Finite Group Databasebeta

← C2×#69241 #10948113

Subgroups — C2×#69241 #10948113

Order 768.

1154 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D3×#135C211yesmaximal, characteristic
2C22×#798C211yesmaximal, characteristic, complemented
3C2×D3×#108C211yesmaximal, characteristic, complemented
4C2×C3×#359C211yesmaximal, characteristic, complemented
5C2×#4567C211yesmaximal, characteristic, complemented
6C2×#4510C211yesmaximal, characteristic, complemented
7C2×#4497C211yesmaximal, characteristic, complemented
8#69241C218yesmaximal, direct product factor
9C2×#2143—31nomaximal, Sylow
10C2×D3×#41—21no—
11C2×C3×#135C2211yescharacteristic
12C2×#950—21no—
13D3×#135—22no—
14D3×#135—22no—
15D3×#135C414yes—
16D3×#135C2214yes—
17C2×#919C411yescharacteristic
18C2×D3×Q16C411yescharacteristic
19C2×#916—21no—
20C2×D3×SD16—21no—
21C2×#912C411yescharacteristic
22C2×D3×D8C411yescharacteristic
23C2×D3×M16—21no—
24C2×#904C2211yescharacteristic
25C22×D3×C8C2211yescharacteristic
26C22×C3×C16C2211yescharacteristic, complemented
27C2×C3×#108C2211yescharacteristic, complemented
28C2×#822—21no—
29C2×#819C2211yescharacteristic, complemented
30C22×C3⋊C16C2211yescharacteristic, complemented
31C2×#798—21no—
32C2×#798—21no—
33C2×#798—22no—
34C2×#798C2214yescomplemented
35D3×#108—22no—
36D3×#108C2214yescomplemented
37C3×#359C2214yescomplemented
38#4567C2214yescomplemented
39#4510C2214yescomplemented
40#4497C2214yescomplemented
41C22×#135—31no—
42C22×M32—31no—
43C22×#108—31no—
44#2143—38no—
45C2×#359—31no—
46C2×#359—31no—
47C2×#359—31no—
48C2×#359D311yescharacteristic, complemented
49C2×C3×#41—21no—
50D3×#41—24no—
51D3×#41—24no—
52D3×#41—42no—
53D3×#41—42no—
54C2×#239—21no—
55C2×D3×Q8—21no—
56C2×#237—21no—
57C2×#237—41no—
58C2×D3×D4—21no—
59C2×D3×D4—41no—
60C2×#235—21no—
61C2×#235—41no—
62C22×C4×D3—41no—
63C22×C4×D3D411yescharacteristic
64C3×#135—21no—
65C3×#135—21no—
66C3×#135C2×C412yes—
67C3×#135C2312yes—
68C2×C3×Q16C2×C411yescharacteristic
69C2×C3×SD16—21no—
70C2×C3×D8C2×C411yescharacteristic
71C2×C3×M16—21no—
72C22×C3×C8C2311yescharacteristic
73#950—24no—
74#950—24no—
75C2×#216—21no—
76C2×#215—21no—
77C2×#214—21no—
78C2×#213—21no—
79C2×#208—21no—
80C22×C3⋊C8D411yescharacteristic
81#919—22no—
82#919C2×C414yes—
83D3×Q16—22no—
84D3×Q16C2×C414yes—
85#916—24no—
86#916—24no—
87D3×SD16—24no—
88D3×SD16—24no—
89#912—22no—
90#912C2×C414yes—
91D3×D8—22no—
92D3×D8C2×C414yes—
93D3×M16—24no—
94D3×M16—42no—
95C2×Dic12C2×C411yescharacteristic
96#904—21no—
97#904—21no—
98#904C2×C412yes—
99#904C2312yes—
100C2×D24C2×C411yescharacteristic
101C2×#204—21no—
102C2×ΓL(1,𝔽25)—41no—
103C2×D3×C8—41no—
104C2×D3×C8C2×C411yescharacteristic
105C2×D3×C8C2×C411yescharacteristic
106C2×D3×C8C2×C412yes—
107C2×D3×C8D412yes—
108C2×D3×C8D411yescharacteristic
109C2×D3×C8D411yescharacteristic
110C2×D3×C8D412yes—
111C2×D3×C8C2312yes—
112C2×C3×C16—21no—
113C2×C3×C16—21no—
114C2×C3×C16C2312yescomplemented
115C3×#108—21no—
116C3×#108C2312yescomplemented
117#822—24no—
118#819—21no—
119#819C2312yescomplemented
120C2×C3⋊C16—21no—
121C2×C3⋊C16—21no—
122C2×C3⋊C16C2312yescomplemented
123#798—24no—
124#798—24no—
125C22×#41—61no—
126C2×#135—34no—
127C2×#135—34no—
128C2×#135—31no—
129C2×#135—31no—
130C2×#135—31no—
131C2×#135—61no—
132C2×#135—62no—
133C2×#135—62no—
134C2×#135—61no—
135C2×#135C2×D311yescharacteristic
136C22×Q16—31no—
137C22×SD16—61no—
138C22×D8—31no—
139C22×M16—61no—
140C23×C8—31no—
141C2×M32—34no—
142C2×M32—61no—
143C2×M32—61no—
144C2×M32—62no—
145C22×C16—31no—
146C22×C16C2×D311yescharacteristic, complemented
147C2×#108—31no—
148C2×#108—34no—
149C2×#108—62no—
150C2×#108—61no—
151C2×#108C2×D311yescharacteristic, complemented
152#359—34no—
153#359—34no—
154#359—34no—
155#359C2×D314yescomplemented
156C23×D3—41no—
157C23×D3SD1611yescharacteristic
158C3×#41—22no—
159C3×#41—22no—
160C3×#41—41no—
161C3×#41—41no—
162C2×C3×Q8—21no—
163C2×C3×D4—21no—
164C2×C3×D4—41no—
165C22×C3×C4—41no—
166C22×C3×C4C2×D411yescharacteristic
167C2×#67—41no—
168C2×#67—41no—
169C2×#67—41no—
170C22×Dic3—41no—
171C22×Dic3SD1611yescharacteristic
172#239—24no—
173#239—42no—
174D3×Q8—24no—
175D3×Q8—42no—
176#237—24no—
177#237—44no—
178#237—44no—
179#237—42no—
180D3×D4—24no—
181D3×D4—44no—
182D3×D4—44no—
183D3×D4—42no—
184#235—22no—
185#235—22no—
186#235—41no—
187#235—41no—
188#235—44no—
189#235—44no—
190C2×D12—21no—
191C2×D12—41no—
192C2×C4×D3—41no—
193C2×C4×D3—41no—
194C2×C4×D3—41no—
195C2×C4×D3—44no—
196C2×C4×D3—44no—
197C2×C4×D3C22⋊C411yescharacteristic
198C2×C4×D3C22⋊C411yescharacteristic
199C2×C4×D3C22⋊C412yes—
200C2×C4×D3D811yescharacteristic
201C2×C4×D3D811yescharacteristic
202C2×C4×D3D812yes—
203C2×C4×D3SD1612yes—
204C2×C4×D3C2×D412yes—
205C2×Dic6—21no—
206C2×Dic6—41no—
207C3×Q16—21no—
208C3×Q16C22×C412yes—
209C3×SD16—22no—
210C3×SD16—22no—
211C3×D8—21no—
212C3×D8C22×C412yes—
213C3×M16—22no—
214C3×M16—41no—
215C2×C3×C8—41no—
216C2×C3×C8C22×C411yescharacteristic
217C2×C3×C8C22×C411yescharacteristic
218C2×C3×C8C2×D411yescharacteristic
219C2×C3×C8C2×D411yescharacteristic
220C2×C3×C8C2×D411yescharacteristic
221C2×C3×C8C2411yescharacteristic
222#216—24no—
223#215—24no—
224#214—24no—
225#213—24no—
226#208—22no—
227#208—41no—
228C2×C3⋊C8—41no—
229C2×C3⋊C8C22⋊C411yescharacteristic
230C2×C3⋊C8C22⋊C411yescharacteristic
231C2×C3⋊C8C22⋊C412yes—
232C2×C3⋊C8C2×D412yes—
233Dic12—21no—
234Dic12C22×C412yes—
235D24—21no—
236D24C22×C412yes—
237#204—22no—
238#204—22no—
239ΓL(1,𝔽25)—44no—
240D3×C8—44no—
241D3×C8C22⋊C412yes—
242D3×C8C22⋊C412yes—
243D3×C8C22⋊C412yes—
244D3×C8C22⋊C412yes—
245D3×C8C22×C412yes—
246D3×C8C22×C412yes—
247D3×C8C2×D412yes—
248D3×C8C2×D412yes—
249C3×C16—22no—
250C3⋊C16—22no—
251C2×#41—21no—
252C2×#41—61no—
253C2×#41—61no—
254C2×#41—64no—
255C2×#41—64no—
256C2×#41—61no—
257C2×#41—121no—
258C2×#41—121no—
259C2×#41—122no—
260C2×#41—122no—
261C22×Q8—61no—
262C22×D4—61no—
263C22×D4—121no—
264C23×C4—31no—
265C23×C4—121no—
266#135—21no—
267#135—21no—
268#135—34no—
269#135—34no—
270#135—32no—
271#135—32no—
272#135—62no—
273#135—64no—
274#135—64no—
275#135—62no—
276#135—64no—
277#135—64no—
278#135—61no—
279#135—61no—
280#135C4×D312yes—
281#135C22×D312yes—
282C2×Q16—34no—
283C2×Q16—31no—
284C2×Q16—62no—
285C2×Q16—61no—
286C2×Q16C4×D311yescharacteristic
287C2×SD16—21no—
288C2×SD16—64no—
289C2×SD16—64no—
290C2×SD16—61no—
291C2×SD16—61no—
292C2×SD16—61no—
293C2×D8—34no—
294C2×D8—31no—
295C2×D8—62no—
296C2×D8—61no—
297C2×D8C4×D311yescharacteristic
298C2×M16—21no—
299C2×M16—64no—
300C2×M16—61no—
301C2×M16—121no—
302C2×M16—122no—
303C22×C8—31no—
304C22×C8—31no—
305C22×C8—31no—
306C22×C8—32no—
307C22×C8—31no—
308C22×C8—32no—
309C22×C8—31no—
310C22×C8—32no—
311C22×C8—32no—
312C22×C8—121no—
313C22×C8C22×D311yescharacteristic
314M32—64no—
315M32—64no—
316C2×C16—21no—
317C2×C16—21no—
318C2×C16—32no—
319C2×C16—61no—
320C2×C16—61no—
321C2×C16C22×D312yescomplemented
322#108—21no—
323#108—32no—
324#108—64no—
325#108—61no—
326#108C22×D312yescomplemented
327C23×C3—41no—
328C23×C3C2×SD1611yescharacteristic
329C22×D3—42no—
330C22×D3—41no—
331C22×D3—44no—
332C22×D3—84no—
333C22×D3#10211yescharacteristic
334C22×D3#10211yescharacteristic
335C22×D3#10212yes—
336C22×D3C2×SD1612yes—
337C3×Q8—22no—
338C3×Q8—41no—
339C3×D4—22no—
340C3×D4—42no—
341C3×D4—42no—
342C3×D4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—42no—
345C2×C3×C4—42no—
346C2×C3×C4C2×C22⋊C411yescharacteristic
347C2×C3×C4C2×C22⋊C411yescharacteristic
348C2×C3×C4C2×D811yescharacteristic
349C2×C3×C4C2×D811yescharacteristic
350C2×C3×C4C2×SD1611yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352#67—42no—
353#67—42no—
354#67—44no—
355#67—44no—
356C2×Dic3—41no—
357C2×Dic3—44no—
358C2×Dic3#10211yescharacteristic
359C2×Dic3#10211yescharacteristic
360C2×Dic3#10212yes—
361C2×Dic3C2×SD1612yes—
362D12—22no—
363D12—44no—
364D12—41no—
365C4×D3—44no—
366C4×D3—42no—
367C4×D3—42no—
368C4×D3—44no—
369C4×D3#10212yes—
370C4×D3#10212yes—
371C4×D3#10212yes—
372C4×D3#10212yes—
373C4×D3C2×C22⋊C412yes—
374C4×D3C2×C22⋊C412yes—
375C4×D3C2×D812yes—
376C4×D3C2×D812yes—
377Dic6—22no—
378Dic6—41no—
379Dic6—44no—
380C3×C8—42no—
381C3×C8C2×C22⋊C411yescharacteristic
382C3×C8C2×C22⋊C411yescharacteristic
383C3×C8C23×C411yescharacteristic
384C3×C8C22×D411yescharacteristic
385C3⋊C8—42no—
386C3⋊C8C2×C22⋊C412yes—
387C3⋊C8C2×C22⋊C412yes—
388C24—31no—
389C24—121no—
390#41—22no—
391#41—22no—
392#41—41no—
393#41—41no—
394#41—64no—
395#41—64no—
396#41—62no—
397#41—62no—
398#41—124no—
399#41—124no—
400#41—124no—
401#41—124no—
402#41—122no—
403#41—122no—
404#41—121no—
405#41—121no—
406C2×Q8—21no—
407C2×Q8—64no—
408C2×Q8—61no—
409C2×Q8—121no—
410C2×Q8—122no—
411C2×D4—21no—
412C2×D4—41no—
413C2×D4—64no—
414C2×D4—61no—
415C2×D4—122no—
416C2×D4—121no—
417C2×D4—121no—
418C2×D4—124no—
419C2×D4—124no—
420C2×D4—121no—
421C2×D4—121no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—31no—
425C22×C4—32no—
426C22×C4—32no—
427C22×C4—31no—
428C22×C4—31no—
429C22×C4—32no—
430C22×C4—32no—
431C22×C4—41no—
432C22×C4—121no—
433C22×C4—124no—
434C22×C4—124no—
435C22×C4—121no—
436C22×C4—121no—
437C22×C4—121no—
438C22×C4D3×D411yescharacteristic
439Q16—21no—
440Q16—32no—
441Q16—64no—
442Q16—61no—
443Q16C2×C4×D312yes—
444SD16—22no—
445SD16—22no—
446SD16—64no—
447SD16—64no—
448SD16—62no—
449SD16—62no—
450D8—21no—
451D8—32no—
452D8—64no—
453D8—61no—
454D8C2×C4×D312yes—
455M16—22no—
456M16—41no—
457M16—62no—
458M16—124no—
459M16—121no—
460C2×C8—31no—
461C2×C8—31no—
462C2×C8—32no—
463C2×C8—32no—
464C2×C8—32no—
465C2×C8—32no—
466C2×C8—32no—
467C2×C8—32no—
468C2×C8—32no—
469C2×C8—32no—
470C2×C8—32no—
471C2×C8—32no—
472C2×C8—41no—
473C2×C8—124no—
474C2×C8—121no—
475C2×C8C2×C4×D311yescharacteristic
476C2×C8C2×C4×D311yescharacteristic
477C2×C8D3×D411yescharacteristic
478C2×C8D3×D411yescharacteristic
479C2×C8D3×D411yescharacteristic
480C2×C8C23×D311yescharacteristic
481C16—22no—
482C16—62no—
483C22×C3—22no—
484C22×C3—42no—
485C22×C3—82no—
486C22×C3C2×#10211yescharacteristic
487C22×C3C2×#10211yescharacteristic
488C22×C3C22×SD1611yescharacteristic
489C2×D3—22no—
490C2×D3—22no—
491C2×D3—42no—
492C2×D3—42no—
493C2×D3—42no—
494C2×D3—88no—
495C2×D3—82no—
496C2×D3C2×#10212yes—
497C2×D3C2×#10212yes—
498C3×C4—42no—
499C3×C4C2×#10211yescharacteristic
500C3×C4C2×#10211yescharacteristic
501C3×C4C22×C22⋊C411yescharacteristic
502C3×C4C22×D811yescharacteristic
503Dic3—42no—
504Dic3C2×#10212yes—
505Dic3C2×#10212yes—
506C23—31no—
507C23—31no—
508C23—32no—
509C23—32no—
510C23—41no—
511C23—122no—
512C23—124no—
513C23—121no—
514C23—244no—
515C23D3×SD1611yescharacteristic
516Q8—22no—
517Q8—41no—
518Q8—62no—
519Q8—124no—
520Q8—121no—
521D4—22no—
522D4—42no—
523D4—42no—
524D4—41no—
525D4—62no—
526D4—124no—
527D4—124no—
528D4—124no—
529D4—122no—
530D4—122no—
531D4—121no—
532C2×C4—32no—
533C2×C4—31no—
534C2×C4—31no—
535C2×C4—32no—
536C2×C4—32no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—32no—
540C2×C4—32no—
541C2×C4—32no—
542C2×C4—32no—
543C2×C4—32no—
544C2×C4—41no—
545C2×C4—42no—
546C2×C4—42no—
547C2×C4—124no—
548C2×C4—124no—
549C2×C4—124no—
550C2×C4—121no—
551C2×C4—122no—
552C2×C4—122no—
553C2×C4D3×C22⋊C411yescharacteristic
554C2×C4D3×C22⋊C411yescharacteristic
555C2×C4D3×D811yescharacteristic
556C2×C4D3×D811yescharacteristic
557C2×C4D3×SD1611yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C8—32no—
560C8—32no—
561C8—42no—
562C8—122no—
563C8D3×C22⋊C411yescharacteristic
564C8D3×C22⋊C411yescharacteristic
565C8C22×C4×D311yescharacteristic
566C8C2×D3×D411yescharacteristic
567C2×C3—21no—
568C2×C3—21no—
569C2×C3—82no—
570C2×C3#214312yescomplemented
571C2×C3C22×#10211yescharacteristic
572D3—22no—
573D3—22no—
574D3—82no—
575C22—22no—
576C22—32no—
577C22—32no—
578C22—42no—
579C22—62no—
580C22—62no—
581C22—82no—
582C22—122no—
583C22—122no—
584C22—122no—
585C22—248no—
586C22—242no—
587C22D3×#10211yescharacteristic
588C22D3×#10211yescharacteristic
589C22C2×D3×SD1611yescharacteristic
590C4—32no—
591C4—32no—
592C4—42no—
593C4—122no—
594C4D3×#10211yescharacteristic
595C4D3×#10211yescharacteristic
596C4C2×D3×C22⋊C411yescharacteristic
597C4C2×D3×D811yescharacteristic
598C3C2×#214311yescharacteristic, complemented, Sylow
599C2—21no—
600C2—21no—
601C2—62no—
602C2—62no—
603C2—82no—
604C2—242no—
605C2#6924112yesdirect product factor
606C2C2×D3×#10211yescharacteristic
subgroupG/H
centerC2×C4D3×D8
commutatorC3×C8C23×C4
FrattiniC2×C8C23×D3
FittingC2×C3×#359C2
socleC22×C3C2×#102
prime radicalC2×D3×#135C2
cosocleC2×C3×C8C24
hypercenterC2×#359D3
hypocenterC3C2×#2143
LCSG ▷ C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C22×C8 ◁ C2×#359

Complements

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

NH ≅ G/Ntype
2. C22×#798C2semidirect
3. C2×D3×#108C2semidirect
4. C2×C3×#359C2semidirect
5. C2×#4567C2semidirect
6. C2×#4510C2semidirect
7. C2×#4497C2semidirect
8. #69241C2direct (N × H)
26. C22×C3×C16C22semidirect
27. C2×C3×#108C22semidirect
29. C2×#819C22semidirect
30. C22×C3⋊C16C22semidirect
34. C2×#798C22semidirect
36. D3×#108C22semidirect
37. C3×#359C22semidirect
38. #4567C22semidirect
39. #4510C22semidirect
40. #4497C22semidirect
48. C2×#359D3semidirect
114. C2×C3×C16C23semidirect
116. C3×#108C23semidirect
119. #819C23semidirect
122. C2×C3⋊C16C23semidirect
146. C22×C16C2×D3semidirect
151. C2×#108C2×D3semidirect
155. #359C2×D3semidirect
321. C2×C16C22×D3semidirect
326. #108C22×D3semidirect
570. C2×C3#2143semidirect
598. C3C2×#2143semidirect
605. C2#69241direct (N × H)

Maximal subgroups

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

subgroupindex
C2×D3×#1352
C22×#7982
C2×D3×#1082
C2×C3×#3592
C2×#45672
C2×#45102
C2×#44972
#692412
C2×#21433