Finite Group Databasebeta

← #10847406

Subgroups — #10847406

Order 768.

1132 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5470C211yesmaximal, characteristic
2C2×#5116C212yesmaximal
3#82075C211yesmaximal, characteristic
4C2×C3×#336C211yesmaximal, characteristic
5#71972C212yesmaximal
6#68689C211yesmaximal, characteristic, complemented
7#68688C211yesmaximal, characteristic, complemented
8#68685C212yesmaximal, complemented
9#68683C212yesmaximal, complemented
10#68632C211yesmaximal, characteristic, complemented
11#68631C211yesmaximal, characteristic, complemented
12#25498—31nomaximal, Sylow
13C23×C3×C8C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#926—21no—
16C2×#924—21no—
17C2×#924C2211yescharacteristic
18C2×#924C2212yes—
19C22×Dic12C2212yes—
20#5470—28no—
21C2×#870—21no—
22C2×#869C2211yescharacteristic
23C2×#869C2211yescharacteristic
24C2×C4×Dic6—21no—
25C2×C4×Dic6C2212yes—
26#5212C2211yescharacteristic
27#5211—21no—
28#5210—21no—
29#5116—28no—
30C2×#818C2212yes—
31C2×#816C2212yes—
32C2×#816C2212yes—
33C2×#816C2212yes—
34C2×#816C2212yes—
35#4933—21no—
36#4933—21no—
37#4929C2211yescharacteristic
38#4929C2211yescharacteristic
39#4928C2211yescharacteristic
40#4928C2211yescharacteristic
41C3×#336C2211yescharacteristic
42C3×#336C2211yescharacteristic
43C3×#336C2212yes—
44#4551C2212yes—
45#4551C2212yes—
46C2×#533—31no—
47C2×#467—32no—
48#2855—31no—
49#2057—31no—
50#2043—31no—
51#2024—31no—
52#2016—31no—
53#1892—32no—
54#1878—32no—
55#1863—32no—
56C2×#336D311yescharacteristic
57C23×C3×C4D411yescharacteristic
58C23×C3×C4D411yescharacteristic
59C23×C3×C4C2311yescharacteristic
60C22×Dic6—21no—
61C22×Dic6C2311yescharacteristic
62C22×Dic6C2312yes—
63C22×C3×C8—24no—
64C22×C3×C8—24no—
65C22×C3×C8—42no—
66C22×C3×C8C2312yes—
67C22×C3×C8C2312yes—
68C2×C3×C42—24no—
69C2×C3×C42D411yescharacteristic
70C2×C3×C42D411yescharacteristic
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73C2×#217—22no—
74C2×#217—21no—
75C2×#217—21no—
76#926—28no—
77C2×C4⋊Dic3—21no—
78C2×C4⋊Dic3—21no—
79C2×C4⋊Dic3—21no—
80C2×C4⋊Dic3C2311yescharacteristic
81C2×C4⋊Dic3C2311yescharacteristic
82C2×C4⋊Dic3C2311yescharacteristic
83C2×C4⋊Dic3C2312yes—
84#924—28no—
85#924—24no—
86#924—24no—
87#924D418yes—
88C2×Dic3⋊C4—22no—
89C2×Dic3⋊C4—22no—
90C2×Dic3⋊C4—21no—
91C2×Dic3⋊C4—21no—
92C2×Dic3⋊C4—21no—
93C2×C4×Dic3—22no—
94C2×Dic12—24no—
95C2×Dic12—24no—
96C2×Dic12—42no—
97#870—28no—
98#869—24no—
99#869—24no—
100C4×Dic6—28no—
101C4×Dic6—24no—
102C4×Dic6—24no—
103C3⋊#95—21no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—21no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109C3⋊#95—21no—
110#818—24no—
111#816—24no—
112#816—24no—
113#816—24no—
114#816—42no—
115C22×Q16—32no—
116C23×C8C2×D312yes—
117#533—68no—
118C2×C4⋊Q8—31no—
119C2×C4⋊Q8—31no—
120C2×#125—61no—
121C2×C22⋊Q8—32no—
122C2×C22⋊Q8—31no—
123C2×C22⋊Q8—61no—
124C2×C4×Q8—32no—
125C2×C4×Q8—61no—
126C2×#117—61no—
127C22×C42C2×D311yescharacteristic
128#467—68no—
129C2×#107—32no—
130C2×#103—32no—
131C2×#103—32no—
132C2×#103—32no—
133C2×#103—32no—
134#391—31no—
135#391—31no—
136#389—31no—
137#389—31no—
138#387—61no—
139#387—61no—
140#385—61no—
141#381—61no—
142#380—31no—
143#340—32no—
144#340—32no—
145#336C2×D311yescharacteristic
146#336C2×D312yes—
147#336C2×D311yescharacteristic
148C24×C3C2×D411yescharacteristic
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—22no—
152C22×C3×C4—21no—
153C22×C3×C4—21no—
154C22×C3×C4—22no—
155C22×C3×C4—22no—
156C22×C3×C4—41no—
157C22×C3×C4Q1614yescomplemented
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2×D411yescharacteristic
161C22×C3×C4C2×D411yescharacteristic
162C22×C3×C4C2×D411yescharacteristic
163C22×C3×C4C2×D414yes—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—22no—
166C22×Dic3—22no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171C2×Dic6—22no—
172C2×Dic6—22no—
173C2×Dic6—22no—
174C2×Dic6—22no—
175C2×Dic6—24no—
176C2×Dic6—22no—
177C2×Dic6—22no—
178C2×Dic6—22no—
179C2×Dic6—21no—
180C2×Dic6—21no—
181C2×Dic6—42no—
182C2×Dic6—42no—
183C2×Dic6C2×D414yes—
184C2×C3×C8—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C2×C3×C8—22no—
188C2×C3×C8—22no—
189C2×C3×C8—48no—
190C2×C3×C8#4114yes—
191C3×C42—22no—
192C3×C42—22no—
193C3×C42—22no—
194C3×C42—22no—
195#217—28no—
196#217—42no—
197#217—42no—
198C4⋊Dic3—24no—
199C4⋊Dic3—24no—
200C4⋊Dic3—24no—
201C4⋊Dic3—22no—
202C4⋊Dic3—22no—
203C4⋊Dic3—22no—
204C4⋊Dic3—21no—
205C4⋊Dic3—21no—
206C4⋊Dic3C2×D414yes—
207Dic3⋊C4—28no—
208Dic3⋊C4—28no—
209Dic3⋊C4—42no—
210Dic3⋊C4—42no—
211Dic3⋊C4—42no—
212C4×Dic3—28no—
213Dic12—48no—
214C22×Q8—32no—
215C22×Q8—31no—
216C22×Q8—61no—
217C23×C4D1211yescharacteristic
218C23×C4D1211yescharacteristic
219C23×C4C22×D311yescharacteristic
220C2×Q16—64no—
221C2×Q16—64no—
222C2×Q16—122no—
223C22×C8—24no—
224C22×C8—24no—
225C22×C8—42no—
226C22×C8C22×D312yes—
227C22×C8C22×D312yes—
228C4⋊Q8—64no—
229C4⋊Q8—64no—
230#125—68no—
231C22⋊Q8—38no—
232C22⋊Q8—68no—
233C22⋊Q8—64no—
234C22⋊Q8—64no—
235C4×Q8—64no—
236C4×Q8—64no—
237C4×Q8—68no—
238#117—68no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—31no—
243C2×C4⋊C4—61no—
244C2×C4⋊C4—61no—
245C2×C4⋊C4—62no—
246C2×C4⋊C4—62no—
247C2×C4⋊C4—61no—
248C2×C4⋊C4—61no—
249C2×C4⋊C4—61no—
250C2×C4⋊C4—61no—
251C2×C22⋊C4—62no—
252C2×C22⋊C4—61no—
253C2×C22⋊C4—61no—
254C2×C42—24no—
255C2×C42—62no—
256C2×C42D1211yescharacteristic
257C2×C42D1211yescharacteristic
258C2×C42C22×D311yescharacteristic
259C2×C42C22×D311yescharacteristic
260#107—64no—
261#103—64no—
262#103—64no—
263#103—64no—
264#103—122no—
265#95—61no—
266#95—61no—
267#95—61no—
268#95—61no—
269#95—61no—
270#95—61no—
271#95—61no—
272C23×C3—21no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3#12012yes—
277C23×C3C2×Q1612yes—
278C23×C3#13712yes—
279C23×C3C22×D411yescharacteristic
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—28no—
289C2×C3×C4—24no—
290C2×C3×C4—24no—
291C2×C3×C4—21no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—44no—
295C2×C3×C4#12012yes—
296C2×C3×C4C2×Q1612yes—
297C2×C3×C4C2×Q1612yes—
298C2×C3×C4#13512yes—
299C2×C3×C4C22×D412yes—
300C2×C3×C4C2×#4112yes—
301C2×C3×C4H(5,𝔽2)11yescharacteristic
302C2×C3×C42−1+411yescharacteristic
303C2×Dic3—24no—
304C2×Dic3—24no—
305C2×Dic3—42no—
306C2×Dic3—42no—
307C2×Dic3—42no—
308C2×Dic3—42no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—41no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318C2×Dic3—41no—
319C2×Dic3—41no—
320C2×Dic3—41no—
321Dic6—24no—
322Dic6—24no—
323Dic6—48no—
324Dic6—42no—
325Dic6—42no—
326Dic6—42no—
327Dic6—42no—
328C3×C8—24no—
329C3×C8—42no—
330C24C2×D1211yescharacteristic
331C2×Q8—34no—
332C2×Q8—64no—
333C2×Q8—62no—
334C2×Q8—62no—
335C2×Q8—62no—
336C2×Q8—62no—
337C2×Q8—62no—
338C2×Q8—62no—
339C2×Q8—61no—
340C2×Q8—62no—
341C2×Q8—61no—
342C2×Q8—122no—
343C2×Q8—122no—
344C22×C4—22no—
345C22×C4—22no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—22no—
350C22×C4—22no—
351C22×C4—41no—
352C22×C4—62no—
353C22×C4—62no—
354C22×C4—61no—
355C22×C4—61no—
356C22×C4—61no—
357C22×C4—61no—
358C22×C4Dic1214yescomplemented
359C22×C4C2×D1211yescharacteristic
360C22×C4C2×D1211yescharacteristic
361C22×C4C2×D1211yescharacteristic
362C22×C4C2×D1211yescharacteristic
363C22×C4C2×D1211yescharacteristic
364C22×C4D3×D414yes—
365C22×C4C23×D311yescharacteristic
366Q16—128no—
367C2×C8—22no—
368C2×C8—22no—
369C2×C8—22no—
370C2×C8—22no—
371C2×C8—22no—
372C2×C8—48no—
373C2×C8#23514yes—
374C4⋊C4—34no—
375C4⋊C4—64no—
376C4⋊C4—64no—
377C4⋊C4—68no—
378C4⋊C4—64no—
379C4⋊C4—68no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—122no—
386C4⋊C4—122no—
387C4⋊C4—122no—
388C22⋊C4—68no—
389C22⋊C4—122no—
390C22⋊C4—122no—
391C42—22no—
392C42—22no—
393C42—22no—
394C42—22no—
395C42—68no—
396C22×C3—22no—
397C22×C3—22no—
398C22×C3—22no—
399C22×C3—22no—
400C22×C3—22no—
401C22×C3—22no—
402C22×C3#45114yes—
403C22×C3C2×#12011yescharacteristic
404C22×C3#53211yescharacteristic
405C22×C3#53711yescharacteristic
406C22×C3C22×Q1611yescharacteristic
407C22×C3C2×#13511yescharacteristic
408C22×C3C2×#13711yescharacteristic
409C22×C3#57511yescharacteristic
410C3×C4—22no—
411C3×C4—22no—
412C3×C4—22no—
413C3×C4—22no—
414C3×C4—41no—
415C3×C4#54612yes—
416C3×C4#54712yes—
417Dic3—44no—
418Dic3—44no—
419Dic3—81no—
420Dic3—81no—
421Dic3—81no—
422Dic3—81no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—21no—
427C23#88212yes—
428C23C2×Dic1212yes—
429C23#90912yes—
430C23C22×D1211yescharacteristic
431Q8—64no—
432Q8—64no—
433Q8—128no—
434Q8—122no—
435Q8—122no—
436Q8—122no—
437Q8—122no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—28no—
446C2×C4—21no—
447C2×C4—24no—
448C2×C4—24no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—21no—
452C2×C4—44no—
453C2×C4—64no—
454C2×C4—64no—
455C2×C4—122no—
456C2×C4—122no—
457C2×C4—122no—
458C2×C4—122no—
459C2×C4—121no—
460C2×C4—121no—
461C2×C4—121no—
462C2×C4—121no—
463C2×C4—121no—
464C2×C4—121no—
465C2×C4—121no—
466C2×C4—121no—
467C2×C4—121no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4#88212yes—
472C2×C4#90412yes—
473C2×C4C2×Dic1212yes—
474C2×C4C2×Dic1212yes—
475C2×C4C2×#23512yes—
476C2×C4C2×D3×D412yes—
477C2×C4#100411yescharacteristic
478C2×C4#100711yescharacteristic
479C8—24no—
480C8—42no—
481C2×C3—24no—
482C2×C3#243011yescharacteristic
483C2×C3C2×#45111yescharacteristic
484C2×C3#300211yescharacteristic
485C2×C3#317011yescharacteristic
486C2×C3#318111yescharacteristic
487C2×C3#324811yescharacteristic
488C2×C3#324911yescharacteristic
489C22—22no—
490C22—22no—
491C22—22no—
492C22—22no—
493C22—22no—
494C22—22no—
495C22#473914yes—
496C22#547711yescharacteristic
497C22C2×#88211yescharacteristic
498C22#551011yescharacteristic
499C22C2×#90411yescharacteristic
500C22C22×Dic1211yescharacteristic
501C22C2×#90911yescharacteristic
502C22#574811yescharacteristic
503C4—22no—
504C4—22no—
505C4—22no—
506C4—22no—
507C4—41no—
508C4—124no—
509C4—124no—
510C4—241no—
511C4—241no—
512C4—241no—
513C4—241no—
514C4#549012yes—
515C4#550712yes—
516C3#2549811yescharacteristic, complemented, Sylow
517C2—24no—
518C2#7337611yescharacteristic
519C2#7847511yescharacteristic
520C2#7850411yescharacteristic
521C2#7861011yescharacteristic
522C2C2×#473911yescharacteristic
523C2#7902911yescharacteristic
524C2#7903711yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#336C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#336D3
hypocenterC3#25498
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#336

Complements

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

NH ≅ G/Ntype
6. #68689C2semidirect
7. #68688C2semidirect
8. #68685C2semidirect
9. #68683C2semidirect
10. #68632C2semidirect
11. #68631C2semidirect
157. C22×C3×C4Q16semidirect
358. C22×C4Dic12semidirect
516. C3#25498semidirect

Maximal subgroups

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

subgroupindex
C2×#54702
C2×#51162
#820752
C2×C3×#3362
#719722
#686892
#686882
#686852
#686832
#686322
#686312
#254983