Finite Group Databasebeta

← #10829237

Subgroups — #10829237

Order 768.

1204 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5563C211yesmaximal, characteristic, complemented
2#82415C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4C2×#4711C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70480C211yesmaximal, characteristic, complemented
7#69961C212yesmaximal, complemented
8#69934C212yesmaximal, complemented
9#69933C212yesmaximal, complemented
10#69929C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24129—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×#938—21no—
15C2×#930C2212yes—
16C2×#930C2212yes—
17C2×#926C2212yes—
18C22×D24C2211yescharacteristic
19#5563—24no—
20#5563—24no—
21C2×#892—21no—
22C2×#885—22no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×C22⋊C8C2212yes—
28C2×C3×C22⋊C8C2212yes—
29C2×C3×#97C2211yescharacteristic
30#5223—21no—
31#5212C2212yes—
32C2×#821C2211yescharacteristic
33C2×#821C2212yes—
34C2×#821C2211yescharacteristic
35C2×#821C2212yes—
36C2×#818C2212yes—
37C2×#817C2211yescharacteristic
38#4992—21no—
39#4956—22no—
40#4936C2211yescharacteristic
41#4846—24no—
42#4711—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#555—31no—
47C2×#474—31no—
48C2×#442—31no—
49#2535—31no—
50#2033—32no—
51#1999—32no—
52#1993—31no—
53#1955D311yescharacteristic, complemented
54#1875—32no—
55#1854—32no—
56#1804—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C22×#67—22no—
61C22×D12C2311yescharacteristic
62C22×D12C2311yescharacteristic
63C22×C4×D3—21no—
64C22×C4×D3—21no—
65C22×Dic6—21no—
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2312yes—
68C22×C3×C8C2311yescharacteristic
69C3×C4×D4—22no—
70C3×C4×D4—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C22⋊C4D412yes—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75C2×#217—22no—
76#938—28no—
77#930—24no—
78#930—24no—
79#930—24no—
80#930—42no—
81C2×(C2×D3)⋊C4—21no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—21no—
85#926—24no—
86#926—24no—
87C2×C4⋊Dic3—21no—
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2311yescharacteristic
92C2×C4×Dic3—22no—
93C2×D24—22no—
94C2×D24—22no—
95C2×D24—22no—
96#892—28no—
97#885—216no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12D414yescomplemented
102#869—22no—
103#869D414yescomplemented
104C3×C4⋊C8—22no—
105C3×C22⋊C8—24no—
106C3×C22⋊C8—42no—
107C3×#97—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111#821—22no—
112#821—22no—
113#821—24no—
114#821—42no—
115#818—24no—
116#817—22no—
117C22×D8—31no—
118#555—64no—
119#555—64no—
120C2×C4⋊Q8—31no—
121C2×#124—62no—
122C2×#121—32no—
123C2×#121—32no—
124C2×#121—61no—
125C2×C4×D4—31no—
126C2×C4×D4—31no—
127C2×C4×D4C2×D311yescharacteristic
128C2×#117—32no—
129C2×#117—61no—
130#474—64no—
131#442—64no—
132C2×#107—32no—
133C2×#106—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#102—31no—
136C2×#102—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8C2×D312yes—
140C2×C22⋊C8C2×D312yes—
141C2×#97C2×D311yescharacteristic
142#397—61no—
143#392—61no—
144#386—62no—
145#382—31no—
146#380—32no—
147#340—32no—
148#340—32no—
149#340—32no—
150C24×C3C2×D412yes—
151C23×D3—21no—
152C23×D3—21no—
153C2×C3×D4—22no—
154C2×C3×D4—42no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D411yescharacteristic
164C22×C3×C4C2411yescharacteristic
165C2×#67—28no—
166C2×#67—44no—
167C2×#67—44no—
168C2×#67—82no—
169C2×#67—82no—
170C2×#67—82no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3—22no—
174C22×Dic3—21no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12—21no—
181C2×D12—21no—
182C2×D12—21no—
183C2×D12—21no—
184C2×D12—21no—
185C2×D12C2×D412yes—
186C2×D12#4112yes—
187C2×C4×D3—24no—
188C2×C4×D3—44no—
189C2×C4×D3—42no—
190C2×C4×D3—82no—
191C2×Dic6—24no—
192C2×Dic6—82no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—22no—
196C2×C3×C8—21no—
197C2×C3×C8—22no—
198C2×C3×C8—22no—
199C2×C3×C8—21no—
200C2×C3×C8—21no—
201C2×C3×C8#4112yes—
202C3×C4⋊C4—22no—
203C3×C22⋊C4—24no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#217—28no—
207#217—82no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—44no—
211(C2×D3)⋊C4—42no—
212C4⋊Dic3—24no—
213C4⋊Dic3—22no—
214C4⋊Dic3—21no—
215C4⋊Dic3—21no—
216C4⋊Dic3—21no—
217C4⋊Dic3—22no—
218C4⋊Dic3—22no—
219C4⋊Dic3C2×D412yes—
220C4⋊Dic3C2×D414yes—
221C4×Dic3—28no—
222D24—28no—
223C22×Q8—61no—
224C22×D4—31no—
225C22×D4—31no—
226C22×D4—62no—
227C22×D4—62no—
228C22×D4D1211yescharacteristic
229C23×C4—61no—
230C23×C4—61no—
231C23×C4C22×D312yes—
232C2×D8—62no—
233C2×D8—62no—
234C2×D8—62no—
235C22×C8C22×D311yescharacteristic
236C22×C8C22×D312yes—
237C22×C8C22×D311yescharacteristic
238C4⋊Q8—34no—
239C4⋊Q8—62no—
240#124—616no—
241#121—64no—
242#121—64no—
243#121—68no—
244#121—64no—
245#121—122no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—34no—
249C4×D4—62no—
250C4×D4—62no—
251C4×D4—62no—
252#117—68no—
253#117—64no—
254#117—64no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4D1211yescharacteristic
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—61no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4D1212yes—
268C2×C42—62no—
269C2×C42C22×D311yescharacteristic
270#107—64no—
271#106—62no—
272C4⋊C8—22no—
273#102—62no—
274#102—64no—
275#102—62no—
276#102—122no—
277C22⋊C8—24no—
278C22⋊C8—42no—
279#97—22no—
280#95—62no—
281#95—61no—
282#95—61no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3—22no—
289C23×C3—22no—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—41no—
296C22×D3—41no—
297C22×D3—42no—
298C22×D3—42no—
299C22×D3—42no—
300C22×D3—42no—
301C3×D4—44no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—42no—
310C2×C3×C4—42no—
311C2×C3×C4#12012yes—
312C2×C3×C4#12012yes—
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C22×D411yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4C2×#4111yescharacteristic
317C2×C3×C4H(5,𝔽2)11yescharacteristic
318C2×C3×C42−1+411yescharacteristic
319#67—88no—
320#67—88no—
321C2×Dic3—22no—
322C2×Dic3—24no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—41no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331D12—22no—
332D12—22no—
333D12—22no—
334D12—22no—
335D12—41no—
336D12—41no—
337D12—41no—
338D12—41no—
339C4×D3—48no—
340C4×D3—84no—
341Dic6—84no—
342C3×C8—22no—
343C3×C8—41no—
344C3×C8—42no—
345C24—61no—
346C24—61no—
347C24C2×D1212yes—
348C2×Q8—64no—
349C2×Q8—242no—
350C2×D4—22no—
351C2×D4—32no—
352C2×D4—32no—
353C2×D4—42no—
354C2×D4—68no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—61no—
360C2×D4—61no—
361C2×D4—61no—
362C2×D4—61no—
363C2×D4—61no—
364C2×D4—61no—
365C2×D4—124no—
366C2×D4—124no—
367C2×D4—242no—
368C2×D4—242no—
369C2×D4—242no—
370C22×C4—22no—
371C22×C4—22no—
372C22×C4—22no—
373C22×C4—22no—
374C22×C4—22no—
375C22×C4—22no—
376C22×C4—61no—
377C22×C4—62no—
378C22×C4—64no—
379C22×C4—62no—
380C22×C4—61no—
381C22×C4—124no—
382C22×C4—122no—
383C22×C4—242no—
384C22×C4C2×D1212yes—
385C22×C4C2×D1211yescharacteristic
386C22×C4C2×D1211yescharacteristic
387C22×C4C23×D311yescharacteristic
388D8—68no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—21no—
395C2×C8—22no—
396C2×C8—21no—
397C2×C8#23712yes—
398C4⋊C4—22no—
399C4⋊C4—32no—
400C4⋊C4—34no—
401C4⋊C4—64no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C22⋊C4—24no—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—64no—
412C22⋊C4—124no—
413C22⋊C4—122no—
414C22⋊C4—242no—
415C42—68no—
416C42D3×D412yes—
417C42D3×D412yes—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3—42no—
421C22×C3—42no—
422C22×C3—42no—
423C22×C3—42no—
424C22×C3—42no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54611yescharacteristic
427C22×C3#54711yescharacteristic
428C22×C3#57512yes—
429C22×C3#57612yes—
430C2×D3—42no—
431C2×D3—42no—
432C2×D3—42no—
433C2×D3—42no—
434C2×D3—44no—
435C2×D3—82no—
436C2×D3—82no—
437C2×D3—81no—
438C2×D3—81no—
439C2×D3—82no—
440C3×C4—22no—
441C3×C4—22no—
442C3×C4—42no—
443C3×C4#52511yescharacteristic
444C3×C4#53711yescharacteristic
445C3×C4#54811yescharacteristic
446C3×C4#55011yescharacteristic
447Dic3—42no—
448Dic3—44no—
449Dic3—81no—
450Dic3—82no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—22no—
456C23—22no—
457C23—22no—
458C23—62no—
459C23—62no—
460C23—121no—
461C23—122no—
462C23—122no—
463C23—122no—
464C23—122no—
465C23—121no—
466C23—121no—
467C23—121no—
468C23C22×D1211yescharacteristic
469Q8—244no—
470D4—44no—
471D4—62no—
472D4—62no—
473D4—62no—
474D4—62no—
475D4—121no—
476D4—121no—
477D4—121no—
478D4—121no—
479D4—248no—
480D4—248no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—22no—
484C2×C4—22no—
485C2×C4—22no—
486C2×C4—42no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—62no—
491C2×C4—64no—
492C2×C4—128no—
493C2×C4—122no—
494C2×C4—122no—
495C2×C4—122no—
496C2×C4—121no—
497C2×C4—122no—
498C2×C4—122no—
499C2×C4—122no—
500C2×C4—122no—
501C2×C4—244no—
502C2×C4#88212yes—
503C2×C4#88212yes—
504C2×C4C2×D3×D411yescharacteristic
505C2×C4C2×D3×D411yescharacteristic
506C2×C4C2×#23711yescharacteristic
507C2×C4#100411yescharacteristic
508C2×C4D3×#4111yescharacteristic
509C2×C4#101011yescharacteristic
510C8—22no—
511C8—42no—
512C8—41no—
513C2×C3—82no—
514C2×C3#240711yescharacteristic
515C2×C3#300511yescharacteristic
516C2×C3#300811yescharacteristic
517C2×C3#331912yes—
518C2×C3#334712yes—
519D3—82no—
520D3—82no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—122no—
529C22—122no—
530C22—124no—
531C22—122no—
532C22—122no—
533C22—241no—
534C22—241no—
535C22—242no—
536C22—242no—
537C22—242no—
538C22C2×#88211yescharacteristic
539C22#555411yescharacteristic
540C22#555511yescharacteristic
541C22#574812yes—
542C22#575212yes—
543C4—22no—
544C4—22no—
545C4—42no—
546C4—122no—
547C4—124no—
548C4—241no—
549C4—242no—
550C4#560611yescharacteristic
551C4#560811yescharacteristic
552C4#567711yescharacteristic
553C4#571811yescharacteristic
554C3#2412911yescharacteristic, complemented, Sylow
555C2—82no—
556C2—242no—
557C2—242no—
558C2#7643511yescharacteristic
559C2#7863811yescharacteristic
560C2#7867011yescharacteristic
561C2#7947312yes—
562C2#7950612yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
hypercenter#1955D3
hypocenterC3#24129
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. C2×#5563C2semidirect
2. #82415C2semidirect
3. C2×#4846C2semidirect
4. C2×#4711C2semidirect
5. C3×#1955C2semidirect
6. #70480C2semidirect
7. #69961C2semidirect
8. #69934C2semidirect
9. #69933C2semidirect
10. #69929C2semidirect
11. #69828C2semidirect
53. #1955D3semidirect
101. C4×D12D4semidirect
103. #869D4semidirect
554. C3#24129semidirect

Maximal subgroups

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

subgroupindex
C2×#55632
#824152
C2×#48462
C2×#47112
C3×#19552
#704802
#699612
#699342
#699332
#699292
#698282
#241293