Finite Group Databasebeta

← #10794458

Subgroups — #10794458

Order 768.

3754 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5964C211yesmaximal, characteristic
2C3×#2311C211yesmaximal, characteristic, complemented
3C2×#5240C211yesmaximal, characteristic, complemented
4C2×#5006C212yesmaximal, complemented
5#82186C212yesmaximal, complemented
6#69598C218yesmaximal, complemented
7#20610—31nomaximal, Sylow
8C22×C3×#41C2211yescharacteristic
9#5964—24no—
10#5964—28no—
11#5964C418yescomplemented
12C2×#1010—22no—
13C2×#1009—22no—
14C2×D3×#41—24no—
15C2×#1007—21no—
16C2×#1004—21no—
17C2×#1004C2214yes—
18C22×#235—22no—
19C22×#235C2212yes—
20C2×C3×#117C2211yescharacteristic, complemented
21C22×C3×C4⋊C4C2211yescharacteristic, complemented
22C2×#926C2212yescomplemented
23C3×#391—21no—
24C3×#390—21no—
25C3×#386C2214yescomplemented
26C3×#380—22no—
27#5240—24no—
28C2×#834C2214yescomplemented
29#5213C2218yescomplemented
30#5006—28no—
31#4988—22no—
32#4987—24no—
33#4973—22no—
34C2×#806C2218yescomplemented
35C2×#585—31no—
36C2×#410—31no—
37C2×#410—32no—
38#2311—32no—
39#2311D311yescharacteristic, complemented
40#1790—38no—
41C2×C3×#41—22no—
42C2×C3×#41—22no—
43C2×C3×#41—24no—
44C2×C3×#41C2×C414yes—
45C2×C3×#41D414yes—
46C22×C3×Q8D411yescharacteristic
47C22×C3×D4D411yescharacteristic
48C22×C3×D4C2312yes—
49C23×C3×C4D412yes—
50C23×C3×C4C2312yes—
51C23×C3×C4C2311yescharacteristic
52C22×#67—24no—
53C22×#67C2314yes—
54#1010—216no—
55#1010—216no—
56#1009—216no—
57#1009—216no—
58D3×#41—232no—
59D3×#41—232no—
60#1007—28no—
61#1007—28no—
62C2×#239—24no—
63C2×D3×Q8—24no—
64#1004—28no—
65#1004—28no—
66#1004—28no—
67#1004—216no—
68#1004C2×C4116yes—
69C2×#237—24no—
70C2×#237—28no—
71C2×D3×D4—24no—
72C2×D3×D4—28no—
73C2×#235—24no—
74C2×#235—24no—
75C2×#235—216no—
76C2×#235—216no—
77C2×#235—22no—
78C2×#235—24no—
79C2×#235—22no—
80C2×#235—22no—
81C2×#235—22no—
82C2×#235—24no—
83C2×#235—24no—
84C2×#235C2×C418yes—
85C2×#235D418yes—
86C22×D12—22no—
87C22×D12D412yes—
88C22×C4×D3—24no—
89C22×C4×D3D414yes—
90C22×Dic6—22no—
91C22×Dic6D412yes—
92C3×#117—24no—
93C2×C3×C4⋊C4—22no—
94C2×C3×C4⋊C4—22no—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4—21no—
97C2×C3×C4⋊C4—21no—
98C2×C3×C22⋊C4—24no—
99C2×C3×C22⋊C4C2312yes—
100C2×C3×C42—21no—
101C2×#217C2314yes—
102C2×(C2×D3)⋊C4—28no—
103#926—24no—
104#926—24no—
105C2×C4⋊Dic3—22no—
106C2×C4×Dic3—22no—
107C3×#95—24no—
108#834—28no—
109C3⋊#95—28no—
110#806—216no—
111#585—38no—
112#585—68no—
113#585—64no—
114C2×2−1+4—62no—
115C2×2−1+4—61no—
116C2×H(5,𝔽2)—34no—
117C2×H(5,𝔽2)—62no—
118C2×H(5,𝔽2)—61no—
119C22×#41—32no—
120C22×#41—64no—
121C22×#41—62no—
122C22×#41C2×D311yescharacteristic
123C2×#117—32no—
124C2×#117C2×D311yescharacteristic, complemented
125C22×C4⋊C4C2×D311yescharacteristic, complemented
126#410—68no—
127#410—64no—
128C2×#99—34no—
129C2×#99—38no—
130#391—21no—
131#391—62no—
132#390—21no—
133#390—62no—
134#386—38no—
135#386C2×D314yescomplemented
136#380—22no—
137#380—64no—
138C24×C3C2×D412yes—
139C24×C3C2411yescharacteristic
140C23×D3—24no—
141C23×D3C2×D414yes—
142C3×#41—28no—
143C3×#41—28no—
144C3×#41—28no—
145C3×#41—28no—
146C2×C3×Q8—22no—
147C2×C3×Q8—22no—
148C2×C3×Q8C22⋊C414yescomplemented
149C2×C3×D4—24no—
150C2×C3×D4—24no—
151C2×C3×D4—22no—
152C2×C3×D4—22no—
153C2×C3×D4C22⋊C414yescomplemented
154C2×C3×D4C22×C414yes—
155C2×C3×D4C2×D414yes—
156C22×C3×C4—22no—
157C22×C3×C4—22no—
158C22×C3×C4—24no—
159C22×C3×C4—24no—
160C22×C3×C4—22no—
161C22×C3×C4—22no—
162C22×C3×C4—82no—
163C22×C3×C4C22⋊C418yescomplemented
164C22×C3×C4C22×C412yes—
165C22×C3×C4C2×D412yes—
166C22×C3×C4C2×D412yes—
167C22×C3×C4C2×D412yes—
168C22×C3×C4C2×D414yes—
169C22×C3×C4C2×D412yes—
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C2×#67—28no—
173C2×#67—216no—
174C2×#67—216no—
175C2×#67—24no—
176C2×#67—24no—
177C2×#67—24no—
178C2×#67—24no—
179C2×#67C22×C418yes—
180C2×#67C2×D418yes—
181C22×Dic3—24no—
182C22×Dic3—24no—
183C22×Dic3C2×D414yes—
184#239—232no—
185D3×Q8—232no—
186#237—232no—
187#237—232no—
188#237—232no—
189D3×D4—232no—
190D3×D4—232no—
191D3×D4—232no—
192#235—216no—
193#235—232no—
194#235—232no—
195#235—28no—
196#235—216no—
197#235—28no—
198#235—28no—
199#235—28no—
200#235—28no—
201#235—28no—
202#235—28no—
203#235—28no—
204C2×D12—24no—
205C2×D12—216no—
206C2×D12—22no—
207C2×D12—22no—
208C2×D12—24no—
209C2×D12C22⋊C418yescomplemented
210C2×C4×D3—28no—
211C2×C4×D3—232no—
212C2×C4×D3—24no—
213C2×C4×D3—24no—
214C2×C4×D3—28no—
215C2×C4×D3C22⋊C4116yescomplemented
216C2×Dic6—24no—
217C2×Dic6—216no—
218C2×Dic6—22no—
219C2×Dic6—22no—
220C2×Dic6—24no—
221C2×Dic6C22⋊C418yescomplemented
222C3×C4⋊C4—41no—
223C3×C4⋊C4—42no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—42no—
226C3×C22⋊C4—24no—
227C3×C22⋊C4—84no—
228C3×C42—42no—
229#217—24no—
230#217—24no—
231(C2×D3)⋊C4—88no—
232C4⋊Dic3—44no—
233C4×Dic3—44no—
2342−1+4—68no—
2352−1+4—616no—
2362−1+4—616no—
2372−1+4—68no—
238H(5,𝔽2)—316no—
239H(5,𝔽2)—616no—
240H(5,𝔽2)—68no—
241H(5,𝔽2)—616no—
242H(5,𝔽2)—616no—
243H(5,𝔽2)—68no—
244H(5,𝔽2)—68no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—24no—
248C2×#41—38no—
249C2×#41—38no—
250C2×#41—64no—
251C2×#41—64no—
252C2×#41—616no—
253C2×#41—68no—
254C2×#41—632no—
255C2×#41—64no—
256C2×#41—64no—
257C2×#41—632no—
258C2×#41—616no—
259C2×#41—64no—
260C2×#41—62no—
261C2×#41—62no—
262C2×#41—64no—
263C2×#41—62no—
264C2×#41—62no—
265C2×#41—64no—
266C2×#41C4×D314yes—
267C2×#41D1214yes—
268C22×Q8—32no—
269C22×Q8—62no—
270C22×Q8—64no—
271C22×Q8#6711yescharacteristic
272C22×D4—34no—
273C22×D4—32no—
274C22×D4—62no—
275C22×D4—68no—
276C22×D4—64no—
277C22×D4—64no—
278C22×D4#6711yescharacteristic
279C22×D4C22×D312yes—
280C23×C4—34no—
281C23×C4—64no—
282C23×C4#6712yes—
283C23×C4C22×D312yes—
284C23×C4C22×D311yescharacteristic
285#117—24no—
286#117—64no—
287#117—64no—
288C2×C4⋊C4—22no—
289C2×C4⋊C4—22no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—62no—
294C2×C22⋊C4—24no—
295C2×C22⋊C4—34no—
296C2×C22⋊C4—68no—
297C2×C22⋊C4C22×D312yes—
298C2×C42—21no—
299C2×C42—62no—
300#99—616no—
301#99—68no—
302#95—24no—
303#95—68no—
304C23×C3—22no—
305C23×C3—22no—
306C23×C3—41no—
307C23×C3—41no—
308C23×C3—42no—
309C23×C3—42no—
310C23×C3C2×C22⋊C414yes—
311C23×C3C23×C411yescharacteristic
312C23×C3C22×D411yescharacteristic
313C23×C3C22×D412yes—
314C23×C3C22×D412yes—
315C23×C3C22×D411yescharacteristic
316C22×D3—24no—
317C22×D3—216no—
318C22×D3—24no—
319C22×D3—44no—
320C22×D3—44no—
321C22×D3—84no—
322C22×D3—84no—
323C22×D3C2×C22⋊C418yes—
324C3×Q8—28no—
325C3×D4—28no—
326C3×D4—28no—
327C3×D4—28no—
328C2×C3×C4—22no—
329C2×C3×C4—22no—
330C2×C3×C4—28no—
331C2×C3×C4—28no—
332C2×C3×C4—48no—
333C2×C3×C4—42no—
334C2×C3×C4—44no—
335C2×C3×C4—42no—
336C2×C3×C4—42no—
337C2×C3×C4—42no—
338C2×C3×C4C2×C22⋊C414yes—
339C2×C3×C4C2×C22⋊C414yes—
340C2×C3×C4C2×C22⋊C414yes—
341C2×C3×C4C2×C22⋊C412yes—
342C2×C3×C4C2×C22⋊C412yes—
343C2×C3×C4#12014yes—
344C2×C3×C4#12014yes—
345C2×C3×C4#12012yes—
346C2×C3×C4#12011yescharacteristic
347C2×C3×C4#12012yes—
348C2×C3×C4#12011yescharacteristic
349C2×C3×C4#12012yes—
350#67—232no—
351#67—28no—
352#67—28no—
353#67—28no—
354#67—28no—
355C2×Dic3—216no—
356C2×Dic3—24no—
357C2×Dic3—24no—
358C2×Dic3—44no—
359C2×Dic3—44no—
360C2×Dic3—44no—
361C2×Dic3C2×C22⋊C418yes—
362D12—216no—
363D12—28no—
364D12—28no—
365C4×D3—232no—
366C4×D3—216no—
367C4×D3—216no—
368Dic6—216no—
369Dic6—28no—
370Dic6—28no—
371C24—34no—
372C24—64no—
373C24C2×#6712yes—
374C24C23×D311yescharacteristic
375#41—28no—
376#41—28no—
377#41—28no—
378#41—28no—
379#41—632no—
380#41—616no—
381#41—632no—
382#41—632no—
383#41—632no—
384#41—632no—
385#41—632no—
386#41—68no—
387#41—68no—
388#41—68no—
389#41—68no—
390#41—68no—
391#41—68no—
392#41—616no—
393#41—68no—
394#41—68no—
395C2×Q8—22no—
396C2×Q8—22no—
397C2×Q8—38no—
398C2×Q8—64no—
399C2×Q8—616no—
400C2×Q8—632no—
401C2×Q8—64no—
402C2×Q8—62no—
403C2×Q8—62no—
404C2×Q8(C2×D3)⋊C414yescomplemented
405C2×D4—24no—
406C2×D4—24no—
407C2×D4—22no—
408C2×D4—22no—
409C2×D4—38no—
410C2×D4—38no—
411C2×D4—38no—
412C2×D4—64no—
413C2×D4—616no—
414C2×D4—632no—
415C2×D4—632no—
416C2×D4—632no—
417C2×D4—68no—
418C2×D4—616no—
419C2×D4—616no—
420C2×D4—64no—
421C2×D4—64no—
422C2×D4—64no—
423C2×D4—64no—
424C2×D4—64no—
425C2×D4—62no—
426C2×D4—62no—
427C2×D4(C2×D3)⋊C414yescomplemented
428C2×D4C2×C4×D314yes—
429C2×D4C2×D1214yes—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—34no—
437C22×C4—316no—
438C22×C4—68no—
439C22×C4—632no—
440C22×C4—64no—
441C22×C4—68no—
442C22×C4—64no—
443C22×C4—64no—
444C22×C4—64no—
445C22×C4—82no—
446C22×C4(C2×D3)⋊C418yescomplemented
447C22×C4C2×C4×D312yes—
448C22×C4C2×D1212yes—
449C22×C4D3×D412yes—
450C22×C4D3×D412yes—
451C22×C4D3×D414yes—
452C22×C4C2×#6712yes—
453C22×C4C2×#6711yescharacteristic
454C22×C4C2×#6711yescharacteristic
455C4⋊C4—41no—
456C4⋊C4—41no—
457C4⋊C4—42no—
458C4⋊C4—42no—
459C4⋊C4—124no—
460C22⋊C4—24no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—84no—
464C22⋊C4—248no—
465C42—42no—
466C42—124no—
467C22×C3—22no—
468C22×C3—21no—
469C22×C3—22no—
470C22×C3—21no—
471C22×C3—24no—
472C22×C3—41no—
473C22×C3—41no—
474C22×C3—42no—
475C22×C3—44no—
476C22×C3—42no—
477C22×C3—42no—
478C22×C3—82no—
479C22×C3C22×C22⋊C412yes—
480C22×C3C22×C22⋊C411yescharacteristic
481C22×C3C2×#12012yes—
482C22×C3C2×#12011yescharacteristic
483C22×C3C2×#12011yescharacteristic
484C2×D3—28no—
485C2×D3—44no—
486C2×D3—44no—
487C2×D3—48no—
488C2×D3—84no—
489C2×D3—816no—
490C3×C4—24no—
491C3×C4—82no—
492C3×C4—82no—
493C3×C4#37912yes—
494C3×C4#37914yes—
495C3×C4#37912yes—
496Dic3—28no—
497Dic3—84no—
498C23—22no—
499C23—22no—
500C23—38no—
501C23—41no—
502C23—41no—
503C23—42no—
504C23—42no—
505C23—616no—
506C23—64no—
507C23—64no—
508C23—124no—
509C23—124no—
510C23—244no—
511C23—244no—
512C23C2×(C2×D3)⋊C414yes—
513C23C22×C4×D311yescharacteristic
514C23C22×D1211yescharacteristic
515C23C2×D3×D412yes—
516C23C2×D3×D412yes—
517C23C22×#6711yescharacteristic
518Q8—28no—
519Q8—616no—
520Q8—68no—
521Q8—68no—
522D4—28no—
523D4—28no—
524D4—28no—
525D4—616no—
526D4—632no—
527D4—68no—
528D4—68no—
529D4—68no—
530D4—68no—
531D4—68no—
532D4—68no—
533C2×C4—22no—
534C2×C4—22no—
535C2×C4—28no—
536C2×C4—28no—
537C2×C4—38no—
538C2×C4—42no—
539C2×C4—44no—
540C2×C4—48no—
541C2×C4—42no—
542C2×C4—42no—
543C2×C4—42no—
544C2×C4—632no—
545C2×C4—616no—
546C2×C4—64no—
547C2×C4—64no—
548C2×C4—616no—
549C2×C4—616no—
550C2×C4—124no—
551C2×C4—124no—
552C2×C4—124no—
553C2×C4D3×C22⋊C414yes—
554C2×C4D3×C22⋊C414yes—
555C2×C4#88214yes—
556C2×C4#88214yes—
557C2×C4C2×(C2×D3)⋊C414yes—
558C2×C4C2×(C2×D3)⋊C412yes—
559C2×C4C2×(C2×D3)⋊C412yes—
560C2×C4#93712yes—
561C2×C4#93712yes—
562C2×C4#93711yescharacteristic
563C2×C4#93711yescharacteristic
564C2×C4#93712yes—
565C2×C3—22no—
566C2×C3—21no—
567C2×C3—22no—
568C2×C3—21no—
569C2×C3—44no—
570C2×C3C2×#37911yescharacteristic
571C2×C3#302711yescharacteristic
572C2×C3#302811yescharacteristic
573D3—48no—
574C22—21no—
575C22—22no—
576C22—21no—
577C22—22no—
578C22—24no—
579C22—41no—
580C22—41no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—42no—
585C22—68no—
586C22—82no—
587C22—128no—
588C22—124no—
589C22—124no—
590C22—244no—
591C22—2416no—
592C22C2×D3×C22⋊C412yes—
593C22C2×#88212yes—
594C22C22×(C2×D3)⋊C411yescharacteristic
595C22C2×#93711yescharacteristic
596C22C2×#93711yescharacteristic
597C4—24no—
598C4—68no—
599C4—82no—
600C4—82no—
601C4—244no—
602C4#495512yes—
603C4#495512yes—
604C4#495514yes—
605C3#2061011yescharacteristic, complemented, Sylow
606C2—21no—
607C2—22no—
608C2—21no—
609C2—22no—
610C2—44no—
611C2—128no—
612C2C2×#495511yescharacteristic
613C2#8339611yescharacteristic
614C2#8341711yescharacteristic
subgroupG/H
centerC22C2×#937
commutatorC23×C3C23×C4
FrattiniC24C23×D3
FittingC3×#2311C2
socleC22×C3C2×#120
prime radicalC2×#5964C2
cosocleC24×C3C24
hypercenter#2311D3
hypocenterC3#20610
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2311

Complements

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

NH ≅ G/Ntype
2. C3×#2311C2semidirect
3. C2×#5240C2semidirect
4. C2×#5006C2semidirect
5. #82186C2semidirect
6. #69598C2semidirect
11. #5964C4semidirect
20. C2×C3×#117C22semidirect
21. C22×C3×C4⋊C4C22semidirect
22. C2×#926C22semidirect
25. C3×#386C22semidirect
28. C2×#834C22semidirect
29. #5213C22semidirect
34. C2×#806C22semidirect
39. #2311D3semidirect
124. C2×#117C2×D3semidirect
125. C22×C4⋊C4C2×D3semidirect
135. #386C2×D3semidirect
148. C2×C3×Q8C22⋊C4semidirect
153. C2×C3×D4C22⋊C4semidirect
163. C22×C3×C4C22⋊C4semidirect
209. C2×D12C22⋊C4semidirect
215. C2×C4×D3C22⋊C4semidirect
221. C2×Dic6C22⋊C4semidirect
404. C2×Q8(C2×D3)⋊C4semidirect
427. C2×D4(C2×D3)⋊C4semidirect
446. C22×C4(C2×D3)⋊C4semidirect
605. C3#20610semidirect

Maximal subgroups

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

subgroupindex
C2×#59642
C3×#23112
C2×#52402
C2×#50062
#821862
#695982
#206103