Finite Group Databasebeta

← #13178271

Subgroups — #13178271

Order 1728.

998 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64088C212yesmaximal
2C3×#64086C211yesmaximal, characteristic
3#11723721C211yesmaximal, characteristic
4#11723721C211yesmaximal, characteristic
5#11723716C212yesmaximal
6C4⋊#6296C211yesmaximal, characteristic, complemented
7C4×#6296C211yesmaximal, characteristic, complemented
8#11723679C211yesmaximal, characteristic, complemented
9#11723679C212yesmaximal, complemented
10#11723669C212yesmaximal, complemented
11#11723665C211yesmaximal, characteristic, complemented
12#10594249—31nomaximal
13#10594247—32nomaximal
14C22×C3×C3⋊Dic3C2211yescharacteristic
15C2×#6296C2211yescharacteristic
16C2×#6296C2211yescharacteristic
17C33×C22⋊C4C2211yescharacteristic
18C3×#3757C2211yescharacteristic
19C3×#3757C2212yes—
20C3×#3757C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×#3750C2211yescharacteristic
23C3×#3750C2211yescharacteristic
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2211yescharacteristic
27C3×C4×C3⋊Dic3C2211yescharacteristic
28C3×C4×C3⋊Dic3C2212yes—
29#88815C2211yescharacteristic
30#88815C2212yes—
31#88815C2212yes—
32#88815C2212yes—
33#88815C2212yes—
34#88815C2211yescharacteristic
35#88815C2212yes—
36#88813C2212yes—
37#64088D312yes—
38#64086D311yescharacteristic
39C3×#879—32no—
40C3×#879—32no—
41C3×#879—62no—
42C3×#877—32no—
43C3×#877—61no—
44#63982—31no—
45#63982—31no—
46#63964—32no—
47#63964—32no—
48#63956—32no—
49#63956—32no—
50#63956—32no—
51#63919—31no—
52#63918—32no—
53C4×#627—32no—
54C4×#627—31no—
55#63870—32no—
56#63870—31no—
57#63867—32no—
58#63867—32no—
59#63867—32no—
60#63847—32no—
61#63846—32no—
62#63845—32no—
63#63838—31no—
64#63837—32no—
65C23×C33C2311yescharacteristic
66C2×C33×C4C2311yescharacteristic
67C2×C33×C4C2311yescharacteristic
68C2×C3×C3⋊Dic3—22no—
69C2×C3×C3⋊Dic3C2311yescharacteristic
70C2×C3×C3⋊Dic3C2311yescharacteristic
71C2×C3×C3⋊Dic3C2312yes—
72C2×C3×C3⋊Dic3C2312yes—
73C2×C3×C3⋊Dic3C2312yes—
74C2×C3×C3⋊Dic3C2311yescharacteristic
75C2×C3×C3⋊Dic3C2312yes—
76C2×C3×C3⋊Dic3C2311yescharacteristic
77#6296—22no—
78#6296—22no—
79#5701—92no—
80#5518—91no—
81C22×C3⋊Dic3C2×D311yescharacteristic
82C22×C3×Dic3—32no—
83C22×C3×Dic3—61no—
84C2×#627—32no—
85C2×#627—31no—
86C2×#627—32no—
87C2×#627—31no—
88C32×C22⋊C4—21no—
89C32×C22⋊C4—22no—
90C32×C22⋊C4—41no—
91C32×C22⋊C4C2×D312yes—
92C32×C22⋊C4C2×D311yescharacteristic
93#3757C2×D311yescharacteristic
94#3757C2×D312yes—
95#3757C2×D312yes—
96C4⋊(C3⋊Dic3)C2×D312yes—
97#3750C2×D311yescharacteristic
98#3750C2×D311yescharacteristic
99#3750C2×D312yes—
100#3750C2×D312yes—
101C4×C3⋊Dic3C2×D311yescharacteristic
102C4×C3⋊Dic3C2×D311yescharacteristic
103C4×C3⋊Dic3C2×D312yes—
104C3×#217—32no—
105C3×#217—32no—
106C3×#217—32no—
107C3×#217—32no—
108C3×#217—32no—
109C3×#217—61no—
110C3×#217—62no—
111C3×#217—62no—
112C3×C4⋊Dic3—32no—
113C3×C4⋊Dic3—32no—
114C3×C4⋊Dic3—62no—
115C3×Dic3⋊C4—32no—
116C3×Dic3⋊C4—32no—
117C3×Dic3⋊C4—32no—
118C3×Dic3⋊C4—32no—
119C3×Dic3⋊C4—32no—
120C3×Dic3⋊C4—32no—
121C3×Dic3⋊C4—61no—
122C3×Dic3⋊C4—61no—
123C3×Dic3⋊C4—62no—
124C3×Dic3⋊C4—62no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—32no—
128C3×C4×Dic3—32no—
129C3×C4×Dic3—61no—
130C3×C4×Dic3—61no—
131C3×C4×Dic3—62no—
132#3724—31no—
133#3724—32no—
134#3724—32no—
135#3724—32no—
136#3724—32no—
137#3724—32no—
138#3724—31no—
139Dic3⋊Dic3—32no—
140Dic3⋊Dic3—32no—
141Dic3⋊Dic3—32no—
142Dic3⋊Dic3—32no—
143Dic3⋊Dic3—32no—
144Dic3⋊Dic3—32no—
145Dic3⋊Dic3—32no—
146Dic3⋊Dic3—32no—
147Dic3⋊Dic3—32no—
148Dic3⋊Dic3—32no—
149Dic3⋊Dic3—32no—
150Dic3⋊Dic3—32no—
151Dic32—32no—
152Dic32—32no—
153Dic32—32no—
154C22×C33—21no—
155C22×C33—21no—
156C22×C33—21no—
157C22×C33C2411yescharacteristic
158C33×C4—21no—
159C33×C4—21no—
160C3×C3⋊Dic3—21no—
161C3×C3⋊Dic3—22no—
162C3×C3⋊Dic3—22no—
163C3×C3⋊Dic3—21no—
164C3×C3⋊Dic3—21no—
165C3×C3⋊Dic3—22no—
166C3×C3⋊Dic3—22no—
167C3×C3⋊Dic3#4112yes—
168C3×#126—92no—
169C3×#117—91no—
170#924—91no—
171#924—91no—
172#890—92no—
173#889—92no—
174#889—92no—
175#889—92no—
176#889—92no—
177#889—92no—
178#888—92no—
179#887—92no—
180#879—32no—
181#879—32no—
182#879—62no—
183#879—92no—
184#879—92no—
185#879—92no—
186#878—92no—
187#878—92no—
188#877—32no—
189#877—61no—
190#870—92no—
191#870—91no—
192#869—91no—
193C4×Dic6—92no—
194C4×Dic6—91no—
195C4×Dic6—91no—
196C23×C32—21no—
197C23×C32—22no—
198C23×C32—41no—
199C23×C32C22×D311yescharacteristic
200C23×C32C22×D312yes—
201C2×C32×C4—21no—
202C2×C32×C4—22no—
203C2×C32×C4—21no—
204C2×C32×C4—22no—
205C2×C32×C4—41no—
206C2×C32×C4—41no—
207C2×C32×C4C22×D311yescharacteristic
208C2×C32×C4C22×D312yes—
209C2×C32×C4C22×D312yes—
210C2×C32×C4C22×D311yescharacteristic
211C2×C3⋊Dic3—22no—
212C2×C3⋊Dic3C22×D311yescharacteristic
213C2×C3⋊Dic3C22×D312yes—
214C2×C3⋊Dic3C22×D312yes—
215C2×C3⋊Dic3C22×D311yescharacteristic
216C2×C3⋊Dic3C22×D311yescharacteristic
217C2×C3⋊Dic3C22×D312yes—
218C2×C3⋊Dic3C22×D312yes—
219C2×C3⋊Dic3C22×D311yescharacteristic
220C2×C3×Dic3—32no—
221C2×C3×Dic3—32no—
222C2×C3×Dic3—32no—
223C2×C3×Dic3—32no—
224C2×C3×Dic3—32no—
225C2×C3×Dic3—32no—
226C2×C3×Dic3—32no—
227C2×C3×Dic3—32no—
228C2×C3×Dic3—32no—
229C2×C3×Dic3—32no—
230C2×C3×Dic3—32no—
231C2×C3×Dic3—32no—
232C2×C3×Dic3—61no—
233C2×C3×Dic3—61no—
234C2×C3×Dic3—61no—
235C2×C3×Dic3—61no—
236C2×C3×Dic3—62no—
237C2×C3×Dic3—62no—
238C2×C3×Dic3—62no—
239C2×C3×Dic3—62no—
240C2×C3×Dic3—64no—
241C2×C3×Dic3—64no—
242#627—64no—
243#627—64no—
244#627—62no—
245#627—62no—
246#541—271noSylow
247C2×C33—41no—
248C2×C33C2×#4111yescharacteristic
249C2×C332−1+411yescharacteristic
250C2×C332−1+411yescharacteristic
251C22×C3×C4—91no—
252C22×Dic3—32no—
253C22×Dic3—61no—
254C2×Dic6—91no—
255C2×Dic6—91no—
256C2×Dic6—92no—
257C2×Dic6—92no—
258C3×C4⋊C4—92no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—91no—
261C3×C4⋊C4—91no—
262C3×C4⋊C4—92no—
263C3×C22⋊C4—21no—
264C3×C22⋊C4—22no—
265C3×C22⋊C4—41no—
266C3×C22⋊C4—92no—
267C3×C22⋊C4—92no—
268C3×C22⋊C4—91no—
269C3×C22⋊C4D3212yes—
270C3×C22⋊C4D3211yescharacteristic
271C3×C42—91no—
272C3×C42—92no—
273C3×C42—91no—
274#217—32no—
275#217—32no—
276#217—32no—
277#217—32no—
278#217—32no—
279#217—61no—
280#217—62no—
281#217—62no—
282C4⋊Dic3—32no—
283C4⋊Dic3—32no—
284C4⋊Dic3—62no—
285C4⋊Dic3—91no—
286C4⋊Dic3—91no—
287C4⋊Dic3—92no—
288C4⋊Dic3—92no—
289C4⋊Dic3—92no—
290C4⋊Dic3—92no—
291C4⋊Dic3—92no—
292Dic3⋊C4—32no—
293Dic3⋊C4—32no—
294Dic3⋊C4—32no—
295Dic3⋊C4—32no—
296Dic3⋊C4—32no—
297Dic3⋊C4—32no—
298Dic3⋊C4—61no—
299Dic3⋊C4—61no—
300Dic3⋊C4—62no—
301Dic3⋊C4—62no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304Dic3⋊C4—92no—
305Dic3⋊C4—92no—
306Dic3⋊C4—92no—
307Dic3⋊C4—92no—
308Dic3⋊C4—92no—
309Dic3⋊C4—92no—
310Dic3⋊C4—92no—
311Dic3⋊C4—92no—
312Dic3⋊C4—92no—
313Dic3⋊C4—92no—
314C4×Dic3—32no—
315C4×Dic3—32no—
316C4×Dic3—32no—
317C4×Dic3—32no—
318C4×Dic3—61no—
319C4×Dic3—61no—
320C4×Dic3—62no—
321C4×Dic3—92no—
322C4×Dic3—92no—
323C4×Dic3—92no—
324C22×C32—21no—
325C22×C32—22no—
326C22×C32—22no—
327C22×C32—21no—
328C22×C32—21no—
329C22×C32—22no—
330C22×C32—22no—
331C22×C32—22no—
332C22×C32—21no—
333C22×C32—41no—
334C22×C32—42no—
335C22×C32—42no—
336C22×C32—41no—
337C22×C32—41no—
338C22×C32—42no—
339C22×C32—42no—
340C22×C32—81no—
341C22×C32—81no—
342C22×C32C23×D311yescharacteristic
343C22×C32C23×D312yes—
344C32×C4—22no—
345C32×C4—22no—
346C32×C4—21no—
347C32×C4—21no—
348C32×C4—42no—
349C32×C4—41no—
350C32×C4—41no—
351C32×C4—42no—
352C32×C4—81no—
353C32×C4—81no—
354C3⋊Dic3—21no—
355C3⋊Dic3—22no—
356C3⋊Dic3—22no—
357C3⋊Dic3—22no—
358C3⋊Dic3—22no—
359C3⋊Dic3—21no—
360C3⋊Dic3—21no—
361C3⋊Dic3#23512yes—
362C3×Dic3—34no—
363C3×Dic3—62no—
364C3×Dic3—62no—
365C3×Dic3—62no—
366C3×Dic3—62no—
367C3×Dic3—62no—
368C3×Dic3—62no—
369C3×Dic3—62no—
370C3×Dic3—62no—
371C3×Dic3—62no—
372C3×Dic3—62no—
373C3×Dic3—62no—
374C3×Dic3—62no—
375C3×Dic3—62no—
376C3×Dic3—122no—
377C3×Dic3—122no—
378C3×Dic3—122no—
379C3×Dic3—121no—
380C3×Dic3—121no—
381C3×Dic3—122no—
382C4⋊Q8—271no—
383#126—92no—
384#126—272no—
385#125—271no—
386#125—272no—
387#125—272no—
388C22⋊Q8—271no—
389C22⋊Q8—271no—
390C4×Q8—271no—
391C4×Q8—271no—
392#117—91no—
393C33#54111yescharacteristic, complemented, Sylow
394C23×C3—21no—
395C23×C3—22no—
396C23×C3—41no—
397C23×C3C2×D3211yescharacteristic
398C23×C3C2×D3212yes—
399C2×C3×C4—21no—
400C2×C3×C4—22no—
401C2×C3×C4—21no—
402C2×C3×C4—22no—
403C2×C3×C4—41no—
404C2×C3×C4—41no—
405C2×C3×C4—91no—
406C2×C3×C4—91no—
407C2×C3×C4—92no—
408C2×C3×C4—92no—
409C2×C3×C4—91no—
410C2×C3×C4—92no—
411C2×C3×C4—91no—
412C2×C3×C4—92no—
413C2×C3×C4—182no—
414C2×C3×C4C2×D3212yes—
415C2×C3×C4C2×D3211yescharacteristic
416C2×C3×C4C2×D3212yes—
417C2×C3×C4C2×D3211yescharacteristic
418C2×Dic3—32no—
419C2×Dic3—32no—
420C2×Dic3—32no—
421C2×Dic3—32no—
422C2×Dic3—32no—
423C2×Dic3—32no—
424C2×Dic3—32no—
425C2×Dic3—32no—
426C2×Dic3—32no—
427C2×Dic3—32no—
428C2×Dic3—32no—
429C2×Dic3—32no—
430C2×Dic3—61no—
431C2×Dic3—61no—
432C2×Dic3—61no—
433C2×Dic3—61no—
434C2×Dic3—62no—
435C2×Dic3—62no—
436C2×Dic3—62no—
437C2×Dic3—62no—
438C2×Dic3—64no—
439C2×Dic3—64no—
440Dic6—184no—
441Dic6—184no—
442Dic6—182no—
443Dic6—182no—
444C2×C32—22no—
445C2×C32—21no—
446C2×C32—22no—
447C2×C32—22no—
448C2×C32—21no—
449C2×C32—21no—
450C2×C32—41no—
451C2×C32—41no—
452C2×C32—41no—
453C2×C32—41no—
454C2×C32—42no—
455C2×C32—42no—
456C2×C32—82no—
457C2×C32—82no—
458C2×C32C2×#23511yescharacteristic
459C2×C32D3×#4112yes—
460C2×C32#101012yes—
461C2×C32#101012yes—
462C2×C32#101011yescharacteristic
463C2×C32#101011yescharacteristic
464C2×Q8—271no—
465C2×Q8—271no—
466C22×C4—91no—
467C4⋊C4—91no—
468C4⋊C4—91no—
469C4⋊C4—92no—
470C4⋊C4—92no—
471C4⋊C4—92no—
472C4⋊C4—271no—
473C4⋊C4—271no—
474C4⋊C4—272no—
475C4⋊C4—272no—
476C4⋊C4—272no—
477C4⋊C4—272no—
478C4⋊C4—272no—
479C22⋊C4—91no—
480C22⋊C4—92no—
481C22⋊C4—92no—
482C22⋊C4#111311yescharacteristic
483C42—91no—
484C42—91no—
485C42—92no—
486C42—272no—
487C22×C3—22no—
488C22×C3—22no—
489C22×C3—21no—
490C22×C3—22no—
491C22×C3—22no—
492C22×C3—21no—
493C22×C3—21no—
494C22×C3—21no—
495C22×C3—22no—
496C22×C3—41no—
497C22×C3—41no—
498C22×C3—42no—
499C22×C3—42no—
500C22×C3—42no—
501C22×C3—42no—
502C22×C3—41no—
503C22×C3—81no—
504C22×C3—81no—
505C22×C3C22×D3212yes—
506C22×C3C22×D3211yescharacteristic
507C3×C4—22no—
508C3×C4—22no—
509C3×C4—21no—
510C3×C4—21no—
511C3×C4—41no—
512C3×C4—42no—
513C3×C4—41no—
514C3×C4—42no—
515C3×C4—81no—
516C3×C4—81no—
517C3×C4—92no—
518C3×C4—182no—
519C3×C4—182no—
520C3×C4—181no—
521C3×C4—182no—
522C3×C4—182no—
523C3×C4—181no—
524C3×C4—181no—
525Dic3—34no—
526Dic3—62no—
527Dic3—62no—
528Dic3—62no—
529Dic3—62no—
530Dic3—62no—
531Dic3—62no—
532Dic3—62no—
533Dic3—62no—
534Dic3—62no—
535Dic3—62no—
536Dic3—62no—
537Dic3—62no—
538Dic3—62no—
539Dic3—122no—
540Dic3—122no—
541Dic3—121no—
542Dic3—121no—
543Dic3—122no—
544Dic3—122no—
545C32—22no—
546C32—21no—
547C32—41no—
548C32#551811yescharacteristic, complemented
549C32#570112yescomplemented
550C23C2×#111311yescharacteristic
551Q8—542no—
552Q8—542no—
553C2×C4—91no—
554C2×C4—91no—
555C2×C4—91no—
556C2×C4—92no—
557C2×C4—92no—
558C2×C4—92no—
559C2×C4—92no—
560C2×C4—91no—
561C2×C4—182no—
562C2×C4C2×#111311yescharacteristic
563C2×C4C2×#111311yescharacteristic
564C2×C3—21no—
565C2×C3—22no—
566C2×C3—21no—
567C2×C3—21no—
568C2×C3—22no—
569C2×C3—22no—
570C2×C3—42no—
571C2×C3—42no—
572C2×C3—41no—
573C2×C3—41no—
574C2×C3—41no—
575C2×C3—41no—
576C2×C3—82no—
577C2×C3—82no—
578C2×C3D3×#23512yes—
579C2×C3#6430611yescharacteristic
580C2×C3#6430912yes—
581C2×C3#6430912yes—
582C2×C3#6430911yescharacteristic
583C2×C3#6430911yescharacteristic
584C22—21no—
585C22—21no—
586C22—21no—
587C22C22×#111311yescharacteristic
588C4—21no—
589C4—21no—
590C4—92no—
591C4—181no—
592C4—182no—
593C4—182no—
594C4—181no—
595C4—181no—
596C4—182no—
597C4—182no—
598C3—21no—
599C3—22no—
600C3—41no—
601C3#1059424712yescomplemented
602C3#1059424911yescharacteristic, complemented
603C2—41no—
604C2#1172461811yescharacteristic
605C2#1172462011yescharacteristic
606C2#1172462011yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C22⋊C4C22
prime radicalC23×C33C23
hypercenterC22⋊C4#1113
hypocenterC33#541
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
6. C4⋊#6296C2semidirect
7. C4×#6296C2semidirect
8. #11723679C2semidirect
9. #11723679C2semidirect
10. #11723669C2semidirect
11. #11723665C2semidirect
393. C33#541semidirect
548. C32#5518semidirect
549. C32#5701semidirect
601. C3#10594247semidirect
602. C3#10594249semidirect

Maximal subgroups

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

subgroupindex
C3×#640882
C3×#640862
#117237212
#117237162
C4⋊#62962
C4×#62962
#117236792
#117236692
#117236652
#105942493
#105942473