Finite Group Databasebeta

← #10823303

Subgroups — #10823303

Order 768.

1074 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#544C211yesmaximal, characteristic
2#79208C212yesmaximal
3#74225C211yesmaximal, characteristic
4C3×#2061C211yesmaximal, characteristic, complemented
5#71900C211yesmaximal, characteristic, complemented
6#70667C212yesmaximal, complemented
7#69543C211yesmaximal, characteristic, complemented
8D3×#343C211yesmaximal, characteristic
9#68978C211yesmaximal, characteristic, complemented
10#68753C212yesmaximal
11#68735C212yesmaximal, complemented
12#23561—31nomaximal, Sylow
13C3×#544C2211yescharacteristic
14C2×D3×M16C2212yes—
15#5729—22no—
16D3×C4⋊Q8—22no—
17#5689—22no—
18D3×#125—22no—
19D3×C22⋊Q8—22no—
20#5623—22no—
21D3×#117C2212yes—
22#5519C2212yes—
23C2×#892C2211yescharacteristic
24C2×D3×C4⋊C4C2211yescharacteristic
25#5500C2211yescharacteristic
26C2×#872C2211yescharacteristic
27C3×#428C2212yes—
28C3×#382C2211yescharacteristic
29#5121C2212yes—
30#4995C2212yes—
31#4977C2211yescharacteristic
32#4975C2211yescharacteristic
33#4964C2212yes—
34#4935C2211yescharacteristic
35#4881—22no—
36#4880—22no—
37#4861—22no—
38#4860—22no—
39#4668—22no—
40#4647—22no—
41C3×#343C2211yescharacteristic
42C3×#339C2212yes—
43#4555C2212yes—
44#4554C2212yes—
45#4532C2212yes—
46#4530C2211yescharacteristic
47#4523C2212yes—
48C2×#544—31no—
49#2911—32no—
50#2317—31no—
51#2061—32no—
52#2061—31no—
53#2061D311yescharacteristic, complemented
54#2060—31no—
55#2060—32no—
56#2060—31no—
57C2×#343—31no—
58#1733—32no—
59C2×D3×Q8—22no—
60C22×C4×D3D412yes—
61C22×C4×D3C2311yescharacteristic
62C2×C3×M16C2312yes—
63C3×C4⋊Q8—22no—
64C3×#125—22no—
65C3×C22⋊Q8—22no—
66C3×#117C2312yes—
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C42C2311yescharacteristic
70#946—22no—
71#944—24no—
72C2×(C2×D3)⋊C4—21no—
73C2×(C2×D3)⋊C4—21no—
74#926C2312yes—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4⋊Dic3C2311yescharacteristic
78#924—22no—
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4D412yes—
82C2×C4×Dic3—21no—
83C2×C4×Dic3C2311yescharacteristic
84C2×#208C2312yes—
85D3×M16—24no—
86D3×M16Q818yescomplemented
87C2×ΓL(1,𝔽25)—22no—
88C2×D3×C8—22no—
89#897—22no—
90#896—24no—
91#892—22no—
92#892—22no—
93#892D412yes—
94#892D412yes—
95D3×C4⋊C4—22no—
96D3×C4⋊C4—24no—
97D3×C4⋊C4—22no—
98D3×C4⋊C4—22no—
99D3×C4⋊C4D412yes—
100D3×C4⋊C4D412yes—
101#890—22no—
102#890—22no—
103#889—24no—
104#889—24no—
105#888—24no—
106#888—22no—
107#888—22no—
108D3×C22⋊C4—22no—
109#878—24no—
110#877—22no—
111#872—22no—
112#872—22no—
113#872Q814yescomplemented
114C42×D3—22no—
115#870—22no—
116#869—22no—
117C3×#107—22no—
118C3×#106—22no—
119C3×#95—22no—
120C3⋊#95—22no—
121#818—22no—
122#817—22no—
123#808—22no—
124#808—22no—
125#807—22no—
126#807—22no—
127C22×M16—32no—
128#544—32no—
129#544—31no—
130#544—62no—
131#544—62no—
132#544—62no—
133#544C2×D311yescharacteristic
134C2×C4⋊Q8—62no—
135C2×#125—62no—
136C2×C22⋊Q8—62no—
137C2×#117—31no—
138C2×#117—32no—
139C2×#117—31no—
140C22×C4⋊C4—31no—
141#428—32no—
142#428—32no—
143#428—32no—
144#428—62no—
145#428—62no—
146#428C2×D312yes—
147#427—62no—
148#427—62no—
149#389—62no—
150#385—62no—
151#384—31no—
152#384—31no—
153#382—31no—
154#382C2×D311yescharacteristic
155#343—32no—
156#343—31no—
157#343C2×D311yescharacteristic
158#339—32no—
159#339—32no—
160#339—32no—
161#339C2×D312yes—
162C23×D3C2×D411yescharacteristic
163C2×C3×Q8—22no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2411yescharacteristic
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3C2×D412yes—
171C22×Dic3C2×D411yescharacteristic
172D3×Q8—84no—
173C2×C4×D3—22no—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—82no—
178C2×C4×D3C2×D412yes—
179C2×C4×D3C2×D412yes—
180C2×C4×D3C2×Q814yes—
181C2×C4×D3C2×Q812yes—
182C2×C4×D3#4112yes—
183C2×Dic6—24no—
184C2×Dic6—22no—
185C3×M16—22no—
186C3×M16C2×Q814yescomplemented
187C2×C3×C8—22no—
188C3×C4⋊C4—24no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—22no—
191C3×C4⋊C4—22no—
192C3×C4⋊C4C2×D412yes—
193C3×C4⋊C4C2×D412yes—
194C3×C22⋊C4—22no—
195C3×C42—22no—
196C3×C42—21no—
197C3×C42C2×Q812yescomplemented
198#217—22no—
199(C2×D3)⋊C4—22no—
200(C2×D3)⋊C4—22no—
201(C2×D3)⋊C4—24no—
202(C2×D3)⋊C4—41no—
203(C2×D3)⋊C4—41no—
204C4⋊Dic3—24no—
205C4⋊Dic3—22no—
206C4⋊Dic3—22no—
207C4⋊Dic3—22no—
208C4⋊Dic3—82no—
209C4⋊Dic3C2×D412yes—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212Dic3⋊C4—24no—
213Dic3⋊C4—24no—
214Dic3⋊C4—22no—
215Dic3⋊C4—22no—
216Dic3⋊C4—24no—
217Dic3⋊C4—24no—
218Dic3⋊C4—82no—
219C4×Dic3—22no—
220C4×Dic3—21no—
221C4×Dic3—22no—
222C4×Dic3—22no—
223C4×Dic3—42no—
224C4×Dic3C2×Q812yescomplemented
225#208—22no—
226#208C2×Q814yescomplemented
227C2×C3⋊C8—22no—
228ΓL(1,𝔽25)—28no—
229D3×C8—28no—
230C22×Q8—62no—
231C23×C4—31no—
232C23×C4—32no—
233C2×M16—32no—
234C2×M16—38no—
235C2×M16—62no—
236C2×M16—64no—
237C2×M16C22×D312yes—
238C22×C8—62no—
239C4⋊Q8—22no—
240C4⋊Q8—62no—
241C4⋊Q8—62no—
242C4⋊Q8—64no—
243C4⋊Q8—64no—
244C4⋊Q8—62no—
245#125—22no—
246#125—62no—
247#125—62no—
248#125—64no—
249#125—64no—
250#125—62no—
251C22⋊Q8—22no—
252C22⋊Q8—64no—
253C22⋊Q8—64no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257#117—34no—
258#117—32no—
259#117—32no—
260#117—32no—
261#117—62no—
262#117—62no—
263#117—62no—
264#117—62no—
265#117—62no—
266#117C22×D312yes—
267C2×C4⋊C4—32no—
268C2×C4⋊C4—32no—
269C2×C4⋊C4—32no—
270C2×C4⋊C4—31no—
271C2×C4⋊C4—31no—
272C2×C4⋊C4—62no—
273C2×C4⋊C4—62no—
274C2×C4⋊C4—62no—
275C2×C4⋊C4—64no—
276C2×C4⋊C4—61no—
277C2×C4⋊C4—62no—
278C2×C4⋊C4—62no—
279C2×C4⋊C4C22×D311yescharacteristic
280C2×C4⋊C4C22×D311yescharacteristic
281C2×C22⋊C4—62no—
282C2×C22⋊C4—61no—
283C2×C22⋊C4—61no—
284C2×C42—31no—
285C2×C42—62no—
286C2×C42—61no—
287C2×C42C22×D311yescharacteristic
288#107—22no—
289#107—62no—
290#107—62no—
291#107—62no—
292#106—22no—
293#106—62no—
294#106—62no—
295#106—62no—
296#95—22no—
297#95—62no—
298C23×C3C22×D411yescharacteristic
299C22×D3—41no—
300C22×D3—41no—
301C22×D3#12011yescharacteristic
302C22×D3#12011yescharacteristic
303C22×D3C22⋊Q812yes—
304C22×D3C4⋊Q812yes—
305C3×Q8—82no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—42no—
313C2×C3×C4—42no—
314C2×C3×C4—41no—
315C2×C3×C4C22×D412yes—
316C2×C3×C4C22×Q812yes—
317C2×C3×C4C22×Q811yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—41no—
328C2×Dic3#12012yes—
329C2×Dic3C22⋊Q812yes—
330C2×Dic3C4⋊Q812yes—
331C4×D3—44no—
332C4×D3—44no—
333C4×D3—44no—
334C4×D3—82no—
335C4×D3C22⋊Q814yes—
336C4×D3C22⋊Q814yes—
337C4×D3C4⋊Q814yes—
338C4×D3C4⋊Q814yes—
339Dic6—82no—
340Dic6—84no—
341C3×C8—24no—
342C3⋊C8—24no—
343C24—31no—
344C2×Q8—22no—
345C2×Q8—64no—
346C2×Q8—62no—
347C2×Q8—244no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—31no—
351C22×C4—32no—
352C22×C4—32no—
353C22×C4—32no—
354C22×C4—34no—
355C22×C4—32no—
356C22×C4—32no—
357C22×C4—62no—
358C22×C4—62no—
359C22×C4—62no—
360C22×C4—62no—
361C22×C4—61no—
362C22×C4—61no—
363C22×C4—242no—
364C22×C4D3×D412yes—
365C22×C4C23×D311yescharacteristic
366M16—22no—
367M16—34no—
368M16—68no—
369M16—62no—
370M16D3×Q814yescomplemented
371C2×C8—22no—
372C2×C8—68no—
373C2×C8—62no—
374C4⋊C4—22no—
375C4⋊C4—24no—
376C4⋊C4—22no—
377C4⋊C4—22no—
378C4⋊C4—32no—
379C4⋊C4—32no—
380C4⋊C4—64no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—64no—
384C4⋊C4—64no—
385C4⋊C4—62no—
386C4⋊C4—64no—
387C4⋊C4—64no—
388C4⋊C4—64no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—242no—
392C4⋊C4—242no—
393C4⋊C4D3×D412yes—
394C4⋊C4D3×D412yes—
395C22⋊C4—22no—
396C22⋊C4—62no—
397C22⋊C4—62no—
398C22⋊C4—64no—
399C22⋊C4—62no—
400C22⋊C4—121no—
401C22⋊C4—121no—
402C42—22no—
403C42—21no—
404C42—32no—
405C42—62no—
406C42—62no—
407C42—62no—
408C42—61no—
409C42—122no—
410C42D3×Q812yescomplemented
411C22×C3—21no—
412C22×C3—21no—
413C22×C3C2×#12011yescharacteristic
414C22×C3C2×C22⋊Q811yescharacteristic
415C22×C3C2×C4⋊Q811yescharacteristic
416C2×D3—21no—
417C2×D3—21no—
418C2×D3—21no—
419C2×D3—21no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3#39512yes—
423C2×D3#39512yes—
424C3×C4—42no—
425C3×C4—42no—
426C3×C4—42no—
427C3×C4—42no—
428C3×C4—81no—
429C3×C4C2×C22⋊Q812yes—
430C3×C4C2×C4⋊Q812yes—
431Dic3—42no—
432Dic3—42no—
433Dic3—42no—
434Dic3—42no—
435Dic3—81no—
436Dic3#39514yes—
437C23—31no—
438C23—31no—
439C23—32no—
440C23—32no—
441C23—121no—
442C23—121no—
443C23C2×D3×D411yescharacteristic
444Q8—82no—
445Q8—242no—
446Q8—244no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—32no—
452C2×C4—32no—
453C2×C4—32no—
454C2×C4—34no—
455C2×C4—34no—
456C2×C4—34no—
457C2×C4—34no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—42no—
461C2×C4—42no—
462C2×C4—41no—
463C2×C4—62no—
464C2×C4—62no—
465C2×C4—62no—
466C2×C4—62no—
467C2×C4—124no—
468C2×C4—124no—
469C2×C4—124no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—122no—
473C2×C4—122no—
474C2×C4—121no—
475C2×C4—242no—
476C2×C4C2×D3×D412yes—
477C2×C4C2×D3×Q811yescharacteristic
478C2×C4C2×D3×Q812yes—
479C2×C4D3×#4111yescharacteristic
480C8—24no—
481C8—64no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3C2×#39511yescharacteristic
485C2×C3#301911yescharacteristic
486C2×C3#302011yescharacteristic
487D3—22no—
488D3—22no—
489C22—21no—
490C22—21no—
491C22—32no—
492C22—32no—
493C22—61no—
494C22—61no—
495C22—61no—
496C22—61no—
497C22—122no—
498C22—122no—
499C22D3×#12011yescharacteristic
500C22D3×C22⋊Q811yescharacteristic
501C22D3×C4⋊Q811yescharacteristic
502C4—34no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—81no—
508C4—122no—
509C4—122no—
510C4—122no—
511C4—122no—
512C4—241no—
513C4D3×C22⋊Q812yes—
514C4D3×C4⋊Q812yes—
515C3#2356111yescharacteristic, complemented, Sylow
516C2—21no—
517C2—21no—
518C2—62no—
519C2—62no—
520C2D3×#39511yescharacteristic
521C2#8028511yescharacteristic
522C2#8029211yescharacteristic
subgroupG/H
centerC22D3×#120
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2061C2
socleC22×C3C2×#120
prime radicalC23×D3C2×D4
hypercenter#2061D3
hypocenterC3#23561
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2061

Complements

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

NH ≅ G/Ntype
4. C3×#2061C2semidirect
5. #71900C2semidirect
6. #70667C2semidirect
7. #69543C2semidirect
9. #68978C2semidirect
11. #68735C2semidirect
53. #2061D3semidirect
86. D3×M16Q8semidirect
113. #872Q8semidirect
186. C3×M16C2×Q8semidirect
197. C3×C42C2×Q8semidirect
224. C4×Dic3C2×Q8semidirect
226. #208C2×Q8semidirect
370. M16D3×Q8semidirect
410. C42D3×Q8semidirect
515. C3#23561semidirect

Maximal subgroups

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

subgroupindex
D3×#5442
#792082
#742252
C3×#20612
#719002
#706672
#695432
D3×#3432
#689782
#687532
#687352
#235613