Finite Group Databasebeta

← #10713913

Subgroups — #10713913

Order 768.

568 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 97 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic, complemented
2#83894C211yesmaximal, characteristic, complemented
3#83742C211yesmaximal, characteristic, complemented
4#72076C211yesmaximal, characteristic, complemented
5#71963C211yesmaximal, characteristic, complemented
6#70656C211yesmaximal, characteristic, complemented
7#67725C211yesmaximal, characteristic, complemented
8#13309—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10C23×C3×C8C2211yescharacteristic
11C3×#525—21no—
12#5791—21no—
13C2×#904C2211yescharacteristic
14C3×#467—21no—
15C3×#466—21no—
16C3×#465—22no—
17C3×#427C2211yescharacteristic
18C3×#416—21no—
19#5195—21no—
20#5194—22no—
21#5193—21no—
22#5179—21no—
23#5147—21no—
24#5116—21no—
25#5115—21no—
26#5114—22no—
27#5113—21no—
28#5106C2211yescharacteristic
29C2×#812C2211yescharacteristic
30C2×#812C2211yescharacteristic
31#4566—22no—
32#4549—22no—
33#4488—22no—
34#3414—31no—
35#3050—31no—
36#3050D311yescharacteristic, complemented
37#2145—31no—
38#2145—31no—
39#2112—31no—
40#1384—31no—
41C2×C3×#41—21no—
42C23×C3×C4D411yescharacteristic
43C2×#235—21no—
44C3×#135—21no—
45C3×#135—21no—
46C3×#135—21no—
47C3×#135—21no—
48C2×C3×Q16D411yescharacteristic, complemented
49C2×C3×SD16—21no—
50C2×C3×D8D411yescharacteristic, complemented
51C22×C3×C8—21no—
52C22×C3×C8—21no—
53C22×C3×C8—21no—
54C22×C3×C8—21no—
55C22×C3×C8—21no—
56C22×C3×C8—21no—
57C22×C3×C8D411yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×#123—41no—
60C3×C22⋊Q8—22no—
61C3×#121—22no—
62C3×#120—41no—
63C3×C4×D4—42no—
64C3×#117—21no—
65#953—41no—
66#930—22no—
67#929—41no—
68C4×#67—42no—
69#926—21no—
70#924—22no—
71C2×Dic12D411yescharacteristic, complemented
72#904—21no—
73#904—21no—
74#904—21no—
75#904—21no—
76C2×D24D411yescharacteristic, complemented
77C2×#204—21no—
78C3×#107—21no—
79C3×#106D412yescomplemented
80C3×#103—22no—
81C3×#102—22no—
82#829—22no—
83#828—22no—
84#827—22no—
85#826—22no—
86#821—22no—
87#818—21no—
88#817D412yescomplemented
89#816—22no—
90#812—21no—
91#812—21no—
92#812—21no—
93#812—21no—
94C2×C3⋊C16—21no—
95C2×C3⋊C16D412yescomplemented
96C2×#135—31no—
97C2×#135C2×D311yescharacteristic
98C23×C8C2×D311yescharacteristic
99#525—21no—
100#525—61no—
101#510—61no—
102#510—61no—
103#509—61no—
104#509—61no—
105#508—62no—
106C2×M32—31no—
107C2×M32—31no—
108#467—21no—
109#467—61no—
110#466—21no—
111#466—61no—
112#465—22no—
113#465—62no—
114#427—31no—
115#427C2×D311yescharacteristic
116#416—21no—
117#416—61no—
118#360—62no—
119#360—62no—
120C22⋊C16—62no—
121C24×C3D811yescharacteristic
122C3×#41—41no—
123C3×#41—41no—
124C3×#41—41no—
125C3×#41—41no—
126C2×C3×Q8—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4D811yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C2×#67—42no—
140C2×#67—41no—
141#235—41no—
142#235—41no—
143#235—41no—
144#235—41no—
145C2×D12—21no—
146C2×C4×D3—41no—
147C2×Dic6—21no—
148C3×Q16—21no—
149C3×Q16—21no—
150C3×SD16—41no—
151C3×SD16—41no—
152C3×D8—21no—
153C3×D8—21no—
154C2×C3×C8—21no—
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—41no—
158C2×C3×C8—41no—
159C2×C3×C8—41no—
160C2×C3×C8—41no—
161C2×C3×C8C2×D411yescharacteristic
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8#4111yescharacteristic
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C22⋊C4—42no—
170C3×C22⋊C4—42no—
171C3×C22⋊C4—41no—
172C3×C42—41no—
173#217—42no—
174#217—41no—
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—22no—
177Dic3⋊C4—42no—
178C4×Dic3—41no—
179Dic12—21no—
180Dic12—21no—
181D24—21no—
182D24—21no—
183#204—41no—
184#204—41no—
185C3⋊C16—22no—
186C3⋊C16—41no—
187C2×#41—21no—
188C2×#41—61no—
189C23×C4#6711yescharacteristic
190#135—21no—
191#135—21no—
192#135—21no—
193#135—21no—
194#135—61no—
195#135—61no—
196#135—61no—
197#135—61no—
198C2×Q16—31no—
199C2×Q16#6711yescharacteristic, complemented
200C2×SD16—21no—
201C2×SD16—61no—
202C2×D8—31no—
203C2×D8#6711yescharacteristic, complemented
204C22×C8—21no—
205C22×C8—21no—
206C22×C8—21no—
207C22×C8—21no—
208C22×C8—21no—
209C22×C8—21no—
210C22×C8#6711yescharacteristic
211C22×C8C22×D311yescharacteristic
212#123—41no—
213#123—121no—
214C22⋊Q8—22no—
215C22⋊Q8—62no—
216#121—22no—
217#121—62no—
218#120—41no—
219#120—121no—
220C4×D4—42no—
221C4×D4—122no—
222#117—21no—
223#117—61no—
224Q32—62no—
225SD32—62no—
226SD32—62no—
227D16—62no—
228M32—61no—
229M32—61no—
230M32—61no—
231M32—61no—
232C2×C16—32no—
233C2×C16—61no—
234#107—21no—
235#107—61no—
236#106—32no—
237#106#6712yescomplemented
238#103—22no—
239#103—62no—
240#102—22no—
241#102—62no—
242C23×C3—21no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—42no—
246C23×C3—41no—
247C23×C3C2×D811yescharacteristic
248C22×D3—41no—
249C3×Q8—41no—
250C3×Q8—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—84no—
254C3×D4—81no—
255C3×D4—81no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—81no—
266C2×C3×C4—81no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12011yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#13511yescharacteristic
271C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
272C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
273#67—84no—
274#67—81no—
275#67—81no—
276C2×Dic3—42no—
277C2×Dic3—41no—
278D12—41no—
279D12—41no—
280C4×D3—81no—
281C4×D3—81no—
282Dic6—41no—
283Dic6—41no—
284C3×C8—41no—
285C3×C8#12011yescharacteristic
286C3×C8#12111yescharacteristic
287C3×C8#12111yescharacteristic
288C3×C8#12711yescharacteristic
289C24#21311yescharacteristic
290#41—41no—
291#41—41no—
292#41—41no—
293#41—41no—
294#41—121no—
295#41—121no—
296#41—121no—
297#41—121no—
298C2×Q8—21no—
299C2×Q8—61no—
300C2×D4—21no—
301C2×D4—42no—
302C2×D4—41no—
303C2×D4—61no—
304C2×D4—122no—
305C2×D4—121no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—41no—
313C22×C4—121no—
314C22×C4#21311yescharacteristic
315C22×C4C2×#6711yescharacteristic
316Q16—21no—
317Q16—21no—
318Q16—61no—
319Q16—61no—
320SD16—41no—
321SD16—41no—
322SD16—121no—
323SD16—121no—
324D8—21no—
325D8—21no—
326D8—61no—
327D8—61no—
328C2×C8—21no—
329C2×C8—21no—
330C2×C8—21no—
331C2×C8—41no—
332C2×C8—41no—
333C2×C8—41no—
334C2×C8—41no—
335C2×C8D3×D411yescharacteristic
336C2×C8D3×D411yescharacteristic
337C2×C8D3×D411yescharacteristic
338C2×C8#23711yescharacteristic
339C2×C8C2×#6711yescharacteristic
340C2×C8C2×#6711yescharacteristic
341C4⋊C4—22no—
342C4⋊C4—42no—
343C4⋊C4—62no—
344C4⋊C4—122no—
345C22⋊C4—42no—
346C22⋊C4—42no—
347C22⋊C4—41no—
348C22⋊C4—122no—
349C22⋊C4—122no—
350C22⋊C4—121no—
351C42—41no—
352C42—121no—
353C16—62no—
354C16—121no—
355C22×C3—22no—
356C22×C3—22no—
357C22×C3—22no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—41no—
362C22×C3—41no—
363C22×C3—82no—
364C22×C3—81no—
365C22×C3#44711yescharacteristic
366C22×C3#44711yescharacteristic
367C22×C3#46611yescharacteristic
368C2×D3—82no—
369C2×D3—81no—
370C3×C4—41no—
371C3×C4—82no—
372C3×C4—81no—
373C3×C4#39211yescharacteristic
374C3×C4#44911yescharacteristic
375C3×C4#44911yescharacteristic
376C3×C4#46911yescharacteristic
377Dic3—82no—
378Dic3—81no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—42no—
383C23—41no—
384C23—121no—
385C23C2×#21311yescharacteristic
386Q8—41no—
387Q8—41no—
388Q8—121no—
389Q8—121no—
390D4—41no—
391D4—41no—
392D4—84no—
393D4—81no—
394D4—81no—
395D4—121no—
396D4—121no—
397D4—244no—
398D4—241no—
399D4—241no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—41no—
409C2×C4—81no—
410C2×C4—81no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—241no—
414C2×C4—241no—
415C2×C4#93211yescharacteristic
416C2×C4#93711yescharacteristic
417C2×C4#93911yescharacteristic
418C2×C4#94911yescharacteristic
419C2×C4#95011yescharacteristic
420C2×C4#95311yescharacteristic
421C8—41no—
422C8#93711yescharacteristic
423C8#93811yescharacteristic
424C8#93911yescharacteristic
425C8#94011yescharacteristic
426C2×C3—21no—
427C2×C3—21no—
428C2×C3—21no—
429C2×C3—42no—
430C2×C3—161no—
431C2×C3#199911yescharacteristic
432D3—161no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—82no—
442C22—81no—
443C22—242no—
444C22—241no—
445C22#503611yescharacteristic
446C22#503811yescharacteristic
447C22#521711yescharacteristic
448C4—41no—
449C4—82no—
450C4—81no—
451C4—242no—
452C4—241no—
453C4#503711yescharacteristic
454C4#503911yescharacteristic
455C4#522311yescharacteristic
456C4#523811yescharacteristic
457C3#1330911yescharacteristic, complemented, Sylow
458C2—21no—
459C2—21no—
460C2—21no—
461C2—42no—
462C2—161no—
463C2—481no—
464C2#7168411yescharacteristic
subgroupG/H
centerC4#5039
commutator, cosocleC22×C3×C8C23
FrattiniC22×C8C22×D3
FittingC3×#3050C2
socleC2×C3#1999
hypercenter#3050D3
hypocenterC3#13309
LCSG ▷ C22×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C23×C8 ◁ #3050

Complements

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

NH ≅ G/Ntype
1. C3×#3050C2semidirect
2. #83894C2semidirect
3. #83742C2semidirect
4. #72076C2semidirect
5. #71963C2semidirect
6. #70656C2semidirect
7. #67725C2semidirect
36. #3050D3semidirect
48. C2×C3×Q16D4semidirect
50. C2×C3×D8D4semidirect
71. C2×Dic12D4semidirect
76. C2×D24D4semidirect
79. C3×#106D4semidirect
88. #817D4semidirect
95. C2×C3⋊C16D4semidirect
199. C2×Q16#67semidirect
203. C2×D8#67semidirect
237. #106#67semidirect
457. C3#13309semidirect

Maximal subgroups

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

subgroupindex
C3×#30502
#838942
#837422
#720762
#719632
#706562
#677252
#133093