Finite Group Databasebeta

← #10829248

Subgroups — #10829248

Order 768.

1184 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5543C211yesmaximal, characteristic, complemented
2#82415C211yesmaximal, characteristic
3C2×#4846C211yesmaximal, characteristic, complemented
4C2×#4710C211yesmaximal, characteristic, complemented
5C3×#1955C211yesmaximal, characteristic, complemented
6#70486C211yesmaximal, characteristic, complemented
7#69962C212yesmaximal, complemented
8#69937C212yesmaximal, complemented
9#69932C212yesmaximal, complemented
10#69927C212yesmaximal, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24130—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×D4×Dic3—21no—
15C2×#930C2212yes—
16C22×C4⋊Dic3C2212yes—
17C2×#924C2212yes—
18C22×#204C2211yescharacteristic
19#5543—24no—
20#5543—24no—
21C2×#890—21no—
22C2×#878—22no—
23C2×C4×D12C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
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×#818C2211yescharacteristic
35C2×#818C2212yes—
36C2×#816C2211yescharacteristic
37C2×#816C2212yes—
38#4992—21no—
39#4956—22no—
40#4936C2211yescharacteristic
41#4846—24no—
42#4710—24no—
43#4551C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2535—31no—
50#2036—32no—
51#2000—32no—
52#1993—31no—
53#1955D311yescharacteristic, complemented
54#1874—32no—
55#1852—32no—
56#1805—31no—
57C22×C3×D4D411yescharacteristic
58C23×C3×C4C2312yes—
59C22×#67—22no—
60C23×Dic3—22no—
61C22×D12C2311yescharacteristic
62C22×C4×D3—21no—
63C22×Dic6—21no—
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C2×C3×C4⋊C4D411yescharacteristic
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—22no—
75D4×Dic3—28no—
76#930—24no—
77#930—42no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×C4⋊Dic3—21no—
81C2×C4⋊Dic3—22no—
82C2×C4⋊Dic3—24no—
83C2×C4⋊Dic3—24no—
84C2×C4⋊Dic3C2312yes—
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2311yescharacteristic
87C2×C4⋊Dic3C2312yes—
88#924—24no—
89#924—24no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—22no—
92C2×C4×Dic3—21no—
93C2×#204—22no—
94C2×#204—22no—
95C2×#204—22no—
96#890—28no—
97#878—216no—
98C4×D12—22no—
99C4×D12—22no—
100#869—22no—
101#869D414yescomplemented
102C4×Dic6—22no—
103C4×Dic6D414yescomplemented
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—24no—
113#818—22no—
114#818—24no—
115#816—22no—
116#816—42no—
117C22×SD16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8—31no—
121C2×#125—61no—
122C2×C22⋊Q8—32no—
123C2×C22⋊Q8—62no—
124C2×#121—32no—
125C2×C4×Q8—31no—
126C2×C4×D4—31no—
127C2×C4×D4—61no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4—32no—
130#474—64no—
131#440—64no—
132C2×#107—31no—
133C2×#107—32no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103—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—
152C2×C3×D4—24no—
153C2×C3×D4—22no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C2×#67—44no—
165C2×#67—44no—
166C2×#67—82no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—22no—
170C22×Dic3—28no—
171C22×Dic3—22no—
172C22×Dic3—21no—
173C22×Dic3—82no—
174C22×Dic3—82no—
175C2×D12—21no—
176C2×D12—21no—
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—21no—
180C2×D12#4112yes—
181C2×C4×D3—42no—
182C2×C4×D3—82no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—24no—
189C2×Dic6—82no—
190C2×Dic6C2×D412yes—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—22no—
195C2×C3×C8—22no—
196C2×C3×C8—22no—
197C2×C3×C8—21no—
198C2×C3×C8—21no—
199C2×C3×C8#4112yes—
200C3×C4⋊C4—22no—
201C3×C22⋊C4—24no—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204#217—28no—
205#217—82no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—42no—
208C4⋊Dic3—28no—
209C4⋊Dic3—24no—
210C4⋊Dic3—21no—
211C4⋊Dic3—22no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D412yes—
217C4⋊Dic3C2×D414yes—
218Dic3⋊C4—28no—
219Dic3⋊C4—24no—
220C4×Dic3—24no—
221#204—28no—
222C22×Q8—31no—
223C22×Q8—61no—
224C22×D4—31no—
225C22×D4—62no—
226C22×D4D1211yescharacteristic
227C23×C4—62no—
228C23×C4—61no—
229C23×C4C22×D312yes—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C22×C8C22×D311yescharacteristic
234C22×C8C22×D312yes—
235C22×C8C22×D311yescharacteristic
236C4⋊Q8—34no—
237C4⋊Q8—62no—
238#125—68no—
239C22⋊Q8—616no—
240C22⋊Q8—64no—
241C22⋊Q8—64no—
242#121—64no—
243#121—122no—
244C4×Q8—34no—
245C4×Q8—62no—
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—68no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—64no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—61no—
260C2×C4⋊C4—62no—
261C2×C4⋊C4D1211yescharacteristic
262C2×C22⋊C4—62no—
263C2×C22⋊C4—61no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4D1212yes—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#107—62no—
270#107—64no—
271C4⋊C8—22no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—64no—
276C22⋊C8—24no—
277C22⋊C8—42no—
278#97—22no—
279#95—62no—
280#95—61no—
281#95—61no—
282C23×C3—22no—
283C23×C3—22no—
284C23×C3—22no—
285C23×C3—22no—
286C23×C3—22no—
287C23×C3—22no—
288C23×C3#13714yes—
289C23×C3C22×D411yescharacteristic
290C22×D3—22no—
291C22×D3—41no—
292C22×D3—41no—
293C22×D3—42no—
294C22×D3—42no—
295C3×D4—28no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4—42no—
305C2×C3×C4#12012yes—
306C2×C3×C4#12012yes—
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C22×D411yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4C2×#4111yescharacteristic
311C2×C3×C4H(5,𝔽2)11yescharacteristic
312C2×C3×C42−1+411yescharacteristic
313#67—88no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—42no—
321C2×Dic3—42no—
322C2×Dic3—41no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325C2×Dic3—42no—
326C2×Dic3—42no—
327C2×Dic3—88no—
328D12—22no—
329D12—22no—
330D12—41no—
331D12—41no—
332C4×D3—84no—
333Dic6—22no—
334Dic6—22no—
335Dic6—41no—
336Dic6—41no—
337Dic6—84no—
338C3×C8—22no—
339C3×C8—41no—
340C3×C8—42no—
341C24—61no—
342C24C2×D1212yes—
343C2×Q8—32no—
344C2×Q8—64no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—242no—
351C2×D4—24no—
352C2×D4—22no—
353C2×D4—32no—
354C2×D4—61no—
355C2×D4—61no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4—61no—
359C2×D4—124no—
360C2×D4—124no—
361C2×D4—242no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—22no—
365C22×C4—22no—
366C22×C4—22no—
367C22×C4—22no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—62no—
371C22×C4—68no—
372C22×C4—62no—
373C22×C4—61no—
374C22×C4—122no—
375C22×C4—242no—
376C22×C4—242no—
377C22×C4—242no—
378C22×C4C2×D1212yes—
379C22×C4C2×D1211yescharacteristic
380C22×C4C2×D1211yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—68no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—22no—
387C2×C8—22no—
388C2×C8—21no—
389C2×C8—22no—
390C2×C8—21no—
391C2×C8#23712yes—
392C4⋊C4—22no—
393C4⋊C4—32no—
394C4⋊C4—34no—
395C4⋊C4—68no—
396C4⋊C4—68no—
397C4⋊C4—64no—
398C4⋊C4—64no—
399C4⋊C4—61no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C22⋊C4—24no—
406C22⋊C4—68no—
407C22⋊C4—124no—
408C22⋊C4—122no—
409C22⋊C4—242no—
410C42—64no—
411C42D3×D412yes—
412C42D3×D412yes—
413C22×C3—24no—
414C22×C3—24no—
415C22×C3—24no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3—42no—
419C22×C3—42no—
420C22×C3C2×#12011yescharacteristic
421C22×C3#54611yescharacteristic
422C22×C3#54711yescharacteristic
423C22×C3C2×#13712yes—
424C22×C3#57612yes—
425C2×D3—42no—
426C2×D3—44no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4#52511yescharacteristic
434C3×C4#53711yescharacteristic
435C3×C4#54811yescharacteristic
436C3×C4#55011yescharacteristic
437Dic3—42no—
438Dic3—44no—
439Dic3—42no—
440Dic3—81no—
441Dic3—82no—
442C23—22no—
443C23—22no—
444C23—22no—
445C23—22no—
446C23—22no—
447C23—22no—
448C23—62no—
449C23—121no—
450C23—122no—
451C23—122no—
452C23—121no—
453C23#90914yes—
454C23C22×D1211yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—121no—
458Q8—121no—
459Q8—244no—
460D4—28no—
461D4—62no—
462D4—62no—
463D4—121no—
464D4—121no—
465D4—248no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—22no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—64no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—121no—
482C2×C4—121no—
483C2×C4—121no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—244no—
489C2×C4—248no—
490C2×C4#88212yes—
491C2×C4#88212yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23711yescharacteristic
495C2×C4#100411yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#101011yescharacteristic
498C8—22no—
499C8—42no—
500C8—41no—
501C2×C3—44no—
502C2×C3#240711yescharacteristic
503C2×C3#300411yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#331412yes—
506C2×C3#334612yes—
507D3—82no—
508C22—24no—
509C22—24no—
510C22—24no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—124no—
516C22—122no—
517C22—241no—
518C22—241no—
519C22—242no—
520C22C2×#88211yescharacteristic
521C22#555411yescharacteristic
522C22#555511yescharacteristic
523C22C2×#90912yes—
524C22#575212yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—122no—
529C4—124no—
530C4—122no—
531C4—241no—
532C4—242no—
533C4#560611yescharacteristic
534C4#560811yescharacteristic
535C4#567711yescharacteristic
536C4#571811yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—44no—
539C2—242no—
540C2#7643511yescharacteristic
541C2#7864511yescharacteristic
542C2#7867011yescharacteristic
543C2#7947212yes—
544C2#7950312yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1955C2
socleC23×C3C22×D4
prime radical#82415C2
hypercenter#1955D3
hypocenterC3#24130
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×#5543C2semidirect
3. C2×#4846C2semidirect
4. C2×#4710C2semidirect
5. C3×#1955C2semidirect
6. #70486C2semidirect
7. #69962C2semidirect
8. #69937C2semidirect
9. #69932C2semidirect
10. #69927C2semidirect
11. #69828C2semidirect
53. #1955D3semidirect
101. #869D4semidirect
103. C4×Dic6D4semidirect
537. C3#24130semidirect

Maximal subgroups

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

subgroupindex
C2×#55432
#824152
C2×#48462
C2×#47102
C3×#19552
#704862
#699622
#699372
#699322
#699272
#698282
#241303