Finite Group Databasebeta

← #13156603

Subgroups — #13156603

Order 1728.

722 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 722 conjugacy classes of subgroups — too many to draw. Only its 108 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, complemented
2C2×#88793C211yesmaximal, characteristic, complemented
3C2×#88792C211yesmaximal, characteristic, complemented
4C3×#63863C211yesmaximal, characteristic, complemented
5C2×C3×#3719C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7C3⋊C8×C3⋊Dic3C211yesmaximal, characteristic, complemented
8#10591609—31nomaximal
9#10591470—31nomaximal
10#10590869—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C2×C3⋊(C3⋊(C3⋊C8))C2211yescharacteristic
16C3×C4×C3⋊Dic3C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88814—22no—
19#88793—22no—
20#88792—22no—
21C3×(C2×D3)⋊Dic3—22no—
22C3×#3719—22no—
23C3×#3714—22no—
24C2×#3755—31no—
25C2×#3754—31no—
26C3×#947—31no—
27C2×C3×#216—31no—
28C3×#936—31no—
29C2×C3×#213—31no—
30C2×C3×Dic12—31no—
31C2×C3×D24—31no—
32#63864—31no—
33#63863—31no—
34#63863D311yescharacteristic, complemented
35#63860—31no—
36C2×#3719D311yescharacteristic, complemented
37C2×#3715—31no—
38C2×#3715—31no—
39C2×#3715—31no—
40C2×#3715—31no—
41C2×#3715—61no—
42C2×#3715—61no—
43C2×#3714D311yescharacteristic
44C8×C3⋊Dic3—31no—
45Dic3×C3⋊C8—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—61no—
48C22×C3×C3⋊D3—21no—
49C22×C32×D3—21no—
50C2×C33×C4C2311yescharacteristic
51C2×C3×C3⋊Dic3—21no—
52C2×C3×C3⋊Dic3D412yes—
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—
62C3⋊(C3⋊(C3⋊C8))D412yes—
63C32×C3⋊C8D412yes—
64#5191—91no—
65#5155—91no—
66#5059—91no—
67C2×C32×Q8—31no—
68C2×C32×D4—31no—
69C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
70C2×#634C2×D311yescharacteristic
71C2×C3×D12—21no—
72C2×C3×D12—31no—
73C2×C3×D12—31no—
74C2×C3×D12—61no—
75C2×C3×D12C2×D311yescharacteristic
76C2×C3×D12C2×D311yescharacteristic
77C2×C3×Dic6—21no—
78C2×C3×Dic6—31no—
79C2×C3×Dic6—31no—
80C2×C3×Dic6—61no—
81C2×C3×Dic6C2×D311yescharacteristic
82C2×C3×Dic6C2×D311yescharacteristic
83C2×C32×C8—31no—
84#3755—62no—
85#3754—62no—
86C4×C3⋊Dic3C2×D311yescharacteristic
87C2×C3⋊(C3⋊C8)—31no—
88C2×C3⋊(C3⋊C8)—31no—
89C2×C3⋊(C3⋊C8)—31no—
90C2×C3⋊(C3⋊C8)—61no—
91C2×C3⋊(C3⋊C8)—61no—
92C2×C3⋊(C3⋊C8)—61no—
93C2×C3⋊(C3⋊C8)—121no—
94C3×#217—62no—
95C3×#216—62no—
96C3×#213—62no—
97C3×(C2×D3)⋊C4—62no—
98C3×C4×Dic3—31no—
99C3×C4×Dic3—31no—
100C3×C4×Dic3—61no—
101C2×C3×C3⋊C8—21no—
102C2×C3×C3⋊C8C2×D311yescharacteristic
103C2×C3×C3⋊C8C2×D311yescharacteristic
104C3×Dic12—62no—
105C3×D24—62no—
106#3722—62no—
107(C2×D3)⋊Dic3—22no—
108(C2×D3)⋊Dic3—62no—
109(C2×D3)⋊Dic3—62no—
110#3719—22no—
111#3715—62no—
112#3715—62no—
113#3715—62no—
114#3715—62no—
115#3715—64no—
116#3715—64no—
117#3714—22no—
118C22×C33C2×D411yescharacteristic
119C2×C3×C3⋊D3—41no—
120C2×C3×C3⋊D3—42no—
121C2×C32×D3—41no—
122C2×C32×D3—42no—
123C33×C4C2×D411yescharacteristic
124C33×C4C2×D411yescharacteristic
125C3×C3⋊Dic3—22no—
126C3×C3⋊Dic3—41no—
127C32×Dic3—41no—
128C2×C3×Q16—91no—
129C2×C3×D8—91no—
130C3×#124—91no—
131#947—31no—
132#947—91no—
133C2×#216—31no—
134C2×#215—91no—
135C2×#215—91no—
136C2×#215—91no—
137C2×#215—181no—
138#936—31no—
139#936—91no—
140C2×#214—91no—
141C2×#214—91no—
142C2×#214—91no—
143C2×#214—181no—
144C2×#213—31no—
145C2×Dic12—31no—
146C2×D24—31no—
147#875—91no—
148C8×Dic3—91no—
149C8×Dic3—91no—
150C8×Dic3—181no—
151C4×C3⋊C8—91no—
152C23×C32—61no—
153C22×C3⋊D3—21no—
154C22×C3×D3—21no—
155C22×C3×D3—21no—
156C22×C3×D3—41no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—121no—
160C32×Q8—61no—
161C32×Q8—61no—
162C32×D4—61no—
163C32×D4—61no—
164C2×C32×C4—21no—
165C2×C32×C4—21no—
166C2×C32×C4—21no—
167C2×C32×C4—41no—
168C2×C32×C4—61no—
169C2×C32×C4C22×D311yescharacteristic
170C2×C32×C4C22×D311yescharacteristic
171C2×C32×C4C22×D311yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3#6712yes—
174C4⋊(C3⋊D3)—21no—
175C4⋊(C3⋊D3)—21no—
176#634—21no—
177#634—21no—
178C2×C3×Dic3—21no—
179C2×C3×Dic3—21no—
180C2×C3×Dic3—32no—
181C2×C3×Dic3—32no—
182C2×C3×Dic3—41no—
183C2×C3×Dic3—62no—
184C2×C3×Dic3—61no—
185C2×C3×Dic3—61no—
186C2×C3×Dic3—121no—
187C3×D12—22no—
188C3×D12—21no—
189C3×D12—21no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—41no—
193C3×D12—62no—
194C3×D12—61no—
195C3×D12—61no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—121no—
199C3×Dic6—22no—
200C3×Dic6—21no—
201C3×Dic6—21no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—41no—
205C3×Dic6—62no—
206C3×Dic6—61no—
207C3×Dic6—61no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—121no—
211C32×C8—32no—
212C3⋊(C3⋊C8)—32no—
213C3⋊(C3⋊C8)—32no—
214C3⋊(C3⋊C8)—32no—
215C3⋊(C3⋊C8)—62no—
216C3⋊(C3⋊C8)—62no—
217C3⋊(C3⋊C8)—62no—
218C3⋊(C3⋊C8)—122no—
219C3×C3⋊C8—22no—
220C3×C3⋊C8#6712yes—
221C3×C3⋊C8#6712yes—
222#495—271noSylow
223C2×C33#12711yescharacteristic
224C2×C33#13512yes—
225C3×C3⋊D3—81no—
226C32×D3—81no—
227C2×C3×Q8—31no—
228C2×C3×Q8—31no—
229C2×C3×Q8—61no—
230C2×C3×Q8—91no—
231C2×C3×D4—31no—
232C2×C3×D4—31no—
233C2×C3×D4—61no—
234C2×C3×D4—91no—
235C2×D12—31no—
236C2×D12—31no—
237C2×D12—61no—
238C2×D12D3211yescharacteristic
239C2×Dic6—31no—
240C2×Dic6—31no—
241C2×Dic6—61no—
242C2×Dic6D3211yescharacteristic
243C3×Q16—182no—
244C3×D8—182no—
245C2×C3×C8—31no—
246C2×C3×C8—31no—
247C2×C3×C8—61no—
248C3×C22⋊C4—182no—
249C3×C42—91no—
250#217—62no—
251#217—182no—
252#216—62no—
253#215—182no—
254#215—182no—
255#215—182no—
256#215—184no—
257#214—182no—
258#214—182no—
259#214—182no—
260#214—184no—
261#213—62no—
262(C2×D3)⋊C4—62no—
263(C2×D3)⋊C4—182no—
264(C2×D3)⋊C4—182no—
265C4×Dic3—31no—
266C4×Dic3—31no—
267C4×Dic3—61no—
268C2×C3⋊C8—91no—
269C2×C3⋊C8—91no—
270C2×C3⋊C8—91no—
271C2×C3⋊C8—181no—
272C2×C3⋊C8—181no—
273C2×C3⋊C8—181no—
274C2×C3⋊C8—361no—
275C2×C3⋊C8D3211yescharacteristic
276Dic12—62no—
277D24—62no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32—121no—
283C22×C32—122no—
284C22×C32D3×D411yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32C2×#6711yescharacteristic
287C2×C3⋊D3—41no—
288C2×C3⋊D3—42no—
289C2×C3×D3—42no—
290C2×C3×D3—42no—
291C2×C3×D3—41no—
292C2×C3×D3—41no—
293C2×C3×D3—42no—
294C2×C3×D3—82no—
295C2×C3×D3—122no—
296C2×C3×D3—122no—
297C2×C3×D3—121no—
298C2×C3×D3—121no—
299C2×C3×D3—122no—
300C2×C3×D3—242no—
301C32×C4—21no—
302C32×C4—21no—
303C32×C4—21no—
304C32×C4—21no—
305C32×C4—21no—
306C32×C4—21no—
307C32×C4—41no—
308C32×C4—41no—
309C32×C4—121no—
310C32×C4D3×D411yescharacteristic
311C32×C4D3×D411yescharacteristic
312C32×C4D3×D411yescharacteristic
313C32×C4D3×D411yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C3⋊Dic3—22no—
317C3⋊Dic3—41no—
318C3×Dic3—42no—
319C3×Dic3—41no—
320C3×Dic3—41no—
321C3×Dic3—64no—
322C3×Dic3—62no—
323C3×Dic3—62no—
324C3×Dic3—122no—
325C3×Dic3—121no—
326C3×Dic3—121no—
327C2×Q16—91no—
328C2×SD16—271no—
329C2×SD16—271no—
330C2×D8—91no—
331#124—91no—
332#124—271no—
333C4×C8—271no—
334C33#49511yescharacteristic, complemented, Sylow
335C23×C3—61no—
336C23×C3—61no—
337C23×C3—121no—
338C23×C3—181no—
339C22×D3—21no—
340C22×D3—61no—
341C22×D3—61no—
342C22×D3—121no—
343C3×Q8—62no—
344C3×Q8—61no—
345C3×Q8—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—121no—
349C3×Q8—181no—
350C3×Q8—181no—
351C3×D4—62no—
352C3×D4—61no—
353C3×D4—61no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—121no—
357C3×D4—181no—
358C3×D4—181no—
359C2×C3×C4—21no—
360C2×C3×C4—21no—
361C2×C3×C4—21no—
362C2×C3×C4—41no—
363C2×C3×C4—61no—
364C2×C3×C4—61no—
365C2×C3×C4—92no—
366C2×C3×C4—121no—
367C2×C3×C4—181no—
368C2×C3×C4C2×D3211yescharacteristic
369C2×C3×C4C2×D3211yescharacteristic
370C2×C3×C4C2×D3211yescharacteristic
371C2×Dic3—21no—
372C2×Dic3—32no—
373C2×Dic3—32no—
374C2×Dic3—62no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—121no—
378D12—21no—
379D12—21no—
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—
394C3×C8—32no—
395C3×C8—32no—
396C3×C8—62no—
397C3⋊C8—92no—
398C3⋊C8—92no—
399C3⋊C8—92no—
400C3⋊C8—182no—
401C3⋊C8—182no—
402C3⋊C8—182no—
403C3⋊C8—362no—
404C3⋊C8#62512yes—
405C2×C32—22no—
406C2×C32—21no—
407C2×C32—22no—
408C2×C32—22no—
409C2×C32—21no—
410C2×C32—21no—
411C2×C32—41no—
412C2×C32—42no—
413C2×C32—241no—
414C2×C32#91212yes—
415C2×C32#91912yes—
416C2×C32#94011yescharacteristic
417C2×C32#94011yescharacteristic
418C2×C32#94011yescharacteristic
419C2×C32#95012yes—
420C3⋊D3—81no—
421C3×D3—82no—
422C3×D3—81no—
423C3×D3—81no—
424C3×D3—242no—
425C3×D3—241no—
426C3×D3—241no—
427C2×Q8—31no—
428C2×Q8—91no—
429C2×D4—31no—
430C2×D4—91no—
431Q16—182no—
432SD16—542no—
433SD16—542no—
434D8—182no—
435C2×C8—31no—
436C2×C8—271no—
437C22⋊C4—182no—
438C22⋊C4—542no—
439C42—91no—
440C22×C3—21no—
441C22×C3—21no—
442C22×C3—21no—
443C22×C3—41no—
444C22×C3—122no—
445C22×C3—122no—
446C22×C3—121no—
447C22×C3—122no—
448C22×C3—121no—
449C22×C3—242no—
450C22×C3—362no—
451C22×C3—361no—
452C22×C3#380211yescharacteristic
453C22×C3D3×#6711yescharacteristic
454C22×C3D3×#6711yescharacteristic
455C2×D3—42no—
456C2×D3—41no—
457C2×D3—122no—
458C2×D3—122no—
459C2×D3—121no—
460C2×D3—122no—
461C2×D3—121no—
462C2×D3—242no—
463C3×C4—21no—
464C3×C4—21no—
465C3×C4—21no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—41no—
470C3×C4—41no—
471C3×C4—122no—
472C3×C4—121no—
473C3×C4—121no—
474C3×C4—182no—
475C3×C4—361no—
476C3×C4#380211yescharacteristic
477C3×C4C2×#62511yescharacteristic
478C3×C4D3×#6711yescharacteristic
479C3×C4D3×#6711yescharacteristic
480C3×C4#381111yescharacteristic
481C3×C4#381111yescharacteristic
482Dic3—41no—
483Dic3—64no—
484Dic3—62no—
485Dic3—62no—
486Dic3—122no—
487Dic3—121no—
488Dic3—121no—
489C32—21no—
490C32—21no—
491C32—21no—
492C32—41no—
493C32#505911yescharacteristic, complemented
494C32#515511yescharacteristic, complemented
495C32#519111yescharacteristic, complemented
496C23—61no—
497C23—181no—
498Q8—61no—
499Q8—61no—
500Q8—181no—
501Q8—181no—
502D4—61no—
503D4—61no—
504D4—181no—
505D4—181no—
506C2×C4—61no—
507C2×C4—92no—
508C2×C4—181no—
509C2×C4D3311yescharacteristic
510C8—32no—
511C8—272no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—22no—
516C2×C3—22no—
517C2×C3—22no—
518C2×C3—41no—
519C2×C3—42no—
520C2×C3—242no—
521C2×C3—241no—
522C2×C3—241no—
523C2×C3—721no—
524C2×C3#6381012yes—
525C2×C3#6391411yescharacteristic
526C2×C3#6393312yes—
527C2×C3#6394912yes—
528C2×C3#6397911yescharacteristic
529C2×C3#6397911yescharacteristic
530D3—81no—
531D3—242no—
532D3—241no—
533D3—241no—
534C22—121no—
535C22—122no—
536C22—361no—
537C22—362no—
538C22#8895311yescharacteristic
539C4—121no—
540C4—182no—
541C4—361no—
542C4D3×#62511yescharacteristic
543C4#8895411yescharacteristic
544C3—21no—
545C3—21no—
546C3—21no—
547C3—41no—
548C3#1059086911yescharacteristic, complemented
549C3#1059147011yescharacteristic, complemented
550C3#1059160911yescharacteristic, complemented
551C2—241no—
552C2—721no—
553C2#1172286012yes—
554C2#1172302911yescharacteristic
subgroupG/H
centerC22#88953
commutator, Fitting, cosocleC2×C33×C4C23
Frattini, hypercenterC2×C4D33
socleC22×C33C2×D4
prime radicalC2×C3×#3714C2
hypocenterC33#495
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. #11723677C2semidirect
2. C2×#88793C2semidirect
3. C2×#88792C2semidirect
4. C3×#63863C2semidirect
5. C2×C3×#3719C2semidirect
7. C3⋊C8×C3⋊Dic3C2semidirect
34. #63863D3semidirect
36. C2×#3719D3semidirect
334. C33#495semidirect
493. C32#5059semidirect
494. C32#5155semidirect
495. C32#5191semidirect
548. C3#10590869semidirect
549. C3#10591470semidirect
550. C3#10591609semidirect

Maximal subgroups

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

subgroupindex
#117236772
C2×#887932
C2×#887922
C3×#638632
C2×C3×#37192
C2×C3×#37142
C3⋊C8×C3⋊Dic32
#105916093
#105914703
#105908693