Finite Group Databasebeta

← #10945057

Subgroups — #10945057

Order 768.

1216 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5516C211yesmaximal, characteristic
2#82263C211yesmaximal, characteristic
3C2×#4739C211yesmaximal, characteristic
4C2×#4723C211yesmaximal, characteristic
5C3×#1818C211yesmaximal, characteristic
6#69963C211yesmaximal, characteristic
7#69926C211yesmaximal, characteristic
8#68685C212yesmaximal, complemented
9#68685C212yesmaximal, complemented
10#68684C212yesmaximal, complemented
11#68683C212yesmaximal, complemented
12#33019—31nomaximal, Sylow
13C23×Dic6C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C2×#924C2211yescharacteristic
17C22×Dic12C2212yes—
18#5516—24no—
19#5516—24no—
20C2×#887—21no—
21C2×#887—21no—
22C2×#877—22no—
23C2×C4×Dic6C2212yes—
24C2×C4×Dic6C2212yes—
25C2×C3×C22⋊C8C2211yescharacteristic
26C2×C3×C22⋊C8C2211yescharacteristic
27#5212C2211yescharacteristic
28C2×#818C2211yescharacteristic
29C2×#817C2211yescharacteristic
30C2×#816C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33C2×#816C2211yescharacteristic
34C2×#816C2211yescharacteristic
35#4985—21no—
36#4983—21no—
37#4952—22no—
38#4929C2212yes—
39#4928C2212yes—
40#4739—24no—
41#4723—24no—
42C3×#336C2212yes—
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#519—31no—
47C2×#485—31no—
48C2×#451—31no—
49#2625—31no—
50#1878—32no—
51#1878—32no—
52#1865—32no—
53#1863—32no—
54#1825—31no—
55#1818D311yescharacteristic
56#1787—31no—
57C23×C3×C4C2311yescharacteristic
58C23×Dic3—21no—
59C22×Dic6—21no—
60C22×Dic6—22no—
61C22×Dic6—24no—
62C22×Dic6—24no—
63C22×Dic6D414yes—
64C22×Dic6C2311yescharacteristic
65C22×Dic6C2312yes—
66C22×C3×C8C2312yes—
67C22×C3×C8C2311yescharacteristic
68C22×C3×C8C2311yescharacteristic
69C3×#117—22no—
70C3×#117—22no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C22⋊C4D412yes—
74C2×C3×C42C2312yes—
75C2×#217—21no—
76C2×#217—21no—
77C2×#217—21no—
78#926—22no—
79#926D414yes—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2311yescharacteristic
84#924—22no—
85#924—22no—
86C2×Dic3⋊C4—22no—
87C2×Dic3⋊C4—22no—
88C2×Dic3⋊C4—21no—
89C2×Dic3⋊C4—22no—
90C2×C4×Dic3—22no—
91C2×C4×Dic3—22no—
92C2×C4×Dic3—21no—
93C2×Dic12—24no—
94C2×Dic12—24no—
95C2×Dic12—24no—
96#887—28no—
97#887—28no—
98#877—216no—
99C4×Dic6—24no—
100C4×Dic6—24no—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3×C22⋊C8—22no—
104C3×C22⋊C8—22no—
105C3⋊#95—22no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—21no—
109#818—22no—
110#817—22no—
111#816—24no—
112#816—24no—
113#816—24no—
114#816—22no—
115#816—22no—
116C23×Q8—31no—
117C22×Q16—32no—
118C2×C22⋊Q8—31no—
119#519—64no—
120#519—64no—
121C2×C4×Q8—32no—
122C2×C4×Q8—32no—
123C2×C4×Q8—61no—
124C2×C4×Q8—61no—
125C2×#117—31no—
126C2×#117—62no—
127C2×#117C2×D311yescharacteristic
128#485—64no—
129#451—64no—
130C2×#107—31no—
131C2×#106—31no—
132C2×#103—31no—
133C2×#103—31no—
134C2×#103—32no—
135C2×#103—32no—
136C2×#103—32no—
137C2×C22⋊C8C2×D311yescharacteristic
138C2×C22⋊C8C2×D311yescharacteristic
139#400—61no—
140#398—61no—
141#397—62no—
142#391—32no—
143#389—32no—
144#380—31no—
145#340—32no—
146#340—32no—
147#336C2×D312yes—
148#336C2×D312yes—
149C24×C3C2×D411yescharacteristic
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4C2×D412yes—
153C22×C3×C4C2×D412yes—
154C22×C3×C4C2×D411yescharacteristic
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D412yes—
157C22×C3×C4C2411yescharacteristic
158C22×Dic3—28no—
159C22×Dic3—24no—
160C22×Dic3—22no—
161C22×Dic3—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164C22×Dic3—21no—
165C22×Dic3—21no—
166C2×Dic6—28no—
167C2×Dic6—28no—
168C2×Dic6—28no—
169C2×Dic6—24no—
170C2×Dic6—22no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—216no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6C2×D414yes—
179C2×Dic6#4114yes—
180C2×Dic6#4114yes—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—22no—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C22⋊C4—24no—
191C3×C42—22no—
192C3×C42—22no—
193#217—24no—
194#217—42no—
195#217—81no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—22no—
201C4⋊Dic3—22no—
202C4⋊Dic3C2×D412yes—
203C4⋊Dic3C2×D412yes—
204Dic3⋊C4—28no—
205Dic3⋊C4—28no—
206Dic3⋊C4—42no—
207Dic3⋊C4—82no—
208C4×Dic3—28no—
209C4×Dic3—28no—
210C4×Dic3—24no—
211Dic12—48no—
212C22×Q8—34no—
213C22×Q8—32no—
214C22×Q8—31no—
215C22×Q8—64no—
216C22×Q8—64no—
217C22×Q8—61no—
218C22×Q8—62no—
219C23×C4—61no—
220C23×C4C22×D311yescharacteristic
221C2×Q16—64no—
222C2×Q16—64no—
223C2×Q16—64no—
224C22×C8C22×D312yes—
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C22⋊Q8—62no—
228C22⋊Q8—62no—
229C4×Q8—64no—
230C4×Q8—64no—
231C4×Q8—64no—
232C4×Q8—64no—
233C4×Q8—68no—
234C4×Q8—68no—
235#117—22no—
236#117—22no—
237#117—34no—
238#117—616no—
239#117—62no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—61no—
247C2×C4⋊C4—62no—
248C2×C4⋊C4D1211yescharacteristic
249C2×C4⋊C4D1211yescharacteristic
250C2×C22⋊C4—61no—
251C2×C22⋊C4—61no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4D1212yes—
254C2×C42—61no—
255C2×C42—62no—
256C2×C42—62no—
257C2×C42C22×D312yes—
258#107—62no—
259#106—62no—
260#103—62no—
261#103—62no—
262#103—64no—
263#103—64no—
264#103—64no—
265C22⋊C8—22no—
266C22⋊C8—22no—
267#95—61no—
268#95—62no—
269#95—62no—
270#95—62no—
271C23×C3—22no—
272C23×C3—21no—
273C23×C3—22no—
274C23×C3—21no—
275C23×C3#12012yes—
276C23×C3C22×D411yescharacteristic
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—22no—
283C2×C3×C4—22no—
284C2×C3×C4—24no—
285C2×C3×C4—22no—
286C2×C3×C4#12012yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4112yes—
290C2×C3×C4C2×#4112yes—
291C2×Dic3—216no—
292C2×Dic3—24no—
293C2×Dic3—24no—
294C2×Dic3—24no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—41no—
298C2×Dic3—42no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—41no—
302C2×Dic3—42no—
303C2×Dic3—42no—
304C2×Dic3—41no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—41no—
308Dic6—216no—
309Dic6—24no—
310Dic6—24no—
311Dic6—24no—
312Dic6—42no—
313Dic6—41no—
314Dic6—41no—
315Dic6#12318yes—
316C3×C8—42no—
317C3×C8—41no—
318C3×C8—41no—
319C24C2×D1211yescharacteristic
320C2×Q8—34no—
321C2×Q8—34no—
322C2×Q8—34no—
323C2×Q8—616no—
324C2×Q8—68no—
325C2×Q8—68no—
326C2×Q8—68no—
327C2×Q8—64no—
328C2×Q8—62no—
329C2×Q8—62no—
330C2×Q8—62no—
331C2×Q8—61no—
332C2×Q8—61no—
333C2×Q8—62no—
334C2×Q8—62no—
335C22×C4—22no—
336C22×C4—22no—
337C22×C4—68no—
338C22×C4—64no—
339C22×C4—62no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—61no—
345C22×C4C2×D1212yes—
346C22×C4C2×D1212yes—
347C22×C4C2×D1211yescharacteristic
348C22×C4D3×D412yes—
349C22×C4D3×D412yes—
350C22×C4C23×D311yescharacteristic
351Q16—128no—
352C2×C8—21no—
353C2×C8—21no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—32no—
362C4⋊C4—32no—
363C4⋊C4—68no—
364C4⋊C4—68no—
365C4⋊C4—61no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—62no—
370C4⋊C4—62no—
371C4⋊C4—122no—
372C4⋊C4—242no—
373C22⋊C4—24no—
374C22⋊C4—64no—
375C22⋊C4—122no—
376C22⋊C4—241no—
377C42—22no—
378C42—22no—
379C42—68no—
380C42—68no—
381C42—64no—
382C22×C3—22no—
383C22×C3—22no—
384C22×C3—22no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3C2×#12011yescharacteristic
388C22×C3#52512yes—
389C22×C3#57711yescharacteristic
390C22×C3#57812yes—
391C22×C3#57811yescharacteristic
392C3×C4—22no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4C2×#12312yes—
396C3×C4#52511yescharacteristic
397C3×C4#52511yescharacteristic
398Dic3—28no—
399Dic3—42no—
400Dic3—42no—
401Dic3—81no—
402Dic3—82no—
403Dic3—81no—
404C23—21no—
405C23—22no—
406C23—21no—
407C23—22no—
408C23#88212yes—
409C23C22×D1211yescharacteristic
410Q8—38no—
411Q8—616no—
412Q8—64no—
413Q8—64no—
414Q8—64no—
415Q8—122no—
416Q8—121no—
417Q8—121no—
418C2×C4—22no—
419C2×C4—24no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—616no—
428C2×C4—64no—
429C2×C4—64no—
430C2×C4—64no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—121no—
434C2×C4—122no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—122no—
439C2×C4—121no—
440C2×C4—122no—
441C2×C4—121no—
442C2×C4—122no—
443C2×C4—122no—
444C2×C4#88212yes—
445C2×C4C2×D3×D411yescharacteristic
446C2×C4C2×D3×D411yescharacteristic
447C2×C4D3×#4112yes—
448C2×C4D3×#4112yes—
449C8—42no—
450C8—41no—
451C8—41no—
452C2×C3—42no—
453C2×C3#284911yescharacteristic
454C2×C3#301111yescharacteristic
455C2×C3#301311yescharacteristic
456C2×C3#312112yes—
457C2×C3#312712yes—
458C22—22no—
459C22—22no—
460C22—22no—
461C22—42no—
462C22—42no—
463C22C2×#88211yescharacteristic
464C22#552412yes—
465C22#575111yescharacteristic
466C22#575312yes—
467C22#575311yescharacteristic
468C4—22no—
469C4—42no—
470C4—42no—
471C4—68no—
472C4—122no—
473C4—122no—
474C4—242no—
475C4—241no—
476C4—241no—
477C4D3×#12312yes—
478C4#565611yescharacteristic
479C4#565611yescharacteristic
480C3#3301911yescharacteristic, complemented, Sylow
481C2—42no—
482C2#7414611yescharacteristic
483C2#7867911yescharacteristic
484C2#7868011yescharacteristic
485C2#7939412yes—
486C2#7939612yes—
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenter#1818D3
hypocenterC3#33019
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
8. #68685C2semidirect
9. #68685C2semidirect
10. #68684C2semidirect
11. #68683C2semidirect
480. C3#33019semidirect

Maximal subgroups

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

subgroupindex
C2×#55162
#822632
C2×#47392
C2×#47232
C3×#18182
#699632
#699262
#686852
#686842
#686832
#330193