Finite Group Databasebeta

← #10830094

Subgroups — #10830094

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×#5585C211yesmaximal, characteristic, complemented
2C2×C3×#440C211yesmaximal, characteristic, complemented
3#82388C211yesmaximal, characteristic, complemented
4C2×#5015C211yesmaximal, characteristic
5#72055C212yesmaximal, complemented
6#71627C211yesmaximal, characteristic, complemented
7#70638C212yesmaximal, complemented
8#70279C212yesmaximal, complemented
9#70208C212yesmaximal, complemented
10#69838C211yesmaximal, characteristic, complemented
11#69823C211yesmaximal, characteristic, complemented
12#24198—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#947C2212yes—
17C22×#213C2212yes—
18#5585—24no—
19#5585—24no—
20C2×#894—22no—
21C2×#893C2212yes—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×Dic6C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32#5221C2212yes—
33#5210—22no—
34C2×#821C2212yes—
35C2×#821C2212yes—
36#5015—24no—
37#4972C2212yes—
38C2×#809C2212yes—
39#4949—21no—
40#4947—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#459—31no—
48C2×#440D311yescharacteristic, complemented
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×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×D12—21no—
61C22×D12—21no—
62C22×D12C2312yes—
63C22×C4×D3—22no—
64C22×Dic6D411yescharacteristic
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4×Q8—22no—
71C3×C4×Q8D414yes—
72C3×C4×D4—22no—
73C3×C4×D4—22no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C22⋊C4—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80C2×#217—22no—
81C2×#217—22no—
82#947—24no—
83#947—24no—
84C2×#213—24no—
85C2×#213—24no—
86C2×#213—42no—
87C2×(C2×D3)⋊C4—22no—
88C2×(C2×D3)⋊C4—22no—
89C2×(C2×D3)⋊C4—22no—
90C2×C4⋊Dic3D411yescharacteristic
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4D412yes—
94C2×C4×Dic3C2312yes—
95C22×C3⋊C8C2312yes—
96#894—216no—
97#893—24no—
98#893—24no—
99#875—28no—
100#874—22no—
101#874D414yes—
102C4×D12—28no—
103C4×Dic6—22no—
104C4×Dic6—22no—
105C3×#107—22no—
106C3×#103—22no—
107C3×#102—22no—
108C3×#97—22no—
109#835—42no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113#821—24no—
114#821—42no—
115#809—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16C2×D311yescharacteristic
118C22×D8—32no—
119#559—64no—
120#559—64no—
121C2×#127—31no—
122C2×#124—32no—
123C2×#124—61no—
124C2×#123—62no—
125C2×C4×Q8—31no—
126C2×C4×Q8C2×D311yescharacteristic
127C2×C4×D4—32no—
128C2×C4×D4—61no—
129C2×C4×D4C2×D311yescharacteristic
130#459—64no—
131#440—24no—
132C2×#107C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103C2×D311yescharacteristic
136C2×#102—32no—
137C2×#102—32no—
138C2×#102—32no—
139C2×#102C2×D311yescharacteristic
140C2×#97C2×D311yescharacteristic
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—
151C23×D3—22no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8C2×D412yes—
158C2×C3×D4—21no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4#4112yes—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4—42no—
168C22×C3×C4—82no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2411yescharacteristic
172C22×Dic3—22no—
173C22×Dic3C2×D412yes—
174C22×Dic3C2×D412yes—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—22no—
178C2×D12—22no—
179C2×D12—24no—
180C2×D12—22no—
181C2×D12—24no—
182C2×D12—44no—
183C2×D12—82no—
184C2×D12C2×D414yes—
185C2×C4×D3—28no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×Dic6—22no—
189C2×Dic6—42no—
190C3×SD16—28no—
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—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4C2×D412yes—
202C3×C22⋊C4—42no—
203C3×C42—24no—
204C3×C42C2×D412yes—
205C3×C42C2×D412yes—
206#217—44no—
207#217—82no—
208#213—48no—
209(C2×D3)⋊C4—28no—
210(C2×D3)⋊C4—28no—
211(C2×D3)⋊C4—28no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215Dic3⋊C4—24no—
216Dic3⋊C4—82no—
217C4×Dic3—22no—
218C4×Dic3—22no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C22×Q8—31no—
223C22×Q8C22×D311yescharacteristic
224C22×D4—32no—
225C22×D4—61no—
226C22×D4—61no—
227C22×D4C22×D311yescharacteristic
228C23×C4—21no—
229C23×C4—62no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—22no—
233C2×D8—64no—
234C2×D8—64no—
235C2×D8—122no—
236C22×C8—32no—
237C22×C8C22×D311yescharacteristic
238C22×C8C22×D311yescharacteristic
239#127—34no—
240#127—62no—
241#124—68no—
242#124—64no—
243#124—64no—
244#123—616no—
245C4×Q8—22no—
246C4×Q8—62no—
247C4×Q8—62no—
248C4×Q8#6714yes—
249C4×D4—22no—
250C4×D4—22no—
251C4×D4—68no—
252C4×D4—64no—
253C4×D4—64no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—31no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—62no—
259C2×C4⋊C4C22×D311yescharacteristic
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—21no—
268C2×C42—32no—
269C2×C42C22×D311yescharacteristic
270#107—22no—
271C4⋊C8—62no—
272#103—22no—
273#103—122no—
274#102—22no—
275#102—64no—
276#102—64no—
277#102—122no—
278#97—22no—
279#95—62no—
280#95—61no—
281#95—61no—
282C23×C3—22no—
283C23×C3—41no—
284C23×C3—41no—
285C23×C3—42no—
286C23×C3—42no—
287C23×C3C22×D411yescharacteristic
288C22×D3—24no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—42no—
293C22×D3—42no—
294C22×D3—42no—
295C3×Q8—22no—
296C3×Q8—22no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×D4—22no—
300C3×D4—22no—
301C3×D4—41no—
302C3×D4—41no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—42no—
310C2×C3×C4—41no—
311C2×C3×C4—42no—
312C2×C3×C4—84no—
313C2×C3×C4#12012yes—
314C2×C3×C4#12012yes—
315C2×C3×C4C22×D411yescharacteristic
316C2×C3×C4C22×D411yescharacteristic
317C2×C3×C4C2×#4111yescharacteristic
318C2×C3×C4C2×#4111yescharacteristic
319C2×C3×C4H(5,𝔽2)11yescharacteristic
320C2×C3×C42−1+411yescharacteristic
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330D12—24no—
331D12—24no—
332D12—48no—
333D12—42no—
334D12—42no—
335D12—84no—
336C4×D3—88no—
337Dic6—44no—
338C3×C8—22no—
339C3×C8—41no—
340C3⋊C8—42no—
341C24—21no—
342C24—62no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—62no—
349C2×Q8—122no—
350C2×Q8C2×#6712yes—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—34no—
357C2×D4—64no—
358C2×D4—64no—
359C2×D4—62no—
360C2×D4—62no—
361C2×D4—62no—
362C2×D4—62no—
363C2×D4—62no—
364C2×D4—124no—
365C2×D4—242no—
366C2×D4#23512yes—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—32no—
371C22×C4—32no—
372C22×C4—42no—
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—28no—
382D8—128no—
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#23512yes—
392C4⋊C4—24no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—62no—
397C4⋊C4—64no—
398C4⋊C4—62no—
399C4⋊C4—62no—
400C4⋊C4—242no—
401C4⋊C4C2×#6712yes—
402C22⋊C4—42no—
403C22⋊C4—68no—
404C22⋊C4—68no—
405C22⋊C4—68no—
406C22⋊C4—124no—
407C22⋊C4—242no—
408C42—24no—
409C42—62no—
410C42—62no—
411C42D3×D412yes—
412C42C2×#6712yes—
413C22×C3—44no—
414C22×C3—42no—
415C22×C3—82no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3#57512yes—
422C22×C3#57612yes—
423C2×D3—44no—
424C2×D3—44no—
425C2×D3—44no—
426C2×D3—82no—
427C2×D3—82no—
428C2×D3—82no—
429C2×D3—82no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—81no—
435C3×C4#52511yescharacteristic
436C3×C4#53711yescharacteristic
437C3×C4#54611yescharacteristic
438C3×C4#54711yescharacteristic
439Dic3—42no—
440Dic3—42no—
441Dic3—82no—
442C23—22no—
443C23—42no—
444C23—42no—
445C23—41no—
446C23—41no—
447C23—64no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23—122no—
453C23—122no—
454C23C2×D3×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—124no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—64no—
465D4—64no—
466D4—128no—
467D4—122no—
468D4—122no—
469D4—244no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—42no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—42no—
478C2×C4—41no—
479C2×C4—62no—
480C2×C4—62no—
481C2×C4—62no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—84no—
486C2×C4—122no—
487C2×C4—122no—
488C2×C4—122no—
489C2×C4—248no—
490C2×C4#93712yes—
491C2×C4#93712yes—
492C2×C4C2×#23511yescharacteristic
493C2×C4C2×#23511yescharacteristic
494C2×C4C2×D3×D411yescharacteristic
495C2×C4#100911yescharacteristic
496C2×C4#101011yescharacteristic
497C2×C4C22×#6711yescharacteristic
498C8—22no—
499C8—41no—
500C8—122no—
501C2×C3—82no—
502C2×C3#240711yescharacteristic
503C2×C3#300511yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#337012yes—
506C2×C3#340312yes—
507D3—84no—
508C22—44no—
509C22—42no—
510C22—81no—
511C22—82no—
512C22—81no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22#555211yescharacteristic
521C22#555311yescharacteristic
522C22#577412yes—
523C22#577712yes—
524C22C2×#93711yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—81no—
530C4—122no—
531C4—122no—
532C4—242no—
533C4#555011yescharacteristic
534C4#555111yescharacteristic
535C4#579111yescharacteristic
536C4#583311yescharacteristic
537C3#2419811yescharacteristic, complemented, Sylow
538C2—82no—
539C2—244no—
540C2#7975312yes—
541C2#7976612yes—
542C2#8240211yescharacteristic
543C2#8394711yescharacteristic
544C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radicalC2×#5015C2
hypercenterC2×#440D3
hypocenterC3#24198
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#440

Complements

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

NH ≅ G/Ntype
1. C2×#5585C2semidirect
2. C2×C3×#440C2semidirect
3. #82388C2semidirect
5. #72055C2semidirect
6. #71627C2semidirect
7. #70638C2semidirect
8. #70279C2semidirect
9. #70208C2semidirect
10. #69838C2semidirect
11. #69823C2semidirect
48. C2×#440D3semidirect
537. C3#24198semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×C3×#4402
#823882
C2×#50152
#720552
#716272
#706382
#702792
#702082
#698382
#698232
#241983