Finite Group Databasebeta

← #10804472

Subgroups — #10804472

Order 768.

1310 conjugacy classes of subgroups.

The subgroup lattice has 574 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5832C211yesmaximal, characteristic
2#78619C211yesmaximal, characteristic, complemented
3#78619C211yesmaximal, characteristic, complemented
4C3×#1785C211yesmaximal, characteristic, complemented
5#72137C211yesmaximal, characteristic, complemented
6#70099C212yesmaximal, complemented
7#70098C212yesmaximal, complemented
8#70065C212yesmaximal, complemented
9#70064C212yesmaximal, complemented
10#69989C211yesmaximal, characteristic, complemented
11#69989C211yesmaximal, characteristic, complemented
12#21800—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C22×#235C2211yescharacteristic
15#5832—22no—
16#5832—22no—
17#5832—22no—
18#5832—22no—
19C2×#947C2211yescharacteristic
20C2×#944C2211yescharacteristic
21C2×#940C2211yescharacteristic
22C2×#939C414yes—
23C2×#936C2211yescharacteristic
24C2×C4×#67C414yes—
25C22×C4×Dic3C2211yescharacteristic
26C2×C3×#101C2212yes—
27C2×C3×#100C2212yes—
28C3×#407C2211yescharacteristic
29C3×#407C2211yescharacteristic
30C2×#836C2212yes—
31C2×#833C2212yes—
32#5207C2211yescharacteristic
33#5207C2211yescharacteristic
34C2×#824C2212yes—
35C2×#824C2212yes—
36C2×#823C2212yes—
37C2×#823C2212yes—
38#5118C2212yes—
39#5118C2212yes—
40Dic3×M16C2212yes—
41Dic3×M16C2212yes—
42#4719—22no—
43#4719—22no—
44#4718—22no—
45#4718—22no—
46C2×#532—31no—
47#2920—31no—
48#2920—31no—
49#1967—32no—
50#1967—32no—
51#1966—32no—
52#1965—32no—
53#1785—31no—
54#1785—31no—
55#1785—31no—
56#1785D311yescharacteristic, complemented
57C2×C3×#41—22no—
58C2×C3×#41—22no—
59C2×C3×#41—42no—
60C2×C3×#41—42no—
61C22×C3×Q8C2311yescharacteristic
62C22×C3×D4C2×C412yes—
63C22×C3×D4C2311yescharacteristic
64C23×C3×C4C2×C412yes—
65C23×C3×C4C2311yescharacteristic
66C22×#67C2×C412yes—
67C22×#67C2×C414yes—
68C23×Dic3C2×C412yes—
69C2×#235—22no—
70C2×#235—22no—
71C2×#235—42no—
72C2×#235—42no—
73C22×D12C2311yescharacteristic
74C22×C4×D3C2×C412yes—
75C22×Dic6C2311yescharacteristic
76C2×C3×M16C2312yes—
77C2×C3×M16C2312yes—
78C2×#217C2×C414yes—
79#947—21no—
80#947—22no—
81#947—21no—
82#944—21no—
83#944—22no—
84#944—21no—
85#940—21no—
86#940—22no—
87#940—21no—
88#939—24no—
89#939—24no—
90#939—24no—
91#939—24no—
92#936—21no—
93#936—22no—
94#936—21no—
95C4×#67—24no—
96C4×#67—24no—
97C4×#67—24no—
98C4×#67—24no—
99C2×(C2×D3)⋊C4C2×C414yes—
100C2×Dic3⋊C4C2×C414yes—
101C2×C4×Dic3C2×C412yes—
102C2×C4×Dic3D412yes—
103C2×C4×Dic3D412yes—
104C2×C4×Dic3D412yes—
105C2×C4×Dic3D412yes—
106C2×C4×Dic3C2311yescharacteristic
107C2×C4×Dic3C2311yescharacteristic
108C2×#208C2312yes—
109C2×#208C2312yes—
110C3×#101—42no—
111C3×#100—42no—
112C3×C22⋊C8—22no—
113C3×C22⋊C8—22no—
114#836—42no—
115#833—42no—
116#830—22no—
117#830—22no—
118#824—42no—
119#824—42no—
120#823—42no—
121#823—42no—
122#815—22no—
123#815—22no—
124Dic3⋊C8—24no—
125Dic3⋊C8—24no—
126C8×Dic3—22no—
127C8×Dic3—22no—
128C22×#41—31no—
129C22×#41C2×D311yescharacteristic
130#532—62no—
131#532—62no—
132#532—62no—
133#532—62no—
134C2×C4⋊Q8—31no—
135C2×#127—31no—
136C2×#124—31no—
137C2×#124—31no—
138C2×#121—34no—
139C2×C4×D4—34no—
140C22×C42—31no—
141#432—62no—
142#432—62no—
143#431—62no—
144#431—62no—
145#423—32no—
146#423—32no—
147C2×#101—32no—
148C2×#101—32no—
149C2×#101—32no—
150C2×#101C2×D312yes—
151C2×#100—32no—
152C2×#100—32no—
153C2×#100—32no—
154C2×#100C2×D312yes—
155#407—31no—
156#407—31no—
157#407C2×D311yescharacteristic
158#407C2×D311yescharacteristic
159C4×M16—32no—
160C4×M16—32no—
161C24×C3C22×C412yes—
162C24×C3C22×C411yescharacteristic
163C23×D3C22×C412yes—
164C3×#41—48no—
165C3×#41—48no—
166C2×C3×Q8—21no—
167C2×C3×Q8—21no—
168C2×C3×Q8—22no—
169C2×C3×Q8—22no—
170C2×C3×D4—24no—
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—22no—
174C2×C3×D4—22no—
175C2×C3×D4—42no—
176C2×C3×D4—42no—
177C22×C3×C4—22no—
178C22×C3×C4—22no—
179C22×C3×C4—24no—
180C22×C3×C4—42no—
181C22×C3×C4—42no—
182C22×C3×C4C22×C412yes—
183C22×C3×C4C22×C411yescharacteristic
184C22×C3×C4C2×D412yes—
185C22×C3×C4C2×D412yes—
186C22×C3×C4C2411yescharacteristic
187C2×#67—24no—
188C2×#67—24no—
189C2×#67—24no—
190C2×#67—42no—
191C2×#67—42no—
192C2×#67—48no—
193C22×Dic3—44no—
194C22×Dic3C22⋊C412yes—
195C22×Dic3C22⋊C412yes—
196C22×Dic3C22⋊C412yes—
197C22×Dic3C22⋊C412yes—
198C22×Dic3C22×C412yes—
199C22×Dic3C22×C412yes—
200C22×Dic3C22×C412yes—
201#235—48no—
202#235—48no—
203C2×D12—21no—
204C2×D12—21no—
205C2×D12—22no—
206C2×D12—22no—
207C2×C4×D3—24no—
208C2×C4×D3—42no—
209C2×C4×D3—42no—
210C2×Dic6—21no—
211C2×Dic6—21no—
212C2×Dic6—22no—
213C2×Dic6—22no—
214C3×M16—22no—
215C3×M16—22no—
216C3×M16—22no—
217C3×M16—22no—
218C2×C3×C8—22no—
219C2×C3×C8—22no—
220#217—24no—
221#217—24no—
222(C2×D3)⋊C4—24no—
223(C2×D3)⋊C4—24no—
224Dic3⋊C4—24no—
225Dic3⋊C4—24no—
226C4×Dic3C22⋊C412yes—
227C4×Dic3C22⋊C412yes—
228C4×Dic3C22⋊C412yes—
229C4×Dic3C22⋊C412yes—
230C4×Dic3C2×D411yescharacteristic
231C4×Dic3C2×D411yescharacteristic
232C4×Dic3C2×D412yes—
233C4×Dic3C2×D412yes—
234C4×Dic3C2×D411yescharacteristic
235C4×Dic3C2×D411yescharacteristic
236#208—22no—
237#208—22no—
238#208—22no—
239#208—22no—
240C2×C3⋊C8—22no—
241C2×C3⋊C8—22no—
242C2×#41—22no—
243C2×#41—22no—
244C2×#41—42no—
245C2×#41—42no—
246C2×#41—62no—
247C2×#41—62no—
248C2×#41—122no—
249C2×#41—122no—
250C22×Q8—31no—
251C22×Q8C22×D311yescharacteristic
252C22×D4—34no—
253C22×D4—32no—
254C22×D4—31no—
255C22×D4C4×D312yes—
256C22×D4C22×D311yescharacteristic
257C23×C4—32no—
258C23×C4—32no—
259C23×C4C4×D312yes—
260C23×C4C22×D311yescharacteristic
261C2×M16—32no—
262C2×M16—32no—
263C2×M16C22×D312yes—
264C2×M16C22×D312yes—
265C4⋊Q8—61no—
266C4⋊Q8—61no—
267C4⋊Q8—62no—
268#127—61no—
269#127—61no—
270#127—62no—
271#124—61no—
272#124—61no—
273#124—62no—
274#124—61no—
275#124—61no—
276#124—62no—
277#121—64no—
278#121—64no—
279#121—64no—
280#121—64no—
281C4×D4—64no—
282C4×D4—64no—
283C4×D4—64no—
284C4×D4—64no—
285C2×C4⋊C4—34no—
286C2×C22⋊C4—34no—
287C2×C22⋊C4—34no—
288C2×C42—32no—
289C2×C42—32no—
290C2×C42—32no—
291C2×C42—32no—
292C2×C42—32no—
293C2×C42—31no—
294C2×C42—31no—
295C4⋊C8—64no—
296C4⋊C8—64no—
297#101—42no—
298#101—122no—
299#101—122no—
300#101—122no—
301#100—42no—
302#100—122no—
303#100—122no—
304#100—122no—
305C22⋊C8—22no—
306C22⋊C8—22no—
307C22⋊C8—62no—
308C22⋊C8—62no—
309#97—62no—
310#97—62no—
311C4×C8—62no—
312C4×C8—62no—
313C23×C3—21no—
314C23×C3—21no—
315C23×C3—22no—
316C23×C3—22no—
317C23×C3—22no—
318C23×C3—42no—
319C23×C3—44no—
320C23×C3C2×C22⋊C412yes—
321C23×C3C23×C411yescharacteristic
322C22×D3—22no—
323C22×D3—22no—
324C22×D3—22no—
325C22×D3—44no—
326C3×Q8—42no—
327C3×Q8—42no—
328C3×D4—48no—
329C3×D4—42no—
330C3×D4—42no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—24no—
334C2×C3×C4—24no—
335C2×C3×C4—22no—
336C2×C3×C4—22no—
337C2×C3×C4—22no—
338C2×C3×C4—48no—
339C2×C3×C4C2×C22⋊C412yes—
340C2×C3×C4C22×D411yescharacteristic
341C2×C3×C4C22×D411yescharacteristic
342C2×C3×C4H(5,𝔽2)12yes—
343C2×C3×C4H(5,𝔽2)12yes—
344#67—48no—
345#67—48no—
346#67—48no—
347C2×Dic3—24no—
348C2×Dic3—24no—
349C2×Dic3—24no—
350C2×Dic3—22no—
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—42no—
354C2×Dic3—42no—
355C2×Dic3—42no—
356C2×Dic3C2×C22⋊C412yes—
357C2×Dic3C2×C22⋊C412yes—
358C2×Dic3C2×C22⋊C412yes—
359C2×Dic3C2×C22⋊C412yes—
360D12—42no—
361D12—42no—
362C4×D3—48no—
363Dic6—42no—
364Dic6—42no—
365C3×C8—42no—
366C3×C8—42no—
367C3⋊C8—42no—
368C3⋊C8—42no—
369C24—32no—
370C24C2×C4×D312yes—
371C24C2×C4×D311yescharacteristic
372#41—48no—
373#41—48no—
374#41—128no—
375#41—128no—
376C2×Q8—21no—
377C2×Q8—21no—
378C2×Q8—22no—
379C2×Q8—22no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—62no—
383C2×Q8—62no—
384C2×D4—24no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—64no—
392C2×D4—64no—
393C2×D4—64no—
394C2×D4—61no—
395C2×D4—61no—
396C2×D4—62no—
397C2×D4—62no—
398C2×D4—122no—
399C2×D4—122no—
400C2×D4—128no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—24no—
404C22×C4—32no—
405C22×C4—32no—
406C22×C4—32no—
407C22×C4—32no—
408C22×C4—32no—
409C22×C4—32no—
410C22×C4—32no—
411C22×C4—42no—
412C22×C4—42no—
413C22×C4—64no—
414C22×C4—122no—
415C22×C4—122no—
416C22×C4—124no—
417C22×C4C2×C4×D312yes—
418C22×C4C2×C4×D311yescharacteristic
419C22×C4D3×D412yes—
420C22×C4D3×D412yes—
421C22×C4C23×D311yescharacteristic
422M16—22no—
423M16—22no—
424M16—22no—
425M16—22no—
426M16—62no—
427M16—62no—
428M16—62no—
429M16—62no—
430C2×C8—22no—
431C2×C8—22no—
432C2×C8—62no—
433C2×C8—62no—
434C4⋊C4—64no—
435C4⋊C4—64no—
436C22⋊C4—64no—
437C22⋊C4—64no—
438C22⋊C4—64no—
439C22⋊C4—64no—
440C42—32no—
441C42—32no—
442C42—32no—
443C42—32no—
444C42—31no—
445C42—31no—
446C42—31no—
447C42—31no—
448C42—32no—
449C42—32no—
450C22×C3—22no—
451C22×C3—22no—
452C22×C3—22no—
453C22×C3—22no—
454C22×C3—42no—
455C22×C3—42no—
456C22×C3—42no—
457C22×C3—44no—
458C22×C3—82no—
459C22×C3—82no—
460C22×C3C22×C22⋊C411yescharacteristic
461C22×C3#51811yescharacteristic
462C22×C3#51811yescharacteristic
463C2×D3—42no—
464C2×D3—44no—
465C2×D3—82no—
466C2×D3—82no—
467C3×C4—22no—
468C3×C4—42no—
469C3×C4#53512yes—
470C3×C4#53512yes—
471Dic3—22no—
472Dic3—22no—
473Dic3—22no—
474Dic3—42no—
475Dic3#41314yes—
476C23—21no—
477C23—21no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—42no—
482C23—44no—
483C23—62no—
484C23—62no—
485C23—62no—
486C23—124no—
487C23D3×C22⋊C412yes—
488C23C22×C4×D311yescharacteristic
489Q8—42no—
490Q8—42no—
491Q8—122no—
492Q8—122no—
493D4—48no—
494D4—42no—
495D4—42no—
496D4—128no—
497D4—128no—
498D4—128no—
499D4—122no—
500D4—122no—
501C2×C4—21no—
502C2×C4—21no—
503C2×C4—24no—
504C2×C4—24no—
505C2×C4—22no—
506C2×C4—22no—
507C2×C4—22no—
508C2×C4—32no—
509C2×C4—32no—
510C2×C4—32no—
511C2×C4—32no—
512C2×C4—48no—
513C2×C4—64no—
514C2×C4—64no—
515C2×C4—64no—
516C2×C4—62no—
517C2×C4—62no—
518C2×C4—62no—
519C2×C4—128no—
520C2×C4—122no—
521C2×C4—122no—
522C2×C4—122no—
523C2×C4D3×C22⋊C412yes—
524C2×C4C2×D3×D411yescharacteristic
525C2×C4C2×D3×D411yescharacteristic
526C2×C4#100412yes—
527C2×C4#100412yes—
528C8—42no—
529C8—42no—
530C8—122no—
531C8—122no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—42no—
535C2×C3—82no—
536C2×C3#230911yescharacteristic
537C2×C3C2×#41312yes—
538D3—82no—
539C22—22no—
540C22—22no—
541C22—22no—
542C22—22no—
543C22—42no—
544C22—42no—
545C22—42no—
546C22—44no—
547C22—82no—
548C22—82no—
549C22—122no—
550C22—124no—
551C22—242no—
552C22—242no—
553C22C2×D3×C22⋊C411yescharacteristic
554C22#548611yescharacteristic
555C22#548611yescharacteristic
556C4—22no—
557C4—34no—
558C4—42no—
559C4—62no—
560C4—62no—
561C4—62no—
562C4—122no—
563C4#559212yes—
564C4#559212yes—
565C3#2180011yescharacteristic, complemented, Sylow
566C2—21no—
567C2—21no—
568C2—42no—
569C2—82no—
570C2—242no—
571C2#7381911yescharacteristic
572C2D3×#41312yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC23×C3C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1785C2
socleC22×C3C22×C22⋊C4
prime radicalC2×#5832C2
cosocleC22×C3×C4C24
hypercenter#1785D3
hypocenterC3#21800
LCSG ▷ C23×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #78619C2semidirect
3. #78619C2semidirect
4. C3×#1785C2semidirect
5. #72137C2semidirect
6. #70099C2semidirect
7. #70098C2semidirect
8. #70065C2semidirect
9. #70064C2semidirect
10. #69989C2semidirect
11. #69989C2semidirect
56. #1785D3semidirect
565. C3#21800semidirect

Maximal subgroups

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

subgroupindex
C2×#58322
#786192
C3×#17852
#721372
#700992
#700982
#700652
#700642
#699892
#218003