Finite Group Databasebeta

← #10801779

Subgroups — #10801779

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×C3×#405C211yesmaximal, characteristic, complemented
2C2×#5115C211yesmaximal, characteristic
3#82236C212yesmaximal, complemented
4C2×#4711C211yesmaximal, characteristic, complemented
5#71990C212yesmaximal, complemented
6#71970C211yesmaximal, characteristic, complemented
7#69825C211yesmaximal, characteristic, complemented
8#69819C211yesmaximal, characteristic, complemented
9#69814C211yesmaximal, characteristic, complemented
10#69768C212yesmaximal, complemented
11C4×#821C212yesmaximal, complemented
12#21529—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16C2×#930C2212yes—
17C22×D24C2211yescharacteristic
18C2×C4×D12C2212yes—
19C3×#405—24no—
20C2×C3×#97C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24#5212C2212yes—
25#5115C418yescomplemented
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×#821C2212yes—
29C2×#818C2211yescharacteristic
30C2×#817C2211yescharacteristic
31C2×#817C2211yescharacteristic
32#4990—22no—
33#4989—22no—
34#4957—22no—
35#4956—22no—
36#4939C2212yes—
37C4×C4⋊Dic3C2212yes—
38#4928C2212yes—
39#4711—24no—
40#4551C2212yes—
41#4551C2212yes—
42#4551C2212yes—
43C2×#466—31no—
44C2×#442—31no—
45C2×#405D311yescharacteristic, complemented
46#2324—32no—
47#1972—31no—
48#1943—31no—
49#1934—32no—
50#1895—32no—
51#1807—31no—
52#1776—31no—
53C4×#102—32no—
54C23×C3×C4C2311yescharacteristic
55C22×#67—22no—
56C22×D12C2312yes—
57C22×C4×D3—22no—
58C2×C3×M16—22no—
59C2×C3×M16—22no—
60C2×C3×M16—22no—
61C22×C3×C8—22no—
62C22×C3×C8—22no—
63C22×C3×C8C2×C414yescomplemented
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C22×C3×C8C2311yescharacteristic
68C3×#117—22no—
69C3×#117—22no—
70C2×C3×C4⋊C4D412yes—
71C2×C3×C22⋊C4D412yes—
72C2×C3×C42C2311yescharacteristic
73C2×C3×C42C2311yescharacteristic
74C2×#217—22no—
75#930—24no—
76#930C2×C418yescomplemented
77C2×(C2×D3)⋊C4—22no—
78C2×(C2×D3)⋊C4—22no—
79C2×(C2×D3)⋊C4—22no—
80C2×(C2×D3)⋊C4—22no—
81C2×C4⋊Dic3C2312yes—
82C2×C4⋊Dic3C2312yes—
83C2×C4⋊Dic3C2312yes—
84C2×Dic3⋊C4—22no—
85C2×C4×Dic3—22no—
86C2×D24—22no—
87C2×D24—22no—
88C2×D24C2×C414yes—
89C4×D12—24no—
90C4×D12—24no—
91C3×#97—22no—
92C3×#97D414yescomplemented
93C3×C4×C8—22no—
94C3×C4×C8—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97C3⋊#95—22no—
98#821—24no—
99#821—24no—
100#821C2×C418yescomplemented
101#818C2×C414yescomplemented
102#817—22no—
103#817—22no—
104C22×D8—31no—
105C22×M16C2×D311yescharacteristic
106C23×C8C2×D311yescharacteristic
107C2×#121—32no—
108C2×C4×D4—32no—
109C2×#117C2×D311yescharacteristic
110#466—38no—
111#442—64no—
112C2×#107—31no—
113C2×#106—31no—
114C2×#106—31no—
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118#405—24no—
119C2×#97C2×D311yescharacteristic
120C2×#97C2×D311yescharacteristic
121C2×C4×C8C2×D311yescharacteristic
122C2×C4×C8C2×D311yescharacteristic
123#390—32no—
124#390—62no—
125#389—32no—
126#388—62no—
127#386—62no—
128#385—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340—32no—
134C24×C3C2×D411yescharacteristic
135C23×D3—22no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4C22×C412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D412yes—
144C22×C3×C4#4112yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—28no—
147C2×#67—82no—
148C2×#67—82no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
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—
161C3×M16—28no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—28no—
169C2×C3×C8C22×C412yes—
170C2×C3×C8C22×C412yes—
171C2×C3×C8C2×D412yes—
172C2×C3×C8C2×D412yes—
173C2×C3×C8#4112yes—
174C2×C3×C8#4112yes—
175C3×C4⋊C4—24no—
176C3×C22⋊C4—24no—
177C3×C42—21no—
178C3×C42—21no—
179C3×C42C2×D412yes—
180C3×C42#4112yes—
181#217—82no—
182(C2×D3)⋊C4—28no—
183(C2×D3)⋊C4—44no—
184(C2×D3)⋊C4—82no—
185(C2×D3)⋊C4—82no—
186C4⋊Dic3—22no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191C4⋊Dic3C22×C414yes—
192Dic3⋊C4—82no—
193C4×Dic3—82no—
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—22no—
203C2×M16—22no—
204C2×M16—22no—
205C22×C8—22no—
206C22×C8—22no—
207C22×C8C4×D314yescomplemented
208C22×C8C22×D311yescharacteristic
209C22×C8C22×D311yescharacteristic
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212#121—38no—
213#121—64no—
214C4×D4—64no—
215C4×D4—64no—
216#117—22no—
217#117—22no—
218C2×C4⋊C4—32no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4—62no—
222C2×C4⋊C4D1212yes—
223C2×C22⋊C4—62no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4D1212yes—
229C2×C42—62no—
230C2×C42C22×D311yescharacteristic
231C2×C42C22×D311yescharacteristic
232#107—34no—
233#106—62no—
234#106—62no—
235#102—38no—
236#102—64no—
237#102—64no—
238#97—22no—
239#97#6714yescomplemented
240C4×C8—22no—
241C4×C8—22no—
242#95—62no—
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—22no—
265C2×C3×C4—22no—
266C2×C3×C4—22no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4C4×D412yes—
270C2×C3×C4#12114yes—
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—24no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283C2×Dic3—42no—
284C2×Dic3—42no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287D12—24no—
288D12—24no—
289D12—42no—
290D12—42no—
291C4×D3—88no—
292C3×C8—22no—
293C3×C8—22no—
294C3×C8—22no—
295C3×C8C4×D414yes—
296C24—62no—
297C24C2×D1211yescharacteristic
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—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—124no—
315C22×C4—242no—
316C22×C4—242no—
317C22×C4C2×C4×D312yes—
318C22×C4C2×D1212yes—
319C22×C4C2×D1211yescharacteristic
320C22×C4C2×D1212yes—
321C22×C4#23512yes—
322C22×C4C23×D311yescharacteristic
323D8—68no—
324M16—28no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—21no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—28no—
332C2×C8C2×C4×D312yes—
333C2×C8C2×C4×D312yes—
334C2×C8#23512yes—
335C2×C8#23512yes—
336C2×C8C2×#6712yes—
337C2×C8C2×#6712yes—
338C4⋊C4—24no—
339C4⋊C4—34no—
340C4⋊C4—62no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—62no—
345C4⋊C4—242no—
346C22⋊C4—24no—
347C22⋊C4—68no—
348C22⋊C4—124no—
349C22⋊C4—242no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C42—21no—
353C42—21no—
354C42—242no—
355C42#23512yes—
356C42C2×#6712yes—
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—24no—
375C3×C4—22no—
376C3×C4—42no—
377C3×C4C2×C4×D411yescharacteristic
378C3×C4C4×#4111yescharacteristic
379C3×C4#52511yescharacteristic
380C3×C4#52911yescharacteristic
381Dic3—44no—
382Dic3—82no—
383Dic3—82no—
384C23—21no—
385C23—22no—
386C23—22no—
387C23—21no—
388C23—64no—
389C23—122no—
390C23—122no—
391C23—122no—
392C23—122no—
393C23—122no—
394C23—122no—
395C23C4×D1212yes—
396C23C22×D1211yescharacteristic
397D4—64no—
398D4—64no—
399D4—122no—
400D4—122no—
401D4—248no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—64no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—122no—
422C2×C4—248no—
423C2×C4C4×D1212yes—
424C2×C4#93014yes—
425C2×C4C22×C4×D311yescharacteristic
426C2×C4C2×#23511yescharacteristic
427C2×C4C2×#23511yescharacteristic
428C2×C4C2×#23511yescharacteristic
429C2×C4C2×#23511yescharacteristic
430C2×C4C22×#6711yescharacteristic
431C8—22no—
432C8—22no—
433C8—22no—
434C8C4×#6714yes—
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×C4×D1211yescharacteristic
455C22#547711yescharacteristic
456C22#575112yes—
457C22#575212yes—
458C22C2×#93011yescharacteristic
459C4—24no—
460C4—22no—
461C4—42no—
462C4—124no—
463C4—242no—
464C4—242no—
465C4C4×#23511yescharacteristic
466C4#547911yescharacteristic
467C4C2×C4×#6711yescharacteristic
468C4#579111yescharacteristic
469C3#2152911yescharacteristic, complemented, Sylow
470C2—42no—
471C2—244no—
472C2#7856111yescharacteristic
473C2#7857411yescharacteristic
474C2#7857812yes—
475C2C4×#93011yescharacteristic
476C2#8380212yes—
subgroupG/H
centerC23C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#405C2
socleC23×C3C22×D4
prime radicalC2×#5115C2
cosocleC22×C3×C4C24
hypercenterC2×#405D3
hypocenterC3#21529
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×C3×#405C2semidirect
3. #82236C2semidirect
4. C2×#4711C2semidirect
5. #71990C2semidirect
6. #71970C2semidirect
7. #69825C2semidirect
8. #69819C2semidirect
9. #69814C2semidirect
10. #69768C2semidirect
11. C4×#821C2semidirect
25. #5115C4semidirect
45. C2×#405D3semidirect
63. C22×C3×C8C2×C4semidirect
76. #930C2×C4semidirect
92. C3×#97D4semidirect
100. #821C2×C4semidirect
101. #818C2×C4semidirect
207. C22×C8C4×D3semidirect
239. #97#67semidirect
469. C3#21529semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4052
C2×#51152
#822362
C2×#47112
#719902
#719702
#698252
#698192
#698142
#697682
C4×#8212
#215293