Finite Group Databasebeta

← #10845928

Subgroups — #10845928

Order 768.

1018 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4887C212yesmaximal, complemented
2C2×#4849C211yesmaximal, characteristic, complemented
3C2×#4834C211yesmaximal, characteristic
4#74369C211yesmaximal, characteristic
5#73800C211yesmaximal, characteristic
6C3×#1947C211yesmaximal, characteristic, complemented
7#70635C212yesmaximal, complemented
8#69579C211yesmaximal, characteristic, complemented
9C4×#810C211yesmaximal, characteristic, complemented
10#68888C212yesmaximal, complemented
11#68882C212yesmaximal, complemented
12#25386—31nomaximal, Sylow
13C22×Dic12C2212yes—
14C22×D3×C8C2212yes—
15C2×#897C2212yes—
16C2×#897C2212yes—
17C2×#897C2212yes—
18C2×C4×D12C2211yescharacteristic
19C2×#869C2211yescharacteristic
20C2×C4×Dic6C2211yescharacteristic
21C2×C3×#107C2212yes—
22C2×C3×C4⋊C8C2211yescharacteristic
23C3×#384C2211yescharacteristic
24C3×C4×C4⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C8C2212yes—
26C2×#816C2212yes—
27#4994—22no—
28#4988—21no—
29#4987C2212yes—
30#4984—21no—
31C2×#810C2212yes—
32C2×#810C2211yescharacteristic
33C2×#810C2212yes—
34C2×#810C2211yescharacteristic
35#4939—21no—
36#4935—21no—
37C2×C4×C3⋊C8C2211yescharacteristic
38#4887—28no—
39#4849—24no—
40#4834—24no—
41#4677—22no—
42C3×#340C2212yes—
43#4529C2211yescharacteristic
44#4529C2211yescharacteristic
45C2×#467—32no—
46C2×#462—31no—
47C2×C8×D4—31no—
48#2542—31no—
49#2538—31no—
50#2036—32no—
51#2001—32no—
52#1956—31no—
53#1947D311yescharacteristic, complemented
54#1892—32no—
55C4×#103—31no—
56C22×D12D411yescharacteristic
57C22×C4×D3C2312yes—
58C22×Dic6C2312yes—
59C22×Dic6C2311yescharacteristic
60C22×C3×C8C2312yes—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C42—21no—
67C2×C3×C42C2311yescharacteristic
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4D412yes—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3—22no—
76C2×C4⋊Dic3—22no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3D411yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—21no—
81C2×C4×Dic3—21no—
82C22×C3⋊C8C2311yescharacteristic
83C22×C3⋊C8C2311yescharacteristic
84C2×Dic12—24no—
85C2×Dic12—24no—
86C2×Dic12—42no—
87C2×D3×C8—24no—
88C2×D3×C8—24no—
89C2×D3×C8—42no—
90#897—24no—
91#897—24no—
92#897—24no—
93#897—42no—
94#897—42no—
95#897—42no—
96C4×D12—22no—
97C4×D12—22no—
98#869—22no—
99#869—22no—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×#107—24no—
103C3×C4⋊C8D414yes—
104C3×#95—21no—
105C3⋊#95—22no—
106(C2×D3)⋊C8—24no—
107#816—24no—
108#810—22no—
109#810—24no—
110#810—42no—
111#810D414yescomplemented
112C4×C3⋊C8—22no—
113C22×Q16—32no—
114C23×C8—32no—
115C2×C4⋊Q8—31no—
116C2×C22⋊Q8—32no—
117C2×C22⋊Q8—32no—
118C2×C22⋊Q8—32no—
119C2×C4×Q8—31no—
120C2×C4×D4—31no—
121#467—68no—
122#462—64no—
123C8×D4—64no—
124C2×#107C2×D312yes—
125C2×C4⋊C8C2×D311yescharacteristic
126C2×#103—31no—
127C2×#103—32no—
128C2×#103—32no—
129C2×#103—31no—
130C2×#103—32no—
131C2×C22⋊C8—32no—
132C2×C4×C8—31no—
133#399—62no—
134#396—62no—
135#390—61no—
136#390—61no—
137#389—61no—
138#384—61no—
139#384C2×D311yescharacteristic
140#380—32no—
141C4×C4⋊C4C2×D311yescharacteristic
142#340—31no—
143#340—31no—
144#340C2×D312yes—
145C23×D3C2×D412yes—
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—22no—
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2411yescharacteristic
152C22×Dic3—21no—
153C22×Dic3—21no—
154C22×Dic3—22no—
155C22×Dic3C2×D412yes—
156C2×D12—22no—
157C2×D12—22no—
158C2×D12Q1614yescomplemented
159C2×C4×D3—22no—
160C2×C4×D3—22no—
161C2×C4×D3—22no—
162C2×C4×D3—22no—
163C2×C4×D3—22no—
164C2×C4×D3—22no—
165C2×Dic6—21no—
166C2×Dic6—21no—
167C2×Dic6—21no—
168C2×Dic6—21no—
169C2×Dic6—21no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—22no—
176C2×Dic6C2×D412yes—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8C2×D414yes—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—22no—
183C3×C4⋊C4—22no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4—81no—
186C3×C4⋊C4C2×D412yes—
187C3×C4⋊C4#4112yes—
188C3×C42—42no—
189C3×C42C2×D412yes—
190C3×C42#4112yes—
191(C2×D3)⋊C4—24no—
192(C2×D3)⋊C4—44no—
193(C2×D3)⋊C4—44no—
194(C2×D3)⋊C4—44no—
195(C2×D3)⋊C4—82no—
196C4⋊Dic3—22no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—44no—
200C4⋊Dic3—44no—
201C4⋊Dic3—44no—
202C4⋊Dic3—42no—
203C4⋊Dic3—42no—
204Dic3⋊C4—42no—
205C4×Dic3—42no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8C2×D412yes—
212Dic12—48no—
213D3×C8—48no—
214C22×Q8—31no—
215C22×Q8—32no—
216C22×D4—31no—
217C23×C4—32no—
218C2×Q16—64no—
219C2×Q16—64no—
220C2×Q16—122no—
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—64no—
224C22×C8—64no—
225C22×C8—122no—
226C22×C8C22×D312yes—
227C4⋊Q8—62no—
228C4⋊Q8—62no—
229C22⋊Q8—64no—
230C22⋊Q8—64no—
231C22⋊Q8—64no—
232C22⋊Q8—122no—
233C22⋊Q8—122no—
234C22⋊Q8—122no—
235C4×Q8—62no—
236C4×Q8—62no—
237C4×D4—62no—
238C4×D4—62no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4C22×D312yes—
250C2×C4⋊C4C22×D311yescharacteristic
251C2×C4⋊C4C22×D311yescharacteristic
252C2×C22⋊C4—32no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C22⋊C4—62no—
257C2×C42—21no—
258C2×C42—61no—
259C2×C42C22×D311yescharacteristic
260#107—24no—
261C4⋊C8D1214yes—
262#103—34no—
263#103—64no—
264#103—64no—
265#103—62no—
266#103—122no—
267C22⋊C8—64no—
268C4×C8—62no—
269#95—21no—
270#95—62no—
271C23×C3C22×D411yescharacteristic
272C22×D3—22no—
273C22×D3—22no—
274C22×D3—22no—
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3C2×Q1614yes—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—42no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4#12112yes—
291C2×C3×C4#12112yes—
292C2×C3×C4C2×Q1612yes—
293C2×C3×C4#13512yes—
294C2×C3×C4C22×D411yescharacteristic
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C2×#4111yescharacteristic
297C2×C3×C4C2×#4111yescharacteristic
298C2×C3×C4H(5,𝔽2)11yescharacteristic
299C2×C3×C4H(5,𝔽2)11yescharacteristic
300C2×Dic3—22no—
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—41no—
305C2×Dic3—41no—
306C2×Dic3—41no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312D12—28no—
313C4×D3—42no—
314C4×D3—42no—
315C4×D3—42no—
316C4×D3—42no—
317Dic6—22no—
318Dic6—22no—
319Dic6—41no—
320Dic6—41no—
321Dic6—42no—
322Dic6—42no—
323Dic6—42no—
324Dic6—42no—
325C3×C8—24no—
326C3⋊C8—22no—
327C3⋊C8—41no—
328C24—32no—
329C2×Q8—32no—
330C2×Q8—61no—
331C2×Q8—61no—
332C2×Q8—62no—
333C2×Q8—62no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—61no—
339C2×Q8—62no—
340C2×Q8—62no—
341C2×D4—34no—
342C2×D4—62no—
343C2×D4—62no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—32no—
348C22×C4—61no—
349C22×C4—62no—
350C22×C4—62no—
351C22×C4—62no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—62no—
355C22×C4—61no—
356C22×C4—62no—
357C22×C4D3×D411yescharacteristic
358C22×C4D3×D411yescharacteristic
359C22×C4C23×D311yescharacteristic
360Q16—128no—
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—32no—
364C2×C8—61no—
365C2×C8—61no—
366C2×C8—61no—
367C2×C8—61no—
368C2×C8—61no—
369C2×C8—128no—
370C2×C8C2×D1214yes—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—22no—
374C4⋊C4—22no—
375C4⋊C4—42no—
376C4⋊C4—62no—
377C4⋊C4—61no—
378C4⋊C4—61no—
379C4⋊C4—81no—
380C4⋊C4—124no—
381C4⋊C4—124no—
382C4⋊C4—124no—
383C4⋊C4—122no—
384C4⋊C4—122no—
385C4⋊C4—122no—
386C4⋊C4D3×D412yes—
387C4⋊C4#23912yes—
388C22⋊C4—64no—
389C22⋊C4—124no—
390C22⋊C4—124no—
391C22⋊C4—124no—
392C22⋊C4—242no—
393C42—42no—
394C42—122no—
395C42C2×D1212yes—
396C42#23912yes—
397C22×C3C2×#12111yescharacteristic
398C22×C3D4211yescharacteristic
399C22×C3#54611yescharacteristic
400C22×C3C22×Q1611yescharacteristic
401C22×C3C2×#13511yescharacteristic
402C22×C3#57711yescharacteristic
403C22×C3#57811yescharacteristic
404C2×D3—24no—
405C2×D3—24no—
406C2×D3—24no—
407C2×D3—42no—
408C2×D3—42no—
409C2×D3—42no—
410C2×D3—42no—
411C3×C4—22no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—82no—
415C3×C4#46714yescomplemented
416C3×C4#53211yescharacteristic
417C3×C4#54011yescharacteristic
418C3×C4D4211yescharacteristic
419C3×C4#55011yescharacteristic
420Dic3—42no—
421Dic3—42no—
422Dic3—82no—
423Dic3—81no—
424C23—34no—
425C23—62no—
426C23—62no—
427C23—62no—
428C23—62no—
429C23—62no—
430C23—62no—
431C23C2×D3×D411yescharacteristic
432Q8—62no—
433Q8—62no—
434Q8—121no—
435Q8—121no—
436Q8—122no—
437Q8—122no—
438Q8—122no—
439Q8—122no—
440D4—68no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—62no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—121no—
457C2×C4—121no—
458C2×C4—122no—
459C2×C4—122no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—121no—
463C2×C4—122no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—122no—
467C2×C4—122no—
468C2×C4#89512yes—
469C2×C4#89512yes—
470C2×C4#91212yes—
471C2×C4D3×Q1612yes—
472C2×C4C22×D1211yescharacteristic
473C2×C4C2×D3×D411yescharacteristic
474C2×C4C2×#23911yescharacteristic
475C2×C4C2×#23911yescharacteristic
476C2×C4#100911yescharacteristic
477C2×C4#100911yescharacteristic
478C8—24no—
479C8—62no—
480C8—121no—
481C2×C3#242811yescharacteristic
482C2×C3C2×#46711yescharacteristic
483C2×C3#306311yescharacteristic
484C2×C3#328411yescharacteristic
485C2×C3D4×Q1611yescharacteristic
486C2×C3#329911yescharacteristic
487C2×C3#330211yescharacteristic
488D3—44no—
489C22—64no—
490C22—64no—
491C22—64no—
492C22—122no—
493C22—122no—
494C22—122no—
495C22—122no—
496C22C2×#89511yescharacteristic
497C22#560911yescharacteristic
498C22#563011yescharacteristic
499C22C2×#91211yescharacteristic
500C22C2×D3×Q1611yescharacteristic
501C22#577511yescharacteristic
502C22#577911yescharacteristic
503C4—22no—
504C4—42no—
505C4—42no—
506C4—82no—
507C4—122no—
508C4—122no—
509C4—241no—
510C4—242no—
511C4#488714yescomplemented
512C4D4×D1211yescharacteristic
513C4#557711yescharacteristic
514C4#569711yescharacteristic
515C4#572511yescharacteristic
516C3#2538611yescharacteristic, complemented, Sylow
517C2—124no—
518C2#7622011yescharacteristic
519C2C2×#488711yescharacteristic
520C2#7921411yescharacteristic
521C2#8035111yescharacteristic
522C2#8049011yescharacteristic
523C2#8098111yescharacteristic
524C2#8111911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1947C2
socleC23×C3C22×D4
prime radicalC22×D12D4
hypercenter#1947D3
hypocenterC3#25386
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
1. C2×#4887C2semidirect
2. C2×#4849C2semidirect
6. C3×#1947C2semidirect
7. #70635C2semidirect
8. #69579C2semidirect
9. C4×#810C2semidirect
10. #68888C2semidirect
11. #68882C2semidirect
53. #1947D3semidirect
111. #810D4semidirect
158. C2×D12Q16semidirect
415. C3×C4#467semidirect
511. C4#4887semidirect
516. C3#25386semidirect

Maximal subgroups

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

subgroupindex
C2×#48872
C2×#48492
C2×#48342
#743692
#738002
C3×#19472
#706352
#695792
C4×#8102
#688882
#688822
#253863