Finite Group Databasebeta

← #13156604

Subgroups — #13156604

Order 1728.

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

normal subgroups only

This group has 746 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
1C2×#88811C211yesmaximal, characteristic, complemented
2#11723436C211yesmaximal, characteristic, complemented
3#11723434C211yesmaximal, characteristic, complemented
4C2×C3×#3719C211yesmaximal, characteristic, complemented
5C2×C3×#3715C211yesmaximal, characteristic, complemented
6C2×C3×#3714C211yesmaximal, characteristic
7#11722018C211yesmaximal, characteristic, complemented
8#10591610—31nomaximal
9#10591474—31nomaximal
10#10590869—31nomaximal
11C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
12C2×C3×#634C2211yescharacteristic
13C2×C32×D12C2211yescharacteristic
14C2×C32×Dic6C2211yescharacteristic
15C4×C3⋊(C3⋊Dic3)C2211yescharacteristic
16C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
17C2×C32×C3⋊C8C2211yescharacteristic
18#88811—22no—
19#88803—22no—
20(C2×D3)⋊(C3⋊Dic3)—22no—
21C3×#3719—22no—
22C3×#3715—22no—
23C3×#3714—22no—
24#64156—31no—
25#64145—31no—
26C2×C3×#216—31no—
27C2×C3×#215—31no—
28C2×C3×#214—31no—
29C2×C3×#213—31no—
30C2×C3×Dic12—31no—
31C2×C3×D24—31no—
32#63863—31no—
33#63863—31no—
34#63863—31no—
35#63863—31no—
36#63863—61no—
37#63863—61no—
38C2×#3719D311yescharacteristic, complemented
39C2×#3717—31no—
40C2×#3716—31no—
41C2×#3715—31no—
42C2×#3715D311yescharacteristic, complemented
43C2×#3714D311yescharacteristic
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—
62C3×C3⋊(C3⋊C8)D412yes—
63C32×C3⋊C8D412yes—
64#5191—91no—
65#5168—91no—
66#5155—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#3757—62no—
85#3752—62no—
86C4×C3⋊Dic3—31no—
87C4×C3⋊Dic3—31no—
88C4×C3⋊Dic3—31no—
89C4×C3⋊Dic3—61no—
90C4×C3⋊Dic3—61no—
91C4×C3⋊Dic3—61no—
92C4×C3⋊Dic3—121no—
93C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
94C3×#216—62no—
95C3×#215—62no—
96C3×#214—62no—
97C3×#213—62no—
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—
105C3×D24—62no—
106(C2×D3)⋊Dic3—62no—
107(C2×D3)⋊Dic3—62no—
108(C2×D3)⋊Dic3—64no—
109(C2×D3)⋊Dic3—62no—
110(C2×D3)⋊Dic3—62no—
111(C2×D3)⋊Dic3—64no—
112#3719—22no—
113#3717—62no—
114#3716—62no—
115#3715—22no—
116#3715—62no—
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×SD16—91no—
130C2×C3×D8—91no—
131#947—91no—
132#947—91no—
133#947—91no—
134#947—181no—
135C2×#216—31no—
136C2×#215—31no—
137C2×#215—91no—
138#936—91no—
139#936—91no—
140#936—91no—
141#936—181no—
142C2×#214—31no—
143C2×#214—91no—
144C2×#213—31no—
145C2×Dic12—31no—
146C2×D24—31no—
147C2×#204—91no—
148C8×Dic3—91no—
149C8×Dic3—91no—
150C8×Dic3—91no—
151C8×Dic3—181no—
152C23×C32—61no—
153C22×C3⋊D3—21no—
154C22×C3×D3—22no—
155C22×C3×D3—21no—
156C22×C3×D3—21no—
157C22×C3×D3—62no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
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—32no—
174C2×C3⋊Dic3—32no—
175C2×C3⋊Dic3—32no—
176C2×C3⋊Dic3—62no—
177C2×C3⋊Dic3—62no—
178C2×C3⋊Dic3—62no—
179C2×C3⋊Dic3—122no—
180C4⋊(C3⋊D3)—21no—
181C4⋊(C3⋊D3)—21no—
182#634—21no—
183#634—21no—
184C2×C3×Dic3—22no—
185C2×C3×Dic3—21no—
186C2×C3×Dic3—21no—
187C2×C3×Dic3—62no—
188C2×C3×Dic3—61no—
189C2×C3×Dic3—61no—
190C3×D12—21no—
191C3×D12—21no—
192C3×D12—21no—
193C3×D12—21no—
194C3×D12—41no—
195C3×D12—41no—
196C3×D12—61no—
197C3×D12—61no—
198C3×D12—61no—
199C3×D12—61no—
200C3×D12—121no—
201C3×D12—121no—
202C3×Dic6—21no—
203C3×Dic6—21no—
204C3×Dic6—21no—
205C3×Dic6—21no—
206C3×Dic6—41no—
207C3×Dic6—41no—
208C3×Dic6—61no—
209C3×Dic6—61no—
210C3×Dic6—61no—
211C3×Dic6—61no—
212C3×Dic6—121no—
213C3×Dic6—121no—
214C32×C8—32no—
215C3⋊(C3⋊C8)#6712yes—
216C3×C3⋊C8—22no—
217C3×C3⋊C8—32no—
218C3×C3⋊C8—32no—
219C3×C3⋊C8—62no—
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×SD16—182no—
245C3×D8—182no—
246C2×C3×C8—31no—
247C2×C3×C8—31no—
248C2×C3×C8—61no—
249C2×C3×C8—91no—
250#217—182no—
251#217—182no—
252#217—182no—
253#217—184no—
254#216—62no—
255#215—62no—
256#215—182no—
257#214—62no—
258#214—182no—
259#213—62no—
260(C2×D3)⋊C4—182no—
261(C2×D3)⋊C4—182no—
262(C2×D3)⋊C4—182no—
263(C2×D3)⋊C4—184no—
264C4×Dic3—91no—
265C4×Dic3—91no—
266C4×Dic3—91no—
267C4×Dic3—181no—
268C4×Dic3—181no—
269C4×Dic3—181no—
270C4×Dic3—361no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—31no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8D3211yescharacteristic
275Dic12—62no—
276D24—62no—
277#204—182no—
278C22×C32—21no—
279C22×C32—21no—
280C22×C32—21no—
281C22×C32—41no—
282C22×C32—122no—
283C22×C32—121no—
284C22×C32D3×D411yescharacteristic
285C22×C32D3×D411yescharacteristic
286C22×C32D3×D411yescharacteristic
287C2×C3⋊D3—42no—
288C2×C3⋊D3—41no—
289C2×C3×D3—42no—
290C2×C3×D3—44no—
291C2×C3×D3—42no—
292C2×C3×D3—42no—
293C2×C3×D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—122no—
296C2×C3×D3—124no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—121no—
300C2×C3×D3—121no—
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×C4C2×#6711yescharacteristic
314C32×C4C2×#6711yescharacteristic
315C32×C4C2×#6711yescharacteristic
316C3⋊Dic3—41no—
317C3⋊Dic3—62no—
318C3⋊Dic3—64no—
319C3⋊Dic3—62no—
320C3⋊Dic3—62no—
321C3⋊Dic3—122no—
322C3⋊Dic3—122no—
323C3⋊Dic3—124no—
324C3×Dic3—42no—
325C3×Dic3—41no—
326C3×Dic3—41no—
327C3×Dic3—122no—
328C3×Dic3—121no—
329C3×Dic3—121no—
330C2×Q16—91no—
331C2×SD16—91no—
332C2×SD16—271no—
333C2×D8—91no—
334#124—271no—
335#124—271no—
336C4×C8—271no—
337C33#49511yescharacteristic, complemented, Sylow
338C23×C3—62no—
339C23×C3—61no—
340C23×C3—61no—
341C23×C3—181no—
342C22×D3—21no—
343C22×D3—62no—
344C22×D3—61no—
345C22×D3—61no—
346C3×Q8—61no—
347C3×Q8—61no—
348C3×Q8—61no—
349C3×Q8—61no—
350C3×Q8—121no—
351C3×Q8—121no—
352C3×Q8—181no—
353C3×Q8—181no—
354C3×D4—61no—
355C3×D4—61no—
356C3×D4—61no—
357C3×D4—61no—
358C3×D4—121no—
359C3×D4—121no—
360C3×D4—181no—
361C3×D4—181no—
362C2×C3×C4—21no—
363C2×C3×C4—21no—
364C2×C3×C4—21no—
365C2×C3×C4—41no—
366C2×C3×C4—62no—
367C2×C3×C4—61no—
368C2×C3×C4—61no—
369C2×C3×C4—181no—
370C2×C3×C4C2×D3211yescharacteristic
371C2×C3×C4C2×D3211yescharacteristic
372C2×C3×C4C2×D3211yescharacteristic
373C2×Dic3—21no—
374C2×Dic3—62no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377C2×Dic3—92no—
378C2×Dic3—92no—
379C2×Dic3—92no—
380C2×Dic3—182no—
381C2×Dic3—182no—
382C2×Dic3—182no—
383C2×Dic3—362no—
384D12—21no—
385D12—21no—
386D12—61no—
387D12—61no—
388D12—61no—
389D12—61no—
390D12—121no—
391D12—121no—
392Dic6—21no—
393Dic6—21no—
394Dic6—61no—
395Dic6—61no—
396Dic6—61no—
397Dic6—61no—
398Dic6—121no—
399Dic6—121no—
400C3×C8—32no—
401C3×C8—32no—
402C3×C8—62no—
403C3×C8—92no—
404C3⋊C8—32no—
405C3⋊C8—32no—
406C3⋊C8—62no—
407C3⋊C8#62512yes—
408C2×C32—22no—
409C2×C32—21no—
410C2×C32—22no—
411C2×C32—22no—
412C2×C32—21no—
413C2×C32—21no—
414C2×C32—41no—
415C2×C32—42no—
416C2×C32—241no—
417C2×C32#91212yes—
418C2×C32#91612yes—
419C2×C32#91912yes—
420C2×C32#94011yescharacteristic
421C2×C32#94011yescharacteristic
422C2×C32#94011yescharacteristic
423C3⋊D3—81no—
424C3×D3—82no—
425C3×D3—81no—
426C3×D3—81no—
427C3×D3—242no—
428C3×D3—241no—
429C3×D3—241no—
430C2×Q8—31no—
431C2×Q8—91no—
432C2×D4—31no—
433C2×D4—91no—
434Q16—182no—
435SD16—182no—
436SD16—542no—
437D8—182no—
438C2×C8—31no—
439C2×C8—91no—
440C22⋊C4—542no—
441C22⋊C4—542no—
442C42—271no—
443C22×C3—21no—
444C22×C3—21no—
445C22×C3—21no—
446C22×C3—41no—
447C22×C3—122no—
448C22×C3—124no—
449C22×C3—122no—
450C22×C3—121no—
451C22×C3—122no—
452C22×C3—121no—
453C22×C3—362no—
454C22×C3—361no—
455C22×C3#380211yescharacteristic
456C22×C3#381111yescharacteristic
457C22×C3#381111yescharacteristic
458C2×D3—42no—
459C2×D3—41no—
460C2×D3—122no—
461C2×D3—124no—
462C2×D3—122no—
463C2×D3—121no—
464C2×D3—122no—
465C2×D3—121no—
466C3×C4—21no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—21no—
472C3×C4—41no—
473C3×C4—41no—
474C3×C4—122no—
475C3×C4—121no—
476C3×C4—121no—
477C3×C4—361no—
478C3×C4#380211yescharacteristic
479C3×C4C2×#62511yescharacteristic
480C3×C4D3×#6711yescharacteristic
481C3×C4D3×#6711yescharacteristic
482C3×C4D3×#6711yescharacteristic
483C3×C4D3×#6711yescharacteristic
484Dic3—41no—
485Dic3—122no—
486Dic3—121no—
487Dic3—121no—
488Dic3—184no—
489Dic3—182no—
490Dic3—182no—
491Dic3—182no—
492Dic3—362no—
493Dic3—362no—
494Dic3—364no—
495C32—21no—
496C32—21no—
497C32—21no—
498C32—41no—
499C32#515511yescharacteristic, complemented
500C32#516811yescharacteristic, complemented
501C32#519111yescharacteristic, complemented
502C23—61no—
503C23—181no—
504Q8—61no—
505Q8—61no—
506Q8—181no—
507Q8—181no—
508D4—61no—
509D4—61no—
510D4—181no—
511D4—181no—
512C2×C4—61no—
513C2×C4—181no—
514C2×C4—272no—
515C2×C4D3311yescharacteristic
516C8—32no—
517C8—92no—
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—242no—
527C2×C3—241no—
528C2×C3—241no—
529C2×C3—721no—
530C2×C3#6381012yes—
531C2×C3#6391411yescharacteristic
532C2×C3#6393612yes—
533C2×C3#6395012yes—
534C2×C3#6397911yescharacteristic
535C2×C3#6397911yescharacteristic
536D3—81no—
537D3—242no—
538D3—241no—
539D3—241no—
540C22—121no—
541C22—122no—
542C22—361no—
543C22—362no—
544C22#8895411yescharacteristic
545C4—121no—
546C4—361no—
547C4—542no—
548C4D3×#62511yescharacteristic
549C4#8895311yescharacteristic
550C3—21no—
551C3—21no—
552C3—21no—
553C3—41no—
554C3#1059086911yescharacteristic, complemented
555C3#1059147411yescharacteristic, complemented
556C3#1059161011yescharacteristic, complemented
557C2—241no—
558C2—721no—
559C2#1172286212yes—
560C2#1172302911yescharacteristic
subgroupG/H
centerC22#88954
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. C2×#88811C2semidirect
2. #11723436C2semidirect
3. #11723434C2semidirect
4. C2×C3×#3719C2semidirect
5. C2×C3×#3715C2semidirect
7. #11722018C2semidirect
38. C2×#3719D3semidirect
42. C2×#3715D3semidirect
337. C33#495semidirect
499. C32#5155semidirect
500. C32#5168semidirect
501. C32#5191semidirect
554. C3#10590869semidirect
555. C3#10591474semidirect
556. C3#10591610semidirect

Maximal subgroups

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

subgroupindex
C2×#888112
#117234362
#117234342
C2×C3×#37192
C2×C3×#37152
C2×C3×#37142
#117220182
#105916103
#105914743
#105908693