530C2×C3#6391111yescharacteristic
531C2×C3#6395112yes—
532C2×C3#6397911yescharacteristic
533D3—242no—
534D3—241no—
535D3—241no—
536C22—361no—
537C22—362no—
538C22#8895511yescharacteristic
539C4—121no—
540C4—182no—
541C4—361no—
542C4—1081no—
543C4D3×#62611yescharacteristic
544C4D3×#62611yescharacteristic
545C3—21no—
546C3—21no—
547C3—21no—
548C3—41no—
549C3#1059089711yescharacteristic, complemented
550C3#1059089911yescharacteristic, complemented
551C3#1059161111yescharacteristic, complemented
552C2—721no—
553C2#1172289612yes—
554C2#1172303011yescharacteristic
subgroupG/H
centerC22#88955
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×C4⋊(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
2. #11723455C2semidirect
3. C2×#88800C2semidirect
5. C2×C3×#3719C2semidirect
7. C3×#63555C2semidirect
40. C2×#3719D3semidirect
47. #63555D3semidirect
339. C33#497semidirect
499. C32#4715semidirect
500. C32#5167semidirect
501. C32#5190semidirect
549. C3#10590897semidirect
550. C3#10590899semidirect
551. C3#10591611semidirect

Maximal subgroups

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

subgroupindex
#117236772
#117234552
C2×#888002
C2×#887962
C2×C3×#37192
C2×C3×#37172
C3×#635552
#105916113
#105908993
#105908973