Finite Group Databasebeta

← #10830090

Subgroups — #10830090

Order 768.

1180 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×#5713C211yesmaximal, characteristic, complemented
2C2×C3×#459C211yesmaximal, characteristic
3#82388C211yesmaximal, characteristic, complemented
4C2×#5018C211yesmaximal, characteristic, complemented
5#72045C212yesmaximal, complemented
6#72037C212yesmaximal, complemented
7#71765C211yesmaximal, characteristic, complemented
8#70486C211yesmaximal, characteristic, complemented
9#70473C211yesmaximal, characteristic, complemented
10#70216C212yesmaximal, complemented
11#70167C212yesmaximal, complemented
12#24198—31nomaximal, Sylow
13C22×C3×D8C2212yes—
14C2×C3×#127C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#936—21no—
17C2×#936C2212yes—
18C2×#935—22no—
19C2×D4×Dic3—21no—
20C2×D4×Dic3C2212yes—
21C22×#214C2211yescharacteristic
22#5713—24no—
23#5713—24no—
24C2×C4×Dic6C2211yescharacteristic
25C2×C4×Dic6C2211yescharacteristic
26C3×#459—24no—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#102C2212yes—
29#5221C2212yes—
30C2×#832C2212yes—
31C2×#832C2211yescharacteristic
32C2×#832C2212yes—
33#5210—22no—
34C2×#816C2212yes—
35#5018—24no—
36#4972C2212yes—
37C2×#810C2211yescharacteristic
38C2×#807C2211yescharacteristic
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×#803C2211yescharacteristic
43#4551C2212yes—
44#4550C2212yes—
45#4550C2212yes—
46C2×#559—31no—
47C2×#459D311yescharacteristic
48C2×#440—31no—
49#2580—31no—
50#2042—32no—
51#2012—32no—
52#1985—31no—
53#1961—31no—
54#1879—32no—
55#1862—32no—
56#1805—31no—
57C22×C3×D4—21no—
58C22×C3×D4—21no—
59C22×C3×D4C2312yes—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×Dic3—22no—
63C22×Dic6D411yescharacteristic
64C22×Dic6C2311yescharacteristic
65C2×C3×D8—24no—
66C2×C3×D8—24no—
67C2×C3×D8—42no—
68C22×C3×C8C2312yes—
69C3×#127—22no—
70C3×#127D414yes—
71C3×C4×D4—22no—
72C3×C4×D4—22no—
73C2×C3×C4⋊C4C2311yescharacteristic
74C2×C3×C22⋊C4—21no—
75C2×C3×C42C2311yescharacteristic
76C2×#217—22no—
77C2×#217—22no—
78C2×#217—22no—
79C2×#217—22no—
80C2×#217—22no—
81#936—28no—
82#936—24no—
83#936—24no—
84#935—216no—
85D4×Dic3—28no—
86D4×Dic3—24no—
87D4×Dic3—24no—
88C2×#214—22no—
89C2×#214—22no—
90C2×#214—22no—
91C2×C4⋊Dic3D411yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2311yescharacteristic
94C2×Dic3⋊C4—21no—
95C2×Dic3⋊C4—22no—
96C2×Dic3⋊C4D412yes—
97C2×C4×Dic3—21no—
98C2×C4×Dic3C2312yes—
99C22×C3⋊C8C2311yescharacteristic
100C22×C3⋊C8C2311yescharacteristic
101C4×Dic6—22no—
102C4×Dic6—22no—
103C4×Dic6—22no—
104C4×Dic6D414yes—
105C3×C4⋊C8—22no—
106C3×#102—24no—
107#832—22no—
108#832—24no—
109#832—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#816—42no—
114#810—22no—
115#807—22no—
116#803—22no—
117C22×SD16—31no—
118C22×D8C2×D312yes—
119#559—64no—
120#559—64no—
121C2×#127C2×D311yescharacteristic
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8—31no—
127C2×C4×D4—32no—
128C2×C4×D4—61no—
129C2×C4×D4C2×D311yescharacteristic
130#459—24no—
131#440—64no—
132C2×#107—31no—
133C2×C4⋊C8C2×D311yescharacteristic
134C2×#103—31no—
135C2×#103—32no—
136C2×#102—32no—
137C2×#102—31no—
138C2×#102—32no—
139C2×#102C2×D312yes—
140C2×#97—31no—
141#399—61no—
142#393—61no—
143#390—32no—
144#389—32no—
145#385—62no—
146#382—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C24×C3—21no—
151C24×C3—22no—
152C2×C3×D4—24no—
153C2×C3×D4—24no—
154C2×C3×D4—21no—
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—22no—
163C2×C3×D4—21no—
164C2×C3×D4—44no—
165C2×C3×D4—82no—
166C2×C3×D4C2×D414yes—
167C2×C3×D4#4112yes—
168C22×C3×C4—21no—
169C22×C3×C4—42no—
170C22×C3×C4—82no—
171C22×C3×C4C2×D411yescharacteristic
172C22×C3×C4C2×D411yescharacteristic
173C22×C3×C4C2411yescharacteristic
174C22×Dic3—21no—
175C22×Dic3—21no—
176C22×Dic3—28no—
177C22×Dic3—22no—
178C22×Dic3—82no—
179C22×Dic3—82no—
180C22×Dic3C2×D412yes—
181C22×Dic3C2×D412yes—
182C2×Dic6—21no—
183C2×Dic6—21no—
184C2×Dic6—21no—
185C2×Dic6—21no—
186C2×Dic6—22no—
187C2×Dic6—21no—
188C2×Dic6—42no—
189C2×Dic6C2×D412yes—
190C3×D8—48no—
191C2×C3×C8—22no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C22⋊C4—42no—
197C3×C42C2×D412yes—
198C3×C42C2×D412yes—
199#217—28no—
200#217—28no—
201#217—28no—
202#217—44no—
203#217—82no—
204#214—28no—
205C4⋊Dic3—22no—
206C4⋊Dic3—21no—
207C4⋊Dic3—22no—
208C4⋊Dic3—22no—
209C4⋊Dic3C2×D412yes—
210Dic3⋊C4—24no—
211Dic3⋊C4—24no—
212Dic3⋊C4—82no—
213C4×Dic3—24no—
214C4×Dic3—22no—
215C4×Dic3—22no—
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—31no—
223C22×Q8—31no—
224C22×D4—21no—
225C22×D4—21no—
226C22×D4C22×D311yescharacteristic
227C22×D4C22×D312yes—
228C23×C4—21no—
229C23×C4—62no—
230C2×SD16—62no—
231C2×SD16—62no—
232C2×SD16—62no—
233C2×D8—24no—
234C2×D8—24no—
235C2×D8—42no—
236C22×C8—31no—
237C22×C8—31no—
238C22×C8C22×D312yes—
239#127—22no—
240#127#6714yes—
241#124—64no—
242#124—68no—
243#124—64no—
244#123—616no—
245C4×Q8—34no—
246C4×Q8—62no—
247C4×Q8—62no—
248C4×Q8—62no—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—64no—
252C4×D4—68no—
253C4×D4—64no—
254C2×C4⋊C4—31no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—62no—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C22⋊C4—21no—
262C2×C22⋊C4—62no—
263C2×C22⋊C4—62no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—32no—
268C2×C42—61no—
269C2×C42C22×D311yescharacteristic
270#107—62no—
271C4⋊C8—22no—
272#103—62no—
273#103—122no—
274#102—24no—
275#102—62no—
276#102—64no—
277#102—122no—
278#97—62no—
279#95—62no—
280#95—61no—
281#95—61no—
282C23×C3—24no—
283C23×C3—22no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3—41no—
288C23×C3—42no—
289C23×C3—42no—
290C23×C3—42no—
291C23×C3—42no—
292C23×C3—42no—
293C23×C3—42no—
294C23×C3C22×D411yescharacteristic
295C3×D4—22no—
296C3×D4—22no—
297C3×D4—24no—
298C3×D4—24no—
299C3×D4—48no—
300C3×D4—42no—
301C3×D4—41no—
302C3×D4—42no—
303C3×D4—41no—
304C3×D4—84no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—42no—
309C2×C3×C4—84no—
310C2×C3×C4#12012yes—
311C2×C3×C4#12012yes—
312C2×C3×C4C22×D411yescharacteristic
313C2×C3×C4C22×D411yescharacteristic
314C2×C3×C4C2×#4111yescharacteristic
315C2×C3×C4C2×#4111yescharacteristic
316C2×C3×C4H(5,𝔽2)11yescharacteristic
317C2×C3×C42−1+411yescharacteristic
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—41no—
327C2×Dic3—41no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—88no—
333Dic6—22no—
334Dic6—22no—
335Dic6—44no—
336Dic6—41no—
337Dic6—41no—
338C3×C8—42no—
339C3⋊C8—22no—
340C3⋊C8—41no—
341C24—21no—
342C24—22no—
343C2×Q8—32no—
344C2×Q8—62no—
345C2×Q8—61no—
346C2×Q8—61no—
347C2×Q8—61no—
348C2×Q8—61no—
349C2×Q8—61no—
350C2×Q8—122no—
351C2×D4—24no—
352C2×D4—24no—
353C2×D4—21no—
354C2×D4—22no—
355C2×D4—22no—
356C2×D4—22no—
357C2×D4—21no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—22no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—44no—
364C2×D4—82no—
365C2×D4#23712yes—
366C2×D4C2×#6714yes—
367C22×C4—21no—
368C22×C4—32no—
369C22×C4—32no—
370C22×C4—42no—
371C22×C4—61no—
372C22×C4—61no—
373C22×C4—68no—
374C22×C4—62no—
375C22×C4—82no—
376C22×C4—242no—
377C22×C4—242no—
378C22×C4D3×D411yescharacteristic
379C22×C4D3×D411yescharacteristic
380C22×C4C23×D311yescharacteristic
381SD16—68no—
382D8—48no—
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—21no—
393C4⋊C4—21no—
394C4⋊C4—32no—
395C4⋊C4—62no—
396C4⋊C4—64no—
397C4⋊C4—64no—
398C4⋊C4—61no—
399C4⋊C4—62no—
400C4⋊C4—62no—
401C4⋊C4—242no—
402C22⋊C4—42no—
403C22⋊C4—68no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—124no—
407C22⋊C4—242no—
408C42—64no—
409C42—62no—
410C42—62no—
411C42D3×D412yes—
412C42C2×#6712yes—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—44no—
416C22×C3—44no—
417C22×C3—44no—
418C22×C3—82no—
419C22×C3—82no—
420C22×C3—82no—
421C22×C3—82no—
422C22×C3—82no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C2×#12011yescharacteristic
426C22×C3#54811yescharacteristic
427C22×C3#55011yescharacteristic
428C22×C3#57512yes—
429C22×C3#57612yes—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—81no—
433C3×C4#52511yescharacteristic
434C3×C4#53711yescharacteristic
435C3×C4#54611yescharacteristic
436C3×C4#54711yescharacteristic
437Dic3—42no—
438Dic3—42no—
439Dic3—42no—
440Dic3—42no—
441Dic3—82no—
442C23—24no—
443C23—22no—
444C23—42no—
445C23—42no—
446C23—42no—
447C23—42no—
448C23—42no—
449C23—42no—
450C23—41no—
451C23—42no—
452C23—42no—
453C23—41no—
454C23C2×D3×D411yescharacteristic
455Q8—62no—
456Q8—62no—
457Q8—124no—
458Q8—121no—
459Q8—121no—
460D4—22no—
461D4—24no—
462D4—24no—
463D4—22no—
464D4—48no—
465D4—41no—
466D4—42no—
467D4—41no—
468D4—42no—
469D4—84no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—42no—
473C2×C4—41no—
474C2×C4—62no—
475C2×C4—62no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—84no—
483C2×C4—122no—
484C2×C4—121no—
485C2×C4—121no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—248no—
490C2×C4#93712yes—
491C2×C4#93712yes—
492C2×C4C2×D3×D411yescharacteristic
493C2×C4C2×#23711yescharacteristic
494C2×C4#100411yescharacteristic
495C2×C4D3×#4111yescharacteristic
496C2×C4#101011yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—42no—
499C8—62no—
500C8—121no—
501C2×C3—82no—
502C2×C3—84no—
503C2×C3#240711yescharacteristic
504C2×C3#300511yescharacteristic
505C2×C3#300811yescharacteristic
506C2×C3#337012yes—
507C2×C3#340312yes—
508C22—42no—
509C22—44no—
510C22—44no—
511C22—44no—
512C22—44no—
513C22—81no—
514C22—82no—
515C22—82no—
516C22—82no—
517C22—82no—
518C22—82no—
519C22—81no—
520C22#559811yescharacteristic
521C22#559911yescharacteristic
522C22#575712yes—
523C22#577312yes—
524C22C2×#93711yescharacteristic
525C4—22no—
526C4—22no—
527C4—81no—
528C4—122no—
529C4—122no—
530C4—122no—
531C4—122no—
532C4—242no—
533C4#560611yescharacteristic
534C4#560811yescharacteristic
535C4#580611yescharacteristic
536C4#582411yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—84no—
539C2—82no—
540C2#8032212yes—
541C2#8032912yes—
542C2#8270111yescharacteristic
543C2#8383311yescharacteristic
544C2#8394511yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#459C2
socleC23×C3C22×D4
hypercenterC2×#459D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#459

Complements

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

NH ≅ G/Ntype
1. C2×#5713C2semidirect
3. #82388C2semidirect
4. C2×#5018C2semidirect
5. #72045C2semidirect
6. #72037C2semidirect
7. #71765C2semidirect
8. #70486C2semidirect
9. #70473C2semidirect
10. #70216C2semidirect
11. #70167C2semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#57132
C2×C3×#4592
#823882
C2×#50182
#720452
#720372
#717652
#704862
#704732
#702162
#701672
#241983