Finite Group Databasebeta

← #10831757

Subgroups — #10831757

Order 768.

1064 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5730C211yesmaximal, characteristic
2C2×C3×#475C211yesmaximal, characteristic, complemented
3#82481C211yesmaximal, characteristic, complemented
4C2×#5026C211yesmaximal, characteristic
5#71788C211yesmaximal, characteristic, complemented
6#70584C212yesmaximal
7#70566C212yesmaximal, complemented
8#70480C211yesmaximal, characteristic
9#70473C211yesmaximal, characteristic, complemented
10#70276C212yesmaximal
11#70232C212yesmaximal, complemented
12#24343—31nomaximal, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C4×Q8C2211yescharacteristic
15C2×#946—22no—
16C22×#215C2211yescharacteristic
17#5730—24no—
18#5730—24no—
19C2×#897—21no—
20C2×#893C2212yes—
21C2×D3×C4⋊C4—21no—
22C2×#890C2212yes—
23C2×C4×D12C2211yescharacteristic
24C2×C4×Dic6C2211yescharacteristic
25C3×#475—24no—
26C2×C3×#106C2212yes—
27C2×C3×C4⋊C8C2211yescharacteristic
28C2×C3×#103C2212yes—
29#5229C2212yes—
30C2×#835C2211yescharacteristic
31C2×#821C2212yes—
32#5026—24no—
33#4986—21no—
34#4979—21no—
35#4972C2212yes—
36C2×#809C2211yescharacteristic
37C2×#807C2211yescharacteristic
38#4936C2211yescharacteristic
39#4933—22no—
40C2×#803C2211yescharacteristic
41#4551C2212yes—
42#4550C2212yes—
43#4550C2212yes—
44#4529C2212yes—
45#4529C2212yes—
46C2×#554—31no—
47C2×#475D311yescharacteristic, complemented
48C2×#440—31no—
49#2581—31no—
50#2059—32no—
51#2026—32no—
52#1993—31no—
53#1985—31no—
54#1877—32no—
55#1864—32no—
56#1804—31no—
57C22×C3×Q8—21no—
58C22×C3×Q8C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C4×D3—21no—
61C22×C4×D3—22no—
62C22×Dic6D411yescharacteristic
63C22×C3×C8C2312yes—
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8D414yes—
66C3×C4×Q8—22no—
67C3×C4×Q8—22no—
68C2×C3×C4⋊C4—21no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4C2311yescharacteristic
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C42—21no—
73C2×C3×C42C2311yescharacteristic
74#946—216no—
75C2×#215—22no—
76C2×#215—22no—
77C2×#215Q814yescomplemented
78C2×(C2×D3)⋊C4—21no—
79C2×(C2×D3)⋊C4—22no—
80C2×C4⋊Dic3—22no—
81C2×C4⋊Dic3D411yescharacteristic
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2311yescharacteristic
84C2×Dic3⋊C4—22no—
85C2×Dic3⋊C4—22no—
86C2×Dic3⋊C4D412yes—
87C2×C4×Dic3C2312yes—
88C22×C3⋊C8C2311yescharacteristic
89C22×C3⋊C8C2311yescharacteristic
90#897—28no—
91#893—24no—
92#893—24no—
93D3×C4⋊C4—28no—
94#890—24no—
95#890—24no—
96C4×D12—22no—
97C4×D12D414yes—
98C4×Dic6—22no—
99C4×Dic6—22no—
100C3×#106—24no—
101C3×C4⋊C8—22no—
102C3×#103—24no—
103#835—22no—
104C3⋊#95—21no—
105C3⋊#95—21no—
106C3⋊#95—22no—
107C3⋊#95—22no—
108C3⋊#95—22no—
109#821—24no—
110#809—22no—
111#807—22no—
112#803—22no—
113C22×SD16—31no—
114#554—64no—
115#554—64no—
116C2×C4⋊Q8C2×D311yescharacteristic
117C2×#125—32no—
118C2×C22⋊Q8—61no—
119C2×C22⋊Q8—62no—
120C2×C4×Q8—31no—
121C2×C4×Q8C2×D311yescharacteristic
122C2×C4×D4—32no—
123C2×C4×D4—31no—
124C22×C4⋊C4—61no—
125#475—24no—
126#440—64no—
127C2×#107—31no—
128C2×#106C2×D312yes—
129C2×C4⋊C8C2×D311yescharacteristic
130C2×#103—31no—
131C2×#103C2×D312yes—
132C2×#102—32no—
133C2×#102—31no—
134C2×#97—31no—
135#400—61no—
136#395—61no—
137#391—32no—
138#389—32no—
139#387—62no—
140#382—31no—
141#340—32no—
142#340—32no—
143#340—32no—
144#336—32no—
145#336—32no—
146C23×D3—21no—
147C2×C3×Q8—24no—
148C2×C3×Q8—21no—
149C2×C3×Q8—21no—
150C2×C3×Q8—21no—
151C2×C3×Q8—21no—
152C2×C3×Q8—21no—
153C2×C3×Q8—82no—
154C2×C3×Q8C2×Q812yes—
155C22×C3×C4—21no—
156C22×C3×C4—22no—
157C22×C3×C4—21no—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D411yescharacteristic
160C22×C3×C4C2411yescharacteristic
161C22×Dic3—21no—
162C22×Dic3—22no—
163C22×Dic3C2×D412yes—
164C22×Dic3C2×D412yes—
165C2×D12—21no—
166C2×D12—21no—
167C2×D12—21no—
168C2×D12C2×D412yes—
169C2×D12C2×Q812yes—
170C2×D12#4112yes—
171C2×C4×D3—24no—
172C2×C4×D3—28no—
173C2×C4×D3—44no—
174C2×C4×D3—84no—
175C2×Dic6—22no—
176C2×Dic6—42no—
177C2×C3×C8—22no—
178C2×C3×C8—22no—
179C2×C3×C8—22no—
180C3×C4⋊C4—24no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—42no—
185C3×C4⋊C4C2×D414yes—
186C3×C42—42no—
187C3×C42C2×D412yes—
188C3×C42C2×D412yes—
189#215—28no—
190(C2×D3)⋊C4—28no—
191(C2×D3)⋊C4—24no—
192C4⋊Dic3—28no—
193C4⋊Dic3—22no—
194C4⋊Dic3—21no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197C4⋊Dic3C2×D412yes—
198Dic3⋊C4—28no—
199Dic3⋊C4—24no—
200Dic3⋊C4—44no—
201C4×Dic3—22no—
202C4×Dic3—22no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8C2×Q812yes—
209C22×Q8—21no—
210C22×Q8—31no—
211C22×Q8C22×D311yescharacteristic
212C22×D4—31no—
213C23×C4—62no—
214C23×C4—61no—
215C2×SD16—34no—
216C2×SD16—62no—
217C2×SD16—62no—
218C22×C8—31no—
219C22×C8—31no—
220C22×C8C22×D312yes—
221C4⋊Q8—22no—
222C4⋊Q8#6714yes—
223#125—64no—
224#125—64no—
225C22⋊Q8—68no—
226C22⋊Q8—616no—
227C4×Q8—22no—
228C4×Q8—22no—
229C4×Q8—62no—
230C4×Q8—62no—
231C4×D4—34no—
232C4×D4—64no—
233C4×D4—64no—
234C4×D4—62no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—32no—
241C2×C4⋊C4—68no—
242C2×C4⋊C4—62no—
243C2×C4⋊C4—62no—
244C2×C4⋊C4—62no—
245C2×C4⋊C4C22×D311yescharacteristic
246C2×C4⋊C4C22×D312yes—
247C2×C22⋊C4—62no—
248C2×C22⋊C4—61no—
249C2×C42—21no—
250C2×C42—32no—
251C2×C42C22×D311yescharacteristic
252#107—62no—
253#106—24no—
254C4⋊C8—22no—
255#103—24no—
256#103—62no—
257#102—62no—
258#102—64no—
259#97—62no—
260#95—62no—
261#95—62no—
262#95—62no—
263#95—61no—
264#95—61no—
265C23×C3C22×D411yescharacteristic
266C22×D3—22no—
267C22×D3—22no—
268C22×D3—22no—
269C22×D3—42no—
270C22×D3—42no—
271C3×Q8—22no—
272C3×Q8—22no—
273C3×Q8—41no—
274C3×Q8—41no—
275C3×Q8—84no—
276C2×C3×C4—22no—
277C2×C3×C4—22no—
278C2×C3×C4—24no—
279C2×C3×C4—22no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—44no—
283C2×C3×C4—41no—
284C2×C3×C4—42no—
285C2×C3×C4#12012yes—
286C2×C3×C4#12012yes—
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C22×Q811yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×C3×C42−1+411yescharacteristic
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—22no—
298C2×Dic3—24no—
299C2×Dic3—42no—
300C2×Dic3—42no—
301C2×Dic3—42no—
302C2×Dic3—42no—
303D12—22no—
304D12—22no—
305D12—22no—
306D12C22⋊Q814yes—
307C4×D3—48no—
308C4×D3—88no—
309Dic6—44no—
310C3×C8—42no—
311C3⋊C8—22no—
312C3⋊C8—41no—
313C24—61no—
314C2×Q8—24no—
315C2×Q8—21no—
316C2×Q8—21no—
317C2×Q8—21no—
318C2×Q8—21no—
319C2×Q8—21no—
320C2×Q8—62no—
321C2×Q8—82no—
322C2×Q8—122no—
323C2×Q8D3×Q812yes—
324C2×D4—32no—
325C2×D4—32no—
326C2×D4—32no—
327C2×D4—61no—
328C2×D4—61no—
329C2×D4—61no—
330C22×C4—21no—
331C22×C4—22no—
332C22×C4—21no—
333C22×C4—32no—
334C22×C4—32no—
335C22×C4—68no—
336C22×C4—61no—
337C22×C4—64no—
338C22×C4—62no—
339C22×C4—124no—
340C22×C4—244no—
341C22×C4D3×D411yescharacteristic
342C22×C4D3×D411yescharacteristic
343C22×C4C23×D311yescharacteristic
344SD16—68no—
345C2×C8—22no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8—32no—
349C2×C8—61no—
350C2×C8—61no—
351C2×C8—61no—
352C2×C8—61no—
353C2×C8—61no—
354C4⋊C4—24no—
355C4⋊C4—22no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—32no—
359C4⋊C4—42no—
360C4⋊C4—68no—
361C4⋊C4—68no—
362C4⋊C4—62no—
363C4⋊C4—64no—
364C4⋊C4—61no—
365C4⋊C4—62no—
366C4⋊C4—62no—
367C4⋊C4—124no—
368C4⋊C4C2×#6714yes—
369C22⋊C4—68no—
370C22⋊C4—64no—
371C42—42no—
372C42—62no—
373C42—62no—
374C42D3×D412yes—
375C42C2×#6712yes—
376C22×C3C2×#12011yescharacteristic
377C22×C3#54811yescharacteristic
378C22×C3D4×Q811yescharacteristic
379C22×C3#57512yes—
380C22×C3#57712yes—
381C2×D3—24no—
382C2×D3—44no—
383C2×D3—44no—
384C2×D3—42no—
385C2×D3—42no—
386C3×C4—22no—
387C3×C4—22no—
388C3×C4—42no—
389C3×C4—44no—
390C3×C4—81no—
391C3×C4C2×C22⋊Q811yescharacteristic
392C3×C4#53611yescharacteristic
393C3×C4#54811yescharacteristic
394C3×C4D4×Q811yescharacteristic
395Dic3—42no—
396Dic3—42no—
397Dic3—42no—
398Dic3—82no—
399C23—62no—
400C23—62no—
401C23—62no—
402C23—122no—
403C23—122no—
404C23C2×D3×D411yescharacteristic
405Q8—22no—
406Q8—22no—
407Q8—41no—
408Q8—41no—
409Q8—84no—
410Q8—124no—
411D4—34no—
412D4—62no—
413D4—62no—
414D4—62no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—44no—
421C2×C4—41no—
422C2×C4—42no—
423C2×C4—41no—
424C2×C4—64no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—128no—
431C2×C4—122no—
432C2×C4—122no—
433C2×C4—122no—
434C2×C4—122no—
435C2×C4—248no—
436C2×C4#93712yes—
437C2×C4#93712yes—
438C2×C4C2×D3×D411yescharacteristic
439C2×C4C2×D3×Q811yescharacteristic
440C2×C4#100711yescharacteristic
441C2×C4D3×#4111yescharacteristic
442C2×C4#101011yescharacteristic
443C2×C4C22×#6711yescharacteristic
444C8—42no—
445C8—62no—
446C8—121no—
447C2×C3#240911yescharacteristic
448C2×C3#300511yescharacteristic
449C2×C3#300911yescharacteristic
450C2×C3#337612yes—
451C2×C3#338812yes—
452D3—44no—
453C22—64no—
454C22—122no—
455C22—124no—
456C22—124no—
457C22—122no—
458C22#563211yescharacteristic
459C22#563311yescharacteristic
460C22#576212yes—
461C22#577712yes—
462C22C2×#93711yescharacteristic
463C4—22no—
464C4—22no—
465C4—42no—
466C4—44no—
467C4—81no—
468C4—122no—
469C4—122no—
470C4—122no—
471C4—242no—
472C4D3×C22⋊Q811yescharacteristic
473C4#562811yescharacteristic
474C4Q8×#6711yescharacteristic
475C4#583111yescharacteristic
476C3#2434311yescharacteristic, complemented, Sylow
477C2—124no—
478C2#8100512yes—
479C2#8100812yes—
480C2#8275111yescharacteristic
481C2#8385911yescharacteristic
482C2#8398011yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#475C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenterC2×#475D3
hypocenterC3#24343
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#475

Complements

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

NH ≅ G/Ntype
2. C2×C3×#475C2semidirect
3. #82481C2semidirect
5. #71788C2semidirect
7. #70566C2semidirect
9. #70473C2semidirect
11. #70232C2semidirect
47. C2×#475D3semidirect
77. C2×#215Q8semidirect
476. C3#24343semidirect

Maximal subgroups

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

subgroupindex
C2×#57302
C2×C3×#4752
#824812
C2×#50262
#717882
#705842
#705662
#704802
#704732
#702762
#702322
#243433