Finite Group Databasebeta

← #10772479

Subgroups — #10772479

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86225C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic, complemented
3#79060C211yesmaximal, characteristic, complemented
4C2×#4681C211yesmaximal, characteristic, complemented
5C3×#1492C211yesmaximal, characteristic, complemented
6#70507C211yesmaximal, characteristic, complemented
7#69734C211yesmaximal, characteristic, complemented
8#68207C214yesmaximal, complemented
9#68206C214yesmaximal, complemented
10#18637—31nomaximal, Sylow
11C3×#532C2211yescharacteristic
12#5833—21no—
13#5827—21no—
14#5728—21no—
15#5716—21no—
16#5682—22no—
17#5618—22no—
18#5598—22no—
19#5563C2212yes—
20#5555—21no—
21#5543C2212yes—
22#5542—21no—
23C4×#235C2211yescharacteristic
24C2×#869C2211yescharacteristic
25C2×#869C2211yescharacteristic
26C3×#423C2211yescharacteristic
27C2×C3×C4×C8C2211yescharacteristic
28#5133—21no—
29#5121—21no—
30C2×#816—22no—
31#5093—21no—
32#5076—21no—
33#5040—22no—
34#5019C2214yes—
35#4925C2211yescharacteristic
36#4851C2212yes—
37#4850—21no—
38#4846C2212yes—
39#4842—21no—
40#4681—22no—
41#4681C2214yes—
42C3×D4⋊C8C2214yes—
43#4481C2214yes—
44#4459C2212yes—
45#4456C2212yes—
46#3511—31no—
47#3206—31no—
48C2×#487—31no—
49#3079—31no—
50#1646—34no—
51#1643—34no—
52#1548—31no—
53#1492D311yescharacteristic, complemented
54#1476—31no—
55C2×C3×#41—21no—
56C2×#237—22no—
57C2×#235D411yescharacteristic
58C22×Dic6—21no—
59C22×Dic6D412yes—
60C2×C3×M16—21no—
61C22×C3×C8—21no—
62C3×C4⋊Q8D411yescharacteristic
63C3×#127D411yescharacteristic
64C3×#124—21no—
65C3×#121—21no—
66C3×#121D412yes—
67C3×C4×D4—21no—
68C3×C4×D4C2312yes—
69C2×C3×C42C2311yescharacteristic
70#946—22no—
71Q8×Dic3—22no—
72#939—24no—
73#938—22no—
74#938—22no—
75#936—22no—
76D4×Dic3—22no—
77D4×Dic3—24no—
78D4×Dic3—22no—
79C2×#214—24no—
80#930—22no—
81C4×#67—21no—
82C4×#67—21no—
83#926—22no—
84#926D411yescharacteristic
85C2×C4⋊Dic3—21no—
86C2×C4⋊Dic3—21no—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3—21no—
89C2×C4⋊Dic3—21no—
90#924—22no—
91#897—22no—
92#892—22no—
93#890—22no—
94#887—22no—
95#886—24no—
96#885—24no—
97#879—24no—
98#878—24no—
99C4×D12D411yescharacteristic
100C4×D12C2311yescharacteristic
101#872—21no—
102C42×D3—21no—
103#869—21no—
104#869—21no—
105#869—21no—
106#869—21no—
107#869—21no—
108#869D412yes—
109#869C2312yes—
110#869C2312yes—
111C4×Dic6D411yescharacteristic
112C4×Dic6C2311yescharacteristic
113C3×C4⋊C8—21no—
114C3×C4⋊C8C2312yes—
115C3×C4×C8—21no—
116C3×C4×C8C2312yes—
117#830—21no—
118#821—22no—
119#818—22no—
120#817—22no—
121#816—22no—
122#816—28no—
123#810—24no—
124C4⋊(C3⋊C8)C2312yes—
125C4×C3⋊C8—21no—
126#556—61no—
127#555—32no—
128#554—32no—
129#553—62no—
130#548—62no—
131#547—61no—
132#542—62no—
133#537—61no—
134#533—61no—
135#532—61no—
136#532C2×D311yescharacteristic
137C2×C4⋊Q8—31no—
138C2×C4⋊Q8—31no—
139#520—61no—
140C4×#41—31no—
141#492—61no—
142#487—34no—
143#487—62no—
144#487—61no—
145#477—61no—
146#476—61no—
147#475—32no—
148#474—32no—
149#461—34no—
150#448—62no—
151#431—31no—
152#428—61no—
153#423C2×D311yescharacteristic
154#417—61no—
155C2×#103—62no—
156C2×C4×C8C2×D311yescharacteristic
157#333—34no—
158Q8⋊C8—32no—
159D4⋊C8—32no—
160D4⋊C8C2×D314yes—
161C3×#41—42no—
162C3×#41—42no—
163C2×C3×Q8—21no—
164C2×C3×D4—22no—
165C2×C3×D4—21no—
166C2×C3×D4—21no—
167C2×C3×D4C2×D412yes—
168C22×C3×C4—22no—
169C22×C3×C4C2×D411yescharacteristic
170C22×C3×C4C2×D411yescharacteristic
171C22×C3×C4C2×D411yescharacteristic
172C2×#67—21no—
173C2×#67—24no—
174C22×Dic3—24no—
175C22×Dic3—22no—
176C22×Dic3—41no—
177#237—84no—
178#235—22no—
179#235—22no—
180C2×D12C2×D411yescharacteristic
181C2×C4×D3—21no—
182C2×C4×D3—21no—
183C2×C4×D3—21no—
184C2×Dic6—22no—
185C2×Dic6—22no—
186C2×Dic6—44no—
187C2×Dic6—41no—
188C2×Dic6—82no—
189C2×Dic6C2×D411yescharacteristic
190C2×Dic6C2×D414yes—
191C3×M16—42no—
192C2×C3×C8—22no—
193C2×C3×C8—22no—
194C2×C3×C8—42no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C2×D412yes—
197C3×C22⋊C4—22no—
198C3×C22⋊C4—22no—
199C3×C42C2×D411yescharacteristic
200C3×C42C2×D411yescharacteristic
201C3×C42C2×D411yescharacteristic
202C3×C42C2411yescharacteristic
203#217—24no—
204#217—24no—
205#217—21no—
206#214—84no—
207(C2×D3)⋊C4—24no—
208(C2×D3)⋊C4—21no—
209(C2×D3)⋊C4—21no—
210C4⋊Dic3—24no—
211C4⋊Dic3—22no—
212C4⋊Dic3—22no—
213C4⋊Dic3—22no—
214C4⋊Dic3—22no—
215C4⋊Dic3—22no—
216C4⋊Dic3—22no—
217C4⋊Dic3—41no—
218C4⋊Dic3—41no—
219C4⋊Dic3C2×D411yescharacteristic
220C4⋊Dic3C2×D411yescharacteristic
221Dic3⋊C4—24no—
222Dic3⋊C4—21no—
223Dic3⋊C4—21no—
224C4×Dic3—24no—
225C4×Dic3—21no—
226C4×Dic3—21no—
227C4×Dic3—21no—
228C2×C3⋊C8—22no—
229C2×#41—21no—
230C2×#41—31no—
231C2×#41—62no—
232C22×Q8—32no—
233C22×Q8—61no—
234C2×SD16—64no—
235C2×M16—21no—
236C22×C8—21no—
237C4⋊Q8—32no—
238C4⋊Q8—32no—
239C4⋊Q8—32no—
240C4⋊Q8—61no—
241C4⋊Q8—61no—
242C4⋊Q8—61no—
243C4⋊Q8—61no—
244C4⋊Q8—61no—
245C4⋊Q8D1211yescharacteristic
246#127D1211yescharacteristic
247#126—64no—
248#125—62no—
249#124—21no—
250#124—64no—
251#124—62no—
252#123—64no—
253C22⋊Q8—64no—
254C22⋊Q8—62no—
255C22⋊Q8—62no—
256C22⋊Q8—62no—
257#121—21no—
258#121—62no—
259#121—62no—
260#121—64no—
261#121—62no—
262#121D1212yes—
263C4×Q8—31no—
264C4×Q8—31no—
265C4×Q8—62no—
266C4×Q8—62no—
267C4×D4—21no—
268C4×D4—31no—
269C4×D4—31no—
270C4×D4—62no—
271C4×D4—64no—
272C4×D4—62no—
273C4×D4—61no—
274C4×D4—61no—
275C4×D4C22×D312yes—
276#117—31no—
277#117—62no—
278#117—62no—
279#117—61no—
280C2×C4⋊C4—62no—
281C2×C4⋊C4—61no—
282C2×C4⋊C4—61no—
283C2×C4⋊C4—61no—
284C2×C4⋊C4—61no—
285C2×C42—61no—
286C2×C42C22×D311yescharacteristic
287#107—62no—
288#106—62no—
289C4⋊C8—21no—
290C4⋊C8—32no—
291C4⋊C8C22×D312yes—
292#103—68no—
293#103—62no—
294#103—64no—
295#102—62no—
296C22⋊C8—61no—
297C4×C8—21no—
298C4×C8—61no—
299C4×C8C22×D312yes—
300C23×C3—22no—
301C23×C3#12011yescharacteristic
302C22×D3—21no—
303C3×Q8—42no—
304C3×D4—24no—
305C3×D4—44no—
306C3×D4—42no—
307C3×D4—42no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—42no—
312C2×C3×C4—42no—
313C2×C3×C4#12011yescharacteristic
314C2×C3×C4#12011yescharacteristic
315C2×C3×C4#12011yescharacteristic
316C2×C3×C4#13512yes—
317C2×C3×C4#13712yes—
318C2×C3×C4C22×D411yescharacteristic
319C2×C3×C4C22×D411yescharacteristic
320C2×C3×C4C22×D411yescharacteristic
321#67—42no—
322#67—84no—
323C2×Dic3—21no—
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—24no—
327C2×Dic3—42no—
328C2×Dic3—82no—
329C2×Dic3—84no—
330C2×Dic3—82no—
331D12—21no—
332D12—21no—
333C4×D3—42no—
334C4×D3—42no—
335C4×D3—42no—
336Dic6—21no—
337Dic6—21no—
338Dic6—44no—
339Dic6—44no—
340Dic6—44no—
341Dic6—82no—
342C3×C8—42no—
343C3×C8—42no—
344C3⋊C8—42no—
345#41—42no—
346#41—42no—
347#41—62no—
348#41—62no—
349#41—244no—
350C2×Q8—21no—
351C2×Q8—31no—
352C2×Q8—34no—
353C2×Q8—62no—
354C2×Q8—62no—
355C2×Q8—124no—
356C2×Q8—121no—
357C2×Q8—242no—
358C2×D4—22no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—31no—
362C2×D4—61no—
363C2×D4—64no—
364C2×D4C2×D1212yes—
365C22×C4—22no—
366C22×C4—64no—
367C22×C4—62no—
368C22×C4—61no—
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—121no—
372C22×C4C2×D1211yescharacteristic
373C22×C4D3×D411yescharacteristic
374C22×C4D3×D411yescharacteristic
375SD16—244no—
376M16—42no—
377C2×C8—22no—
378C2×C8—22no—
379C2×C8—42no—
380C2×C8—62no—
381C4⋊C4—21no—
382C4⋊C4—31no—
383C4⋊C4—31no—
384C4⋊C4—64no—
385C4⋊C4—64no—
386C4⋊C4—62no—
387C4⋊C4—62no—
388C4⋊C4—61no—
389C4⋊C4—62no—
390C4⋊C4—62no—
391C4⋊C4—61no—
392C4⋊C4—62no—
393C4⋊C4—62no—
394C4⋊C4—121no—
395C4⋊C4—121no—
396C4⋊C4C2×D1212yes—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—64no—
400C22⋊C4—64no—
401C22⋊C4—64no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C42—64no—
406C42—61no—
407C42—61no—
408C42—61no—
409C42C2×D1211yescharacteristic
410C42D3×D411yescharacteristic
411C42D3×D411yescharacteristic
412C42C23×D311yescharacteristic
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—42no—
416C22×C3—42no—
417C22×C3—42no—
418C22×C3#45212yes—
419C22×C3C2×#12011yescharacteristic
420C2×D3—41no—
421C2×D3—41no—
422C2×D3—41no—
423C3×C4—21no—
424C3×C4—21no—
425C3×C4—42no—
426C3×C4#45212yes—
427C3×C4#45512yes—
428C3×C4C2×#13511yescharacteristic
429C3×C4C2×#13711yescharacteristic
430C3×C4#57712yes—
431Dic3—41no—
432Dic3—41no—
433Dic3—44no—
434Dic3—41no—
435Dic3—82no—
436C23—22no—
437C23—61no—
438C23#88211yescharacteristic
439Q8—42no—
440Q8—61no—
441Q8—61no—
442Q8—124no—
443Q8—124no—
444Q8—124no—
445Q8—242no—
446D4—24no—
447D4—44no—
448D4—42no—
449D4—42no—
450D4—61no—
451D4—61no—
452D4—122no—
453D4—244no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—61no—
460C2×C4—61no—
461C2×C4—61no—
462C2×C4—64no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—122no—
466C2×C4—122no—
467C2×C4—242no—
468C2×C4—244no—
469C2×C4—242no—
470C2×C4#88211yescharacteristic
471C2×C4#88211yescharacteristic
472C2×C4#88211yescharacteristic
473C2×C4#91212yes—
474C2×C4#91512yes—
475C2×C4C22×D1211yescharacteristic
476C2×C4C2×D3×D411yescharacteristic
477C2×C4C2×D3×D411yescharacteristic
478C8—42no—
479C8—42no—
480C8—122no—
481C2×C3—22no—
482C2×C3—82no—
483C2×C3C2×#45211yescharacteristic
484C2×C3#301011yescharacteristic
485C2×C3C2×#45511yescharacteristic
486D3—81no—
487C22—21no—
488C22—21no—
489C22—42no—
490C22—42no—
491C22—42no—
492C22—121no—
493C22—121no—
494C22—121no—
495C22#478312yes—
496C22C2×#88211yescharacteristic
497C4—21no—
498C4—21no—
499C4—42no—
500C4—121no—
501C4—121no—
502C4—124no—
503C4—121no—
504C4—242no—
505C4#478312yes—
506C4#482512yes—
507C4#575112yes—
508C4C2×#91211yescharacteristic
509C4C2×#91511yescharacteristic
510C3#1863711yescharacteristic, complemented, Sylow
511C2—22no—
512C2—82no—
513C2—241no—
514C2#7867811yescharacteristic
515C2C2×#478311yescharacteristic
516C2C2×#482511yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1492C2
socleC22×C3C2×#120
prime radical#86225C2
hypercenter#1492D3
hypocenterC3#18637
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #83317C2semidirect
3. #79060C2semidirect
4. C2×#4681C2semidirect
5. C3×#1492C2semidirect
6. #70507C2semidirect
7. #69734C2semidirect
8. #68207C2semidirect
9. #68206C2semidirect
53. #1492D3semidirect
510. C3#18637semidirect

Maximal subgroups

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

subgroupindex
#862252
#833172
#790602
C2×#46812
C3×#14922
#705072
#697342
#682072
#682062
#186373