Finite Group Databasebeta

← #10829990

Subgroups — #10829990

Order 768.

1146 conjugacy classes of subgroups.

The subgroup lattice has 550 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5585C211yesmaximal, characteristic
2C2×C3×C4×Q16C211yesmaximal, characteristic, complemented
3#82482C211yesmaximal, characteristic, complemented
4C2×#5027C211yesmaximal, characteristic
5#72072C212yesmaximal, complemented
6#71626C211yesmaximal, characteristic, complemented
7#70638C212yesmaximal, complemented
8#70279C212yesmaximal, complemented
9#70276C212yesmaximal, complemented
10#69788C211yesmaximal, characteristic, complemented
11#69772C211yesmaximal, characteristic
12#24190—31nomaximal, Sylow
13C22×C3×Q16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×Q8C2211yescharacteristic
16C2×#947C2212yes—
17C22×#215C2212yes—
18#5585—24no—
19#5585—24no—
20C2×#894—22no—
21C2×#893C2212yes—
22C2×#875—21no—
23C2×#874C2211yescharacteristic
24C2×C4×D12—21no—
25C2×C4×Dic6C2211yescharacteristic
26C3×C4×Q16—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#103C2211yescharacteristic
30C2×C3×C4×C8C2211yescharacteristic
31#5229C2212yes—
32C2×#835C2212yes—
33C2×#821C2212yes—
34C2×#821C2212yes—
35#5027—24no—
36#4982—22no—
37#4972C2212yes—
38C2×#809C2212yes—
39#4935C2211yescharacteristic
40#4929—21no—
41#4928—21no—
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4550C2212yes—
44#4550C2212yes—
45#4529C2212yes—
46C2×#559—31no—
47C2×#460—31no—
48C2×C4×Q16D311yescharacteristic, complemented
49#2582—31no—
50#2042—32no—
51#2015—32no—
52#1984—31no—
53#1957—31no—
54#1877—32no—
55#1862—32no—
56#1803—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×Q8C2311yescharacteristic
59C22×D12—21no—
60C22×D12—21no—
61C22×D12C2312yes—
62C22×C4×D3—22no—
63C22×Dic6D411yescharacteristic
64C2×C3×Q16—22no—
65C2×C3×Q16—22no—
66C2×C3×Q16—22no—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×C4×Q8—22no—
70C3×C4×Q8—22no—
71C3×C4×Q8—22no—
72C3×C4×Q8D414yes—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—21no—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C42—21no—
78C2×C3×C42—21no—
79C2×C3×C42C2311yescharacteristic
80#947—24no—
81#947—24no—
82C2×#215—24no—
83C2×#215—24no—
84C2×#215—42no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
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×#103—22no—
108C3×C4×C8—22no—
109#835—42no—
110C3⋊#95—22no—
111C3⋊#95—22no—
112C3⋊#95—22no—
113#821—24no—
114#821—42no—
115#809—24no—
116C4⋊(C3⋊C8)—22no—
117C22×Q16C2×D311yescharacteristic
118C22×SD16—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×Q8C2×D311yescharacteristic
128C2×C4×D4—32no—
129C2×C4×D4—61no—
130#460—64no—
131C4×Q16—24no—
132C2×#107C2×D311yescharacteristic
133C2×C4⋊C8—31no—
134C2×#103—32no—
135C2×#103C2×D311yescharacteristic
136C2×#103C2×D311yescharacteristic
137C2×#102—32no—
138C2×#102—32no—
139C2×#102—32no—
140C2×C4×C8C2×D311yescharacteristic
141#398—62no—
142#391—32no—
143#391—61no—
144#389—32no—
145#389—61no—
146#384—31no—
147#340—32no—
148#336—32no—
149#336—32no—
150C23×D3—22no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—21no—
154C2×C3×Q8—21no—
155C2×C3×Q8—21no—
156C2×C3×Q8—21no—
157C2×C3×Q8—21no—
158C2×C3×Q8—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—21no—
161C2×C3×Q8C2×D412yes—
162C2×C3×Q8#4112yes—
163C22×C3×C4—21no—
164C22×C3×C4—21no—
165C22×C3×C4—21no—
166C22×C3×C4—21no—
167C22×C3×C4C2×D411yescharacteristic
168C22×C3×C4C2×D411yescharacteristic
169C22×C3×C4C2411yescharacteristic
170C22×Dic3—22no—
171C22×Dic3C2×D412yes—
172C22×Dic3C2×D412yes—
173C2×D12—22no—
174C2×D12—22no—
175C2×D12—22no—
176C2×D12—22no—
177C2×D12—24no—
178C2×D12—22no—
179C2×D12—24no—
180C2×D12—42no—
181C2×D12—42no—
182C2×D12—81no—
183C2×D12—81no—
184C2×D12C2×D414yes—
185C2×C4×D3—28no—
186C2×C4×D3—82no—
187C2×C4×D3—82no—
188C2×Dic6—22no—
189C2×Dic6—22no—
190C2×Dic6—22no—
191C3×Q16—28no—
192C2×C3×C8—21no—
193C2×C3×C8—21no—
194C2×C3×C8—21no—
195C2×C3×C8—21no—
196C2×C3×C8—21no—
197C2×C3×C8#4112yes—
198C3×C4⋊C4—24no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—21no—
202C3×C4⋊C4—42no—
203C3×C4⋊C4C2×D412yes—
204C3×C42—24no—
205C3×C42—42no—
206C3×C42C2×D412yes—
207C3×C42C2×D412yes—
208#215—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—
215C4⋊Dic3—42no—
216C4⋊Dic3—81no—
217Dic3⋊C4—24no—
218Dic3⋊C4—82no—
219C4×Dic3—22no—
220C4×Dic3—22no—
221C2×C3⋊C8—22no—
222C2×C3⋊C8—22no—
223C2×C3⋊C8—22no—
224C22×Q8—31no—
225C22×Q8C22×D311yescharacteristic
226C22×Q8C22×D311yescharacteristic
227C22×D4—32no—
228C22×D4—61no—
229C22×D4—61no—
230C23×C4—62no—
231C2×Q16—22no—
232C2×Q16—22no—
233C2×Q16—22no—
234C2×SD16—64no—
235C2×SD16—64no—
236C2×SD16—122no—
237C22×C8—32no—
238C22×C8C22×D311yescharacteristic
239C22×C8C22×D311yescharacteristic
240#127—34no—
241#127—62no—
242#124—68no—
243#124—64no—
244#124—64no—
245#123—616no—
246C4×Q8—22no—
247C4×Q8—22no—
248C4×Q8—22no—
249C4×Q8—62no—
250C4×Q8—62no—
251C4×Q8#6714yes—
252C4×D4—68no—
253C4×D4—64no—
254C4×D4—64no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4—21no—
257C2×C4⋊C4—31no—
258C2×C4⋊C4—32no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—61no—
261C2×C4⋊C4—62no—
262C2×C4⋊C4—61no—
263C2×C4⋊C4C22×D311yescharacteristic
264C2×C4⋊C4C22×D311yescharacteristic
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C22⋊C4—62no—
268C2×C42—21no—
269C2×C42—21no—
270C2×C42—32no—
271C2×C42C22×D311yescharacteristic
272#107—22no—
273C4⋊C8—62no—
274#103—22no—
275#103—22no—
276#103—122no—
277#102—64no—
278#102—64no—
279#102—122no—
280C4×C8—22no—
281#95—62no—
282#95—62no—
283#95—62no—
284C23×C3C22×D411yescharacteristic
285C22×D3—24no—
286C22×D3—42no—
287C22×D3—42no—
288C22×D3—42no—
289C22×D3—42no—
290C22×D3—42no—
291C22×D3—42no—
292C3×Q8—22no—
293C3×Q8—22no—
294C3×Q8—22no—
295C3×Q8—22no—
296C3×Q8—41no—
297C3×Q8—41no—
298C3×Q8—41no—
299C3×Q8—41no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—22no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—41no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4#12012yes—
315C2×C3×C4#12012yes—
316C2×C3×C4#13512yes—
317C2×C3×C4#13712yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C2×#4111yescharacteristic
321C2×C3×C4C2×#4111yescharacteristic
322C2×C3×C4H(5,𝔽2)11yescharacteristic
323C2×C3×C42−1+411yescharacteristic
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—42no—
331C2×Dic3—42no—
332C2×Dic3—42no—
333D12—24no—
334D12—24no—
335D12—42no—
336D12—42no—
337D12—48no—
338D12—84no—
339C4×D3—88no—
340Dic6—44no—
341C3×C8—22no—
342C3×C8—41no—
343C3⋊C8—42no—
344C24—62no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—21no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—21no—
354C2×Q8—21no—
355C2×Q8—62no—
356C2×Q8—62no—
357C2×Q8—62no—
358C2×Q8#23512yes—
359C2×Q8C2×#6712yes—
360C2×D4—34no—
361C2×D4—64no—
362C2×D4—64no—
363C2×D4—62no—
364C2×D4—62no—
365C2×D4—62no—
366C2×D4—62no—
367C2×D4—62no—
368C2×D4—122no—
369C2×D4—122no—
370C2×D4—241no—
371C2×D4—241no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4—21no—
376C22×C4—32no—
377C22×C4—32no—
378C22×C4—68no—
379C22×C4—62no—
380C22×C4—242no—
381C22×C4—242no—
382C22×C4D3×D411yescharacteristic
383C22×C4D3×D411yescharacteristic
384C22×C4C23×D311yescharacteristic
385Q16—28no—
386SD16—128no—
387C2×C8—21no—
388C2×C8—21no—
389C2×C8—21no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—62no—
393C2×C8—62no—
394C2×C8—62no—
395C2×C8#23512yes—
396C4⋊C4—24no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—42no—
401C4⋊C4—62no—
402C4⋊C4—64no—
403C4⋊C4—62no—
404C4⋊C4—62no—
405C4⋊C4—122no—
406C4⋊C4—241no—
407C4⋊C4—242no—
408C4⋊C4C2×#6712yes—
409C22⋊C4—68no—
410C22⋊C4—68no—
411C22⋊C4—68no—
412C42—24no—
413C42—42no—
414C42—62no—
415C42—62no—
416C42D3×D412yes—
417C42C2×#6712yes—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54811yescharacteristic
420C22×C3#55011yescharacteristic
421C22×C3C2×#13511yescharacteristic
422C22×C3C2×#13711yescharacteristic
423C22×C3#57611yescharacteristic
424C22×C3#57711yescharacteristic
425C2×D3—44no—
426C2×D3—44no—
427C2×D3—44no—
428C2×D3—82no—
429C2×D3—82no—
430C2×D3—82no—
431C2×D3—82no—
432C3×C4—22no—
433C3×C4—42no—
434C3×C4—42no—
435C3×C4—42no—
436C3×C4—81no—
437C3×C4#45214yes—
438C3×C4#52511yescharacteristic
439C3×C4#53711yescharacteristic
440C3×C4#54611yescharacteristic
441C3×C4#54711yescharacteristic
442Dic3—42no—
443Dic3—42no—
444Dic3—82no—
445C23—64no—
446C23—122no—
447C23—122no—
448C23—122no—
449C23—122no—
450C23—122no—
451C23—122no—
452C23C2×D3×D411yescharacteristic
453Q8—22no—
454Q8—22no—
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—41no—
461Q8—124no—
462D4—64no—
463D4—64no—
464D4—128no—
465D4—122no—
466D4—122no—
467D4—244no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—22no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—41no—
476C2×C4—41no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—41no—
482C2×C4—62no—
483C2×C4—62no—
484C2×C4—62no—
485C2×C4—62no—
486C2×C4—62no—
487C2×C4—62no—
488C2×C4—122no—
489C2×C4—122no—
490C2×C4—122no—
491C2×C4—248no—
492C2×C4#91812yes—
493C2×C4#91912yes—
494C2×C4#93712yes—
495C2×C4#93712yes—
496C2×C4C2×#23511yescharacteristic
497C2×C4C2×#23511yescharacteristic
498C2×C4C2×D3×D411yescharacteristic
499C2×C4#100911yescharacteristic
500C2×C4#101011yescharacteristic
501C2×C4C22×#6711yescharacteristic
502C8—22no—
503C8—41no—
504C8—122no—
505C2×C3#240711yescharacteristic
506C2×C3C2×#45211yescharacteristic
507C2×C3#301211yescharacteristic
508C2×C3#336511yescharacteristic
509C2×C3#337511yescharacteristic
510C2×C3#339611yescharacteristic
511C2×C3#340011yescharacteristic
512D3—84no—
513C22—124no—
514C22—124no—
515C22—124no—
516C22—242no—
517C22—242no—
518C22—242no—
519C22—242no—
520C22#555211yescharacteristic
521C22#555311yescharacteristic
522C22C2×#91811yescharacteristic
523C22C2×#91911yescharacteristic
524C22#576911yescharacteristic
525C22#577011yescharacteristic
526C22C2×#93711yescharacteristic
527C4—22no—
528C4—42no—
529C4—42no—
530C4—42no—
531C4—81no—
532C4—122no—
533C4—122no—
534C4—242no—
535C4#518714yes—
536C4#555011yescharacteristic
537C4#555111yescharacteristic
538C4#579111yescharacteristic
539C4#583311yescharacteristic
540C3#2419011yescharacteristic, complemented, Sylow
541C2—244no—
542C2#7970111yescharacteristic
543C2#7970311yescharacteristic
544C2#7971211yescharacteristic
545C2#7971311yescharacteristic
546C2#8240211yescharacteristic
547C2C2×#518711yescharacteristic
548C2#8391211yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×C4×Q16C2
socleC23×C3C22×D4
prime radicalC2×#874C22
hypercenterC2×C4×Q16D3
hypocenterC3#24190
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×Q16

Complements

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

NH ≅ G/Ntype
2. C2×C3×C4×Q16C2semidirect
3. #82482C2semidirect
5. #72072C2semidirect
6. #71626C2semidirect
7. #70638C2semidirect
8. #70279C2semidirect
9. #70276C2semidirect
10. #69788C2semidirect
48. C2×C4×Q16D3semidirect
540. C3#24190semidirect

Maximal subgroups

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

subgroupindex
C2×#55852
C2×C3×C4×Q162
#824822
C2×#50272
#720722
#716262
#706382
#702792
#702762
#697882
#697722
#241903