Finite Group Databasebeta

← #10798784

Subgroups — #10798784

Order 768.

1642 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×#5832C211yesmaximal, characteristic
2C2×C3×#408C211yesmaximal, characteristic, complemented
3C2×#5234C211yesmaximal, characteristic, complemented
4C2×#5129C211yesmaximal, characteristic, complemented
5C2×Dic3×M16C211yesmaximal, characteristic, complemented
6C2×#5110C211yesmaximal, characteristic, complemented
7C22×Dic3⋊C8C211yesmaximal, characteristic, complemented
8#69917C214yesmaximal, complemented
9#69916C214yesmaximal, complemented
10#21189—31nomaximal, Sylow
11C22×C3×#41C2211yescharacteristic
12C22×#235C2211yescharacteristic
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15#5832—24no—
16#5832—24no—
17C2×#947C411yescharacteristic
18C2×#944C411yescharacteristic
19C2×#940C411yescharacteristic
20C2×#939C412yes—
21C2×#939C412yes—
22C2×#936C411yescharacteristic
23C2×C4×#67C2212yes—
24C2×C4×#67C2212yes—
25C22×C4×Dic3C2211yescharacteristic
26C22×#208C2211yescharacteristic
27C23×C3⋊C8C2211yescharacteristic
28C3×#408—24no—
29C2×C3×C22⋊C8C2214yes—
30#5234—24no—
31C2×#830C2214yes—
32#5129—24no—
33Dic3×M16—24no—
34#5110—24no—
35C2×(C2×D3)⋊C8C2214yes—
36C2×(C2×D3)⋊C8C2214yes—
37C2×#815C2212yes—
38C2×Dic3⋊C8—24no—
39C2×Dic3⋊C8C2212yes—
40C2×Dic3⋊C8C2212yes—
41C2×C8×Dic3C2212yes—
42C2×#532—31no—
43C22×C4⋊C8—31no—
44C2×#408—31no—
45C2×#408—31no—
46C2×#408—31no—
47C2×#408D311yescharacteristic, complemented
48C2×C4×M16—31no—
49#1955—34no—
50#1954—34no—
51C2×C3×#41—22no—
52C2×C3×#41—22no—
53C2×C3×#41—22no—
54C2×C3×#41—22no—
55C2×C3×#41—22no—
56C2×C3×#41—22no—
57C22×C3×Q8C2×C411yescharacteristic
58C22×C3×D4C2×C411yescharacteristic
59C22×C3×D4C2×C412yes—
60C23×C3×C4C2312yes—
61C23×C3×C4C2311yescharacteristic
62C22×#67C2×C412yes—
63C22×#67C2×C412yes—
64C22×#67C2×C412yes—
65C23×Dic3C2×C411yescharacteristic
66C23×Dic3C2×C411yescharacteristic
67C2×#235—22no—
68C2×#235—22no—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—22no—
72C2×#235—22no—
73C22×D12C2×C411yescharacteristic
74C22×C4×D3C2312yes—
75C22×Dic6C2×C411yescharacteristic
76C2×C3×M16—22no—
77C2×C3×M16—22no—
78C2×C3×M16D414yescomplemented
79C22×C3×C8—22no—
80C22×C3×C8—22no—
81C22×C3×C8—22no—
82C22×C3×C8C2312yes—
83C22×C3×C8C2312yes—
84C2×#217C2×C412yes—
85C2×#217C2×C412yes—
86#947—24no—
87#944—24no—
88#940—24no—
89#939—28no—
90#939—28no—
91#936—24no—
92C4×#67—24no—
93C4×#67—24no—
94C4×#67—24no—
95C4×#67—24no—
96C2×(C2×D3)⋊C4C2×C412yes—
97C2×(C2×D3)⋊C4C2×C412yes—
98C2×Dic3⋊C4C2×C412yes—
99C2×Dic3⋊C4C2×C412yes—
100C2×C4×Dic3C2×C411yescharacteristic
101C2×C4×Dic3C2×C411yescharacteristic
102C2×C4×Dic3D414yes—
103C2×C4×Dic3D414yes—
104C2×C4×Dic3C2311yescharacteristic
105C2×C4×Dic3C2311yescharacteristic
106C2×#208—22no—
107C2×#208—22no—
108C2×#208D414yescomplemented
109C22×C3⋊C8—22no—
110C22×C3⋊C8—22no—
111C22×C3⋊C8—22no—
112C22×C3⋊C8C2312yes—
113C22×C3⋊C8C2312yes—
114C3×C22⋊C8—28no—
115#830—28no—
116(C2×D3)⋊C8—28no—
117(C2×D3)⋊C8—28no—
118#815—24no—
119Dic3⋊C8—24no—
120Dic3⋊C8—24no—
121C8×Dic3—24no—
122C22×#41—31no—
123C22×#41C2×D311yescharacteristic
124C22×M16—31no—
125C22×M16C2×D311yescharacteristic
126C23×C8—31no—
127C23×C8C2×D311yescharacteristic
128#532—64no—
129#532—64no—
130C2×C4⋊Q8—31no—
131C2×#127—31no—
132C2×#124—31no—
133C2×#124—31no—
134C2×#121—32no—
135C2×#121—32no—
136C2×C4×D4—32no—
137C2×C4×D4—32no—
138C22×C42—31no—
139C2×C4⋊C8—32no—
140C2×C4⋊C8—32no—
141C2×C4⋊C8—64no—
142#408—24no—
143#408—64no—
144#408—64no—
145#408—64no—
146C2×C22⋊C8—34no—
147C2×C22⋊C8—34no—
148C2×C22⋊C8—34no—
149C2×C22⋊C8C2×D314yes—
150C4×M16—64no—
151C2×#97—32no—
152C2×C4×C8—32no—
153C24×C3C22×C412yes—
154C24×C3C22×C411yescharacteristic
155C23×D3C22×C412yes—
156C3×#41—44no—
157C3×#41—44no—
158C3×#41—44no—
159C3×#41—44no—
160C2×C3×Q8—22no—
161C2×C3×Q8—22no—
162C2×C3×Q8—22no—
163C2×C3×D4—24no—
164C2×C3×D4—24no—
165C2×C3×D4—24no—
166C2×C3×D4—22no—
167C2×C3×D4—22no—
168C2×C3×D4—22no—
169C22×C3×C4—22no—
170C22×C3×C4—22no—
171C22×C3×C4—22no—
172C22×C3×C4—22no—
173C22×C3×C4—22no—
174C22×C3×C4—22no—
175C22×C3×C4C22×C412yes—
176C22×C3×C4C22×C411yescharacteristic
177C22×C3×C4C2×D412yes—
178C22×C3×C4C2×D412yes—
179C22×C3×C4C2×D412yes—
180C22×C3×C4C2×D412yes—
181C22×C3×C4#4112yes—
182C22×C3×C4#4112yes—
183C22×C3×C4C2411yescharacteristic
184C2×#67—24no—
185C2×#67—24no—
186C2×#67—24no—
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—28no—
190C2×#67—44no—
191C22×Dic3—24no—
192C22×Dic3—42no—
193C22×Dic3C22⋊C414yes—
194C22×Dic3C22⋊C414yes—
195C22×Dic3C22×C412yes—
196C22×Dic3C22×C411yescharacteristic
197C22×Dic3C22×C411yescharacteristic
198C22×Dic3C22×C411yescharacteristic
199C22×Dic3C22×C411yescharacteristic
200#235—44no—
201#235—44no—
202#235—44no—
203#235—44no—
204C2×D12—22no—
205C2×D12—22no—
206C2×D12—22no—
207C2×C4×D3—22no—
208C2×C4×D3—22no—
209C2×C4×D3—22no—
210C2×C4×D3—22no—
211C2×C4×D3—22no—
212C2×C4×D3—22no—
213C2×Dic6—22no—
214C2×Dic6—22no—
215C2×Dic6—22no—
216C3×M16—28no—
217C2×C3×C8—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220C2×C3×C8—22no—
221C2×C3×C8—22no—
222C2×C3×C8—44no—
223C2×C3×C8C2×D414yes—
224#217—24no—
225#217—24no—
226(C2×D3)⋊C4—24no—
227(C2×D3)⋊C4—24no—
228Dic3⋊C4—24no—
229Dic3⋊C4—24no—
230C4×Dic3C22⋊C414yes—
231C4×Dic3C22⋊C414yes—
232C4×Dic3C2×D412yes—
233C4×Dic3C2×D412yes—
234C4×Dic3C2×D412yes—
235C4×Dic3C2×D412yes—
236#208—28no—
237C2×C3⋊C8—22no—
238C2×C3⋊C8—22no—
239C2×C3⋊C8—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×C3⋊C8—44no—
243C2×C3⋊C8C2×D414yes—
244C2×#41—22no—
245C2×#41—22no—
246C2×#41—22no—
247C2×#41—22no—
248C2×#41—22no—
249C2×#41—22no—
250C2×#41—62no—
251C2×#41—62no—
252C2×#41—62no—
253C2×#41—62no—
254C2×#41—62no—
255C2×#41—62no—
256C22×Q8—31no—
257C22×Q8C4×D311yescharacteristic
258C22×D4—32no—
259C22×D4—32no—
260C22×D4—32no—
261C22×D4—31no—
262C22×D4C4×D312yes—
263C22×D4C4×D311yescharacteristic
264C23×C4—31no—
265C23×C4—31no—
266C23×C4—32no—
267C23×C4C22×D312yes—
268C23×C4C22×D311yescharacteristic
269C2×M16—22no—
270C2×M16—22no—
271C2×M16—34no—
272C2×M16—62no—
273C2×M16—62no—
274C2×M16#6714yescomplemented
275C22×C8—22no—
276C22×C8—22no—
277C22×C8—22no—
278C22×C8—32no—
279C22×C8—32no—
280C22×C8—62no—
281C22×C8—62no—
282C22×C8—62no—
283C22×C8C22×D312yes—
284C22×C8C22×D312yes—
285C4⋊Q8—64no—
286#127—64no—
287#124—64no—
288#124—64no—
289#121—68no—
290#121—68no—
291C4×D4—64no—
292C4×D4—64no—
293C4×D4—64no—
294C4×D4—64no—
295C2×C4⋊C4—32no—
296C2×C4⋊C4—32no—
297C2×C22⋊C4—32no—
298C2×C22⋊C4—32no—
299C2×C22⋊C4—32no—
300C2×C22⋊C4—32no—
301C2×C42—31no—
302C2×C42—31no—
303C2×C42—34no—
304C2×C42—34no—
305C2×C42—31no—
306C2×C42—31no—
307C4⋊C8—64no—
308C4⋊C8—64no—
309C22⋊C8—28no—
310C22⋊C8—68no—
311C22⋊C8—68no—
312C22⋊C8—68no—
313#97—64no—
314C4×C8—64no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—22no—
319C23×C3—22no—
320C23×C3—24no—
321C23×C3—24no—
322C23×C3C2×C22⋊C412yes—
323C23×C3C4×D412yes—
324C23×C3C4×D412yes—
325C23×C3C23×C411yescharacteristic
326C22×D3—22no—
327C22×D3—22no—
328C22×D3—22no—
329C22×D3—24no—
330C22×D3—24no—
331C3×Q8—44no—
332C3×D4—48no—
333C3×D4—44no—
334C2×C3×C4—22no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—42no—
338C2×C3×C4—42no—
339C2×C3×C4—42no—
340C2×C3×C4—42no—
341C2×C3×C4C2×C22⋊C412yes—
342C2×C3×C4C4×D412yes—
343C2×C3×C4C4×D412yes—
344C2×C3×C4#12114yes—
345C2×C3×C4#12114yes—
346C2×C3×C4#12414yes—
347C2×C3×C4#12714yes—
348C2×C3×C4C22×D411yescharacteristic
349C2×C3×C4C22×D411yescharacteristic
350C2×C3×C4C22×D411yescharacteristic
351C2×C3×C4C22×D411yescharacteristic
352C2×C3×C4C2×#4111yescharacteristic
353C2×C3×C4C2×#4111yescharacteristic
354#67—48no—
355#67—48no—
356#67—48no—
357C2×Dic3—22no—
358C2×Dic3—22no—
359C2×Dic3—22no—
360C2×Dic3—28no—
361C2×Dic3—22no—
362C2×Dic3—22no—
363C2×Dic3—22no—
364C2×Dic3—44no—
365C2×Dic3C2×C22⋊C412yes—
366C2×Dic3C2×C22⋊C412yes—
367C2×Dic3C2×C22⋊C412yes—
368C2×Dic3C2×C22⋊C412yes—
369C2×Dic3#13114yes—
370D12—44no—
371C4×D3—42no—
372C4×D3—42no—
373C4×D3—42no—
374C4×D3—42no—
375Dic6—44no—
376C3×C8—24no—
377C3×C8—42no—
378C3⋊C8—24no—
379C3⋊C8—42no—
380C24—32no—
381C24C2×C4×D312yes—
382C24C2×C4×D311yescharacteristic
383#41—44no—
384#41—44no—
385#41—44no—
386#41—44no—
387#41—124no—
388#41—124no—
389#41—124no—
390#41—124no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—62no—
395C2×Q8—62no—
396C2×Q8—62no—
397C2×D4—24no—
398C2×D4—24no—
399C2×D4—24no—
400C2×D4—22no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—64no—
404C2×D4—64no—
405C2×D4—68no—
406C2×D4—64no—
407C2×D4—64no—
408C2×D4—64no—
409C2×D4—62no—
410C2×D4—62no—
411C2×D4—62no—
412C2×D4—124no—
413C22×C4—22no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—31no—
420C22×C4—34no—
421C22×C4—31no—
422C22×C4—31no—
423C22×C4—31no—
424C22×C4—34no—
425C22×C4—32no—
426C22×C4—64no—
427C22×C4—62no—
428C22×C4—62no—
429C22×C4—62no—
430C22×C4—62no—
431C22×C4—62no—
432C22×C4—62no—
433C22×C4—122no—
434C22×C4C2×C4×D312yes—
435C22×C4C2×C4×D311yescharacteristic
436C22×C4#23512yes—
437C22×C4D3×D412yes—
438C22×C4D3×D412yes—
439C22×C4D3×D412yes—
440C22×C4#23712yes—
441C22×C4C2×#6712yes—
442C22×C4C23×D311yescharacteristic
443M16—28no—
444M16—68no—
445C2×C8—22no—
446C2×C8—22no—
447C2×C8—22no—
448C2×C8—22no—
449C2×C8—22no—
450C2×C8—34no—
451C2×C8—44no—
452C2×C8—62no—
453C2×C8—62no—
454C2×C8—62no—
455C2×C8—62no—
456C2×C8—62no—
457C2×C8—124no—
458C2×C8C2×#6714yes—
459C4⋊C4—64no—
460C4⋊C4—64no—
461C22⋊C4—64no—
462C22⋊C4—64no—
463C22⋊C4—64no—
464C22⋊C4—64no—
465C42—34no—
466C42—34no—
467C42—32no—
468C42—32no—
469C42—32no—
470C42—32no—
471C22×C3—22no—
472C22×C3—22no—
473C22×C3—24no—
474C22×C3—24no—
475C22×C3—42no—
476C22×C3—42no—
477C22×C3—44no—
478C22×C3—44no—
479C22×C3—42no—
480C22×C3#39014yes—
481C22×C3C22×C22⋊C411yescharacteristic
482C22×C3C2×C4×D411yescharacteristic
483C22×C3C2×C4×D411yescharacteristic
484C22×C3C2×#13112yes—
485C22×C3#56812yes—
486C2×D3—42no—
487C2×D3—42no—
488C2×D3—44no—
489C2×D3—44no—
490C2×D3—42no—
491C3×C4—42no—
492C3×C4#39014yes—
493C3×C4C2×#12111yescharacteristic
494C3×C4C2×#12111yescharacteristic
495C3×C4C2×#12411yescharacteristic
496C3×C4C2×#12711yescharacteristic
497Dic3—22no—
498Dic3—22no—
499Dic3—22no—
500Dic3—42no—
501Dic3#40814yescomplemented
502C23—22no—
503C23—22no—
504C23—22no—
505C23—22no—
506C23—24no—
507C23—24no—
508C23—22no—
509C23—62no—
510C23—62no—
511C23—64no—
512C23—64no—
513C23—62no—
514C23D3×C22⋊C412yes—
515C23#67⋊C412yes—
516C23C4×#6712yes—
517C23C22×C4×D311yescharacteristic
518Q8—44no—
519Q8—124no—
520D4—48no—
521D4—44no—
522D4—128no—
523D4—128no—
524D4—128no—
525D4—124no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—22no—
529C2×C4—32no—
530C2×C4—34no—
531C2×C4—32no—
532C2×C4—32no—
533C2×C4—32no—
534C2×C4—42no—
535C2×C4—42no—
536C2×C4—42no—
537C2×C4—42no—
538C2×C4—68no—
539C2×C4—62no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—124no—
546C2×C4—122no—
547C2×C4—122no—
548C2×C4—122no—
549C2×C4—122no—
550C2×C4D3×C22⋊C412yes—
551C2×C4#67⋊C412yes—
552C2×C4#88414yes—
553C2×C4#88514yes—
554C2×C4C4×#6712yes—
555C2×C4#93914yes—
556C2×C4#94014yes—
557C2×C4C2×#23511yescharacteristic
558C2×C4C2×D3×D411yescharacteristic
559C2×C4C2×D3×D411yescharacteristic
560C2×C4C2×D3×D411yescharacteristic
561C2×C4C2×#23711yescharacteristic
562C2×C4C22×#6711yescharacteristic
563C8—24no—
564C8—42no—
565C8—64no—
566C8—122no—
567C2×C3—24no—
568C2×C3—82no—
569C2×C3C2×#39011yescharacteristic
570C2×C3C2×#40811yescharacteristic
571C2×C3#287911yescharacteristic
572C2×C3#293512yes—
573C2×C3#293512yes—
574D3—82no—
575C22—24no—
576C22—22no—
577C22—22no—
578C22—24no—
579C22—44no—
580C22—42no—
581C22—42no—
582C22—44no—
583C22—42no—
584C22—124no—
585C22—122no—
586C22—122no—
587C22—124no—
588C22—122no—
589C22#495714yes—
590C22C2×D3×C22⋊C411yescharacteristic
591C22C2×#67⋊C411yescharacteristic
592C22D3×#13112yes—
593C22#575012yes—
594C22C2×C4×#6711yescharacteristic
595C4—34no—
596C4—42no—
597C4—62no—
598C4—62no—
599C4—62no—
600C4—122no—
601C4#495714yes—
602C4C2×#88411yescharacteristic
603C4C2×#88511yescharacteristic
604C4C2×#93911yescharacteristic
605C4C2×#94011yescharacteristic
606C3#2118911yescharacteristic, complemented, Sylow
607C2—24no—
608C2—82no—
609C2—242no—
610C2#7866012yes—
611C2D3×#40811yescharacteristic
612C2#7866211yescharacteristic
613C2C2×#495711yescharacteristic
614C2#8378812yes—
subgroupG/H
center, FrattiniC22×C4C23×D3
commutator, socleC23×C3C23×C4
FittingC2×C3×#408C2
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenterC2×#408D3
hypocenterC3#21189
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C22×C4 ◁ C2×#408

Complements

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

NH ≅ G/Ntype
2. C2×C3×#408C2semidirect
3. C2×#5234C2semidirect
4. C2×#5129C2semidirect
5. C2×Dic3×M16C2semidirect
6. C2×#5110C2semidirect
7. C22×Dic3⋊C8C2semidirect
8. #69917C2semidirect
9. #69916C2semidirect
47. C2×#408D3semidirect
78. C2×C3×M16D4semidirect
108. C2×#208D4semidirect
274. C2×M16#67semidirect
501. Dic3#408semidirect
606. C3#21189semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
C2×C3×#4082
C2×#52342
C2×#51292
C2×Dic3×M162
C2×#51102
C22×Dic3⋊C82
#699172
#699162
#211893