Finite Group Databasebeta

← #13162009

Subgroups — #13162009

Order 1728.

762 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C3×#64156C211yesmaximal, characteristic, complemented
2C2×C3×#3756C211yesmaximal, characteristic, complemented
3C2×C3×#3755C211yesmaximal, characteristic, complemented
4#11723677C211yesmaximal, characteristic, complemented
5C2×#88811C211yesmaximal, characteristic, complemented
6C2×#88810C211yesmaximal, characteristic
7#11722070C211yesmaximal, characteristic, complemented
8#10591610—31nomaximal
9#10591609—31nomaximal
10#10591606—31nomaximal
11C2×C33×Q8C2211yescharacteristic
12C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×C4⋊(C3⋊D3)C2211yescharacteristic
14C2×C3×#634C2211yescharacteristic
15C3×#3756—22no—
16C3×#3755—22no—
17C3×#3752—22no—
18C3×C4×C3⋊Dic3C2211yescharacteristic
19C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
20C2×C3×C3⋊(C3⋊C8)C2211yescharacteristic
21#88814—22no—
22#88811—22no—
23#88810—22no—
24#64156D311yescharacteristic, complemented
25C2×#3756D311yescharacteristic, complemented
26C2×#3755D311yescharacteristic, complemented
27C3×#947—31no—
28C3×#947—31no—
29C3×#947—61no—
30C2×C3×#216—31no—
31C2×C3×#216—31no—
32C2×C3×#216—61no—
33C2×C3×#215—31no—
34C2×C3×#215—31no—
35C2×C3×#215—61no—
36#63864—31no—
37#63863—31no—
38#63860—31no—
39C2×#3717—31no—
40C2×#3716—31no—
41C2×#3715—31no—
42C2×#3714—31no—
43C2×#3714—31no—
44C2×#3713—31no—
45#63553—31no—
46Dic3×C3⋊C8—31no—
47Dic3×C3⋊C8—31no—
48C22×C3×C3⋊D3—21no—
49C22×C3×C3⋊D3—21no—
50C33×Q8—21no—
51C33×Q8—21no—
52C2×C33×C4—21no—
53C2×C33×C4C2311yescharacteristic
54C2×C3×C3⋊Dic3—21no—
55C2×C3×C3⋊Dic3D412yes—
56C3×C4⋊(C3⋊D3)—21no—
57C3×C4⋊(C3⋊D3)—21no—
58C3×C4⋊(C3⋊D3)—21no—
59C3×C4⋊(C3⋊D3)—21no—
60C3×#634—21no—
61C3×#634—21no—
62C3×C3⋊(C3⋊C8)D412yes—
63C3×C3⋊(C3⋊C8)D412yes—
64#5191—91no—
65#5168—91no—
66#5059—91no—
67C2×C32×Q8—21no—
68C2×C32×Q8—21no—
69C2×C32×Q8—21no—
70C2×C32×Q8—41no—
71C2×C32×Q8C2×D311yescharacteristic
72C2×C32×Q8C2×D311yescharacteristic
73C2×C32×Q8C2×D311yescharacteristic
74C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
75C2×C4⋊(C3⋊D3)C2×D311yescharacteristic
76C2×#634C2×D311yescharacteristic
77C2×C3×D12—31no—
78C2×C3×D12—31no—
79C2×C3×D12—31no—
80C2×C3×D12—31no—
81C2×C3×D12—61no—
82C2×C3×D12—61no—
83C2×C3×Dic6—31no—
84C2×C3×Dic6—31no—
85C2×C3×Dic6—61no—
86#3756—22no—
87#3755—22no—
88#3752—22no—
89C4×C3⋊Dic3C2×D311yescharacteristic
90C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
91C2×C3⋊(C3⋊C8)C2×D311yescharacteristic
92C3×#216—62no—
93C3×#216—62no—
94C3×#216—64no—
95C3×#215—62no—
96C3×#215—62no—
97C3×#215—64no—
98C3×(C2×D3)⋊C4—62no—
99C3×(C2×D3)⋊C4—62no—
100C3×(C2×D3)⋊C4—64no—
101C3×C4×Dic3—31no—
102C3×C4×Dic3—31no—
103C3×C4×Dic3—61no—
104C2×C3×C3⋊C8—31no—
105C2×C3×C3⋊C8—31no—
106C2×C3×C3⋊C8—31no—
107C2×C3×C3⋊C8—31no—
108C2×C3×C3⋊C8—61no—
109C2×C3×C3⋊C8—61no—
110#3722—62no—
111(C2×D3)⋊Dic3—62no—
112(C2×D3)⋊Dic3—62no—
113#3717—62no—
114#3716—62no—
115#3715—62no—
116#3714—62no—
117#3714—62no—
118#3713—62no—
119C22×C33C2×D411yescharacteristic
120C2×C3×C3⋊D3—41no—
121C2×C3×C3⋊D3—41no—
122C2×C3×C3⋊D3—42no—
123C2×C3×C3⋊D3—42no—
124C33×C4—41no—
125C33×C4C2×D411yescharacteristic
126C33×C4C2×D411yescharacteristic
127C3×C3⋊Dic3—22no—
128C3×C3⋊Dic3—41no—
129C2×C3×Q16—91no—
130C2×C3×SD16—91no—
131C3×#124—91no—
132#947—31no—
133#947—31no—
134#947—61no—
135#947—91no—
136C2×#216—31no—
137C2×#216—31no—
138C2×#216—61no—
139C2×#215—31no—
140C2×#215—31no—
141C2×#215—61no—
142C2×#215—91no—
143#936—91no—
144C2×#214—91no—
145C2×#213—91no—
146C2×#213—91no—
147C2×D24—91no—
148C2×#204—91no—
149#875—91no—
150C8×Dic3—91no—
151C8×Dic3—91no—
152C4×C3⋊C8—91no—
153C22×C3⋊D3—21no—
154C22×C3⋊D3—21no—
155C22×C3×D3—62no—
156C22×C3×D3—61no—
157C22×C3×D3—61no—
158C22×C3×D3—61no—
159C22×C3×D3—61no—
160C22×C3×D3—121no—
161C32×Q8—21no—
162C32×Q8—21no—
163C32×Q8—22no—
164C32×Q8—22no—
165C32×Q8—21no—
166C32×Q8—21no—
167C32×Q8—21no—
168C32×Q8—21no—
169C32×Q8—41no—
170C32×Q8—41no—
171C32×Q8—41no—
172C32×Q8—41no—
173C32×Q8—42no—
174C32×Q8—42no—
175C2×C32×C4—21no—
176C2×C32×C4—22no—
177C2×C32×C4—21no—
178C2×C32×C4—21no—
179C2×C32×C4—21no—
180C2×C32×C4—21no—
181C2×C32×C4—21no—
182C2×C32×C4—41no—
183C2×C32×C4—41no—
184C2×C32×C4—41no—
185C2×C32×C4—42no—
186C2×C32×C4C22×D311yescharacteristic
187C2×C32×C4C22×D311yescharacteristic
188C2×C32×C4C22×D311yescharacteristic
189C2×C3⋊Dic3—21no—
190C2×C3⋊Dic3#6712yes—
191C4⋊(C3⋊D3)—21no—
192C4⋊(C3⋊D3)—21no—
193C4⋊(C3⋊D3)—21no—
194C4⋊(C3⋊D3)—21no—
195#634—21no—
196#634—21no—
197C2×C3×Dic3—32no—
198C2×C3×Dic3—32no—
199C2×C3×Dic3—62no—
200C2×C3×Dic3—61no—
201C2×C3×Dic3—61no—
202C2×C3×Dic3—121no—
203C3×D12—62no—
204C3×D12—61no—
205C3×D12—61no—
206C3×D12—61no—
207C3×D12—61no—
208C3×D12—61no—
209C3×D12—61no—
210C3×D12—61no—
211C3×D12—61no—
212C3×D12—121no—
213C3×D12—121no—
214C3×D12—121no—
215C3×Dic6—62no—
216C3×Dic6—61no—
217C3×Dic6—61no—
218C3×Dic6—61no—
219C3×Dic6—61no—
220C3×Dic6—121no—
221C3⋊(C3⋊C8)#6712yes—
222C3⋊(C3⋊C8)#6712yes—
223C3×C3⋊C8—32no—
224C3×C3⋊C8—32no—
225C3×C3⋊C8—32no—
226C3×C3⋊C8—32no—
227C3×C3⋊C8—62no—
228C3×C3⋊C8—62no—
229#495—271noSylow
230C2×C33#12711yescharacteristic
231C2×C33#13512yes—
232C3×C3⋊D3—81no—
233C3×C3⋊D3—81no—
234C2×C3×Q8—21no—
235C2×C3×Q8—21no—
236C2×C3×Q8—21no—
237C2×C3×Q8—41no—
238C2×C3×Q8—91no—
239C2×C3×Q8D3211yescharacteristic
240C2×C3×Q8D3211yescharacteristic
241C2×C3×Q8D3211yescharacteristic
242C2×C3×D4—91no—
243C2×C3×D4—91no—
244C2×D12—31no—
245C2×D12—31no—
246C2×D12—31no—
247C2×D12—31no—
248C2×D12—61no—
249C2×D12—61no—
250C2×Dic6—31no—
251C2×Dic6—31no—
252C2×Dic6—61no—
253C3×Q16—182no—
254C3×SD16—182no—
255C2×C3×C8—91no—
256C2×C3×C8—91no—
257C3×C22⋊C4—182no—
258C3×C42—91no—
259#217—182no—
260#216—62no—
261#216—62no—
262#216—64no—
263#215—62no—
264#215—62no—
265#215—64no—
266#215—182no—
267#214—182no—
268#213—182no—
269#213—182no—
270(C2×D3)⋊C4—62no—
271(C2×D3)⋊C4—62no—
272(C2×D3)⋊C4—64no—
273(C2×D3)⋊C4—182no—
274(C2×D3)⋊C4—182no—
275C4×Dic3—31no—
276C4×Dic3—31no—
277C4×Dic3—61no—
278C2×C3⋊C8—31no—
279C2×C3⋊C8—31no—
280C2×C3⋊C8—31no—
281C2×C3⋊C8—31no—
282C2×C3⋊C8—61no—
283C2×C3⋊C8—61no—
284D24—182no—
285#204—182no—
286C22×C32—21no—
287C22×C32—21no—
288C22×C32—21no—
289C22×C32—41no—
290C22×C32D3×D411yescharacteristic
291C22×C32D3×D411yescharacteristic
292C22×C32C2×#6711yescharacteristic
293C2×C3⋊D3—41no—
294C2×C3⋊D3—41no—
295C2×C3⋊D3—42no—
296C2×C3⋊D3—42no—
297C2×C3×D3—122no—
298C2×C3×D3—122no—
299C2×C3×D3—124no—
300C2×C3×D3—122no—
301C2×C3×D3—122no—
302C2×C3×D3—122no—
303C2×C3×D3—121no—
304C2×C3×D3—121no—
305C2×C3×D3—121no—
306C2×C3×D3—121no—
307C2×C3×D3—122no—
308C2×C3×D3—242no—
309C32×C4—21no—
310C32×C4—21no—
311C32×C4—21no—
312C32×C4—21no—
313C32×C4—21no—
314C32×C4—21no—
315C32×C4—41no—
316C32×C4—41no—
317C32×C4—42no—
318C32×C4—42no—
319C32×C4—42no—
320C32×C4—44no—
321C32×C4—41no—
322C32×C4—41no—
323C32×C4—41no—
324C32×C4D3×D411yescharacteristic
325C32×C4D3×D411yescharacteristic
326C32×C4D3×D411yescharacteristic
327C32×C4D3×D411yescharacteristic
328C32×C4C2×#6711yescharacteristic
329C32×C4C2×#6711yescharacteristic
330C3⋊Dic3—22no—
331C3⋊Dic3—41no—
332C3×Dic3—62no—
333C3×Dic3—62no—
334C3×Dic3—122no—
335C3×Dic3—122no—
336C3×Dic3—121no—
337C3×Dic3—121no—
338C2×Q16—91no—
339C2×SD16—91no—
340C2×SD16—271no—
341C2×D8—271no—
342#124—91no—
343#124—271no—
344C4×C8—271no—
345C33#49511yescharacteristic, complemented, Sylow
346C23×C3—181no—
347C23×C3—181no—
348C22×D3—62no—
349C22×D3—61no—
350C22×D3—61no—
351C22×D3—61no—
352C22×D3—61no—
353C22×D3—121no—
354C3×Q8—22no—
355C3×Q8—22no—
356C3×Q8—21no—
357C3×Q8—21no—
358C3×Q8—21no—
359C3×Q8—21no—
360C3×Q8—21no—
361C3×Q8—21no—
362C3×Q8—41no—
363C3×Q8—41no—
364C3×Q8—41no—
365C3×Q8—41no—
366C3×Q8—42no—
367C3×Q8—42no—
368C3×Q8—181no—
369C3×Q8—181no—
370C3×D4—181no—
371C3×D4—181no—
372C3×D4—181no—
373C3×D4—181no—
374C2×C3×C4—22no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—21no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—41no—
382C2×C3×C4—41no—
383C2×C3×C4—42no—
384C2×C3×C4—41no—
385C2×C3×C4—92no—
386C2×C3×C4—181no—
387C2×C3×C4C2×D3211yescharacteristic
388C2×C3×C4C2×D3211yescharacteristic
389C2×C3×C4C2×D3211yescharacteristic
390C2×Dic3—32no—
391C2×Dic3—32no—
392C2×Dic3—62no—
393C2×Dic3—61no—
394C2×Dic3—61no—
395C2×Dic3—121no—
396D12—62no—
397D12—61no—
398D12—61no—
399D12—61no—
400D12—61no—
401D12—61no—
402D12—61no—
403D12—61no—
404D12—61no—
405D12—121no—
406D12—121no—
407D12—121no—
408Dic6—62no—
409Dic6—61no—
410Dic6—61no—
411Dic6—61no—
412Dic6—61no—
413Dic6—121no—
414C3×C8—92no—
415C3×C8—92no—
416C3⋊C8—32no—
417C3⋊C8—32no—
418C3⋊C8—32no—
419C3⋊C8—32no—
420C3⋊C8—62no—
421C3⋊C8—62no—
422C2×C32—22no—
423C2×C32—22no—
424C2×C32—22no—
425C2×C32—21no—
426C2×C32—21no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—42no—
430C2×C32#91612yes—
431C2×C32#91912yes—
432C2×C32#94011yescharacteristic
433C2×C32#94011yescharacteristic
434C2×C32#94011yescharacteristic
435C2×C32#95012yes—
436C3⋊D3—81no—
437C3⋊D3—81no—
438C3×D3—242no—
439C3×D3—242no—
440C3×D3—241no—
441C3×D3—241no—
442C3×D3—241no—
443C3×D3—241no—
444C2×Q8—91no—
445C2×Q8#111311yescharacteristic
446C2×D4—91no—
447C2×D4—91no—
448Q16—182no—
449SD16—182no—
450SD16—542no—
451D8—542no—
452C2×C8—91no—
453C2×C8—91no—
454C22⋊C4—182no—
455C22⋊C4—542no—
456C42—91no—
457C22×C3—21no—
458C22×C3—21no—
459C22×C3—21no—
460C22×C3—41no—
461C22×C3—362no—
462C22×C3—362no—
463C22×C3—361no—
464C22×C3—361no—
465C22×C3D3×#6711yescharacteristic
466C22×C3D3×#6711yescharacteristic
467C22×C3#381111yescharacteristic
468C2×D3—122no—
469C2×D3—122no—
470C2×D3—124no—
471C2×D3—122no—
472C2×D3—122no—
473C2×D3—122no—
474C2×D3—122no—
475C2×D3—121no—
476C2×D3—121no—
477C2×D3—121no—
478C2×D3—121no—
479C2×D3—242no—
480C3×C4—21no—
481C3×C4—21no—
482C3×C4—21no—
483C3×C4—21no—
484C3×C4—21no—
485C3×C4—21no—
486C3×C4—42no—
487C3×C4—42no—
488C3×C4—42no—
489C3×C4—44no—
490C3×C4—41no—
491C3×C4—41no—
492C3×C4—41no—
493C3×C4—41no—
494C3×C4—41no—
495C3×C4—182no—
496C3×C4—361no—
497C3×C4D3×#6711yescharacteristic
498C3×C4D3×#6711yescharacteristic
499C3×C4D3×#6711yescharacteristic
500C3×C4D3×#6711yescharacteristic
501C3×C4#381111yescharacteristic
502C3×C4#381111yescharacteristic
503Dic3—62no—
504Dic3—62no—
505Dic3—122no—
506Dic3—122no—
507Dic3—121no—
508Dic3—121no—
509C32—21no—
510C32—21no—
511C32—21no—
512C32—41no—
513C32#505911yescharacteristic, complemented
514C32#516811yescharacteristic, complemented
515C32#519111yescharacteristic, complemented
516C23—181no—
517C23—181no—
518Q8—21no—
519Q8—21no—
520Q8—181no—
521Q8—181no—
522D4—181no—
523D4—181no—
524D4—181no—
525D4—181no—
526C2×C4—21no—
527C2×C4—92no—
528C2×C4—181no—
529C2×C4C2×#111311yescharacteristic
530C8—92no—
531C8—92no—
532C2×C3—22no—
533C2×C3—22no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3—21no—
538C2×C3—42no—
539C2×C3—41no—
540C2×C3—721no—
541C2×C3—721no—
542C2×C3#6394712yes—
543C2×C3#6394912yes—
544C2×C3#6395012yes—
545C2×C3#6397911yescharacteristic
546C2×C3#6397911yescharacteristic
547C2×C3#6397911yescharacteristic
548D3—242no—
549D3—242no—
550D3—241no—
551D3—241no—
552D3—241no—
553D3—241no—
554C22—362no—
555C22—362no—
556C22—361no—
557C22—361no—
558C22#8904911yescharacteristic
559C4—41no—
560C4—182no—
561C4—361no—
562C4#8904911yescharacteristic
563C4#8904911yescharacteristic
564C3—21no—
565C3—21no—
566C3—21no—
567C3—41no—
568C3#1059160611yescharacteristic, complemented
569C3#1059160911yescharacteristic, complemented
570C3#1059161011yescharacteristic, complemented
571C2—721no—
572C2—721no—
573C2#1172371312yes—
574C2#1172372811yescharacteristic
subgroupG/H
centerC22#89049
commutator, cosocleC2×C33×C4C23
FrattiniC2×C4C2×#1113
FittingC2×C33×Q8C22
socleC22×C33C2×D4
prime radicalC2×#88810C2
hypercenterC2×Q8#1113
hypocenterC33#495
LCSG ▷ C2×C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×Q8

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
1. C3×#64156C2semidirect
2. C2×C3×#3756C2semidirect
3. C2×C3×#3755C2semidirect
4. #11723677C2semidirect
5. C2×#88811C2semidirect
7. #11722070C2semidirect
24. #64156D3semidirect
25. C2×#3756D3semidirect
26. C2×#3755D3semidirect
345. C33#495semidirect
513. C32#5059semidirect
514. C32#5168semidirect
515. C32#5191semidirect
568. C3#10591606semidirect
569. C3#10591609semidirect
570. C3#10591610semidirect

Maximal subgroups

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

subgroupindex
C3×#641562
C2×C3×#37562
C2×C3×#37552
#117236772
C2×#888112
C2×#888102
#117220702
#105916103
#105916093
#105916063