Finite Group Databasebeta

← #10829250

Subgroups — #10829250

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×C3×#440C211yesmaximal, characteristic, complemented
3#82416C211yesmaximal, characteristic
4C2×#5007C211yesmaximal, characteristic, complemented
5#72054C212yesmaximal, complemented
6#71627C211yesmaximal, characteristic, complemented
7#70632C212yesmaximal, complemented
8#70275C212yesmaximal, complemented
9#70200C212yesmaximal, complemented
10#69837C211yesmaximal, characteristic, complemented
11#69828C211yesmaximal, characteristic, complemented
12#24130—31nomaximal, Sylow
13C22×C3×SD16C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×#946C2212yes—
17C2×#938C2212yes—
18#5579—24no—
19#5579—24no—
20C2×#896—22no—
21C2×D3×C4⋊C4C2212yes—
22C2×C4×D12—21no—
23C2×C4×D12C2211yescharacteristic
24C2×#870—21no—
25C2×#869C2211yescharacteristic
26C3×#440—24no—
27C2×C3×#107C2211yescharacteristic
28C2×C3×#103C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C2×C3×#97C2211yescharacteristic
31C2×#835C2212yes—
32C2×#832C2212yes—
33#5213—22no—
34C2×(C2×D3)⋊C8C2212yes—
35C2×(C2×D3)⋊C8C2212yes—
36#5007—24no—
37#4987C2212yes—
38C2×#807C2212yes—
39#4960—21no—
40#4953—21no—
41#4936C2211yescharacteristic
42C2×C4⋊(C3⋊C8)C2211yescharacteristic
43#4551C2212yes—
44#4551C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#474—31no—
48C2×#440D311yescharacteristic, complemented
49#2535—31no—
50#2036—32no—
51#2000—32no—
52#1993—31no—
53#1955—31no—
54#1874—32no—
55#1852—32no—
56#1805—31no—
57C22×C3×Q8C2311yescharacteristic
58C22×C3×D4C2311yescharacteristic
59C23×C3×C4—21no—
60C22×#67—22no—
61C22×D12D411yescharacteristic
62C22×C4×D3—22no—
63C22×C4×D3C2312yes—
64C22×Dic6—21no—
65C2×C3×SD16—22no—
66C2×C3×SD16—22no—
67C2×C3×SD16—22no—
68C22×C3×C8C2311yescharacteristic
69C22×C3×C8C2311yescharacteristic
70C3×C4×Q8—22no—
71C3×C4×Q8D414yescomplemented
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—
81#946—24no—
82#946—24no—
83#938—24no—
84#938—42no—
85C2×(C2×D3)⋊C4—22no—
86C2×(C2×D3)⋊C4—22no—
87C2×(C2×D3)⋊C4D412yes—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3D411yescharacteristic
90C2×C4⋊Dic3C2312yes—
91C2×C4⋊Dic3C2312yes—
92C2×Dic3⋊C4—22no—
93C2×Dic3⋊C4—22no—
94C22×C3⋊C8C2312yes—
95#896—216no—
96D3×C4⋊C4—24no—
97D3×C4⋊C4—24no—
98C4×D12—28no—
99C4×D12—22no—
100C4×D12—22no—
101#870—28no—
102#869—22no—
103#869D414yescomplemented
104C3×#107—22no—
105C3×#103—22no—
106C3×#102—22no—
107C3×#97—22no—
108#835—42no—
109#832—24no—
110C3⋊#95—21no—
111C3⋊#95—21no—
112C3⋊#95—22no—
113(C2×D3)⋊C8—24no—
114(C2×D3)⋊C8—42no—
115#807—24no—
116C4⋊(C3⋊C8)—22no—
117C22×SD16C2×D311yescharacteristic
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—61no—
128C2×C4×D4C2×D311yescharacteristic
129C22×C4⋊C4—32no—
130#474—64no—
131#440—24no—
132C2×#107—32no—
133C2×#107C2×D311yescharacteristic
134C2×C4⋊C8—31no—
135C2×#103—32no—
136C2×#103C2×D311yescharacteristic
137C2×#102—32no—
138C2×#102C2×D311yescharacteristic
139C2×C22⋊C8—32no—
140C2×C22⋊C8—32no—
141C2×#97C2×D311yescharacteristic
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—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
172C2×#67—44no—
173C2×#67—44no—
174C2×#67—82no—
175C22×Dic3—22no—
176C22×Dic3—22no—
177C22×Dic3C2×D412yes—
178C2×D12—24no—
179C2×D12—22no—
180C2×C4×D3—22no—
181C2×C4×D3—22no—
182C2×C4×D3—22no—
183C2×C4×D3—22no—
184C2×C4×D3—22no—
185C2×C4×D3—22no—
186C2×C4×D3—28no—
187C2×C4×D3—82no—
188C2×C4×D3—82no—
189C2×Dic6—24no—
190C2×Dic6—82no—
191C3×SD16—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⋊C4C2×D412yes—
203C3×C22⋊C4—42no—
204C3×C42—24no—
205C3×C42C2×D412yes—
206C3×C42C2×D412yes—
207#217—44no—
208(C2×D3)⋊C4—28no—
209(C2×D3)⋊C4—24no—
210(C2×D3)⋊C4—82no—
211C4⋊Dic3—24no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3C2×D414yes—
217Dic3⋊C4—28no—
218Dic3⋊C4—28no—
219C2×C3⋊C8—22no—
220C2×C3⋊C8—22no—
221C2×C3⋊C8—22no—
222C22×Q8—61no—
223C22×Q8C22×D311yescharacteristic
224C22×D4—31no—
225C22×D4—62no—
226C22×D4C22×D311yescharacteristic
227C23×C4—21no—
228C23×C4—32no—
229C23×C4—62no—
230C2×SD16—22no—
231C2×SD16—22no—
232C2×SD16—22no—
233C22×C8—32no—
234C22×C8C22×D311yescharacteristic
235C22×C8C22×D311yescharacteristic
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×Q8#6714yescomplemented
246C4×D4—22no—
247C4×D4—22no—
248C4×D4—68no—
249C4×D4—62no—
250C4×D4—62no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—31no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—64no—
256C2×C4⋊C4—62no—
257C2×C4⋊C4—62no—
258C2×C4⋊C4—61no—
259C2×C4⋊C4—64no—
260C2×C4⋊C4C22×D311yescharacteristic
261C2×C4⋊C4C22×D311yescharacteristic
262C2×C22⋊C4—21no—
263C2×C22⋊C4—32no—
264C2×C22⋊C4—62no—
265C2×C22⋊C4—62no—
266C2×C22⋊C4—62no—
267C2×C42—21no—
268C2×C42C22×D311yescharacteristic
269#107—22no—
270#107—64no—
271C4⋊C8—62no—
272#103—22no—
273#103—122no—
274#102—22no—
275#102—64no—
276C22⋊C8—64no—
277C22⋊C8—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—22no—
289C22×D3—22no—
290C22×D3—22no—
291C22×D3—22no—
292C22×D3—22no—
293C22×D3—22no—
294C22×D3#13714yes—
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
321#67—88no—
322C2×Dic3—22no—
323C2×Dic3—24no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—42no—
327C2×Dic3—42no—
328C2×Dic3—42no—
329C2×Dic3—42no—
330C2×Dic3—42no—
331D12—28no—
332C4×D3—42no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336C4×D3—88no—
337Dic6—84no—
338C3×C8—22no—
339C3×C8—41no—
340C3⋊C8—42no—
341C24—21no—
342C24—32no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—21no—
346C2×Q8—21no—
347C2×Q8—21no—
348C2×Q8—64no—
349C2×Q8—242no—
350C2×Q8C2×#6712yes—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—64no—
357C2×D4—62no—
358C2×D4—124no—
359C2×D4—124no—
360C2×D4—242no—
361C2×D4#23512yes—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—32no—
366C22×C4—42no—
367C22×C4—68no—
368C22×C4—62no—
369C22×C4—62no—
370C22×C4—62no—
371C22×C4—62no—
372C22×C4—62no—
373C22×C4—62no—
374C22×C4—62no—
375C22×C4—62no—
376C22×C4—82no—
377C22×C4—242no—
378C22×C4—242no—
379C22×C4D3×D411yescharacteristic
380C22×C4D3×D411yescharacteristic
381C22×C4C23×D311yescharacteristic
382SD16—28no—
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—34no—
397C4⋊C4—68no—
398C4⋊C4—68no—
399C4⋊C4—62no—
400C4⋊C4—64no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—62no—
404C4⋊C4C2×#6712yes—
405C22⋊C4—42no—
406C22⋊C4—68no—
407C22⋊C4—64no—
408C22⋊C4—124no—
409C22⋊C4—242no—
410C42—24no—
411C42D3×D412yes—
412C42C2×#6712yes—
413C22×C3—42no—
414C22×C3—44no—
415C22×C3—82no—
416C22×C3—81no—
417C22×C3—81no—
418C22×C3C2×#12011yescharacteristic
419C22×C3#54611yescharacteristic
420C22×C3#54711yescharacteristic
421C22×C3C2×#13712yes—
422C22×C3#57612yes—
423C2×D3—24no—
424C2×D3—24no—
425C2×D3—24no—
426C2×D3—42no—
427C2×D3—42no—
428C2×D3—42no—
429C2×D3—42no—
430C3×C4—22no—
431C3×C4—22no—
432C3×C4—42no—
433C3×C4—42no—
434C3×C4—81no—
435C3×C4#52511yescharacteristic
436C3×C4#53711yescharacteristic
437C3×C4#54811yescharacteristic
438C3×C4#55011yescharacteristic
439Dic3—42no—
440Dic3—44no—
441Dic3—82no—
442C23—22no—
443C23—34no—
444C23—42no—
445C23—42no—
446C23—41no—
447C23—41no—
448C23—62no—
449C23—62no—
450C23—62no—
451C23—62no—
452C23—62no—
453C23—62no—
454C23C2×D3×D411yescharacteristic
455Q8—22no—
456Q8—22no—
457Q8—41no—
458Q8—41no—
459Q8—244no—
460D4—22no—
461D4—22no—
462D4—41no—
463D4—41no—
464D4—68no—
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—41no—
475C2×C4—64no—
476C2×C4—62no—
477C2×C4—62no—
478C2×C4—62no—
479C2×C4—84no—
480C2×C4—122no—
481C2×C4—122no—
482C2×C4—122no—
483C2×C4—122no—
484C2×C4—122no—
485C2×C4—122no—
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#300411yescharacteristic
504C2×C3#300811yescharacteristic
505C2×C3#331412yes—
506C2×C3#334612yes—
507D3—44no—
508C22—44no—
509C22—42no—
510C22—64no—
511C22—64no—
512C22—64no—
513C22—81no—
514C22—82no—
515C22—81no—
516C22—122no—
517C22—122no—
518C22—122no—
519C22—122no—
520C22#555011yescharacteristic
521C22#555111yescharacteristic
522C22#577412yes—
523C22D3×#13712yes—
524C22C2×#93711yescharacteristic
525C4—22no—
526C4—22no—
527C4—42no—
528C4—42no—
529C4—81no—
530C4—122no—
531C4—124no—
532C4—242no—
533C4#555211yescharacteristic
534C4#555311yescharacteristic
535C4#579111yescharacteristic
536C4#583311yescharacteristic
537C3#2413011yescharacteristic, complemented, Sylow
538C2—82no—
539C2—124no—
540C2#7980212yes—
541C2#7980712yes—
542C2#8240211yescharacteristic
543C2#8394711yescharacteristic
544C2#8397911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#440C2
socleC23×C3C22×D4
prime radical#82416C2
hypercenterC2×#440D3
hypocenterC3#24130
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×#5579C2semidirect
2. C2×C3×#440C2semidirect
4. C2×#5007C2semidirect
5. #72054C2semidirect
6. #71627C2semidirect
7. #70632C2semidirect
8. #70275C2semidirect
9. #70200C2semidirect
10. #69837C2semidirect
11. #69828C2semidirect
48. C2×#440D3semidirect
71. C3×C4×Q8D4semidirect
103. #869D4semidirect
245. C4×Q8#67semidirect
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×C3×#4402
#824162
C2×#50072
#720542
#716272
#706322
#702752
#702002
#698372
#698282
#241303