Finite Group Databasebeta

← #10949658

Subgroups — #10949658

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#509C211yesmaximal, characteristic, complemented
2#83914C211yesmaximal, characteristic, complemented
3C2×#5157C211yesmaximal, characteristic, complemented
4C2×D3×M32C211yesmaximal, characteristic, complemented
5#72078C211yesmaximal, characteristic, complemented
6#72033C211yesmaximal, characteristic, complemented
7#71488C212yesmaximal, complemented
8#71487C212yesmaximal, complemented
9#71470C212yesmaximal, complemented
10#71469C212yesmaximal, complemented
11#71428C211yesmaximal, characteristic, complemented
12#33605—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C22×C3×D8C2211yescharacteristic
15#5828—21no—
16C2×#938—21no—
17C22×D3×C8C2211yescharacteristic
18C3×#509—24no—
19C2×C3×SD32C2212yescomplemented
20C2×C3×D16C2212yescomplemented
21C2×C3×M32C2211yescharacteristic
22#5233—21no—
23C2×#832—21no—
24#5188C2211yescharacteristic, complemented
25#5188C2211yescharacteristic, complemented
26#5171—21no—
27#5171—21no—
28#5157—21no—
29#5157—21no—
30#5157C2212yes—
31#5157C2212yes—
32#5157C2211yescharacteristic, complemented
33#5157C2211yescharacteristic, complemented
34#5106C2211yescharacteristic
35C2×#818C2211yescharacteristic
36C2×#812C2211yescharacteristic
37D3×M32—24no—
38C2×#798C2212yescomplemented
39C2×D3×C16C2212yescomplemented
40#4564C2212yescomplemented
41#4562C2212yescomplemented
42#4562C2212yescomplemented
43#4562C2212yescomplemented
44#4505C2212yescomplemented
45#4504C2212yescomplemented
46C2×#509D311yescharacteristic, complemented
47C22×M32—31no—
48#3050—31no—
49C2×#466—31no—
50#2220—32no—
51#2218—32no—
52#2217—32no—
53#2215—32no—
54#2159—31no—
55#2140—31no—
56#2139—31no—
57C2×C3×#41—21no—
58C22×C3×D4—21no—
59C22×#67—41no—
60C22×C4×D3D411yescharacteristic
61C3×#135—22no—
62C3×#135—22no—
63C2×C3×Q16C2311yescharacteristic
64C2×C3×SD16—21no—
65C2×C3×D8—22no—
66C2×C3×D8—21no—
67C2×C3×D8—22no—
68C2×C3×D8C2311yescharacteristic
69C2×C3×D8C2312yes—
70C22×C3×C8C2311yescharacteristic
71C2×#217—41no—
72#946—21no—
73#946—21no—
74#938—22no—
75#938—22no—
76#938—21no—
77#938—21no—
78#938—41no—
79#938—41no—
80#937—41no—
81#935—41no—
82C4×#67—41no—
83C4×#67—41no—
84#926—21no—
85C2×C4⋊Dic3—21no—
86C22×C3⋊C8D411yescharacteristic
87C2×D3×C8—22no—
88C2×D3×C8—22no—
89C2×D3×C8D411yescharacteristic
90C2×D3×C8D411yescharacteristic
91C2×D3×C8C2311yescharacteristic
92C2×D3×C8C2311yescharacteristic
93C3×SD32—24no—
94C3×D16—24no—
95C3×M32D414yescomplemented
96C2×C3×C16C2312yes—
97#835—21no—
98#835—21no—
99#832—22no—
100#832—22no—
101#832—21no—
102#832—21no—
103#818—21no—
104#818C2312yes—
105#818C2311yescharacteristic
106#818C2311yescharacteristic
107#817—21no—
108#812—22no—
109C2×C3⋊C16C2312yes—
110#798—24no—
111D3×C16—24no—
112C2×#135C2×D311yescharacteristic
113C22×D8C2×D311yescharacteristic
114C23×C8—31no—
115#525—61no—
116C2×#121—61no—
117#509—24no—
118C2×SD32C2×D312yescomplemented
119C2×D16C2×D312yescomplemented
120C2×M32—31no—
121C2×M32—32no—
122C2×M32—64no—
123C2×M32C2×D311yescharacteristic
124C22×C16—32no—
125#467—31no—
126#467—31no—
127#466—32no—
128#466—32no—
129#466—31no—
130#466—31no—
131#466—61no—
132#466—61no—
133#465—61no—
134#465—61no—
135#427—31no—
136C2×#107—31no—
137#416—61no—
138C2×#102—61no—
139#367—32no—
140#366—32no—
141#358—32no—
142#357—32no—
143#357—32no—
144#357—32no—
145C24×C3—41no—
146C23×D3D811yescharacteristic
147C3×#41—42no—
148C3×#41—42no—
149C2×C3×Q8—21no—
150C2×C3×D4—22no—
151C2×C3×D4—21no—
152C2×C3×D4—41no—
153C2×C3×D4—42no—
154C2×C3×D4—41no—
155C2×C3×D4—42no—
156C22×C3×C4—41no—
157C22×C3×C4C2×D411yescharacteristic
158C2×#67—41no—
159C2×#67—41no—
160C2×#67—42no—
161C2×#67—42no—
162C2×#67—84no—
163C22×Dic3—41no—
164C22×Dic3D811yescharacteristic
165C2×C4×D3—22no—
166C2×C4×D3—22no—
167C2×C4×D3D811yescharacteristic
168C2×C4×D3D811yescharacteristic
169C2×C4×D3C2×D411yescharacteristic
170C2×C4×D3C2×D411yescharacteristic
171C3×Q16—21no—
172C3×Q16—21no—
173C3×SD16—42no—
174C3×D8—21no—
175C3×D8—22no—
176C3×D8—22no—
177C3×D8—21no—
178C3×D8—42no—
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D412yes—
181C2×C3×C8#4112yes—
182C2×C3×C8C2411yescharacteristic
183#217—41no—
184#217—41no—
185#217—41no—
186#217—42no—
187#217—42no—
188(C2×D3)⋊C4—41no—
189(C2×D3)⋊C4—41no—
190C4⋊Dic3—21no—
191C4⋊Dic3—21no—
192C4⋊Dic3—22no—
193C4⋊Dic3—41no—
194Dic3⋊C4—41no—
195Dic3⋊C4—41no—
196C4×Dic3—41no—
197C2×C3⋊C8—22no—
198C2×C3⋊C8C2×D411yescharacteristic
199C2×C3⋊C8C2×D411yescharacteristic
200D3×C8—22no—
201D3×C8—22no—
202D3×C8—21no—
203D3×C8—21no—
204D3×C8—21no—
205D3×C8—21no—
206C3×C16C2×D414yescomplemented
207C3⋊C16—22no—
208C2×#41—21no—
209C22×D4—21no—
210C22×D4—121no—
211C23×C4—31no—
212#135—22no—
213#135—22no—
214C2×Q16C22×D311yescharacteristic
215C2×SD16—21no—
216C2×D8—21no—
217C2×D8—22no—
218C2×D8—22no—
219C2×D8C22×D312yes—
220C2×D8C22×D311yescharacteristic
221C22×C8—31no—
222C22×C8—31no—
223C22×C8—31no—
224C22×C8—31no—
225C22×C8—31no—
226C22×C8—62no—
227C22×C8—62no—
228C22×C8C22×D311yescharacteristic
229#123—121no—
230C22⋊Q8—61no—
231C22⋊Q8—61no—
232#121—62no—
233#121—62no—
234#121—61no—
235#121—61no—
236#121—121no—
237#121—121no—
238#120—121no—
239C4×D4—121no—
240C4×D4—121no—
241#117—61no—
242C2×C4⋊C4—61no—
243C2×C22⋊C4—121no—
244SD32—24no—
245D16—24no—
246M32—62no—
247M32—64no—
248M32#6714yescomplemented
249C2×C16—32no—
250C2×C16—64no—
251C2×C16C22×D312yes—
252#107—31no—
253#107—31no—
254#107—32no—
255#107—61no—
256#106—61no—
257#103—61no—
258#103—61no—
259#102—62no—
260#102—62no—
261#102—61no—
262#102—61no—
263C23×C3—41no—
264C23×C3—42no—
265C23×C3—82no—
266C23×C3—82no—
267C23×C3—82no—
268C23×C3C2×D811yescharacteristic
269C22×D3—22no—
270C22×D3—22no—
271C22×D3—41no—
272C22×D3C2×D811yescharacteristic
273C22×D3C2×D811yescharacteristic
274C3×Q8—41no—
275C3×Q8—41no—
276C3×D4—42no—
277C3×D4—41no—
278C3×D4—42no—
279C3×D4—41no—
280C3×D4—82no—
281C3×D4—82no—
282C2×C3×C4—41no—
283C2×C3×C4—82no—
284C2×C3×C4#12112yes—
285C2×C3×C4C2×D811yescharacteristic
286C2×C3×C4#13512yes—
287C2×C3×C4C22×D411yescharacteristic
288#67—82no—
289#67—84no—
290#67—82no—
291#67—84no—
292C2×Dic3—22no—
293C2×Dic3—41no—
294C2×Dic3—41no—
295C2×Dic3—42no—
296C2×Dic3—82no—
297C2×Dic3C2×D811yescharacteristic
298C2×Dic3C2×D811yescharacteristic
299C4×D3—21no—
300C4×D3—21no—
301C4×D3—21no—
302C4×D3—21no—
303C4×D3—22no—
304C4×D3—22no—
305C3×C8#12112yes—
306C3×C8C22×D411yescharacteristic
307C3×C8C2×#4111yescharacteristic
308C3⋊C8—21no—
309C3⋊C8—21no—
310C24—31no—
311C24—41no—
312#41—42no—
313#41—42no—
314C2×Q8—21no—
315C2×D4—22no—
316C2×D4—21no—
317C2×D4—41no—
318C2×D4—41no—
319C2×D4—42no—
320C2×D4—42no—
321C2×D4—121no—
322C2×D4—121no—
323C2×D4—122no—
324C2×D4—122no—
325C2×D4—244no—
326C22×C4—31no—
327C22×C4—31no—
328C22×C4—31no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—41no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—121no—
335C22×C4D3×D411yescharacteristic
336Q16—21no—
337Q16—21no—
338SD16—42no—
339D8—22no—
340D8—21no—
341D8—21no—
342D8—22no—
343D8—42no—
344C2×C8—31no—
345C2×C8—31no—
346C2×C8—62no—
347C2×C8—62no—
348C2×C8—62no—
349C2×C8—61no—
350C2×C8—61no—
351C2×C8—61no—
352C2×C8—61no—
353C2×C8D3×D411yescharacteristic
354C2×C8#23712yes—
355C2×C8C2×#6712yes—
356C2×C8C23×D311yescharacteristic
357C4⋊C4—62no—
358C4⋊C4—61no—
359C4⋊C4—61no—
360C4⋊C4—121no—
361C4⋊C4—121no—
362C4⋊C4—121no—
363C22⋊C4—122no—
364C22⋊C4—122no—
365C22⋊C4—121no—
366C22⋊C4—121no—
367C22⋊C4—121no—
368C22⋊C4—121no—
369C22⋊C4—121no—
370C42—121no—
371C16—62no—
372C16C2×#6714yescomplemented
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—82no—
376C22×C3—81no—
377C22×C3—81no—
378C22×C3—82no—
379C22×C3—81no—
380C22×C3—82no—
381C22×C3—164no—
382C22×C3#46612yes—
383C22×C3C22×D811yescharacteristic
384C2×D3—21no—
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—21no—
389C2×D3—44no—
390C2×D3#50912yescomplemented
391C3×C4—81no—
392C3×C4#46612yes—
393C3×C4C2×#12111yescharacteristic
394C3×C4C2×#13511yescharacteristic
395Dic3—21no—
396Dic3—21no—
397Dic3—82no—
398Dic3—81no—
399Dic3—81no—
400C23—31no—
401C23—31no—
402C23—41no—
403C23—42no—
404C23—62no—
405C23—62no—
406C23—82no—
407C23—82no—
408C23—82no—
409C23—121no—
410C23D3×D811yescharacteristic
411Q8—41no—
412Q8—41no—
413D4—42no—
414D4—41no—
415D4—42no—
416D4—41no—
417D4—82no—
418D4—82no—
419D4—244no—
420D4—242no—
421D4—244no—
422D4—242no—
423C2×C4—31no—
424C2×C4—31no—
425C2×C4—41no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—82no—
434C2×C4—121no—
435C2×C4—121no—
436C2×C4—122no—
437C2×C4—242no—
438C2×C4D3×D811yescharacteristic
439C2×C4#91212yes—
440C2×C4#93812yes—
441C2×C4C2×D3×D411yescharacteristic
442C8—61no—
443C8—61no—
444C8#93812yes—
445C8C2×#23711yescharacteristic
446C8C22×#6711yescharacteristic
447C2×C3—22no—
448C2×C3—161no—
449C2×C3—162no—
450C2×C3C2×#46611yescharacteristic
451C2×C3C2×#50911yescharacteristic
452C2×C3#341411yescharacteristic
453D3—22no—
454D3—41no—
455C22—21no—
456C22—21no—
457C22—32no—
458C22—61no—
459C22—61no—
460C22—61no—
461C22—61no—
462C22—61no—
463C22—82no—
464C22—81no—
465C22—81no—
466C22—82no—
467C22—81no—
468C22—82no—
469C22—124no—
470C22—164no—
471C22#515712yes—
472C22C2×D3×D811yescharacteristic
473C4—61no—
474C4—61no—
475C4—81no—
476C4—242no—
477C4—241no—
478C4—241no—
479C4#515712yes—
480C4C2×#91211yescharacteristic
481C4C2×#93811yescharacteristic
482C3#3360511yescharacteristic, complemented, Sylow
483C2—22no—
484C2—62no—
485C2—121no—
486C2—162no—
487C2—161no—
488C2D3×#50911yescharacteristic
489C2#8203511yescharacteristic
490C2C2×#515711yescharacteristic
subgroupG/H
centerC22C2×D3×D8
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×#509C2
socleC22×C3C22×D8
hypercenterC2×#509D3
hypocenterC3#33605
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#509

Complements

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

NH ≅ G/Ntype
1. C2×C3×#509C2semidirect
2. #83914C2semidirect
3. C2×#5157C2semidirect
4. C2×D3×M32C2semidirect
5. #72078C2semidirect
6. #72033C2semidirect
7. #71488C2semidirect
8. #71487C2semidirect
9. #71470C2semidirect
10. #71469C2semidirect
11. #71428C2semidirect
19. C2×C3×SD32C22semidirect
20. C2×C3×D16C22semidirect
24. #5188C22semidirect
25. #5188C22semidirect
32. #5157C22semidirect
33. #5157C22semidirect
38. C2×#798C22semidirect
39. C2×D3×C16C22semidirect
40. #4564C22semidirect
41. #4562C22semidirect
42. #4562C22semidirect
43. #4562C22semidirect
44. #4505C22semidirect
45. #4504C22semidirect
46. C2×#509D3semidirect
95. C3×M32D4semidirect
118. C2×SD32C2×D3semidirect
119. C2×D16C2×D3semidirect
206. C3×C16C2×D4semidirect
248. M32#67semidirect
372. C16C2×#67semidirect
390. C2×D3#509semidirect
482. C3#33605semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#5092
#839142
C2×#51572
C2×D3×M322
#720782
#720332
#714882
#714872
#714702
#714692
#714282
#336053