Finite Group Databasebeta

← #10901425

Subgroups — #10901425

Order 768.

1304 conjugacy classes of subgroups.

The subgroup lattice has 562 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5477C211yesmaximal, characteristic
2#82074C211yesmaximal, characteristic, complemented
3C2×#4847C212yesmaximal, complemented
4C3×#1797C211yesmaximal, characteristic, complemented
5#70480C212yesmaximal, complemented
6#69935C212yesmaximal, complemented
7#69934C212yesmaximal, complemented
8#68633C211yesmaximal, characteristic, complemented
9#68632C211yesmaximal, characteristic, complemented
10#68629C211yesmaximal, characteristic, complemented
11#68629C211yesmaximal, characteristic, complemented
12#29694—31nomaximal, Sylow
13C22×#235—21no—
14C22×C3×C42C2211yescharacteristic
15C2×#930—21no—
16C2×#930C2212yes—
17#5477—28no—
18C2×#875—21no—
19C2×#874C2211yescharacteristic
20C2×C4×D12—21no—
21C2×C4×D12C2212yes—
22C2×#870C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C3×C4⋊C8C2212yes—
25C2×C3×C22⋊C8C2212yes—
26#5212C2211yescharacteristic
27#5212C2211yescharacteristic
28#5210—21no—
29#5210—21no—
30C2×#821C2212yes—
31C2×#821C2212yes—
32C2×#821C2212yes—
33C2×#821C2212yes—
34C2×#818C2212yes—
35C2×#817C2212yes—
36#4936C2211yescharacteristic
37#4933—21no—
38#4933—21no—
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41#4847—28no—
42C3×#336C2211yescharacteristic
43C3×#336C2211yescharacteristic
44#4551C2212yes—
45#4551C2212yes—
46C2×#532—31no—
47C2×#478—32no—
48#2851—31no—
49#2058—31no—
50#2057—31no—
51#2041—31no—
52#2041—31no—
53#2034—32no—
54#2033—32no—
55#1804—32no—
56#1797D311yescharacteristic, complemented
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×#67—22no—
61C2×#235—28no—
62C2×#235—44no—
63C2×#235—44no—
64C22×D12—21no—
65C22×D12—21no—
66C22×D12C2312yes—
67C22×C4×D3—22no—
68C22×Dic6—21no—
69C22×C3×C8C2312yes—
70C22×C3×C8C2312yes—
71C2×C3×C42—24no—
72C2×C3×C42D411yescharacteristic
73C2×C3×C42D411yescharacteristic
74C2×C3×C42C2311yescharacteristic
75C2×C3×C42C2311yescharacteristic
76C2×#217—21no—
77C2×#217—21no—
78#930—28no—
79#930—24no—
80#930—24no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3C2311yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2311yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×Dic3⋊C4—21no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93#875—28no—
94#874—24no—
95C4×D12—28no—
96C4×D12—24no—
97C4×D12D418yes—
98#870—24no—
99#869—24no—
100C3×C4⋊C8—24no—
101C3×C22⋊C8—42no—
102C3⋊#95—21no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108#821—24no—
109#821—24no—
110#821—42no—
111#821—42no—
112#818—24no—
113#817—24no—
114C22×#41—61no—
115#532—68no—
116C2×C4⋊Q8—31no—
117C2×#127—31no—
118C2×#125—31no—
119C2×#124—61no—
120C2×#121—32no—
121C2×#121—61no—
122C2×C4×D4—32no—
123C2×C4×D4—61no—
124C22×C42C2×D311yescharacteristic
125#478—68no—
126C2×#107—32no—
127C2×#106—32no—
128C2×C4⋊C8C2×D312yes—
129C2×#102—32no—
130C2×#102—32no—
131C2×#102—32no—
132C2×#102—32no—
133C2×C22⋊C8C2×D312yes—
134#389—31no—
135#389—31no—
136#387—61no—
137#387—61no—
138#385—61no—
139#385—61no—
140#382—31no—
141#380—31no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336C2×D311yescharacteristic
146#336C2×D311yescharacteristic
147C24×C3C2×D411yescharacteristic
148C23×D3—22no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4—22no—
158C22×C3×C4—41no—
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—28no—
166C2×#67—44no—
167C2×#67—44no—
168C22×Dic3—22no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173#235—432no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—24no—
179C2×D12—24no—
180C2×D12—22no—
181C2×D12—42no—
182C2×D12—42no—
183C2×D12—42no—
184C2×D12—42no—
185C2×D12C2×D414yes—
186C2×C4×D3—28no—
187C2×C4×D3—44no—
188C2×C4×D3—44no—
189C2×Dic6—24no—
190C2×Dic6—42no—
191C2×Dic6—42no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—22no—
198C3×C42—22no—
199C3×C42—22no—
200C3×C42—22no—
201C3×C42C2×D414yes—
202#217—42no—
203#217—42no—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—28no—
206C4⋊Dic3—24no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C2×D414yes—
215Dic3⋊C4—42no—
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218Dic3⋊C4—42no—
219C2×#41—68no—
220C2×#41—124no—
221C2×#41—124no—
222C22×Q8—61no—
223C22×D4—32no—
224C22×D4—62no—
225C22×D4—61no—
226C22×D4—61no—
227C23×C4—62no—
228C23×C4D1211yescharacteristic
229C23×C4D1211yescharacteristic
230C23×C4C22×D311yescharacteristic
231C22×C8C22×D312yes—
232C22×C8C22×D312yes—
233C4⋊Q8—64no—
234#127—64no—
235#125—64no—
236#124—68no—
237#121—64no—
238#121—64no—
239#121—68no—
240C4×D4—38no—
241C4×D4—68no—
242C4×D4—64no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—31no—
245C2×C4⋊C4—31no—
246C2×C4⋊C4—31no—
247C2×C4⋊C4—31no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C4⋊C4—61no—
252C2×C4⋊C4—61no—
253C2×C22⋊C4—62no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—61no—
256C2×C22⋊C4—61no—
257C2×C42—24no—
258C2×C42D1211yescharacteristic
259C2×C42D1211yescharacteristic
260C2×C42C22×D311yescharacteristic
261C2×C42C22×D311yescharacteristic
262#107—64no—
263#106—64no—
264C4⋊C8—24no—
265#102—64no—
266#102—64no—
267#102—122no—
268#102—122no—
269C22⋊C8—42no—
270#95—61no—
271#95—61no—
272#95—61no—
273#95—61no—
274#95—61no—
275#95—61no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3—21no—
279C23×C3—21no—
280C23×C3—21no—
281C23×C3—21no—
282C23×C3AGL(1,ℤ/8)12yes—
283C23×C3C22×D411yescharacteristic
284C22×D3—24no—
285C22×D3—42no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C2×C3×C4—28no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—44no—
304C2×C3×C4—42no—
305C2×C3×C4—42no—
306C2×C3×C4#12012yes—
307C2×C3×C4#12012yes—
308C2×C3×C4#13512yes—
309C2×C3×C4AGL(1,ℤ/8)12yes—
310C2×C3×C4C22×D412yes—
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C2×C3×C42−1+411yescharacteristic
313C2×C3×C42−1+412yes—
314#67—416no—
315C2×Dic3—24no—
316C2×Dic3—42no—
317C2×Dic3—42no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325C2×Dic3—41no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330D12—24no—
331D12—24no—
332D12—48no—
333D12—48no—
334D12—42no—
335D12—42no—
336C4×D3—416no—
337Dic6—48no—
338C3×C8—42no—
339C3×C8—42no—
340C24—62no—
341C24C2×D1211yescharacteristic
342#41—1232no—
343C2×Q8—64no—
344C2×Q8—122no—
345C2×Q8—122no—
346C2×D4—34no—
347C2×D4—64no—
348C2×D4—64no—
349C2×D4—68no—
350C2×D4—62no—
351C2×D4—62no—
352C2×D4—62no—
353C2×D4—62no—
354C2×D4—62no—
355C2×D4—122no—
356C2×D4—122no—
357C2×D4—124no—
358C2×D4—124no—
359C2×D4—122no—
360C2×D4—122no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—22no—
370C22×C4—41no—
371C22×C4—62no—
372C22×C4—68no—
373C22×C4—61no—
374C22×C4—61no—
375C22×C4—61no—
376C22×C4—61no—
377C22×C4—124no—
378C22×C4—124no—
379C22×C4C2×D1211yescharacteristic
380C22×C4C2×D1211yescharacteristic
381C22×C4C2×D1211yescharacteristic
382C22×C4C2×D1211yescharacteristic
383C22×C4C2×D1211yescharacteristic
384C22×C4C23×D311yescharacteristic
385C2×C8—22no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C4⋊C4—34no—
392C4⋊C4—64no—
393C4⋊C4—62no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4—61no—
397C4⋊C4—61no—
398C4⋊C4—61no—
399C4⋊C4—61no—
400C4⋊C4—122no—
401C4⋊C4—122no—
402C4⋊C4—122no—
403C4⋊C4—122no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—122no—
407C22⋊C4—122no—
408C42—22no—
409C42—22no—
410C42—22no—
411C42D3×D414yes—
412C22×C3—22no—
413C22×C3—22no—
414C22×C3—22no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3C2×#12011yescharacteristic
420C22×C3#53611yescharacteristic
421C22×C3#53711yescharacteristic
422C22×C3C2×#13511yescharacteristic
423C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
424C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
425C22×C3#57511yescharacteristic
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—44no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—82no—
432C2×D3—82no—
433C3×C4—22no—
434C3×C4—22no—
435C3×C4—22no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4#53612yes—
439C3×C4#53712yes—
440Dic3—44no—
441Dic3—81no—
442Dic3—81no—
443Dic3—81no—
444Dic3—81no—
445C23—21no—
446C23—21no—
447C23—21no—
448C23—21no—
449C23—21no—
450C23—21no—
451C23—64no—
452C23—122no—
453C23—122no—
454C23—122no—
455C23—122no—
456C23—122no—
457C23—122no—
458C23#90812yes—
459C23C22×D1211yescharacteristic
460Q8—128no—
461D4—64no—
462D4—64no—
463D4—128no—
464D4—128no—
465D4—1216no—
466D4—122no—
467D4—122no—
468C2×C4—28no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—21no—
474C2×C4—21no—
475C2×C4—21no—
476C2×C4—21no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—21no—
480C2×C4—44no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—64no—
484C2×C4—1216no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—121no—
488C2×C4—121no—
489C2×C4—121no—
490C2×C4—121no—
491C2×C4—121no—
492C2×C4—121no—
493C2×C4—121no—
494C2×C4—121no—
495C2×C4—121no—
496C2×C4—121no—
497C2×C4—121no—
498C2×C4—121no—
499C2×C4#88212yes—
500C2×C4#88212yes—
501C2×C4#90412yes—
502C2×C4#90812yes—
503C2×C4C2×D3×D412yes—
504C2×C4#100411yescharacteristic
505C2×C4#100711yescharacteristic
506C2×C4#101012yes—
507C8—42no—
508C8—42no—
509C2×C3—42no—
510C2×C3#261611yescharacteristic
511C2×C3#300211yescharacteristic
512C2×C3#300311yescharacteristic
513C2×C3#322811yescharacteristic
514C2×C3#323411yescharacteristic
515C2×C3#324711yescharacteristic
516C2×C3#325111yescharacteristic
517D3—84no—
518C22—22no—
519C22—22no—
520C22—22no—
521C22—41no—
522C22—41no—
523C22—41no—
524C22—41no—
525C22—124no—
526C22—124no—
527C22—124no—
528C22—242no—
529C22—242no—
530C22—242no—
531C22—242no—
532C22C2×#88211yescharacteristic
533C22#551011yescharacteristic
534C22#552611yescharacteristic
535C22C2×#90411yescharacteristic
536C22C2×#90811yescharacteristic
537C22C2×#90811yescharacteristic
538C22#574811yescharacteristic
539C4—22no—
540C4—22no—
541C4—22no—
542C4—22no—
543C4—41no—
544C4—124no—
545C4—241no—
546C4—241no—
547C4—241no—
548C4—241no—
549C4#560812yes—
550C4#562812yes—
551C3#2969411yescharacteristic, complemented, Sylow
552C2—42no—
553C2—244no—
554C2#7395011yescharacteristic
555C2#7861011yescharacteristic
556C2#7863711yescharacteristic
557C2#7902811yescharacteristic
558C2#7906211yescharacteristic
559C2#7927011yescharacteristic
560C2#7931811yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1797C2
socleC23×C3C22×D4
prime radicalC2×#5477C2
hypercenter#1797D3
hypocenterC3#29694
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
2. #82074C2semidirect
3. C2×#4847C2semidirect
4. C3×#1797C2semidirect
5. #70480C2semidirect
6. #69935C2semidirect
7. #69934C2semidirect
8. #68633C2semidirect
9. #68632C2semidirect
10. #68629C2semidirect
11. #68629C2semidirect
56. #1797D3semidirect
551. C3#29694semidirect

Maximal subgroups

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

subgroupindex
C2×#54772
#820742
C2×#48472
C3×#17972
#704802
#699352
#699342
#686332
#686322
#686292
#296943