Finite Group Databasebeta

← #10691652

Subgroups — #10691652

Order 768.

750 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5040C211yesmaximal, characteristic
2#73876C211yesmaximal, characteristic, complemented
3C3×#1519C211yesmaximal, characteristic
4#71686C211yesmaximal, characteristic
5#69963C211yesmaximal, characteristic, complemented
6#67678C211yesmaximal, characteristic, complemented
7#67625C211yesmaximal, characteristic, complemented
8#9262—31nomaximal, Sylow
9C23×Dic6C2211yescharacteristic
10C2×C3×#121C2211yescharacteristic
11C22×#214C412yescomplemented
12C2×#887—21no—
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×#834—21no—
16C2×#834—21no—
17#5212C2211yescharacteristic
18C4×#217C2211yescharacteristic
19C2×#830C2211yescharacteristic
20C2×#816—22no—
21#5040—21no—
22#5040—22no—
23#5040—21no—
24#4973—21no—
25C2×#810C412yescomplemented
26#4942—21no—
27Dic3×C22⋊C4—21no—
28#4646—22no—
29C3×#327—22no—
30C2×#448—31no—
31Q8×C22⋊C4—31no—
32#1964—31no—
33#1787—31no—
34#1519D311yescharacteristic
35#1289—31no—
36#1282—31no—
37C22×C3×D4—21no—
38C22×C3×D4C2×C411yescharacteristic
39C23×C3×C4C2311yescharacteristic
40C23×Dic3—21no—
41C22×Dic6—21no—
42C22×Dic6—21no—
43C22×Dic6—21no—
44C22×Dic6—44no—
45C22×Dic6C2×C412yes—
46C22×Dic6D411yescharacteristic
47C22×Dic6D412yes—
48C22×Dic6D411yescharacteristic
49C22×C3×C8—21no—
50C3×#121—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×#121—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4C2×C411yescharacteristic
56C2×C3×C22⋊C4—21no—
57C2×C3×C22⋊C4—21no—
58C2×#217—21no—
59C2×#217D411yescharacteristic
60C2×#217D411yescharacteristic
61C2×#214—22no—
62C2×#214—22no—
63C2×#214—44no—
64C2×C4⋊Dic3—21no—
65C2×Dic3⋊C4—22no—
66C2×C4×Dic3—22no—
67C2×C4×Dic3—21no—
68C22×C3⋊C8C2×C412yes—
69#887—44no—
70C3×C22⋊C8—21no—
71C3×C22⋊C8—21no—
72C3×#95D412yes—
73#834—42no—
74#834—42no—
75C3⋊#95—22no—
76C3⋊#95—22no—
77C3⋊#95—21no—
78C3⋊#95—21no—
79#830—21no—
80#830—21no—
81#816—44no—
82#810—22no—
83#810—22no—
84C23×Q8—31no—
85C22×SD16—32no—
86C2×#121C2×D311yescharacteristic
87C2×C4×Q8—61no—
88#448—61no—
89#448—62no—
90#448—61no—
91C2×#103—32no—
92C2×#103—62no—
93C2×#99—61no—
94C2×#99—61no—
95C2×C22⋊C8—31no—
96C2×C22⋊C8C2×D311yescharacteristic
97#391—61no—
98#391—62no—
99#380—31no—
100#380—61no—
101#380C2×D311yescharacteristic
102C4×C22⋊C4—31no—
103C4×C22⋊C4—61no—
104#327—22no—
105C24×C3—21no—
106C24×C3C2×D411yescharacteristic
107C2×C3×D4—21no—
108C2×C3×D4—21no—
109C2×C3×D4—42no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C22×C3×C4—21no—
117C22×C3×C4—22no—
118C22×C3×C4—42no—
119C22×C3×C4C22×C411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C22×Dic3—21no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128C22×Dic3—21no—
129C22×Dic3—22no—
130C22×Dic3—21no—
131C22×Dic3—44no—
132C22×Dic3—41no—
133C2×Dic6—21no—
134C2×Dic6—24no—
135C2×Dic6—24no—
136C2×Dic6—24no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—21no—
140C2×Dic6—42no—
141C2×Dic6—42no—
142C2×Dic6—42no—
143C2×Dic6—44no—
144C2×Dic6—44no—
145C2×Dic6—44no—
146C2×Dic6C22⋊C412yes—
147C2×Dic6C22⋊C412yes—
148C2×C3×C8—42no—
149C2×C3×C8—41no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—42no—
153C3×C22⋊C4—41no—
154C3×C22⋊C4—41no—
155C3×C22⋊C4—41no—
156C3×C22⋊C4—41no—
157#217—21no—
158#217—21no—
159#217—21no—
160#217—41no—
161#217—41no—
162#217SD1612yescomplemented
163#214—44no—
164#214—44no—
165C4⋊Dic3—42no—
166Dic3⋊C4—44no—
167C4×Dic3—44no—
168C4×Dic3—42no—
169C2×C3⋊C8—22no—
170C2×C3⋊C8—42no—
171C22×Q8—32no—
172C22×Q8—31no—
173C22×Q8—32no—
174C22×Q8—31no—
175C22×Q8—61no—
176C22×Q8—61no—
177C22×Q8—61no—
178C22×Q8—124no—
179C22×D4—21no—
180C22×D4C4×D311yescharacteristic
181C23×C4—61no—
182C23×C4C22×D311yescharacteristic
183C2×SD16—62no—
184C2×SD16—62no—
185C2×SD16—124no—
186C22×C8—21no—
187C22×C8—32no—
188#121—21no—
189#121—21no—
190#121—21no—
191#121—21no—
192C4×Q8—124no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—62no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4C4×D311yescharacteristic
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—31no—
200C2×C22⋊C4—31no—
201C2×C22⋊C4—61no—
202C2×C42—62no—
203C2×C42—61no—
204#103—62no—
205#103—62no—
206#103—124no—
207#99—122no—
208#99—122no—
209C22⋊C8—21no—
210C22⋊C8—21no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#95—62no—
214#95—62no—
215#95—61no—
216#95—61no—
217#95#6712yes—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3—41no—
221C23×C3—41no—
222C23×C3—41no—
223C23×C3—81no—
224C23×C3—81no—
225C23×C3C4×D411yescharacteristic
226C23×C3#12011yescharacteristic
227C23×C3#12111yescharacteristic
228C23×C3C2×SD1611yescharacteristic
229C23×C3#13511yescharacteristic
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C23×C3#13711yescharacteristic
232C3×D4—42no—
233C3×D4—81no—
234C3×D4—81no—
235C3×D4—81no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4C2×C22⋊C411yescharacteristic
251C2×C3×C4C4×D411yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12311yescharacteristic
254C2×Dic3—24no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—44no—
259C2×Dic3—44no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—81no—
267C2×Dic3—81no—
268C2×Dic3—81no—
269Dic6—24no—
270Dic6—22no—
271Dic6—22no—
272Dic6—22no—
273Dic6—44no—
274Dic6—44no—
275Dic6—42no—
276Dic6#9914yescomplemented
277C3×C8—81no—
278C3⋊C8—42no—
279C24—21no—
280C24D3×D411yescharacteristic
281C2×Q8—32no—
282C2×Q8—32no—
283C2×Q8—64no—
284C2×Q8—64no—
285C2×Q8—64no—
286C2×Q8—61no—
287C2×Q8—61no—
288C2×Q8—62no—
289C2×Q8—62no—
290C2×Q8—122no—
291C2×Q8—122no—
292C2×Q8—122no—
293C2×Q8—124no—
294C2×Q8—124no—
295C2×Q8—124no—
296C2×D4—21no—
297C2×D4—21no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—41no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—41no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—42no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—124no—
315C22×C4—121no—
316C22×C4C2×C4×D311yescharacteristic
317C22×C4#23511yescharacteristic
318C22×C4D3×D411yescharacteristic
319C22×C4D3×D411yescharacteristic
320C22×C4#23711yescharacteristic
321C22×C4C2×#6711yescharacteristic
322SD16—124no—
323SD16—124no—
324C2×C8—42no—
325C2×C8—41no—
326C2×C8—62no—
327C2×C8—122no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—42no—
331C4⋊C4—122no—
332C4⋊C4—124no—
333C22⋊C4—32no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—41no—
338C22⋊C4—61no—
339C22⋊C4—61no—
340C22⋊C4—61no—
341C22⋊C4—121no—
342C22⋊C4—121no—
343C42—122no—
344C42—124no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—21no—
348C22×C3—21no—
349C22×C3—21no—
350C22×C3—21no—
351C22×C3—41no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—41no—
355C22×C3—81no—
356C22×C3—81no—
357C22×C3—81no—
358C22×C3—161no—
359C22×C3—161no—
360C22×C3#38611yescharacteristic
361C22×C3C4×SD1611yescharacteristic
362C22×C3#44011yescharacteristic
363C22×C3#44811yescharacteristic
364C22×C3#44911yescharacteristic
365C22×C3#46011yescharacteristic
366C22×C3#46411yescharacteristic
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—82no—
371C3×C4—81no—
372C3×C4C2×#9912yes—
373Dic3—24no—
374Dic3—42no—
375Dic3—42no—
376Dic3—81no—
377Dic3—161no—
378C23—41no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—41no—
383C23—81no—
384C23—81no—
385C23#67⋊C411yescharacteristic
386C23#88411yescharacteristic
387C23#91111yescharacteristic
388C23#91211yescharacteristic
389C23D3×SD1611yescharacteristic
390C23#91511yescharacteristic
391C23#93711yescharacteristic
392Q8—34no—
393Q8—64no—
394Q8—62no—
395Q8—62no—
396Q8—62no—
397Q8—124no—
398Q8—124no—
399Q8—122no—
400D4—42no—
401D4—81no—
402D4—81no—
403D4—81no—
404D4—81no—
405D4—81no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—64no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—124no—
423C2×C4—124no—
424C2×C4—121no—
425C2×C4—122no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—241no—
431C2×C4—241no—
432C2×C4—241no—
433C2×C4D3×C22⋊C411yescharacteristic
434C2×C4#88311yescharacteristic
435C2×C4C4×#6711yescharacteristic
436C2×C4#93811yescharacteristic
437C8—81no—
438C8—122no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—161no—
446C2×C3#188011yescharacteristic
447C2×C3#189811yescharacteristic
448C2×C3#190511yescharacteristic
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—81no—
460C22—81no—
461C22—81no—
462C22—161no—
463C22—161no—
464C22#475711yescharacteristic
465C22#475811yescharacteristic
466C22#476211yescharacteristic
467C22#476711yescharacteristic
468C22#495611yescharacteristic
469C22#515811yescharacteristic
470C22#517011yescharacteristic
471C4—21no—
472C4—21no—
473C4—21no—
474C4—64no—
475C4—82no—
476C4—81no—
477C4—122no—
478C4—122no—
479C4—241no—
480C4—481no—
481C4D3×#9912yes—
482C3#926211yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—21no—
486C2—21no—
487C2—21no—
488C2—21no—
489C2—161no—
490C2#7003611yescharacteristic
491C2#7021311yescharacteristic
492C2#7041011yescharacteristic
subgroupG/H
centerC22#4956
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1519C2
socleC22×C3#386
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#1519D3
hypocenterC3#9262
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1519

Complements

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

NH ≅ G/Ntype
2. #73876C2semidirect
5. #69963C2semidirect
6. #67678C2semidirect
7. #67625C2semidirect
11. C22×#214C4semidirect
25. C2×#810C4semidirect
162. #217SD16semidirect
276. Dic6#99semidirect
482. C3#9262semidirect

Maximal subgroups

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

subgroupindex
C2×#50402
#738762
C3×#15192
#716862
#699632
#676782
#676252
#92623