Finite Group Databasebeta

← #13178317

Subgroups — #13178317

Order 1728.

1036 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#64106C211yesmaximal, characteristic
2C3×#64098C212yesmaximal, complemented
3C4⋊#6296C212yesmaximal, complemented
4#11723692C212yesmaximal
5#11723683C212yesmaximal
6#11723682C211yesmaximal, characteristic
7#11723671C211yesmaximal, characteristic
8#11723669C212yesmaximal, complemented
9#11723666C212yesmaximal, complemented
10#10594401—31nomaximal
11#10594389—32nomaximal
12C2×C3×C4×C3⋊D3C2211yescharacteristic
13C2×C3×#634C2211yescharacteristic
14C2×#6296C2212yes—
15C2×#6296C2211yescharacteristic
16C2×#6294C2211yescharacteristic
17C2×#6294C2211yescharacteristic
18C33×C4⋊C4C2211yescharacteristic
19C3×#3752C2212yes—
20C3×C4⋊(C3⋊Dic3)C2212yes—
21C3×C4⋊(C3⋊Dic3)C2212yes—
22C3×#3750C2212yes—
23C3×#3750C2212yes—
24C3×#3750C2212yes—
25C3×#3750C2212yes—
26C3×C4×C3⋊Dic3C2212yes—
27#88815C2212yes—
28#88815C2212yes—
29#88815C2211yescharacteristic
30#88815C2212yes—
31#88814C2212yes—
32#88813C2212yes—
33#64106D311yescharacteristic
34#64098D312yescomplemented
35C3×#897—32no—
36C3×#897—61no—
37C3×#889—32no—
38C3×#889—32no—
39C3×#889—62no—
40#63919—32no—
41#63918—32no—
42#63918—32no—
43#63899—32no—
44#63895—32no—
45#63893—32no—
46#63883—31no—
47#63878—32no—
48#63877—32no—
49D3×Dic3⋊C4—32no—
50#63865—31no—
51#63851—32no—
52#63847—32no—
53#63846—32no—
54#63845—32no—
55#63840—32no—
56#63839—32no—
57#63839—32no—
58#63836—32no—
59C22×C3×C3⋊D3C2311yescharacteristic
60C2×C33×C4C2312yes—
61C2×C33×C4C2311yescharacteristic
62C2×C3×C3⋊Dic3C2312yes—
63C2×C3×C3⋊Dic3C2312yes—
64C2×C3×C3⋊Dic3C2312yes—
65C2×C3×C3⋊Dic3C2311yescharacteristic
66C2×C3×C3⋊Dic3C2312yes—
67C2×C3×C3⋊Dic3C2312yes—
68C3×C4×C3⋊D3—41no—
69C3×#634—41no—
70#6296—42no—
71#6296—41no—
72#6294—41no—
73#6294Q814yescomplemented
74#5689—92no—
75#5623—91no—
76C2×C4×C3⋊D3C2×D311yescharacteristic
77C2×#634C2×D311yescharacteristic
78C2×C3×C4×D3—32no—
79C2×C3×C4×D3—61no—
80C2×C3×Dic6—32no—
81C2×C3×Dic6—61no—
82C2×#627—32no—
83C2×#627—32no—
84C2×#627—32no—
85C2×#627—32no—
86C2×#627—31no—
87C2×#624—31no—
88C2×#624—31no—
89C2×D3×Dic3—32no—
90C2×D3×Dic3—32no—
91C32×C4⋊C4—22no—
92C32×C4⋊C4—21no—
93C32×C4⋊C4—41no—
94C32×C4⋊C4C2×D311yescharacteristic
95C32×C4⋊C4C2×D312yes—
96#3752C2×D312yes—
97C4⋊(C3⋊Dic3)C2×D312yes—
98C4⋊(C3⋊Dic3)C2×D312yes—
99#3750C2×D312yes—
100#3750C2×D312yes—
101#3750C2×D312yes—
102#3750C2×D312yes—
103C4×C3⋊Dic3C2×D312yes—
104C3×(C2×D3)⋊C4—32no—
105C3×(C2×D3)⋊C4—32no—
106C3×(C2×D3)⋊C4—62no—
107C3×C4⋊Dic3—32no—
108C3×C4⋊Dic3—32no—
109C3×C4⋊Dic3—32no—
110C3×C4⋊Dic3—32no—
111C3×C4⋊Dic3—62no—
112C3×C4⋊Dic3—62no—
113C3×Dic3⋊C4—32no—
114C3×Dic3⋊C4—32no—
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—62no—
122C3×Dic3⋊C4—62no—
123C3×Dic3⋊C4—62no—
124C3×Dic3⋊C4—62no—
125C3×C4×Dic3—32no—
126C3×C4×Dic3—32no—
127C3×C4×Dic3—62no—
128#3724—32no—
129#3724—32no—
130#3724—31no—
131#3724—32no—
132Dic3⋊Dic3—32no—
133Dic3⋊Dic3—32no—
134Dic3⋊Dic3—32no—
135Dic3⋊Dic3—32no—
136Dic3⋊Dic3—32no—
137Dic3⋊Dic3—32no—
138Dic3⋊Dic3—32no—
139#3722—32no—
140(C2×D3)⋊Dic3—32no—
141(C2×D3)⋊Dic3—32no—
142Dic32—32no—
143Dic32—32no—
144Dic32—32no—
145C22×C33C2411yescharacteristic
146C2×C3×C3⋊D3—21no—
147C2×C3×C3⋊D3—21no—
148C2×C3×C3⋊D3C2×Q812yes—
149C33×C4—21no—
150C33×C4—22no—
151C3×C3⋊Dic3—22no—
152C3×C3⋊Dic3—22no—
153C3×C3⋊Dic3—22no—
154C3×C3⋊Dic3—21no—
155C3×C3⋊Dic3—22no—
156C3×C3⋊Dic3C2×Q814yes—
157C3×#125—92no—
158C3×C22⋊Q8—91no—
159#944—92no—
160#924—91no—
161C2×Dic3⋊C4—91no—
162#897—32no—
163#897—61no—
164#897—92no—
165#896—92no—
166#896—92no—
167#892—92no—
168D3×C4⋊C4—92no—
169#890—92no—
170#889—32no—
171#889—32no—
172#889—62no—
173#889—92no—
174#889—92no—
175#888—92no—
176#888—92no—
177#888—92no—
178#888—92no—
179#878—92no—
180#877—92no—
181#872—92no—
182#869—92no—
183C22×C3⋊D3C22×D311yescharacteristic
184C22×C3×D3—32no—
185C22×C3×D3—61no—
186C2×C32×C4—21no—
187C2×C32×C4—22no—
188C2×C32×C4—22no—
189C2×C32×C4—22no—
190C2×C32×C4—22no—
191C2×C32×C4—41no—
192C2×C32×C4—42no—
193C2×C32×C4C22×D312yes—
194C2×C32×C4C22×D312yes—
195C2×C32×C4C22×D311yescharacteristic
196C2×C32×C4C22×D312yes—
197C2×C32×C4C22×D312yes—
198C2×C3⋊Dic3C22×D312yes—
199C2×C3⋊Dic3C22×D312yes—
200C2×C3⋊Dic3C22×D312yes—
201C2×C3⋊Dic3C22×D311yescharacteristic
202C2×C3⋊Dic3C22×D312yes—
203C2×C3⋊Dic3C22×D312yes—
204C4×C3⋊D3—41no—
205#634—41no—
206C2×C3×Dic3—32no—
207C2×C3×Dic3—32no—
208C2×C3×Dic3—32no—
209C2×C3×Dic3—32no—
210C2×C3×Dic3—32no—
211C2×C3×Dic3—32no—
212C2×C3×Dic3—32no—
213C2×C3×Dic3—32no—
214C2×C3×Dic3—32no—
215C2×C3×Dic3—32no—
216C2×C3×Dic3—32no—
217C2×C3×Dic3—62no—
218C2×C3×Dic3—62no—
219C2×C3×Dic3—62no—
220C2×C3×Dic3—62no—
221C2×C3×Dic3—61no—
222C2×C3×Dic3—62no—
223C3×C4×D3—122no—
224C3×C4×D3—122no—
225C3×Dic6—122no—
226C3×Dic6—122no—
227#627—122no—
228#627—121no—
229#627—122no—
230#627—122no—
231#627—122no—
232#624—34no—
233#624—121no—
234D3×Dic3—38no—
235D3×Dic3—122no—
236#544—271noSylow
237C2×C33C22×Q811yescharacteristic
238C2×C33H(5,𝔽2)11yescharacteristic
239C2×C332−1+411yescharacteristic
240C3×C3⋊D3—22no—
241C2×C3×Q8—91no—
242C22×C3×C4—91no—
243C22×Dic3—91no—
244C22×Dic3—91no—
245C2×C4×D3—32no—
246C2×C4×D3—61no—
247C2×C4×D3—92no—
248C2×C4×D3—92no—
249C2×Dic6—32no—
250C2×Dic6—61no—
251C2×Dic6—92no—
252C2×Dic6—92no—
253C2×Dic6—92no—
254C2×Dic6—92no—
255C2×Dic6—91no—
256C3×C4⋊C4—22no—
257C3×C4⋊C4—21no—
258C3×C4⋊C4—41no—
259C3×C4⋊C4—92no—
260C3×C4⋊C4—92no—
261C3×C4⋊C4—92no—
262C3×C4⋊C4—92no—
263C3×C4⋊C4—92no—
264C3×C4⋊C4—92no—
265C3×C4⋊C4D3211yescharacteristic
266C3×C4⋊C4D3212yes—
267C3×C22⋊C4—92no—
268C3×C42—92no—
269#217—92no—
270(C2×D3)⋊C4—32no—
271(C2×D3)⋊C4—32no—
272(C2×D3)⋊C4—62no—
273(C2×D3)⋊C4—92no—
274(C2×D3)⋊C4—92no—
275C4⋊Dic3—32no—
276C4⋊Dic3—32no—
277C4⋊Dic3—32no—
278C4⋊Dic3—32no—
279C4⋊Dic3—62no—
280C4⋊Dic3—62no—
281C4⋊Dic3—92no—
282C4⋊Dic3—92no—
283C4⋊Dic3—92no—
284C4⋊Dic3—91no—
285Dic3⋊C4—32no—
286Dic3⋊C4—32no—
287Dic3⋊C4—32no—
288Dic3⋊C4—32no—
289Dic3⋊C4—32no—
290Dic3⋊C4—32no—
291Dic3⋊C4—32no—
292Dic3⋊C4—32no—
293Dic3⋊C4—62no—
294Dic3⋊C4—62no—
295Dic3⋊C4—62no—
296Dic3⋊C4—62no—
297Dic3⋊C4—92no—
298Dic3⋊C4—92no—
299Dic3⋊C4—92no—
300Dic3⋊C4—92no—
301Dic3⋊C4—92no—
302Dic3⋊C4—92no—
303Dic3⋊C4—92no—
304C4×Dic3—32no—
305C4×Dic3—32no—
306C4×Dic3—62no—
307C4×Dic3—92no—
308C4×Dic3—92no—
309C4×Dic3—92no—
310C22×C32—22no—
311C22×C32—21no—
312C22×C32—41no—
313C22×C32C23×D311yescharacteristic
314C22×C32C23×D312yes—
315C2×C3⋊D3—21no—
316C2×C3⋊D3—21no—
317C2×C3⋊D3D3×Q812yes—
318C2×C3×D3—34no—
319C2×C3×D3—62no—
320C2×C3×D3—62no—
321C2×C3×D3—62no—
322C2×C3×D3—121no—
323C2×C3×D3—121no—
324C32×C4—21no—
325C32×C4—22no—
326C32×C4—22no—
327C32×C4—22no—
328C32×C4—22no—
329C32×C4—22no—
330C32×C4—42no—
331C32×C4—42no—
332C32×C4—42no—
333C32×C4—42no—
334C32×C4—42no—
335C32×C4—82no—
336C3⋊Dic3—22no—
337C3⋊Dic3—21no—
338C3⋊Dic3—22no—
339C3⋊Dic3—22no—
340C3⋊Dic3—22no—
341C3⋊Dic3D3×Q814yes—
342C3×Dic3—34no—
343C3×Dic3—34no—
344C3×Dic3—64no—
345C3×Dic3—62no—
346C3×Dic3—62no—
347C3×Dic3—62no—
348C3×Dic3—62no—
349C3×Dic3—62no—
350C3×Dic3—62no—
351C3×Dic3—62no—
352C3×Dic3—62no—
353C3×Dic3—62no—
354C3×Dic3—122no—
355C3×Dic3—121no—
356C3×Dic3—122no—
357C3×Dic3—122no—
358C3×Dic3—122no—
359C4⋊Q8—272no—
360C4⋊Q8—272no—
361#125—92no—
362#125—272no—
363C22⋊Q8—91no—
364C22⋊Q8—272no—
365C22⋊Q8—271no—
366#117—272no—
367C2×C4⋊C4—271no—
368C33#54411yescharacteristic, complemented, Sylow
369C23×C3—91no—
370C22×D3—32no—
371C22×D3—61no—
372C3×Q8—361no—
373C2×C3×C4—21no—
374C2×C3×C4—22no—
375C2×C3×C4—22no—
376C2×C3×C4—22no—
377C2×C3×C4—22no—
378C2×C3×C4—41no—
379C2×C3×C4—42no—
380C2×C3×C4—92no—
381C2×C3×C4—92no—
382C2×C3×C4—92no—
383C2×C3×C4—91no—
384C2×C3×C4—92no—
385C2×C3×C4—92no—
386C2×C3×C4—361no—
387C2×C3×C4C2×D3212yes—
388C2×C3×C4C2×D3212yes—
389C2×C3×C4C2×D3211yescharacteristic
390C2×C3×C4C2×D3212yes—
391C2×C3×C4C2×D3212yes—
392C2×Dic3—32no—
393C2×Dic3—32no—
394C2×Dic3—32no—
395C2×Dic3—32no—
396C2×Dic3—32no—
397C2×Dic3—32no—
398C2×Dic3—32no—
399C2×Dic3—32no—
400C2×Dic3—32no—
401C2×Dic3—32no—
402C2×Dic3—32no—
403C2×Dic3—62no—
404C2×Dic3—62no—
405C2×Dic3—62no—
406C2×Dic3—62no—
407C2×Dic3—61no—
408C2×Dic3—62no—
409C2×Dic3—94no—
410C2×Dic3—361no—
411C4×D3—98no—
412C4×D3—122no—
413C4×D3—122no—
414C4×D3—362no—
415Dic6—122no—
416Dic6—122no—
417Dic6—362no—
418Dic6—362no—
419Dic6—362no—
420Dic6—361no—
421Dic6—362no—
422C2×C32—22no—
423C2×C32—21no—
424C2×C32—22no—
425C2×C32—21no—
426C2×C32—22no—
427C2×C32—21no—
428C2×C32—41no—
429C2×C32—41no—
430C2×C32—41no—
431C2×C32#100411yescharacteristic
432C2×C32C2×D3×Q812yes—
433C2×C32C2×D3×Q811yescharacteristic
434C2×C32#100912yes—
435C2×C32#101012yes—
436C2×C32#101011yescharacteristic
437C3⋊D3—22no—
438C3×D3—64no—
439C3×D3—122no—
440C2×Q8—91no—
441C2×Q8—271no—
442C2×Q8—272no—
443C22×C4—91no—
444C22×C4—271no—
445C22×C4—271no—
446C4⋊C4—92no—
447C4⋊C4—92no—
448C4⋊C4—92no—
449C4⋊C4—92no—
450C4⋊C4—92no—
451C4⋊C4—92no—
452C4⋊C4—272no—
453C4⋊C4—272no—
454C4⋊C4—271no—
455C4⋊C4—272no—
456C4⋊C4#111311yescharacteristic
457C22⋊C4—92no—
458C22⋊C4—272no—
459C42—92no—
460C42—272no—
461C22×C3—22no—
462C22×C3—21no—
463C22×C3—41no—
464C22×C3—92no—
465C22×C3—181no—
466C22×C3—181no—
467C22×C3C22×D3212yes—
468C22×C3C22×D3211yescharacteristic
469C2×D3—34no—
470C2×D3—62no—
471C2×D3—62no—
472C2×D3—62no—
473C2×D3—121no—
474C2×D3—121no—
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—21no—
478C3×C4—22no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—42no—
482C3×C4—42no—
483C3×C4—42no—
484C3×C4—42no—
485C3×C4—42no—
486C3×C4—82no—
487C3×C4—94no—
488C3×C4—182no—
489C3×C4—182no—
490C3×C4—181no—
491C3×C4—182no—
492C3×C4—182no—
493Dic3—34no—
494Dic3—34no—
495Dic3—64no—
496Dic3—62no—
497Dic3—62no—
498Dic3—62no—
499Dic3—62no—
500Dic3—62no—
501Dic3—62no—
502Dic3—62no—
503Dic3—62no—
504Dic3—62no—
505Dic3—122no—
506Dic3—121no—
507Dic3—122no—
508Dic3—122no—
509Dic3—122no—
510C32—22no—
511C32—21no—
512C32—41no—
513C32#562311yescharacteristic, complemented
514C32#568912yescomplemented
515C23—91no—
516Q8—361no—
517Q8—1081no—
518Q8—1082no—
519C2×C4—92no—
520C2×C4—92no—
521C2×C4—92no—
522C2×C4—92no—
523C2×C4—92no—
524C2×C4—91no—
525C2×C4—274no—
526C2×C4—361no—
527C2×C4—1081no—
528C2×C4C2×#111312yes—
529C2×C4C2×#111311yescharacteristic
530C2×C3—21no—
531C2×C3—22no—
532C2×C3—22no—
533C2×C3—21no—
534C2×C3—22no—
535C2×C3—21no—
536C2×C3—41no—
537C2×C3—41no—
538C2×C3—41no—
539C2×C3—182no—
540C2×C3#6430712yes—
541C2×C3#6430811yescharacteristic
542C2×C3#6430912yes—
543C2×C3#6430911yescharacteristic
544C2×C3D32×Q812yes—
545C2×C3D32×Q811yescharacteristic
546D3—64no—
547D3—122no—
548C22—92no—
549C22—181no—
550C22—181no—
551C22C22×#111311yescharacteristic
552C4—21no—
553C4—22no—
554C4—94no—
555C4—181no—
556C4—182no—
557C4—182no—
558C4—182no—
559C4—182no—
560C3—21no—
561C3—22no—
562C3—41no—
563C3#1059438912yescomplemented
564C3#1059440111yescharacteristic, complemented
565C2—182no—
566C2#1172461911yescharacteristic
567C2#1172462011yescharacteristic
568C2Q8×#111311yescharacteristic
subgroupG/H
center, FrattiniC22C22×#1113
commutator, socle, cosocleC22×C33C24
FittingC33×C4⋊C4C22
prime radicalC22×C3×C3⋊D3C23
hypercenterC4⋊C4#1113
hypocenterC33#544
LCSG ▷ C22×C33 ▷ C33
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#64098C2semidirect
3. C4⋊#6296C2semidirect
8. #11723669C2semidirect
9. #11723666C2semidirect
34. #64098D3semidirect
73. #6294Q8semidirect
368. C33#544semidirect
513. C32#5623semidirect
514. C32#5689semidirect
563. C3#10594389semidirect
564. C3#10594401semidirect

Maximal subgroups

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

subgroupindex
C3×#641062
C3×#640982
C4⋊#62962
#117236922
#117236832
#117236822
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