Finite Group Databasebeta

← #10829263

Subgroups — #10829263

Order 768.

1140 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5732C211yesmaximal, characteristic, complemented
2C2×C3×#476C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic
4C2×#5018C211yesmaximal, characteristic, complemented
5#71788C211yesmaximal, characteristic, complemented
6#70582C212yesmaximal, complemented
7#70580C212yesmaximal, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70471C211yesmaximal, characteristic, complemented
10#70205C212yesmaximal, complemented
11#70201C212yesmaximal, complemented
12#24131—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×#947—21no—
16C2×Q8×Dic3—21no—
17C2×#938C2212yes—
18C22×#214C2211yescharacteristic
19#5732—24no—
20#5732—24no—
21C2×#898—22no—
22C2×#897C2212yes—
23C2×#892C2212yes—
24C2×C4×D12C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#476—24no—
27C2×C3×#106C2212yes—
28C2×C3×C4⋊C8C2211yescharacteristic
29C2×C3×#102C2212yes—
30C2×#832C2211yescharacteristic
31#5213—22no—
32C2×(C2×D3)⋊C8C2212yes—
33C2×(C2×D3)⋊C8C2212yes—
34C2×#816C2212yes—
35#5018—24no—
36#4987C2212yes—
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4960—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#556—31no—
47C2×#476D311yescharacteristic, complemented
48C2×#440—31no—
49#2535—31no—
50#2035—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1875—32no—
55#1854—32no—
56#1805—31no—
57C22×C3×Q8—21no—
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C22×D12D411yescharacteristic
62C22×C4×D3C2312yes—
63C22×Dic6C2311yescharacteristic
64C22×C3×C8C2312yes—
65C3×C4⋊Q8—22no—
66C3×C4⋊Q8D414yescomplemented
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—21no—
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75#947—28no—
76Q8×Dic3—28no—
77#938—24no—
78#938—42no—
79C2×#214—22no—
80C2×#214—22no—
81C2×#214—22no—
82C2×(C2×D3)⋊C4—22no—
83C2×(C2×D3)⋊C4—22no—
84C2×(C2×D3)⋊C4—22no—
85C2×(C2×D3)⋊C4D412yes—
86C2×C4⋊Dic3—22no—
87C2×C4⋊Dic3D411yescharacteristic
88C2×C4⋊Dic3C2312yes—
89C2×C4⋊Dic3C2311yescharacteristic
90C2×Dic3⋊C4—21no—
91C2×C4×Dic3—21no—
92C2×C4×Dic3—22no—
93C22×C3⋊C8C2311yescharacteristic
94C22×C3⋊C8C2311yescharacteristic
95#898—216no—
96#897—24no—
97#897—24no—
98#892—24no—
99#892—24no—
100C4×D12—22no—
101C4×D12—22no—
102C4×Dic6—22no—
103C4×Dic6D414yescomplemented
104C3×#106—24no—
105C3×C4⋊C8—22no—
106C3×#102—24no—
107#832—22no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110C3⋊#95—22no—
111(C2×D3)⋊C8—24no—
112(C2×D3)⋊C8—42no—
113#816—42no—
114#810—22no—
115#807—22no—
116#803—22no—
117C22×SD16—31no—
118#556—64no—
119#556—64no—
120C2×C4⋊Q8C2×D311yescharacteristic
121C2×#126—62no—
122C2×#124—61no—
123C2×C22⋊Q8—32no—
124C2×#121—32no—
125C2×C4×Q8—31no—
126C2×C4×Q8—61no—
127C2×C4×D4—31no—
128C2×C4×D4C2×D311yescharacteristic
129C2×#117—32no—
130#476—24no—
131#440—64no—
132C2×#107—31no—
133C2×#106C2×D312yes—
134C2×C4⋊C8C2×D311yescharacteristic
135C2×#103—31no—
136C2×#103—32no—
137C2×#102—31no—
138C2×#102C2×D312yes—
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97—31no—
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×C3—21no—
151C23×D3C2×D412yes—
152C2×C3×Q8—24no—
153C2×C3×Q8—44no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—21no—
157C2×C3×D4—21no—
158C2×C3×D4—21no—
159C2×C3×D4#4112yes—
160C22×C3×C4—22no—
161C22×C3×C4—21no—
162C22×C3×C4—42no—
163C22×C3×C4—82no—
164C22×C3×C4C2×D411yescharacteristic
165C22×C3×C4C2×D411yescharacteristic
166C22×C3×C4C2411yescharacteristic
167C2×#67—44no—
168C2×#67—44no—
169C2×#67—82no—
170C22×Dic3—21no—
171C22×Dic3—21no—
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C2×D12—22no—
175C2×D12—42no—
176C2×C4×D3—22no—
177C2×C4×D3—22no—
178C2×C4×D3—22no—
179C2×C4×D3—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—21no—
187C2×Dic6C2×D412yes—
188C2×C3×C8—22no—
189C2×C3×C8—22no—
190C2×C3×C8—22no—
191C3×C4⋊C4—24no—
192C3×C4⋊C4—22no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4C2×D414yes—
196C3×C22⋊C4—42no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—44no—
200#214—28no—
201(C2×D3)⋊C4—28no—
202(C2×D3)⋊C4—28no—
203(C2×D3)⋊C4—24no—
204(C2×D3)⋊C4—82no—
205C4⋊Dic3—28no—
206C4⋊Dic3—22no—
207C4⋊Dic3—21no—
208C4⋊Dic3—22no—
209C4⋊Dic3—22no—
210C4⋊Dic3C2×D412yes—
211Dic3⋊C4—24no—
212C4×Dic3—28no—
213C4×Dic3—24no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C2×C3⋊C8—21no—
217C2×C3⋊C8—21no—
218C2×C3⋊C8—21no—
219C2×C3⋊C8#4112yes—
220C22×Q8—21no—
221C22×Q8—31no—
222C22×D4—31no—
223C22×D4—62no—
224C22×D4C22×D311yescharacteristic
225C23×C4—21no—
226C23×C4—32no—
227C2×SD16—62no—
228C2×SD16—62no—
229C2×SD16—62no—
230C22×C8—31no—
231C22×C8—31no—
232C22×C8C22×D312yes—
233C4⋊Q8—22no—
234C4⋊Q8#6714yescomplemented
235#126—616no—
236#124—68no—
237C22⋊Q8—64no—
238C22⋊Q8—64no—
239#121—64no—
240#121—122no—
241C4×Q8—34no—
242C4×Q8—68no—
243C4×Q8—62no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—62no—
247C4×D4—62no—
248#117—64no—
249#117—64no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—31no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—62no—
255C2×C4⋊C4—61no—
256C2×C4⋊C4C22×D311yescharacteristic
257C2×C4⋊C4C22×D312yes—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—62no—
261C2×C22⋊C4—62no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C42—62no—
265C2×C42—61no—
266C2×C42C22×D311yescharacteristic
267#107—62no—
268#106—24no—
269C4⋊C8—22no—
270#103—62no—
271#103—122no—
272#102—24no—
273#102—62no—
274C22⋊C8—64no—
275C22⋊C8—122no—
276#97—62no—
277#95—62no—
278#95—61no—
279#95—61no—
280C23×C3—22no—
281C23×C3—41no—
282C23×C3—41no—
283C23×C3—42no—
284C23×C3—42no—
285C23×C3C22×D411yescharacteristic
286C22×D3—22no—
287C22×D3—22no—
288C22×D3—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C3×Q8—48no—
294C3×D4—22no—
295C3×D4—22no—
296C3×D4—41no—
297C3×D4—41no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—24no—
301C2×C3×C4—44no—
302C2×C3×C4—41no—
303C2×C3×C4—42no—
304C2×C3×C4—84no—
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—22no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324C2×Dic3—42no—
325D12—44no—
326C4×D3—42no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—42no—
330Dic6—22no—
331Dic6—22no—
332Dic6—41no—
333Dic6—41no—
334C3×C8—42no—
335C3⋊C8—22no—
336C3⋊C8—41no—
337C24—21no—
338C24—32no—
339C2×Q8—24no—
340C2×Q8—32no—
341C2×Q8—44no—
342C2×Q8—61no—
343C2×Q8—61no—
344C2×Q8—61no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×D4—21no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—62no—
353C2×D4—122no—
354C2×D4—124no—
355C2×D4—124no—
356C2×D4—242no—
357C2×D4#23912yes—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—32no—
361C22×C4—42no—
362C22×C4—61no—
363C22×C4—61no—
364C22×C4—62no—
365C22×C4—62no—
366C22×C4—62no—
367C22×C4—62no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—82no—
372C22×C4D3×D411yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4C23×D311yescharacteristic
375SD16—68no—
376C2×C8—22no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—32no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—61no—
383C2×C8—61no—
384C2×C8—61no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—21no—
388C4⋊C4—21no—
389C4⋊C4—32no—
390C4⋊C4—68no—
391C4⋊C4—62no—
392C4⋊C4—64no—
393C4⋊C4—61no—
394C4⋊C4—62no—
395C4⋊C4—62no—
396C4⋊C4C2×#6714yes—
397C22⋊C4—42no—
398C22⋊C4—68no—
399C22⋊C4—68no—
400C22⋊C4—64no—
401C22⋊C4—124no—
402C22⋊C4—242no—
403C42—68no—
404C42—64no—
405C42D3×D412yes—
406C42C2×#6712yes—
407C22×C3—42no—
408C22×C3—44no—
409C22×C3—82no—
410C22×C3—81no—
411C22×C3—81no—
412C22×C3C2×#12011yescharacteristic
413C22×C3#54611yescharacteristic
414C22×C3#54711yescharacteristic
415C22×C3#57512yes—
416C22×C3#57712yes—
417C2×D3—42no—
418C2×D3—42no—
419C2×D3—42no—
420C2×D3—42no—
421C2×D3—42no—
422C2×D3—42no—
423C2×D3—42no—
424C3×C4—22no—
425C3×C4—22no—
426C3×C4—44no—
427C3×C4—81no—
428C3×C4#52511yescharacteristic
429C3×C4#53711yescharacteristic
430C3×C4#54811yescharacteristic
431C3×C4#55011yescharacteristic
432Dic3—42no—
433Dic3—42no—
434Dic3—42no—
435Dic3—82no—
436C23—22no—
437C23—42no—
438C23—42no—
439C23—41no—
440C23—41no—
441C23—62no—
442C23—62no—
443C23—62no—
444C23—62no—
445C23—62no—
446C23—62no—
447C23—62no—
448C23C2×D3×D411yescharacteristic
449Q8—48no—
450Q8—62no—
451Q8—62no—
452Q8—121no—
453Q8—121no—
454D4—22no—
455D4—22no—
456D4—41no—
457D4—41no—
458D4—124no—
459D4—248no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—24no—
463C2×C4—44no—
464C2×C4—42no—
465C2×C4—41no—
466C2×C4—62no—
467C2×C4—62no—
468C2×C4—62no—
469C2×C4—62no—
470C2×C4—62no—
471C2×C4—84no—
472C2×C4—122no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—122no—
476C2×C4—122no—
477C2×C4—122no—
478C2×C4—122no—
479C2×C4—122no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4#93712yes—
483C2×C4#93712yes—
484C2×C4C2×D3×D411yescharacteristic
485C2×C4C2×#23911yescharacteristic
486C2×C4#100711yescharacteristic
487C2×C4D3×#4111yescharacteristic
488C2×C4#100911yescharacteristic
489C2×C4C22×#6711yescharacteristic
490C8—42no—
491C8—62no—
492C8—121no—
493C2×C3—82no—
494C2×C3#240711yescharacteristic
495C2×C3#300511yescharacteristic
496C2×C3#301011yescharacteristic
497C2×C3#331112yes—
498C2×C3#335012yes—
499D3—82no—
500C22—42no—
501C22—44no—
502C22—81no—
503C22—82no—
504C22—81no—
505C22—122no—
506C22—122no—
507C22—122no—
508C22—122no—
509C22—122no—
510C22—122no—
511C22—122no—
512C22#563011yescharacteristic
513C22#563111yescharacteristic
514C22#576212yes—
515C22#577312yes—
516C22C2×#93711yescharacteristic
517C4—22no—
518C4—22no—
519C4—44no—
520C4—81no—
521C4—122no—
522C4—122no—
523C4—122no—
524C4—242no—
525C4#562711yescharacteristic
526C4#562911yescharacteristic
527C4#581711yescharacteristic
528C4#583411yescharacteristic
529C3#2413111yescharacteristic, complemented, Sylow
530C2—82no—
531C2—242no—
532C2#8107812yes—
533C2#8108412yes—
534C2#8281211yescharacteristic
535C2#8386011yescharacteristic
536C2#8394511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#476C2
socleC23×C3C22×D4
prime radical#82416C2
hypercenterC2×#476D3
hypocenterC3#24131
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#5732C2semidirect
2. C2×C3×#476C2semidirect
4. C2×#5018C2semidirect
5. #71788C2semidirect
6. #70582C2semidirect
7. #70580C2semidirect
8. #70486C2semidirect
9. #70471C2semidirect
10. #70205C2semidirect
11. #70201C2semidirect
47. C2×#476D3semidirect
66. C3×C4⋊Q8D4semidirect
103. C4×Dic6D4semidirect
234. C4⋊Q8#67semidirect
529. C3#24131semidirect

Maximal subgroups

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

subgroupindex
C2×#57322
C2×C3×#4762
#824162
C2×#50182
#717882
#705822
#705802
#704862
#704712
#702052
#702012
#241313