Finite Group Databasebeta

← #10801772

Subgroups — #10801772

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5115C211yesmaximal, characteristic
2C2×#5101C211yesmaximal, characteristic, complemented
3C2×#4869C211yesmaximal, characteristic, complemented
4#73872C212yesmaximal, complemented
5C3×#1807C211yesmaximal, characteristic, complemented
6#70576C211yesmaximal, characteristic, complemented
7#70561C211yesmaximal, characteristic, complemented
8#70553C211yesmaximal, characteristic, complemented
9#69517C212yesmaximal, complemented
10#68852C212yesmaximal, complemented
11#68780C212yesmaximal, complemented
12#21529—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14C2×#930C2212yes—
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C22×D24C2211yescharacteristic
18C2×#893C2212yes—
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C3×#380C2212yes—
22#5115C418yescomplemented
23C2×#821C2212yes—
24C2×#818C2211yescharacteristic
25#5101—24no—
26C2×#815C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29C2×C8×Dic3C2211yescharacteristic
30#4988C2212yes—
31Dic3×C4⋊C4C2212yes—
32#4972C2212yes—
33C2×#809C2212yes—
34C2×#809C2212yes—
35#4957—22no—
36#4956—22no—
37#4869—24no—
38#4670—22no—
39#4668—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#466—31no—
44C2×#442—31no—
45C2×#405—31no—
46#2324—32no—
47#1972—31no—
48#1943—31no—
49#1934—32no—
50#1895—32no—
51#1807D311yescharacteristic, complemented
52#1776—31no—
53C4×#102—32no—
54C23×C3×C4C2311yescharacteristic
55C22×#67—22no—
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C22×C3×C8C2×C414yescomplemented
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C22⋊C4—22no—
66C2×#217D412yes—
67#930—24no—
68#930C2×C418yescomplemented
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73#926—22no—
74#926—22no—
75C2×C4⋊Dic3D412yes—
76C2×C4⋊Dic3C2312yes—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3C2311yescharacteristic
80C2×C4×Dic3C2311yescharacteristic
81C2×#208—22no—
82C2×#208—22no—
83C2×#208—22no—
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2311yescharacteristic
86C2×D24—22no—
87C2×D24—22no—
88C2×D24C2×C414yes—
89#893—24no—
90#893—24no—
91C3×#106—22no—
92C3×#106—22no—
93C3×#95—22no—
94C3⋊#95—22no—
95C3⋊#95—22no—
96#821C2×C418yescomplemented
97#818C2×C414yescomplemented
98#815—22no—
99#815D414yescomplemented
100C8×Dic3—22no—
101C8×Dic3—22no—
102#809—24no—
103#809—24no—
104C22×D8—31no—
105C22×M16—31no—
106C23×C8C2×D311yescharacteristic
107C2×#121—32no—
108C2×C4×D4—32no—
109C2×#117—31no—
110#466—38no—
111#442—64no—
112C2×#107—31no—
113C2×#106C2×D311yescharacteristic
114C2×#106C2×D311yescharacteristic
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118#405—64no—
119C2×#97—31no—
120C2×#97—31no—
121C2×C4×C8—31no—
122C2×C4×C8—31no—
123#390—32no—
124#390—62no—
125#389—32no—
126#388—62no—
127#386—62no—
128#385—62no—
129#380C2×D312yes—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134C24×C3C2×D411yescharacteristic
135C23×D3—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4C22×C412yes—
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4112yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—28no—
147C2×#67—82no—
148C2×#67—82no—
149C22×Dic3—22no—
150C22×Dic3C2×D412yes—
151C22×Dic3C2×D412yes—
152C2×D12—22no—
153C2×D12—22no—
154C2×D12—22no—
155C2×D12—22no—
156C2×D12—22no—
157C2×D12C22×C414yes—
158C2×C4×D3—44no—
159C2×C4×D3—82no—
160C2×C4×D3—82no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—28no—
164C2×C3×C8C22×C412yes—
165C2×C3×C8C22×C412yes—
166C2×C3×C8C2×D412yes—
167C2×C3×C8#4112yes—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—22no—
172C3×C22⋊C4—82no—
173#217—24no—
174(C2×D3)⋊C4—28no—
175(C2×D3)⋊C4—44no—
176(C2×D3)⋊C4—82no—
177(C2×D3)⋊C4—82no—
178C4⋊Dic3—22no—
179C4⋊Dic3—24no—
180C4⋊Dic3C22×C414yes—
181Dic3⋊C4—82no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3—82no—
185C4×Dic3C2×D412yes—
186C4×Dic3#4112yes—
187#208—28no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8C2×D412yes—
193C2×C3⋊C8#4112yes—
194D24—28no—
195C22×D4—32no—
196C22×D4—62no—
197C23×C4—62no—
198C23×C4C22×D311yescharacteristic
199C2×D8—34no—
200C2×D8—62no—
201C2×D8—62no—
202C2×M16—62no—
203C2×M16—62no—
204C2×M16—62no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8—31no—
208C22×C8—31no—
209C22×C8C4×D314yescomplemented
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212#121—38no—
213#121—64no—
214C4×D4—64no—
215C4×D4—64no—
216#117—62no—
217#117—62no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—62no—
221C2×C4⋊C4C22×D312yes—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—22no—
224C2×C22⋊C4—32no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—31no—
230C2×C42—31no—
231C2×C42—62no—
232#107—34no—
233#106—22no—
234#106—22no—
235#102—38no—
236#102—64no—
237#102—64no—
238#97—34no—
239#97—62no—
240C4×C8—62no—
241C4×C8—62no—
242#95—22no—
243#95—62no—
244#95—62no—
245C23×C3—21no—
246C23×C3—21no—
247C23×C3—22no—
248C23×C3—22no—
249C23×C3C4×D412yes—
250C23×C3C22×D411yescharacteristic
251C22×D3—24no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3—42no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4C4×D412yes—
271C2×C3×C4C23×C411yescharacteristic
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277#67—88no—
278C2×Dic3—22no—
279C2×Dic3—22no—
280C2×Dic3—24no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3#12114yes—
287D12—24no—
288D12—24no—
289D12—42no—
290D12—42no—
291C4×D3—88no—
292C3×C8—22no—
293C3×C8C4×D414yes—
294C3⋊C8—22no—
295C3⋊C8—22no—
296C24—62no—
297C24D3×D411yescharacteristic
298C2×D4—34no—
299C2×D4—68no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—242no—
306C2×D4—242no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—22no—
312C22×C4—22no—
313C22×C4—32no—
314C22×C4—32no—
315C22×C4—62no—
316C22×C4—124no—
317C22×C4—242no—
318C22×C4—242no—
319C22×C4C2×C4×D312yes—
320C22×C4D3×D411yescharacteristic
321C22×C4#23912yes—
322C22×C4C23×D311yescharacteristic
323D8—68no—
324M16—68no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—28no—
328C2×C8—32no—
329C2×C8—32no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8C2×C4×D312yes—
335C2×C8C2×C4×D312yes—
336C2×C8D3×D412yes—
337C2×C8#23712yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—34no—
343C4⋊C4—64no—
344C4⋊C4—62no—
345C4⋊C4—242no—
346C22⋊C4—64no—
347C22⋊C4—68no—
348C22⋊C4—82no—
349C22⋊C4—124no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C42—32no—
353C42—32no—
354C42—61no—
355C42—61no—
356C42—242no—
357C22×C3—22no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3C2×C4×D411yescharacteristic
363C22×C3C2×#12111yescharacteristic
364C22×C3#53211yescharacteristic
365C22×C3#57612yes—
366C22×C3#57712yes—
367C2×D3—44no—
368C2×D3—44no—
369C2×D3—44no—
370C2×D3—82no—
371C2×D3—82no—
372C2×D3—82no—
373C2×D3—82no—
374C3×C4—22no—
375C3×C4—82no—
376C3×C4—82no—
377C3×C4C2×C4×D411yescharacteristic
378C3×C4C4×#4111yescharacteristic
379C3×C4#52511yescharacteristic
380C3×C4#52911yescharacteristic
381Dic3—24no—
382Dic3—44no—
383Dic3—42no—
384C23—22no—
385C23—22no—
386C23—21no—
387C23—21no—
388C23—64no—
389C23—122no—
390C23—122no—
391C23—122no—
392C23—122no—
393C23—122no—
394C23—122no—
395C23#89312yes—
396C23C2×D3×D411yescharacteristic
397D4—64no—
398D4—64no—
399D4—122no—
400D4—122no—
401D4—248no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—34no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—42no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—64no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—248no—
424C2×C4#89312yes—
425C2×C4C22×C4×D311yescharacteristic
426C2×C4C2×D3×D411yescharacteristic
427C2×C4C2×#23711yescharacteristic
428C2×C4C2×#23911yescharacteristic
429C2×C4D3×#4111yescharacteristic
430C2×C4D3×#4111yescharacteristic
431C8—22no—
432C8—62no—
433C8—62no—
434C8#67⋊C414yes—
435C2×C3—42no—
436C2×C3C4×#12111yescharacteristic
437C2×C3#294811yescharacteristic
438C2×C3#295211yescharacteristic
439C2×C3#306212yes—
440C2×C3#317912yes—
441D3—84no—
442C22—22no—
443C22—22no—
444C22—22no—
445C22—42no—
446C22—42no—
447C22—124no—
448C22—124no—
449C22—124no—
450C22—242no—
451C22—242no—
452C22—242no—
453C22—242no—
454C22C2×#89311yescharacteristic
455C22#559611yescharacteristic
456C22D3×#12111yescharacteristic
457C22#576212yes—
458C22#576912yes—
459C4—22no—
460C4—64no—
461C4—82no—
462C4—82no—
463C4—122no—
464C4—124no—
465C4C2×#67⋊C411yescharacteristic
466C4#552211yescharacteristic
467C4#562411yescharacteristic
468C4#562711yescharacteristic
469C3#2152911yescharacteristic, complemented, Sylow
470C2—42no—
471C2—244no—
472C2#7386511yescharacteristic
473C2#7910611yescharacteristic
474C2#7916711yescharacteristic
475C2#8063712yes—
476C2#8066112yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1807C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
cosocleC22×C3×C4C24
hypercenter#1807D3
hypocenterC3#21529
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1807

Complements

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

NH ≅ G/Ntype
2. C2×#5101C2semidirect
3. C2×#4869C2semidirect
4. #73872C2semidirect
5. C3×#1807C2semidirect
6. #70576C2semidirect
7. #70561C2semidirect
8. #70553C2semidirect
9. #69517C2semidirect
10. #68852C2semidirect
11. #68780C2semidirect
22. #5115C4semidirect
51. #1807D3semidirect
60. C22×C3×C8C2×C4semidirect
68. #930C2×C4semidirect
96. #821C2×C4semidirect
97. #818C2×C4semidirect
99. #815D4semidirect
209. C22×C8C4×D3semidirect
469. C3#21529semidirect

Maximal subgroups

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

subgroupindex
C2×#51152
C2×#51012
C2×#48692
#738722
C3×#18072
#705762
#705612
#705532
#695172
#688522
#687802
#215293