Finite Group Databasebeta

← #10818116

Subgroups — #10818116

Order 768.

892 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#472C211yesmaximal, characteristic
2#83930C212yesmaximal, complemented
3#83823C211yesmaximal, characteristic, complemented
4#83784C211yesmaximal, characteristic, complemented
5#72088C212yesmaximal, complemented
6#72039C211yesmaximal, characteristic, complemented
7#72009C211yesmaximal, characteristic, complemented
8#70703C211yesmaximal, characteristic, complemented
9#70700C211yesmaximal, characteristic, complemented
10#70072C212yesmaximal, complemented
11#70068C212yesmaximal, complemented
12#23048—31nomaximal, Sylow
13C2×C3×AGL(1,ℤ/8)C2212yes—
14C22×C3×D8C2211yescharacteristic
15C2×C3×#131C2211yescharacteristic
16#5827C2212yes—
17#5798C2212yes—
18C3×#472C418yescomplemented
19C2×C3×#108C2211yescharacteristic
20C2×C3×#100C2212yes—
21C2×#837C2211yescharacteristic
22#5233C2211yescharacteristic
23#5221C2212yes—
24#5215C2212yes—
25C2×#832C2212yes—
26#5164—22no—
27Dic3×SD16—22no—
28#5152—22no—
29#5152—21no—
30#5152—21no—
31Dic3×D8—22no—
32Dic3×D8—22no—
33#5122C2212yes—
34#5121—22no—
35#5121C2212yes—
36#5121C2212yes—
37#5106—21no—
38#5106C2211yescharacteristic
39C2×#817—21no—
40C2×#817C2211yescharacteristic
41#5101C2211yescharacteristic
42C2×#815C2211yescharacteristic
43#4558C2211yescharacteristic
44#4555C2212yes—
45#4555C2211yescharacteristic
46#4554C2212yes—
47C2×#472D311yescharacteristic
48#2948—31no—
49#2946—32no—
50#2912—31no—
51#1972—31no—
52#1951—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1773—31no—
57#1765—31no—
58C2×C3×#41C2311yescharacteristic
59C22×C3×D4C2312yes—
60C3×AGL(1,ℤ/8)—24no—
61C3×AGL(1,ℤ/8)C2×C418yescomplemented
62C2×C3×SD16—22no—
63C2×C3×D8—22no—
64C2×C3×D8—21no—
65C2×C3×D8—21no—
66C2×C3×D8—22no—
67C2×C3×D8C2×C414yescomplemented
68C3×#131—22no—
69C3×#131C2×C414yescomplemented
70C2×C3×M16—21no—
71C2×C3×M16C2311yescharacteristic
72C2×C3×M16C2312yes—
73C22×C3×C8—21no—
74C22×C3×C8C2311yescharacteristic
75C2×#217—42no—
76C2×#217—42no—
77Q8×Dic3—22no—
78D4×Dic3—22no—
79D4×Dic3—22no—
80D4×Dic3—22no—
81D4×Dic3—22no—
82D4×Dic3—22no—
83D4×Dic3—42no—
84#926—22no—
85#926C2311yescharacteristic
86#926C2312yes—
87C2×C4⋊Dic3—22no—
88C2×C4⋊Dic3C2312yes—
89C2×C4×Dic3C2311yescharacteristic
90C2×#208C2311yescharacteristic
91C22×C3⋊C8C2311yescharacteristic
92C3×#108C2×C414yescomplemented
93C3×#100C2×C418yescomplemented
94#837—22no—
95#835D412yes—
96#832—22no—
97#832—22no—
98#832—22no—
99#832—22no—
100#832D412yes—
101#818—22no—
102#818—22no—
103#818D412yes—
104#817—21no—
105#817—21no—
106#817—22no—
107#817—22no—
108#817D411yescharacteristic
109#817D411yescharacteristic
110#815—21no—
111#815—22no—
112#815—22no—
113C8×Dic3—21no—
114C8×Dic3—22no—
115C2×AGL(1,ℤ/8)C2×D312yes—
116C22×D8C2×D311yescharacteristic
117C2×#131C2×D311yescharacteristic
118#520—32no—
119#518—32no—
120#472Dic318yescomplemented
121#442—62no—
122#442—61no—
123#442—61no—
124#440—62no—
125C4×SD16—62no—
126C4×D8—62no—
127C4×D8—62no—
128C2×#108C2×D311yescharacteristic
129#428—32no—
130#428—32no—
131#428—62no—
132#427—31no—
133#427—61no—
134C2×#106—31no—
135C2×#106—61no—
136C2×#104—31no—
137#418—32no—
138#416—31no—
139C2×#102—32no—
140C2×#100C2×D312yes—
141#405—32no—
142#405—31no—
143C2×#97—31no—
144#390—32no—
145#343—32no—
146#341—31no—
147#339—31no—
148#339—32no—
149C24×C3—22no—
150C3×#41—21no—
151C3×#41—21no—
152C3×#41C22×C412yes—
153C3×#41#4112yes—
154C2×C3×Q8C2×D411yescharacteristic
155C2×C3×D4—21no—
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—22no—
159C2×C3×D4—22no—
160C2×C3×D4—22no—
161C2×C3×D4C22×C414yes—
162C2×C3×D4C2×D411yescharacteristic
163C22×C3×C4—21no—
164C22×C3×C4C2411yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—22no—
167C22×Dic3—42no—
168C22×Dic3—42no—
169C22×Dic3—42no—
170C3×SD16—28no—
171C3×D8—28no—
172C3×D8—24no—
173C3×D8—24no—
174C3×M16—21no—
175C3×M16—22no—
176C3×M16—24no—
177C3×M16C22×C412yes—
178C3×M16C22×C414yes—
179C2×C3×C8—22no—
180C2×C3×C8—21no—
181C2×C3×C8—24no—
182C2×C3×C8C22×C412yes—
183C2×C3×C8C2×D411yescharacteristic
184C2×C3×C8C2×D411yescharacteristic
185C2×C3×C8#4112yes—
186#217—22no—
187#217—21no—
188#217—42no—
189#217—42no—
190#217—42no—
191#217—84no—
192C4⋊Dic3—22no—
193C4⋊Dic3—22no—
194C4⋊Dic3—42no—
195C4⋊Dic3C2×D412yes—
196C4⋊Dic3C2×D412yes—
197C4⋊Dic3C2×D412yes—
198C4×Dic3—21no—
199C4×Dic3—22no—
200C4×Dic3—21no—
201C4×Dic3—21no—
202C4×Dic3—42no—
203#208—21no—
204#208—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8C2×D412yes—
209C2×#41C22×D311yescharacteristic
210C22×D4C22×D312yes—
211AGL(1,ℤ/8)—24no—
212AGL(1,ℤ/8)C2×Dic318yescomplemented
213C2×SD16—22no—
214C2×D8—22no—
215C2×D8—21no—
216C2×D8—21no—
217C2×D8—22no—
218C2×D8C2×Dic314yescomplemented
219#131—22no—
220#131C2×Dic314yescomplemented
221C2×M16—21no—
222C2×M16—31no—
223C2×M16C22×D311yescharacteristic
224C2×M16C22×D312yes—
225C22×C8—21no—
226C22×C8—31no—
227C22×C8C22×D311yescharacteristic
228C4×Q8—62no—
229C4×D4—62no—
230C4×D4—62no—
231C4×D4—62no—
232C4×D4—62no—
233C4×D4—62no—
234C4×D4—122no—
235#117—31no—
236#117—32no—
237#117—62no—
238C2×C4⋊C4—32no—
239C2×C4⋊C4—62no—
240C2×C22⋊C4—122no—
241C2×C22⋊C4—122no—
242C2×C42—31no—
243#108C2×Dic314yescomplemented
244#107—32no—
245#107—62no—
246#107—62no—
247#106—31no—
248#106—31no—
249#106—62no—
250#106—62no—
251#106—61no—
252#106—61no—
253#104—62no—
254#103—32no—
255#102—32no—
256#102—62no—
257#102—62no—
258#102—62no—
259#102—62no—
260#100C2×Dic318yescomplemented
261#97—62no—
262#97—61no—
263#97—62no—
264C4×C8—62no—
265C4×C8—61no—
266C23×C3—21no—
267C23×C3—24no—
268C23×C3—42no—
269C23×C3—42no—
270C23×C3—82no—
271C23×C3—82no—
272C23×C3H(5,𝔽2)11yescharacteristic
273C3×Q8—21no—
274C3×Q8C4×D412yes—
275C3×D4—22no—
276C3×D4—22no—
277C3×D4—24no—
278C3×D4—24no—
279C3×D4—21no—
280C3×D4—42no—
281C3×D4—42no—
282C3×D4C4×D412yes—
283C2×C3×C4—21no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4C23×C411yescharacteristic
287C2×C3×C4C22×D411yescharacteristic
288C2×C3×C4C22×D411yescharacteristic
289C2×C3×C4C2×#4111yescharacteristic
290C2×C3×C4C2×#4111yescharacteristic
291C2×C3×C42−1+411yescharacteristic
292C2×Dic3—22no—
293C2×Dic3—22no—
294C2×Dic3—22no—
295C2×Dic3—42no—
296C2×Dic3—42no—
297C2×Dic3—41no—
298C2×Dic3—82no—
299C2×Dic3—162no—
300C2×Dic3—162no—
301C2×Dic3—162no—
302C3×C8—22no—
303C3×C8—24no—
304C3×C8C4×D412yes—
305C3×C8C4×D412yes—
306C3⋊C8—22no—
307C3⋊C8—41no—
308C24—22no—
309#41—21no—
310#41—21no—
311#41#23712yes—
312#41C22×Dic312yes—
313C2×Q8D3×D411yescharacteristic
314C2×D4—21no—
315C2×D4—22no—
316C2×D4—22no—
317C2×D4—22no—
318C2×D4—22no—
319C2×D4—22no—
320C2×D4D3×D411yescharacteristic
321C2×D4C22×Dic314yes—
322C22×C4—21no—
323C22×C4—61no—
324C22×C4—62no—
325C22×C4—122no—
326C22×C4—122no—
327C22×C4—122no—
328C22×C4C23×D311yescharacteristic
329SD16—28no—
330D8—28no—
331D8—24no—
332D8—24no—
333M16—21no—
334M16—22no—
335M16—24no—
336M16—61no—
337M16—61no—
338M16C22×Dic312yes—
339M16C22×Dic314yes—
340C2×C8—22no—
341C2×C8—21no—
342C2×C8—24no—
343C2×C8—32no—
344C2×C8—61no—
345C2×C8—61no—
346C2×C8—61no—
347C2×C8D3×D411yescharacteristic
348C2×C8D3×D411yescharacteristic
349C2×C8#23712yes—
350C2×C8C22×Dic312yes—
351C4⋊C4—32no—
352C4⋊C4—32no—
353C4⋊C4—32no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C22⋊C4—61no—
358C22⋊C4—62no—
359C22⋊C4—122no—
360C22⋊C4—122no—
361C22⋊C4—122no—
362C22⋊C4—244no—
363C42—61no—
364C42—62no—
365C42—61no—
366C42—61no—
367C42—122no—
368C22×C3—22no—
369C22×C3—22no—
370C22×C3—42no—
371C22×C3—44no—
372C22×C3—84no—
373C22×C3—82no—
374C22×C3—82no—
375C22×C3—162no—
376C22×C3#52011yescharacteristic
377C22×C3D4211yescharacteristic
378C22×C3#55311yescharacteristic
379C3×C4—22no—
380C3×C4C2×C4×D411yescharacteristic
381C3×C4C2×C4×D411yescharacteristic
382C3×C4#54711yescharacteristic
383C3×C4#54811yescharacteristic
384Dic3—42no—
385Dic3—42no—
386Dic3—42no—
387Dic3—81no—
388Dic3—82no—
389C23—21no—
390C23—24no—
391C23—42no—
392C23—42no—
393C23—82no—
394C23—82no—
395C23#100911yescharacteristic
396Q8—21no—
397Q8D4×Dic312yes—
398D4—22no—
399D4—22no—
400D4—24no—
401D4—24no—
402D4—21no—
403D4—42no—
404D4—42no—
405D4D4×Dic312yes—
406C2×C4—21no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—62no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—121no—
415C2×C4—242no—
416C2×C4—482no—
417C2×C4—482no—
418C2×C4—482no—
419C2×C4C2×D3×D411yescharacteristic
420C2×C4C2×D3×D411yescharacteristic
421C2×C4C2×#23711yescharacteristic
422C2×C4C2×#23711yescharacteristic
423C2×C4#101011yescharacteristic
424C2×C4C23×Dic311yescharacteristic
425C8—22no—
426C8—24no—
427C8—62no—
428C8—121no—
429C8D4×Dic312yes—
430C8D4×Dic312yes—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—42no—
434C2×C3—84no—
435C2×C3#236111yescharacteristic
436C2×C3#328812yes—
437C22—22no—
438C22—22no—
439C22—42no—
440C22—44no—
441C22—84no—
442C22—82no—
443C22—82no—
444C22—162no—
445C22#560911yescharacteristic
446C22#561811yescharacteristic
447C22#582711yescharacteristic
448C4—22no—
449C4—122no—
450C4—122no—
451C4—122no—
452C4—241no—
453C4—242no—
454C4#559811yescharacteristic
455C4#561111yescharacteristic
456C4C2×D4×Dic311yescharacteristic
457C4C2×D4×Dic311yescharacteristic
458C3#2304811yescharacteristic, complemented, Sylow
459C2—21no—
460C2—21no—
461C2—42no—
462C2—84no—
463C2#8093912yes—
464C2#8263611yescharacteristic
subgroupG/H
centerC22#5609
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
Fitting, prime radicalC2×C3×#472C2
socleC22×C3D42
cosocleC22×C3×C4C24
hypercenterC2×#472D3
hypocenterC3#23048
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#472

Complements

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

NH ≅ G/Ntype
2. #83930C2semidirect
3. #83823C2semidirect
4. #83784C2semidirect
5. #72088C2semidirect
6. #72039C2semidirect
7. #72009C2semidirect
8. #70703C2semidirect
9. #70700C2semidirect
10. #70072C2semidirect
11. #70068C2semidirect
18. C3×#472C4semidirect
61. C3×AGL(1,ℤ/8)C2×C4semidirect
67. C2×C3×D8C2×C4semidirect
69. C3×#131C2×C4semidirect
92. C3×#108C2×C4semidirect
93. C3×#100C2×C4semidirect
120. #472Dic3semidirect
212. AGL(1,ℤ/8)C2×Dic3semidirect
218. C2×D8C2×Dic3semidirect
220. #131C2×Dic3semidirect
243. #108C2×Dic3semidirect
260. #100C2×Dic3semidirect
458. C3#23048semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4722
#839302
#838232
#837842
#720882
#720392
#720092
#707032
#707002
#700722
#700682
#230483