Finite Group Databasebeta

← #10829251

Subgroups — #10829251

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×#5579C211yesmaximal, characteristic, complemented
2C2×#5026C211yesmaximal, characteristic, complemented
3C2×#4846C211yesmaximal, characteristic, complemented
4#74239C211yesmaximal, characteristic
5C3×#1805C211yesmaximal, characteristic, complemented
6#70471C211yesmaximal, characteristic, complemented
7#70272C212yesmaximal, complemented
8#70244C212yesmaximal, complemented
9#69555C211yesmaximal, characteristic, complemented
10#68880C212yesmaximal, complemented
11#68874C212yesmaximal, complemented
12#24130—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C2×#946C2212yes—
15C22×#215C2211yescharacteristic
16#5579—24no—
17#5579—24no—
18C2×#896—22no—
19C2×#895C2212yes—
20C2×D3×C4⋊C4C2212yes—
21C2×C4×D12—21no—
22C2×C4×D12C2211yescharacteristic
23C2×C4×D12C2211yescharacteristic
24C2×#870—21no—
25C2×#869C2211yescharacteristic
26C2×C3×C4⋊C8C2211yescharacteristic
27C2×C3×#103C2212yes—
28C3×#382C2211yescharacteristic
29C2×#835C2211yescharacteristic
30C2×#821C2212yes—
31C2×(C2×D3)⋊C8C2212yes—
32C2×(C2×D3)⋊C8C2212yes—
33C2×#818C2212yes—
34#5026—24no—
35#4987C2212yes—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38C2×#803C2211yescharacteristic
39#4846—24no—
40#4674—21no—
41#4671—21no—
42#4665—22no—
43C3×#340C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#440—31no—
49#2535—31no—
50#2036—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1852—32no—
56#1805D311yescharacteristic, complemented
57C22×C3×Q8C2311yescharacteristic
58C22×D12—22no—
59C22×D12D411yescharacteristic
60C22×D12C2311yescharacteristic
61C22×C4×D3—22no—
62C22×C4×D3—21no—
63C22×C4×D3C2312yes—
64C22×Dic6—21no—
65C22×C3×C8C2312yes—
66C3×C4×Q8—22no—
67C3×C4×Q8D414yescomplemented
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C42—21no—
72C2×C3×C42C2311yescharacteristic
73#946—24no—
74#946—24no—
75C2×#215—22no—
76C2×#215—22no—
77C2×#215—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—21no—
80C2×(C2×D3)⋊C4—22no—
81C2×(C2×D3)⋊C4—22no—
82C2×(C2×D3)⋊C4D412yes—
83C2×C4⋊Dic3—21no—
84C2×C4⋊Dic3D411yescharacteristic
85C2×C4⋊Dic3C2311yescharacteristic
86C2×C4⋊Dic3C2312yes—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—22no—
89C22×C3⋊C8C2311yescharacteristic
90C22×C3⋊C8C2311yescharacteristic
91#896—216no—
92#895—24no—
93#895—42no—
94D3×C4⋊C4—24no—
95D3×C4⋊C4—24no—
96C4×D12—28no—
97C4×D12—22no—
98C4×D12—22no—
99C4×D12—22no—
100C4×D12—22no—
101#870—28no—
102#869—22no—
103#869D414yescomplemented
104C3×C4⋊C8—22no—
105C3×#103—42no—
106C3×#95—21no—
107C3×#95—21no—
108#835—22no—
109C3⋊#95—22no—
110#821—24no—
111(C2×D3)⋊C8—24no—
112(C2×D3)⋊C8—42no—
113#818—24no—
114#809—22no—
115#807—22no—
116#803—22no—
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×Q8C2×D311yescharacteristic
126C2×C4×D4—31no—
127C2×C4×D4—31no—
128C2×C4×D4—61no—
129C22×C4⋊C4—32no—
130#474—64no—
131#440—64no—
132C2×#107—32no—
133C2×#107—31no—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103C2×D312yes—
137C2×#102—32no—
138C2×#102—31no—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
142#397—61no—
143#392—61no—
144#386—62no—
145#382C2×D311yescharacteristic
146#380—32no—
147#340—32no—
148#340—32no—
149#340C2×D312yes—
150C23×D3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—22no—
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2411yescharacteristic
164C22×Dic3—22no—
165C22×Dic3—21no—
166C22×Dic3C2×D412yes—
167C2×D12—24no—
168C2×D12—22no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—21no—
173C2×D12—21no—
174C2×D12—44no—
175C2×D12—44no—
176C2×D12—82no—
177C2×D12#4112yes—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—28no—
185C2×C4×D3—42no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×C4×D3—82no—
189C2×Dic6—24no—
190C2×Dic6—82no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—24no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4—22no—
197C3×C4⋊C4—22no—
198C3×C4⋊C4C2×D412yes—
199C3×C42—24no—
200C3×C42C2×D412yes—
201C3×C42C2×D412yes—
202#215—28no—
203(C2×D3)⋊C4—28no—
204(C2×D3)⋊C4—24no—
205(C2×D3)⋊C4—44no—
206(C2×D3)⋊C4—42no—
207(C2×D3)⋊C4—82no—
208C4⋊Dic3—24no—
209C4⋊Dic3—22no—
210C4⋊Dic3—22no—
211C4⋊Dic3—21no—
212C4⋊Dic3—21no—
213C4⋊Dic3C2×D414yes—
214Dic3⋊C4—28no—
215Dic3⋊C4—28no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8—21no—
220C2×C3⋊C8—21no—
221C2×C3⋊C8#4112yes—
222C22×Q8—61no—
223C22×Q8C22×D311yescharacteristic
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—31no—
234C22×C8—31no—
235C22×C8C22×D312yes—
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—22no—
245C4×Q8D1214yescomplemented
246C4×D4—68no—
247C4×D4—62no—
248C4×D4—62no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—32no—
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×D311yescharacteristic
262C2×C22⋊C4—32no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—61no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42C22×D311yescharacteristic
269#107—64no—
270#107—62no—
271C4⋊C8—22no—
272#103—42no—
273#103—62no—
274#102—64no—
275#102—62no—
276C22⋊C8—64no—
277C22⋊C8—122no—
278#97—62no—
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—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—42no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4#12012yes—
310C2×C3×C4#12012yes—
311C2×C3×C4C22×D411yescharacteristic
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C2×#4111yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4H(5,𝔽2)11yescharacteristic
316C2×C3×C42−1+411yescharacteristic
317C2×Dic3—22no—
318C2×Dic3—24no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—41no—
324C2×Dic3—42no—
325D12—28no—
326D12—22no—
327D12—22no—
328D12—41no—
329D12—41no—
330D12—88no—
331C4×D3—42no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—88no—
336C4×D3—84no—
337Dic6—84no—
338C3×C8—42no—
339C3⋊C8—22no—
340C3⋊C8—41no—
341C24—32no—
342C24—61no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—64no—
349C2×Q8—242no—
350C2×Q8C2×D1212yes—
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—21no—
363C22×C4—21no—
364C22×C4—22no—
365C22×C4—32no—
366C22×C4—68no—
367C22×C4—62no—
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—22no—
384C2×C8—22no—
385C2×C8—22no—
386C2×C8—32no—
387C2×C8—61no—
388C2×C8—61no—
389C2×C8—61no—
390C2×C8—61no—
391C2×C8—61no—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—22no—
395C4⋊C4—22no—
396C4⋊C4—34no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—62no—
400C4⋊C4—64no—
401C4⋊C4—62no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4C2×D1212yes—
405C22⋊C4—68no—
406C22⋊C4—64no—
407C22⋊C4—124no—
408C22⋊C4—122no—
409C22⋊C4—242no—
410C42—24no—
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—42no—
433C3×C4—42no—
434C3×C4—82no—
435C3×C4#52511yescharacteristic
436C3×C4#53711yescharacteristic
437C3×C4#54811yescharacteristic
438C3×C4#55011yescharacteristic
439Dic3—42no—
440Dic3—44no—
441Dic3—81no—
442C23—34no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23—62no—
449C23—62no—
450C23—121no—
451C23—122no—
452C23—122no—
453C23—121no—
454C23C2×D3×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—244no—
460D4—68no—
461D4—62no—
462D4—62no—
463D4—121no—
464D4—121no—
465D4—248no—
466C2×C4—22no—
467C2×C4—22no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—42no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—42no—
476C2×C4—64no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—122no—
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×C4#100411yescharacteristic
495C2×C4#100711yescharacteristic
496C2×C4D3×#4111yescharacteristic
497C2×C4D3×#4111yescharacteristic
498C8—42no—
499C8—62no—
500C8—121no—
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—122no—
516C22—124no—
517C22—242no—
518C22—241no—
519C22—241no—
520C22C2×#88211yescharacteristic
521C22#561211yescharacteristic
522C22#563111yescharacteristic
523C22#576912yes—
524C22D3×#13712yes—
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—82no—
530C4—122no—
531C4—124no—
532C4—241no—
533C4#551011yescharacteristic
534C4#552411yescharacteristic
535C4#559911yescharacteristic
536C4#563311yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—124no—
539C2—242no—
540C2#7401511yescharacteristic
541C2#7883411yescharacteristic
542C2#7895811yescharacteristic
543C2#8061912yes—
544C2#8125612yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1805C2
socleC23×C3C22×D4
prime radical#74239C2
hypercenter#1805D3
hypocenterC3#24130
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1805

Complements

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

NH ≅ G/Ntype
1. C2×#5579C2semidirect
2. C2×#5026C2semidirect
3. C2×#4846C2semidirect
5. C3×#1805C2semidirect
6. #70471C2semidirect
7. #70272C2semidirect
8. #70244C2semidirect
9. #69555C2semidirect
10. #68880C2semidirect
11. #68874C2semidirect
56. #1805D3semidirect
67. C3×C4×Q8D4semidirect
103. #869D4semidirect
245. 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×#55792
C2×#50262
C2×#48462
#742392
C3×#18052
#704712
#702722
#702442
#695552
#688802
#688742
#241303