Finite Group Databasebeta

← #10824339

Subgroups — #10824339

Order 768.

924 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4852C212yesmaximal
2#75961C211yesmaximal, characteristic
3#74320C211yesmaximal, characteristic, complemented
4C3×#2084C211yesmaximal, characteristic, complemented
5#70589C212yesmaximal, complemented
6#69633C211yesmaximal, characteristic, complemented
7#69585C211yesmaximal, characteristic
8#68863C212yesmaximal
9#68781C212yesmaximal, complemented
10#68687C211yesmaximal, characteristic
11#68602C211yesmaximal, characteristic, complemented
12#23677—31nomaximal, Sylow
13C22×#204C2212yes—
14C2×#893C2211yescharacteristic
15C2×#892—21no—
16C2×#887C2212yes—
17C2×#869C2211yescharacteristic
18C2×C3×#106C2212yes—
19C3×#389C2211yescharacteristic
20C3×#384C2211yescharacteristic
21C2×#816C2212yes—
22C2×Dic3⋊C8C2212yes—
23C2×C8×Dic3C2212yes—
24#4990—21no—
25#4988C2211yescharacteristic
26#4984C2211yescharacteristic
27Dic3×C4⋊C4—21no—
28Dic3×C4⋊C4—21no—
29Dic3×C4⋊C4C2211yescharacteristic
30#4973C2212yes—
31#4972—21no—
32#4972—21no—
33C2×#810C2212yes—
34C2×#810C2212yes—
35C2×#809C2212yes—
36#4939C2211yescharacteristic
37C42×Dic3C2211yescharacteristic
38#4852—28no—
39#4670—21no—
40#4666—21no—
41C3×#340C2212yes—
42C3×#336C2211yescharacteristic
43#4529C2212yes—
44#4524C2211yescharacteristic
45C2×C4×SD16—32no—
46C4×C4⋊Q8—31no—
47#2293—31no—
48#2084D311yescharacteristic, complemented
49#2071—31no—
50#2066—32no—
51#2055—32no—
52#2043—31no—
53#2013—31no—
54C4×C4⋊C8—31no—
55C4×#103—32no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—21no—
58C22×Dic6C2312yes—
59C22×C3×C8C2312yes—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×(C2×D3)⋊C4—21no—
68C2×(C2×D3)⋊C4—21no—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—21no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×Dic3⋊C4—22no—
77C2×C4×Dic3—21no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3D411yescharacteristic
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3D411yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C2×C4×Dic3C2311yescharacteristic
85C22×C3⋊C8C2312yes—
86C2×#204—24no—
87C2×#204—24no—
88C2×#204—24no—
89#893—22no—
90#893D414yes—
91#892—44no—
92#887—24no—
93#887—24no—
94#869—24no—
95C3×#106—24no—
96C3×#95—21no—
97C3⋊#95—21no—
98C3⋊#95—21no—
99C3⋊#95—22no—
100C3⋊#95—22no—
101#816—24no—
102Dic3⋊C8—24no—
103C8×Dic3—24no—
104#810—24no—
105#810—24no—
106#809—24no—
107C22×SD16—32no—
108C2×C4⋊Q8—31no—
109C2×C4×Q8—32no—
110C2×C4×D4—31no—
111C2×#117—61no—
112C4×SD16—68no—
113C2×#106C2×D312yes—
114C2×C4⋊C8—32no—
115C2×#103—32no—
116C2×#103—32no—
117C2×#103—32no—
118C2×#102—32no—
119C2×C4×C8—32no—
120#391—32no—
121#390—31no—
122#390—31no—
123#389—31no—
124#389—61no—
125#389—61no—
126#389C2×D311yescharacteristic
127#388—61no—
128#388—61no—
129#384C2×D311yescharacteristic
130C4×C4⋊C4—31no—
131C4×C4⋊C4—61no—
132C4×C4⋊C4—61no—
133C4×C22⋊C4—61no—
134C43—31no—
135#340—32no—
136#340C2×D312yes—
137#336—31no—
138#336C2×D311yescharacteristic
139C23×D3—21no—
140C22×C3×C4—21no—
141C22×C3×C4—22no—
142C22×C3×C4—21no—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C22×Dic3—22no—
147C22×Dic3—21no—
148C22×Dic3C2×D411yescharacteristic
149C22×Dic3C2×D411yescharacteristic
150C22×Dic3C2×D411yescharacteristic
151C22×Dic3C2×D411yescharacteristic
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—21no—
155C2×D12C2×D412yes—
156C2×C4×D3—24no—
157C2×C4×D3—42no—
158C2×C4×D3—42no—
159C2×Dic6—22no—
160C2×Dic6—22no—
161C2×Dic6—22no—
162C2×Dic6—22no—
163C2×Dic6—22no—
164C2×Dic6#4114yes—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8#4114yes—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C4⋊C4—42no—
173C3×C4⋊C4—42no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4C2×D412yes—
176C3×C42—22no—
177(C2×D3)⋊C4—24no—
178(C2×D3)⋊C4—42no—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—42no—
181(C2×D3)⋊C4—42no—
182C4⋊Dic3—24no—
183C4⋊Dic3—24no—
184C4⋊Dic3—22no—
185C4⋊Dic3—42no—
186Dic3⋊C4—44no—
187C4×Dic3—22no—
188C4×Dic3—22no—
189C4×Dic3—22no—
190C4×Dic3—41no—
191C4×Dic3—44no—
192C4×Dic3—42no—
193C4×Dic3SD1614yescomplemented
194C4×Dic3C2×D412yes—
195C4×Dic3#4112yes—
196C2×C3⋊C8—22no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8—22no—
199#204—216no—
200C22×Q8—32no—
201C22×D4—31no—
202C23×C4—61no—
203C2×SD16—64no—
204C2×SD16—64no—
205C2×SD16—64no—
206C22×C8—32no—
207C22×C8C22×D312yes—
208C4⋊Q8—64no—
209C4×Q8—64no—
210C4×Q8—64no—
211C4×D4—34no—
212C4×D4—62no—
213#117—124no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—31no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—61no—
220C2×C4⋊C4—61no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C4⋊C4C22×D311yescharacteristic
225C2×C22⋊C4—61no—
226C2×C22⋊C4—61no—
227C2×C22⋊C4—61no—
228C2×C22⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42—31no—
232C2×C42—31no—
233C2×C42—31no—
234C2×C42—31no—
235C2×C42—31no—
236C2×C42—61no—
237C2×C42—62no—
238C2×C42C22×D311yescharacteristic
239#106—24no—
240C4⋊C8—64no—
241#103—64no—
242#103—64no—
243#103—64no—
244#102—64no—
245C4×C8—64no—
246#95—21no—
247#95—61no—
248#95—61no—
249#95—62no—
250#95—62no—
251C23×C3C22×D411yescharacteristic
252C22×D3—22no—
253C22×D3—41no—
254C22×D3—41no—
255C22×D3—41no—
256C22×D3—41no—
257C22×D3—41no—
258C22×D3—41no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—42no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4—42no—
272C2×C3×C4C2×SD1612yes—
273C2×C3×C4#13512yes—
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4112yes—
276C2×C3×C4C2×#4112yes—
277C2×C3×C4C2×#4111yescharacteristic
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—21no—
282C2×Dic3—24no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—21no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—41no—
289C2×Dic3—41no—
290C2×Dic3—41no—
291C2×Dic3#12112yes—
292C2×Dic3#12112yes—
293C2×Dic3C2×SD1612yes—
294C2×Dic3C2×SD1612yes—
295C2×Dic3#13712yes—
296D12—24no—
297D12—42no—
298C4×D3—48no—
299Dic6—24no—
300Dic6—24no—
301Dic6—42no—
302Dic6—42no—
303C3×C8—24no—
304C3⋊C8—42no—
305C24—61no—
306C2×Q8—34no—
307C2×Q8—62no—
308C2×Q8—62no—
309C2×Q8—62no—
310C2×Q8—62no—
311C2×Q8—62no—
312C2×D4—32no—
313C2×D4—62no—
314C2×D4—62no—
315C2×D4—61no—
316C22×C4—21no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—31no—
320C22×C4—31no—
321C22×C4—31no—
322C22×C4—31no—
323C22×C4—64no—
324C22×C4—62no—
325C22×C4—61no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4D3×D411yescharacteristic
329C22×C4D3×D411yescharacteristic
330C22×C4C23×D311yescharacteristic
331SD16—616no—
332C2×C8—22no—
333C2×C8—22no—
334C2×C8—62no—
335C2×C8—62no—
336C2×C8—62no—
337C2×C8#23914yes—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—42no—
344C4⋊C4—42no—
345C4⋊C4—64no—
346C4⋊C4—64no—
347C4⋊C4—62no—
348C4⋊C4—122no—
349C4⋊C4—124no—
350C4⋊C4D3×D412yes—
351C22⋊C4—64no—
352C22⋊C4—122no—
353C22⋊C4—122no—
354C22⋊C4—122no—
355C22⋊C4—122no—
356C42—22no—
357C42—34no—
358C42—32no—
359C42—32no—
360C42—62no—
361C42—62no—
362C42—62no—
363C42—121no—
364C42—122no—
365C42—124no—
366C22×C3C2×#12111yescharacteristic
367C22×C3#52511yescharacteristic
368C22×C3#53211yescharacteristic
369C22×C3C22×SD1611yescharacteristic
370C22×C3C2×#13511yescharacteristic
371C22×C3C2×#13711yescharacteristic
372C22×C3#57511yescharacteristic
373C2×D3—42no—
374C2×D3—42no—
375C2×D3—42no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C2×D3—81no—
380C3×C4—22no—
381C3×C4—22no—
382C3×C4—42no—
383C3×C4—81no—
384C3×C4—82no—
385C3×C4#52512yes—
386C3×C4#52912yes—
387Dic3—22no—
388Dic3—22no—
389Dic3—44no—
390Dic3—41no—
391Dic3—81no—
392Dic3#46114yes—
393C23—62no—
394C23—121no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23—121no—
399C23—121no—
400C23C2×D3×D411yescharacteristic
401Q8—64no—
402Q8—64no—
403Q8—122no—
404Q8—122no—
405D4—64no—
406D4—122no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—22no—
412C2×C4—32no—
413C2×C4—32no—
414C2×C4—32no—
415C2×C4—32no—
416C2×C4—32no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—42no—
424C2×C4—42no—
425C2×C4—61no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—61no—
429C2×C4—64no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—128no—
434C2×C4—122no—
435C2×C4—122no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4D3×SD1612yes—
440C2×C4#91612yes—
441C2×C4C2×D3×D411yescharacteristic
442C2×C4C2×#23912yes—
443C2×C4D3×#4111yescharacteristic
444C2×C4D3×#4112yes—
445C8—24no—
446C8—122no—
447C2×C3#239211yescharacteristic
448C2×C3C2×#46111yescharacteristic
449C2×C3#303511yescharacteristic
450C2×C3#310311yescharacteristic
451C2×C3#310811yescharacteristic
452C2×C3#316711yescharacteristic
453C2×C3#317211yescharacteristic
454D3—82no—
455C22—122no—
456C22—122no—
457C22—122no—
458C22—241no—
459C22—241no—
460C22—241no—
461C22—241no—
462C22D3×#12111yescharacteristic
463C22#565611yescharacteristic
464C22#572511yescharacteristic
465C22C2×D3×SD1611yescharacteristic
466C22C2×#91611yescharacteristic
467C22#577311yescharacteristic
468C22D3×#13711yescharacteristic
469C4—22no—
470C4—22no—
471C4—34no—
472C4—42no—
473C4—62no—
474C4—62no—
475C4—81no—
476C4—82no—
477C4—121no—
478C4—124no—
479C4—241no—
480C4#562712yes—
481C4#569112yes—
482C3#2367711yescharacteristic, complemented, Sylow
483C2—242no—
484C2#7773711yescharacteristic
485C2#8038211yescharacteristic
486C2D3×#46111yescharacteristic
487C2#8130811yescharacteristic
488C2#8141311yescharacteristic
489C2#8188611yescharacteristic
490C2#8195911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2084C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2084D3
hypocenterC3#23677
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2084

Complements

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

NH ≅ G/Ntype
3. #74320C2semidirect
4. C3×#2084C2semidirect
5. #70589C2semidirect
6. #69633C2semidirect
9. #68781C2semidirect
11. #68602C2semidirect
48. #2084D3semidirect
193. C4×Dic3SD16semidirect
482. C3#23677semidirect

Maximal subgroups

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

subgroupindex
C2×#48522
#759612
#743202
C3×#20842
#705892
#696332
#695852
#688632
#687812
#686872
#686022
#236773