Finite Group Databasebeta

← #10829253

Subgroups — #10829253

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×#5730C211yesmaximal, characteristic, complemented
2C2×#5087C211yesmaximal, characteristic, complemented
3C2×#4710C211yesmaximal, characteristic, complemented
4#74239C211yesmaximal, characteristic
5C3×#1993C211yesmaximal, characteristic, complemented
6#69837C211yesmaximal, characteristic, complemented
7#69559C211yesmaximal, characteristic, complemented
8#68887C212yesmaximal, complemented
9#68880C212yesmaximal, complemented
10#68874C212yesmaximal, complemented
11#68825C212yesmaximal, complemented
12#24130—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×#946—22no—
15C22×#204C2211yescharacteristic
16#5730—24no—
17#5730—24no—
18C2×#897C2212yes—
19C2×#895C2212yes—
20C2×#893—21no—
21C2×D3×C4⋊C4C2212yes—
22C2×#890—21no—
23C2×C4×D12C2211yescharacteristic
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C2×C3×#97C2211yescharacteristic
27C3×#382C2211yescharacteristic
28C2×#821C2211yescharacteristic
29C2×(C2×D3)⋊C8C2212yes—
30C2×(C2×D3)⋊C8C2212yes—
31C2×#818C2211yescharacteristic
32C2×#816C2211yescharacteristic
33#5087—24no—
34#4987C2212yes—
35C2×#810C2212yes—
36C2×#809C2212yes—
37C2×#807C2212yes—
38C2×C4⋊(C3⋊C8)C2211yescharacteristic
39#4710—24no—
40#4674—21no—
41#4671—21no—
42#4665—22no—
43C3×#340C2212yes—
44C3×#340C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2535—31no—
50#2036—32no—
51#2000—32no—
52#1993D311yescharacteristic, complemented
53#1955—31no—
54#1874—32no—
55#1852—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×D12—22no—
59C22×D12D411yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—22no—
62C22×C4×D3—21no—
63C22×C4×D3C2312yes—
64C22×Dic6C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×C4⋊Q8—22no—
68C3×C4⋊Q8D414yescomplemented
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42C2311yescharacteristic
73#946—216no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—21no—
76C2×(C2×D3)⋊C4—22no—
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4D412yes—
79C2×C4⋊Dic3—22no—
80C2×C4⋊Dic3D411yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×Dic3⋊C4—21no—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—21no—
86C22×C3⋊C8C2312yes—
87C2×#204—22no—
88C2×#204—22no—
89C2×#204—22no—
90#897—24no—
91#897—24no—
92#895—24no—
93#895—42no—
94#893—28no—
95D3×C4⋊C4—24no—
96D3×C4⋊C4—24no—
97#890—28no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101C4×D12—22no—
102C4×Dic6—22no—
103C4×Dic6D414yescomplemented
104C3×#97—22no—
105C3×#95—21no—
106C3×#95—21no—
107C3⋊#95—22no—
108#821—22no—
109(C2×D3)⋊C8—24no—
110(C2×D3)⋊C8—42no—
111#818—22no—
112#816—22no—
113#810—42no—
114#809—24no—
115#807—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16—31no—
118#554—64no—
119#554—64no—
120C2×C4⋊Q8C2×D311yescharacteristic
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—31no—
128C2×C4×D4—61no—
129C22×C4⋊C4—32no—
130#474—64no—
131#440—64no—
132C2×#107—31no—
133C2×#107—32no—
134C2×C4⋊C8—31no—
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
142#397—61no—
143#392—61no—
144#386—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340C2×D312yes—
149#340C2×D312yes—
150C23×D3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—82no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C22×Dic3—21no—
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3C2×D412yes—
163C2×D12—24no—
164C2×D12—22no—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—44no—
171C2×D12—44no—
172C2×D12—82no—
173C2×D12#4112yes—
174C2×C4×D3—22no—
175C2×C4×D3—22no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—28no—
181C2×C4×D3—42no—
182C2×C4×D3—82no—
183C2×C4×D3—82no—
184C2×C4×D3—82no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6—21no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6C2×D412yes—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8#4112yes—
197C3×C4⋊C4—24no—
198C3×C4⋊C4—22no—
199C3×C4⋊C4—22no—
200C3×C4⋊C4—22no—
201C3×C4⋊C4C2×D414yes—
202C3×C42C2×D412yes—
203C3×C42C2×D412yes—
204(C2×D3)⋊C4—28no—
205(C2×D3)⋊C4—24no—
206(C2×D3)⋊C4—44no—
207(C2×D3)⋊C4—42no—
208(C2×D3)⋊C4—82no—
209C4⋊Dic3—28no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3—21no—
214C4⋊Dic3C2×D412yes—
215Dic3⋊C4—28no—
216Dic3⋊C4—24no—
217C4×Dic3—24no—
218C2×C3⋊C8—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221#204—28no—
222C22×Q8—21no—
223C22×Q8—31no—
224C22×D4—31no—
225C22×D4—31no—
226C22×D4—62no—
227C23×C4—32no—
228C23×C4—62no—
229C23×C4—61no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C22×C8—32no—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
236C4⋊Q8—22no—
237C4⋊Q8D1214yescomplemented
238#125—68no—
239C22⋊Q8—64no—
240C22⋊Q8—616no—
241C22⋊Q8—64no—
242#121—64no—
243#121—122no—
244C4×Q8—34no—
245C4×Q8—62no—
246C4×D4—68no—
247C4×D4—62no—
248C4×D4—62no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4C22×D312yes—
261C2×C4⋊C4C22×D312yes—
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C42—61no—
268C2×C42C22×D311yescharacteristic
269#107—62no—
270#107—64no—
271C4⋊C8—62no—
272#103—62no—
273#103—122no—
274#102—62no—
275#102—64no—
276C22⋊C8—64no—
277C22⋊C8—122no—
278#97—22no—
279#95—21no—
280#95—21no—
281#95—62no—
282C23×C3C22×D411yescharacteristic
283C22×D3—22no—
284C22×D3—22no—
285C22×D3—22no—
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—41no—
291C22×D3—41no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3#13714yes—
295C3×Q8—84no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—24no—
299C2×C3×C4—42no—
300C2×C3×C4—42no—
301C2×C3×C4—42no—
302C2×C3×C4—42no—
303C2×C3×C4—42no—
304C2×C3×C4#12012yes—
305C2×C3×C4#12012yes—
306C2×C3×C4C22×D411yescharacteristic
307C2×C3×C4C22×D411yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310C2×C3×C4H(5,𝔽2)11yescharacteristic
311C2×C3×C42−1+411yescharacteristic
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—42no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322D12—28no—
323D12—22no—
324D12—22no—
325D12—41no—
326D12—41no—
327D12—88no—
328C4×D3—42no—
329C4×D3—42no—
330C4×D3—42no—
331C4×D3—42no—
332C4×D3—88no—
333C4×D3—84no—
334Dic6—22no—
335Dic6—22no—
336Dic6—41no—
337Dic6—41no—
338C3×C8—22no—
339C3×C8—41no—
340C3⋊C8—42no—
341C24—32no—
342C24—61no—
343C2×Q8—24no—
344C2×Q8—32no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—82no—
351C2×D4—32no—
352C2×D4—64no—
353C2×D4—62no—
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—32no—
365C22×C4—68no—
366C22×C4—61no—
367C22×C4—61no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—61no—
375C22×C4—122no—
376C22×C4—242no—
377C22×C4—242no—
378C22×C4—242no—
379C22×C4D3×D411yescharacteristic
380C22×C4D3×D411yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—68no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8—21no—
388C2×C8—62no—
389C2×C8—62no—
390C2×C8—62no—
391C2×C8#23912yes—
392C4⋊C4—24no—
393C4⋊C4—22no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—32no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—62no—
400C4⋊C4—64no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4C2×D1214yes—
405C22⋊C4—68no—
406C22⋊C4—64no—
407C22⋊C4—124no—
408C22⋊C4—122no—
409C22⋊C4—242no—
410C42—64no—
411C42C2×D1212yes—
412C42D3×D412yes—
413C22×C3C2×#12011yescharacteristic
414C22×C3#54611yescharacteristic
415C22×C3#54711yescharacteristic
416C22×C3C2×#13712yes—
417C22×C3#57612yes—
418C2×D3—24no—
419C2×D3—24no—
420C2×D3—24no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C2×D3—42no—
425C2×D3—42no—
426C2×D3—44no—
427C2×D3—82no—
428C2×D3—81no—
429C2×D3—81no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—44no—
433C3×C4—82no—
434C3×C4#52511yescharacteristic
435C3×C4#53711yescharacteristic
436C3×C4#54811yescharacteristic
437C3×C4#55011yescharacteristic
438Dic3—42no—
439Dic3—42no—
440Dic3—42no—
441Dic3—81no—
442C23—34no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—121no—
451C23—121no—
452C23—122no—
453C23—122no—
454C23C2×D3×D411yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—84no—
458Q8—121no—
459Q8—121no—
460D4—68no—
461D4—62no—
462D4—62no—
463D4—121no—
464D4—121no—
465D4—248no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—24no—
469C2×C4—42no—
470C2×C4—42no—
471C2×C4—42no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
486C2×C4—121no—
487C2×C4—122no—
488C2×C4—244no—
489C2×C4—248no—
490C2×C4#88212yes—
491C2×C4#88212yes—
492C2×C4C22×D1211yescharacteristic
493C2×C4C2×D3×D411yescharacteristic
494C2×C4C2×#23911yescharacteristic
495C2×C4#100711yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4#100911yescharacteristic
498C8—22no—
499C8—41no—
500C8—122no—
501C2×C3#240711yescharacteristic
502C2×C3#300411yescharacteristic
503C2×C3#300811yescharacteristic
504C2×C3#331412yes—
505C2×C3#334612yes—
506D3—44no—
507D3—82no—
508C22—64no—
509C22—64no—
510C22—64no—
511C22—122no—
512C22—122no—
513C22—122no—
514C22—122no—
515C22—124no—
516C22—122no—
517C22—242no—
518C22—241no—
519C22—241no—
520C22C2×#88211yescharacteristic
521C22#567611yescharacteristic
522C22#572611yescharacteristic
523C22#577412yes—
524C22D3×#13712yes—
525C4—22no—
526C4—22no—
527C4—44no—
528C4—82no—
529C4—122no—
530C4—122no—
531C4—122no—
532C4—241no—
533C4#557611yescharacteristic
534C4#557711yescharacteristic
535C4#562711yescharacteristic
536C4#562911yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—124no—
539C2—242no—
540C2#7661811yescharacteristic
541C2#7892811yescharacteristic
542C2#7895811yescharacteristic
543C2#8170012yes—
544C2#8199312yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1993C2
socleC23×C3C22×D4
prime radical#74239C2
hypercenter#1993D3
hypocenterC3#24130
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1993

Complements

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

NH ≅ G/Ntype
1. C2×#5730C2semidirect
2. C2×#5087C2semidirect
3. C2×#4710C2semidirect
5. C3×#1993C2semidirect
6. #69837C2semidirect
7. #69559C2semidirect
8. #68887C2semidirect
9. #68880C2semidirect
10. #68874C2semidirect
11. #68825C2semidirect
52. #1993D3semidirect
68. C3×C4⋊Q8D4semidirect
103. C4×Dic6D4semidirect
237. C4⋊Q8D12semidirect
537. C3#24130semidirect

Maximal subgroups

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

subgroupindex
C2×#57302
C2×#50872
C2×#47102
#742392
C3×#19932
#698372
#695592
#688872
#688802
#688742
#688252
#241303