Finite Group Databasebeta

← #10831713

Subgroups — #10831713

Order 768.

1112 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic, complemented
2C2×C3×#474C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#5014C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#70638C212yesmaximal, complemented
7#70618C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic, complemented
9#70473C211yesmaximal, characteristic, complemented
10#70208C212yesmaximal
11#70164C212yesmaximal, complemented
12#24339—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#946—22no—
16C22×#213C2211yescharacteristic
17#5730—24no—
18#5730—24no—
19C2×#897—21no—
20C2×#893C2212yes—
21C2×D3×C4⋊C4—21no—
22C2×#890C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#474—24no—
26C2×C3×#107C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29#5221C2212yes—
30C2×#832C2211yescharacteristic
31#5210—22no—
32C2×#821C2212yes—
33#5014—24no—
34#4972C2212yes—
35C2×#809C2211yescharacteristic
36C2×#808C2211yescharacteristic
37#4949—21no—
38#4947—21no—
39#4936C2211yescharacteristic
40C2×#803C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474D311yescharacteristic, complemented
48C2×#442—31no—
49#2580—31no—
50#2059—32no—
51#2025—32no—
52#1993—31no—
53#1985—31no—
54#1879—32no—
55#1862—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12C2311yescharacteristic
61C22×C4×D3—21no—
62C22×C4×D3—22no—
63C22×Dic6D411yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yes—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76#946—216no—
77C2×#213—22no—
78C2×#213—22no—
79C2×#213Q814yescomplemented
80C2×(C2×D3)⋊C4—21no—
81C2×(C2×D3)⋊C4—22no—
82C2×C4⋊Dic3—22no—
83C2×C4⋊Dic3D411yescharacteristic
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4D412yes—
89C2×C4×Dic3C2312yes—
90C22×C3⋊C8C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92#897—28no—
93#893—24no—
94#893—24no—
95D3×C4⋊C4—28no—
96#890—24no—
97#890—24no—
98C4×D12—22no—
99C4×D12D414yes—
100C4×Dic6—22no—
101C4×Dic6—22no—
102C3×#107—24no—
103C3×C4⋊C8—22no—
104C3×#102—24no—
105#832—22no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—22no—
109#821—24no—
110#809—22no—
111#808—22no—
112#803—22no—
113C22×D8—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—32no—
118C2×C22⋊Q8—61no—
119C2×C22⋊Q8—62no—
120C2×C4×Q8—31no—
121C2×C4×D4—32no—
122C2×C4×D4—31no—
123C2×C4×D4C2×D311yescharacteristic
124C22×C4⋊C4—61no—
125#474—24no—
126#442—64no—
127C2×#107C2×D312yes—
128C2×#106—31no—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#102—31no—
131C2×#102—32no—
132C2×#102—31no—
133C2×#102C2×D312yes—
134C2×#97—31no—
135#399—61no—
136#393—61no—
137#390—32no—
138#389—32no—
139#385—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C24×C3—21no—
147C23×D3—21no—
148C2×C3×Q8—24no—
149C2×C3×Q8—82no—
150C2×C3×D4—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—21no—
155C2×C3×D4C2×Q812yes—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4—42no—
159C22×C3×C4—82no—
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2411yescharacteristic
163C22×Dic3—21no—
164C22×Dic3—22no—
165C22×Dic3C2×D412yes—
166C22×Dic3C2×D412yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12C2×D412yes—
171C2×D12C2×Q812yes—
172C2×D12#4112yes—
173C2×C4×D3—24no—
174C2×C4×D3—28no—
175C2×C4×D3—44no—
176C2×C4×D3—82no—
177C2×C4×D3—82no—
178C2×Dic6—22no—
179C2×Dic6—42no—
180C2×C3×C8—22no—
181C2×C3×C8—22no—
182C2×C3×C8—22no—
183C3×C4⋊C4—24no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4C2×D414yes—
188C3×C22⋊C4—42no—
189C3×C42C2×D412yes—
190C3×C42C2×D412yes—
191#217—44no—
192#217—82no—
193#213—28no—
194(C2×D3)⋊C4—28no—
195(C2×D3)⋊C4—24no—
196C4⋊Dic3—28no—
197C4⋊Dic3—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—22no—
200C4⋊Dic3—22no—
201C4⋊Dic3C2×D412yes—
202Dic3⋊C4—28no—
203Dic3⋊C4—24no—
204Dic3⋊C4—82no—
205C4×Dic3—22no—
206C4×Dic3—22no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—21no—
210C2×C3⋊C8—21no—
211C2×C3⋊C8—21no—
212C2×C3⋊C8C2×Q812yes—
213C22×Q8—21no—
214C22×Q8—31no—
215C22×D4—31no—
216C22×D4C22×D311yescharacteristic
217C23×C4—21no—
218C23×C4—62no—
219C23×C4—61no—
220C2×D8—34no—
221C2×D8—62no—
222C2×D8—62no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8C22×D312yes—
226C4⋊Q8—22no—
227C4⋊Q8#6714yes—
228#125—64no—
229#125—64no—
230C22⋊Q8—68no—
231C22⋊Q8—616no—
232C4×Q8—62no—
233C4×Q8—62no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—34no—
237C4×D4—64no—
238C4×D4—64no—
239C4×D4—62no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C4⋊C4—68no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4—62no—
249C2×C4⋊C4C22×D311yescharacteristic
250C2×C4⋊C4C22×D312yes—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—62no—
253C2×C22⋊C4—61no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256C2×C42—32no—
257C2×C42C22×D311yescharacteristic
258#107—24no—
259#106—62no—
260C4⋊C8—22no—
261#102—24no—
262#102—62no—
263#102—62no—
264#102—64no—
265#97—62no—
266#95—62no—
267#95—61no—
268#95—61no—
269C23×C3—22no—
270C23×C3—41no—
271C23×C3—41no—
272C23×C3—42no—
273C23×C3—42no—
274C23×C3C22×D411yescharacteristic
275C22×D3—22no—
276C22×D3—22no—
277C22×D3—22no—
278C22×D3—42no—
279C22×D3—42no—
280C3×Q8—84no—
281C3×D4—22no—
282C3×D4—22no—
283C3×D4—41no—
284C3×D4—41no—
285C2×C3×C4—22no—
286C2×C3×C4—22no—
287C2×C3×C4—24no—
288C2×C3×C4—42no—
289C2×C3×C4—42no—
290C2×C3×C4—41no—
291C2×C3×C4—42no—
292C2×C3×C4—84no—
293C2×C3×C4#12012yes—
294C2×C3×C4#12012yes—
295C2×C3×C4C22×D411yescharacteristic
296C2×C3×C4C22×D411yescharacteristic
297C2×C3×C4C22×Q811yescharacteristic
298C2×C3×C4C2×#4111yescharacteristic
299C2×C3×C42−1+411yescharacteristic
300C2×C3×C42−1+411yescharacteristic
301C2×Dic3—22no—
302C2×Dic3—22no—
303C2×Dic3—22no—
304C2×Dic3—22no—
305C2×Dic3—22no—
306C2×Dic3—22no—
307C2×Dic3—22no—
308C2×Dic3—42no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—42no—
312D12—22no—
313D12—22no—
314D12—22no—
315D12C22⋊Q814yes—
316C4×D3—48no—
317C4×D3—88no—
318Dic6—44no—
319C3×C8—42no—
320C3⋊C8—22no—
321C3⋊C8—41no—
322C24—21no—
323C24—61no—
324C2×Q8—24no—
325C2×Q8—62no—
326C2×Q8—82no—
327C2×Q8—122no—
328C2×D4—21no—
329C2×D4—21no—
330C2×D4—21no—
331C2×D4—21no—
332C2×D4—21no—
333C2×D4—32no—
334C2×D4—32no—
335C2×D4—32no—
336C2×D4—61no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4D3×Q812yes—
340C22×C4—22no—
341C22×C4—21no—
342C22×C4—32no—
343C22×C4—32no—
344C22×C4—42no—
345C22×C4—68no—
346C22×C4—61no—
347C22×C4—64no—
348C22×C4—62no—
349C22×C4—82no—
350C22×C4—124no—
351C22×C4—242no—
352C22×C4—242no—
353C22×C4D3×D411yescharacteristic
354C22×C4D3×D411yescharacteristic
355C22×C4C23×D311yescharacteristic
356D8—68no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—22no—
360C2×C8—32no—
361C2×C8—61no—
362C2×C8—61no—
363C2×C8—61no—
364C2×C8—61no—
365C2×C8—61no—
366C4⋊C4—24no—
367C4⋊C4—22no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—32no—
371C4⋊C4—68no—
372C4⋊C4—68no—
373C4⋊C4—62no—
374C4⋊C4—64no—
375C4⋊C4—61no—
376C4⋊C4—62no—
377C4⋊C4—62no—
378C4⋊C4—242no—
379C4⋊C4C2×#6714yes—
380C22⋊C4—42no—
381C22⋊C4—68no—
382C22⋊C4—64no—
383C22⋊C4—124no—
384C22⋊C4—242no—
385C42—62no—
386C42—62no—
387C42D3×D412yes—
388C42C2×#6712yes—
389C22×C3—44no—
390C22×C3—42no—
391C22×C3—82no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3C2×#12011yescharacteristic
395C22×C3#54811yescharacteristic
396C22×C3D4×Q811yescharacteristic
397C22×C3#57512yes—
398C22×C3#57612yes—
399C2×D3—24no—
400C2×D3—44no—
401C2×D3—44no—
402C2×D3—42no—
403C2×D3—42no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4—44no—
407C3×C4—81no—
408C3×C4C2×C22⋊Q811yescharacteristic
409C3×C4#53611yescharacteristic
410C3×C4#54811yescharacteristic
411C3×C4D4×Q811yescharacteristic
412Dic3—42no—
413Dic3—42no—
414Dic3—42no—
415Dic3—82no—
416C23—22no—
417C23—41no—
418C23—42no—
419C23—42no—
420C23—41no—
421C23—62no—
422C23—62no—
423C23—62no—
424C23—122no—
425C23—122no—
426C23C2×D3×D411yescharacteristic
427Q8—84no—
428Q8—124no—
429D4—22no—
430D4—22no—
431D4—34no—
432D4—41no—
433D4—41no—
434D4—62no—
435D4—62no—
436D4—62no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—24no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—62no—
450C2×C4—62no—
451C2×C4—84no—
452C2×C4—128no—
453C2×C4—122no—
454C2×C4—122no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—248no—
458C2×C4#93712yes—
459C2×C4#93712yes—
460C2×C4C2×D3×D411yescharacteristic
461C2×C4C2×D3×Q811yescharacteristic
462C2×C4#100711yescharacteristic
463C2×C4D3×#4111yescharacteristic
464C2×C4#101011yescharacteristic
465C2×C4C22×#6711yescharacteristic
466C8—42no—
467C8—62no—
468C8—121no—
469C2×C3—82no—
470C2×C3#240911yescharacteristic
471C2×C3#300511yescharacteristic
472C2×C3#300811yescharacteristic
473C2×C3#337012yes—
474C2×C3#338912yes—
475D3—44no—
476C22—42no—
477C22—44no—
478C22—64no—
479C22—81no—
480C22—82no—
481C22—81no—
482C22—122no—
483C22—124no—
484C22—124no—
485C22—122no—
486C22#563211yescharacteristic
487C22#563311yescharacteristic
488C22#575712yes—
489C22#577712yes—
490C22C2×#93711yescharacteristic
491C4—22no—
492C4—22no—
493C4—44no—
494C4—81no—
495C4—122no—
496C4—122no—
497C4—122no—
498C4—242no—
499C4D3×C22⋊Q811yescharacteristic
500C4#562811yescharacteristic
501C4Q8×#6711yescharacteristic
502C4#583111yescharacteristic
503C3#2433911yescharacteristic, complemented, Sylow
504C2—82no—
505C2—124no—
506C2#8095512yes—
507C2#8096212yes—
508C2#8275111yescharacteristic
509C2#8383311yescharacteristic
510C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#474C2
socleC23×C3C22×D4
prime radicalC22×#213C22
hypercenterC2×#474D3
hypocenterC3#24339
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×C3×#474C2semidirect
3. #82388C2semidirect
5. #71788C2semidirect
6. #70638C2semidirect
7. #70618C2semidirect
8. #70480C2semidirect
9. #70473C2semidirect
11. #70164C2semidirect
47. C2×#474D3semidirect
79. C2×#213Q8semidirect
503. C3#24339semidirect

Maximal subgroups

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

subgroupindex
C2×#57302
C2×C3×#4742
#823882
C2×#50142
#717882
#706382
#706182
#704802
#704732
#702082
#701642
#243393