Finite Group Databasebeta

← #10949456

Subgroups — #10949456

Order 768.

718 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4904C212yesmaximal, complemented
2#78467C211yesmaximal, characteristic, complemented
3C4×#904C211yesmaximal, characteristic, complemented
4C2×C3×C4⋊C16C211yesmaximal, characteristic, complemented
5#71412C212yesmaximal, complemented
6#69294C212yesmaximal, complemented
7#69293C211yesmaximal, characteristic, complemented
8#69292C211yesmaximal, characteristic, complemented
9#69288C212yesmaximal, complemented
10#69287C211yesmaximal, characteristic, complemented
11#69286C211yesmaximal, characteristic, complemented
12#33579—31nomaximal, Sylow
13C2×#904—21no—
14C2×#904C2212yes—
15C2×#904C2211yescharacteristic
16#5477—21no—
17C4×#235—21no—
18C22×C3×C16C2212yes—
19C2×C3×C4×C8C2211yescharacteristic
20#5113—21no—
21#5106C2211yescharacteristic
22C2×Dic24C2212yes—
23#4904—28no—
24C2×#800C2212yes—
25C2×#800C2212yes—
26C2×D48C2212yes—
27#4699C2211yescharacteristic
28C4×Dic12C2211yescharacteristic
29C4×Dic12C2211yescharacteristic
30#4696—21no—
31#4696C2212yes—
32#4695C2211yescharacteristic
33#4693—21no—
34C4×D24C2211yescharacteristic
35C4×D24C2211yescharacteristic
36C4×#204—21no—
37C4×#204—21no—
38C3×C4⋊C16C2211yescharacteristic
39C3×C4⋊C16C2211yescharacteristic
40C3×C4⋊C16C2212yes—
41#4509C2212yes—
42#4509C2212yes—
43#4503C2212yes—
44#4503C2212yes—
45C2×#508—32no—
46#3147—31no—
47C4×#135—31no—
48#2211—32no—
49#2210—32no—
50#2209—31no—
51#2208—31no—
52#2207—31no—
53#2206—31no—
54C2×C4⋊C16D311yescharacteristic, complemented
55#2138—32no—
56C2×#235—21no—
57C2×#235—22no—
58C22×C3×C8D411yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C2×C3×C42D411yescharacteristic
61#930—41no—
62#930—41no—
63C4×#67—41no—
64C4×#67—41no—
65#926—21no—
66C2×Dic12—21no—
67C2×Dic12C2312yes—
68C2×Dic12C2311yescharacteristic
69#904—22no—
70#904—24no—
71#904—24no—
72#904—44no—
73#904D414yescomplemented
74C2×D24—21no—
75C2×D24C2312yes—
76C2×D24C2311yescharacteristic
77C2×#204—21no—
78C2×#204—22no—
79C2×#204—22no—
80#875—22no—
81#874—21no—
82C4×D12—21no—
83C4×D12—21no—
84C4×D12—41no—
85C4×D12—41no—
86#872—41no—
87C42×D3—41no—
88#869—21no—
89C4×Dic6—21no—
90C4×Dic6—21no—
91C2×C3×C16—24no—
92C2×C3×C16C2312yes—
93C2×C3×C16C2312yes—
94C3×C4×C8D411yescharacteristic
95C3×C4×C8D411yescharacteristic
96C3×C4×C8C2311yescharacteristic
97C3×C4×C8C2311yescharacteristic
98#821—21no—
99#821—21no—
100#818C2311yescharacteristic
101#818C2311yescharacteristic
102#817—21no—
103#816—21no—
104#816—21no—
105Dic24—24no—
106#800—24no—
107#800—24no—
108D48—24no—
109C2×#135—31no—
110C2×#135—32no—
111C2×#135—61no—
112#532—61no—
113C4×#41—61no—
114#508—68no—
115C2×Q32—32no—
116C2×SD32—32no—
117C2×SD32—32no—
118C2×D16—32no—
119C22×C16C2×D312yes—
120#495—32no—
121#495—61no—
122#494—31no—
123#493—31no—
124#492—61no—
125C4×Q16—31no—
126C4×Q16—31no—
127C4×SD16—61no—
128C4×SD16—61no—
129C4×D8—31no—
130C4×D8—31no—
131#427—31no—
132#416—61no—
133C2×C4×C8C2×D311yescharacteristic
134C4⋊C16C2×D311yescharacteristic
135C4⋊C16C2×D311yescharacteristic
136C4⋊C16C2×D312yes—
137#358—32no—
138#358—32no—
139#357—32no—
140#357—32no—
141C22×C3×C4D811yescharacteristic
142C22×C3×C4D811yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C2×#67—41no—
145C2×#67—42no—
146#235—24no—
147#235—42no—
148#235—44no—
149#235—44no—
150C2×D12—21no—
151C2×D12—22no—
152C2×D12—41no—
153C2×D12—41no—
154C2×C4×D3—41no—
155C2×C4×D3—41no—
156C2×C4×D3—41no—
157C2×C4×D3—42no—
158C2×Dic6—21no—
159C2×Dic6—22no—
160C2×C3×C8—22no—
161C2×C3×C8C2×D411yescharacteristic
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2×D412yes—
165C2×C3×C8#4112yes—
166C2×C3×C8C2411yescharacteristic
167C3×C42D811yescharacteristic
168C3×C42D811yescharacteristic
169C3×C42C2×D411yescharacteristic
170C3×C42C2×D411yescharacteristic
171#217—41no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174(C2×D3)⋊C4—42no—
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—21no—
177C4⋊Dic3—21no—
178C4⋊Dic3—41no—
179C4⋊Dic3—41no—
180Dic3⋊C4—41no—
181Dic3⋊C4—41no—
182C4×Dic3—41no—
183C4×Dic3—41no—
184C4×Dic3—41no—
185Dic12—22no—
186Dic12—21no—
187Dic12—22no—
188Dic12—42no—
189Dic12C2×D412yes—
190D24—22no—
191D24—21no—
192D24—22no—
193D24—42no—
194D24C2×D412yes—
195#204—24no—
196#204—44no—
197#204—44no—
198C3×C16—22no—
199C3×C16—22no—
200C2×#41—61no—
201C2×#41—62no—
202#135—34no—
203#135—64no—
204#135—62no—
205#135—64no—
206#135—124no—
207C2×Q16—31no—
208C2×Q16—32no—
209C2×Q16—61no—
210C2×SD16—61no—
211C2×SD16—62no—
212C2×SD16—62no—
213C2×D8—31no—
214C2×D8—32no—
215C2×D8—61no—
216C22×C8D1211yescharacteristic
217C22×C8C22×D311yescharacteristic
218C4⋊Q8—61no—
219#127—61no—
220#124—62no—
221#121—121no—
222#121—121no—
223C4×Q8—61no—
224C4×Q8—61no—
225C4×D4—61no—
226C4×D4—61no—
227C4×D4—121no—
228C4×D4—121no—
229C4×D4—121no—
230C4×D4—121no—
231#117—61no—
232#117—121no—
233C2×C42—121no—
234C2×C42D1211yescharacteristic
235Q32—64no—
236SD32—64no—
237SD32—64no—
238D16—64no—
239C2×C16—24no—
240C2×C16C22×D312yes—
241C2×C16C22×D312yes—
242#107—31no—
243#107—31no—
244#106—61no—
245#103—61no—
246#103—61no—
247#102—61no—
248#102—61no—
249C4×C8D1211yescharacteristic
250C4×C8D1211yescharacteristic
251C4×C8C22×D311yescharacteristic
252C4×C8C22×D311yescharacteristic
253C23×C3C2×D811yescharacteristic
254C22×D3—41no—
255C22×D3—42no—
256C2×C3×C4—22no—
257C2×C3×C4—41no—
258C2×C3×C4#12112yes—
259C2×C3×C4C2×D811yescharacteristic
260C2×C3×C4C2×D811yescharacteristic
261C2×C3×C4C2×D811yescharacteristic
262C2×C3×C4C2×D811yescharacteristic
263C2×C3×C4C2×D811yescharacteristic
264C2×C3×C4AGL(1,ℤ/8)12yes—
265C2×C3×C4C22×D411yescharacteristic
266#67—44no—
267#67—84no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—41no—
271C2×Dic3—42no—
272D12—22no—
273D12—42no—
274D12—41no—
275D12—42no—
276D12—82no—
277D12—82no—
278C4×D3—44no—
279C4×D3—82no—
280C4×D3—84no—
281C4×D3—82no—
282Dic6—22no—
283Dic6—41no—
284Dic6—42no—
285Dic6—42no—
286C3×C8—21no—
287C3×C8—21no—
288C3×C8#12112yes—
289C3×C8C22×D411yescharacteristic
290C3×C8C2×#4111yescharacteristic
291#41—64no—
292#41—122no—
293#41—124no—
294#41—124no—
295C2×Q8—61no—
296C2×Q8—62no—
297C2×D4—61no—
298C2×D4—62no—
299C2×D4—121no—
300C2×D4—122no—
301C2×D4—121no—
302C2×D4—121no—
303C22×C4—121no—
304C22×C4—121no—
305C22×C4—121no—
306C22×C4—122no—
307C22×C4D2411yescharacteristic
308C22×C4D2411yescharacteristic
309C22×C4C2×D1211yescharacteristic
310Q16—32no—
311Q16—62no—
312Q16—61no—
313Q16—62no—
314Q16—122no—
315SD16—64no—
316SD16—124no—
317SD16—124no—
318D8—32no—
319D8—62no—
320D8—61no—
321D8—62no—
322D8—122no—
323C2×C8—22no—
324C2×C8C2×D1211yescharacteristic
325C2×C8C2×D1211yescharacteristic
326C2×C8C2×D1211yescharacteristic
327C2×C8D3×D412yes—
328C2×C8#23912yes—
329C2×C8C23×D311yescharacteristic
330C4⋊C4—61no—
331C4⋊C4—61no—
332C4⋊C4—121no—
333C4⋊C4—121no—
334C4⋊C4—121no—
335C4⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—121no—
338C22⋊C4—121no—
339C22⋊C4—122no—
340C22⋊C4—122no—
341C42—121no—
342C42—121no—
343C42—121no—
344C42D2411yescharacteristic
345C42D2411yescharacteristic
346C42C2×D1211yescharacteristic
347C42C2×D1211yescharacteristic
348C16—22no—
349C16—22no—
350C22×C3—21no—
351C22×C3—21no—
352C22×C3#45912yes—
353C22×C3C22×D811yescharacteristic
354C2×D3—42no—
355C2×D3—81no—
356C2×D3—81no—
357C2×D3—82no—
358C2×D3—82no—
359C2×D3—82no—
360C3×C4—21no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4#45912yes—
364C3×C4#50812yescomplemented
365C3×C4C2×#12111yescharacteristic
366C3×C4C2×AGL(1,ℤ/8)11yescharacteristic
367Dic3—42no—
368Dic3—81no—
369Dic3—81no—
370Dic3—82no—
371C23—121no—
372C23—122no—
373C23C2×D2411yescharacteristic
374Q8—62no—
375Q8—121no—
376Q8—122no—
377Q8—122no—
378D4—62no—
379D4—124no—
380D4—121no—
381D4—122no—
382D4—122no—
383D4—242no—
384D4—244no—
385D4—242no—
386C2×C4—22no—
387C2×C4—41no—
388C2×C4—121no—
389C2×C4—124no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—122no—
393C2×C4—242no—
394C2×C4—242no—
395C2×C4—244no—
396C2×C4#89512yes—
397C2×C4C2×D2411yescharacteristic
398C2×C4C2×D2411yescharacteristic
399C2×C4C2×D2411yescharacteristic
400C2×C4C2×D2411yescharacteristic
401C2×C4C2×D2411yescharacteristic
402C2×C4#90812yes—
403C2×C4C22×D1211yescharacteristic
404C8—21no—
405C8—21no—
406C8#89512yes—
407C8C2×D3×D411yescharacteristic
408C8C2×#23911yescharacteristic
409C2×C3—22no—
410C2×C3C2×#45911yescharacteristic
411C2×C3C2×#50811yescharacteristic
412C2×C3#341411yescharacteristic
413D3—82no—
414D3—162no—
415C22—21no—
416C22—21no—
417C22—122no—
418C22—241no—
419C22—241no—
420C22—242no—
421C22—242no—
422C22—242no—
423C22#484312yes—
424C22C22×D2411yescharacteristic
425C4—21no—
426C4—21no—
427C4—21no—
428C4—122no—
429C4—241no—
430C4—241no—
431C4—242no—
432C4#484312yes—
433C4#490412yescomplemented
434C4C2×#89511yescharacteristic
435C4C2×#90811yescharacteristic
436C3#3357911yescharacteristic, complemented, Sylow
437C2—22no—
438C2—242no—
439C2—482no—
440C2C2×#484311yescharacteristic
441C2C2×#490411yescharacteristic
442C2#8200911yescharacteristic
subgroupG/H
centerC2×C4C2×D24
commutator, cosocleC2×C3×C8C24
FrattiniC2×C8C23×D3
FittingC2×C3×C4⋊C16C2
socleC22×C3C22×D8
hypercenterC2×C4⋊C16D3
hypocenterC3#33579
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4⋊C16

Complements

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

NH ≅ G/Ntype
1. C2×#4904C2semidirect
2. #78467C2semidirect
3. C4×#904C2semidirect
4. C2×C3×C4⋊C16C2semidirect
5. #71412C2semidirect
6. #69294C2semidirect
7. #69293C2semidirect
8. #69292C2semidirect
9. #69288C2semidirect
10. #69287C2semidirect
11. #69286C2semidirect
54. C2×C4⋊C16D3semidirect
73. #904D4semidirect
364. C3×C4#508semidirect
433. C4#4904semidirect
436. C3#33579semidirect

Maximal subgroups

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

subgroupindex
C2×#49042
#784672
C4×#9042
C2×C3×C4⋊C162
#714122
#692942
#692932
#692922
#692882
#692872
#692862
#335793