Finite Group Databasebeta

← #10824862

Subgroups — #10824862

Order 768.

816 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D8C211yesmaximal, characteristic, complemented
2#79070C211yesmaximal, characteristic, complemented
3#78914C212yesmaximal, complemented
4C3×#2045C211yesmaximal, characteristic, complemented
5#72046C211yesmaximal, characteristic
6#70532C211yesmaximal, characteristic, complemented
7#70317C212yesmaximal, complemented
8#70092C212yesmaximal, complemented
9Dic3×#100C212yesmaximal, complemented
10#68921C211yesmaximal, characteristic, complemented
11#68916C211yesmaximal, characteristic, complemented
12#23716—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×D4×Dic3C2212yes—
15C2×#932C2212yes—
16#5522C2211yescharacteristic
17C3×#424C2211yescharacteristic
18C3×#418C2212yes—
19C2×C3×#100C2212yes—
20C2×#833C2212yes—
21C2×#832C2212yes—
22Dic3×D8—22no—
23Dic3×D8—22no—
24#5133C2212yes—
25#5129C2211yescharacteristic
26C2×#822C2212yes—
27Dic3×M16C2212yes—
28Dic3×M16C2211yescharacteristic
29#5110C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4999C2211yescharacteristic
33#4995C2212yes—
34#4840—22no—
35#4833—22no—
36#4785—22no—
37#4758—22no—
38#4757—22no—
39#4756—22no—
40C3×#341C2211yescharacteristic
41#4554C2212yes—
42#4554C2212yes—
43#4552C2211yescharacteristic
44C4×AGL(1,ℤ/8)—32no—
45C2×C4×D8—31no—
46#2922—31no—
47#2089—31no—
48#2078—31no—
49#2063—32no—
50#2051—32no—
51#2045D311yescharacteristic, complemented
52#2018—31no—
53C4×#108—31no—
54C4×#100—32no—
55C22×C3×D4C2312yes—
56C23×Dic3—22no—
57C2×#235C2311yescharacteristic
58C2×C3×D8—24no—
59C2×C3×D8—22no—
60C2×C3×M16C2312yes—
61C2×C3×M16C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#117C2311yescharacteristic
64C2×#217—22no—
65D4×Dic3—22no—
66D4×Dic3—22no—
67D4×Dic3—22no—
68D4×Dic3—22no—
69C2×#214—22no—
70#932—24no—
71#932—24no—
72C2×#213—22no—
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3C2311yescharacteristic
75C2×#208C2311yescharacteristic
76C2×#208C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78#893D411yescharacteristic
79#893D411yescharacteristic
80#887D411yescharacteristic
81#887D411yescharacteristic
82#67⋊C4—21no—
83#67⋊C4—21no—
84#872—21no—
85C42×D3—21no—
86C3×C4⋊C8—22no—
87C3×#102—22no—
88C3×#102—22no—
89C3×#100—24no—
90#833—24no—
91#832—24no—
92#822—24no—
93#821—22no—
94(C2×D3)⋊C8—22no—
95(C2×D3)⋊C8—22no—
96#818—22no—
97#816—22no—
98#815—22no—
99#815—21no—
100C8×Dic3—22no—
101C8×Dic3—21no—
102C8×Dic3D412yes—
103C8×Dic3D412yes—
104#808—22no—
105#807—22no—
106C4⋊(C3⋊C8)—22no—
107C2×AGL(1,ℤ/8)—32no—
108C22×D8C2×D311yescharacteristic
109C4×#41—31no—
110C2×C4×D4—32no—
111#442—62no—
112#440—62no—
113C4×SD16—62no—
114C4×D8—62no—
115C4×D8—62no—
116C4×D8—62no—
117#433—62no—
118#431—62no—
119C2×#108—32no—
120#428—32no—
121C2×#107—31no—
122#424—31no—
123#424C2×D311yescharacteristic
124#418C2×D312yes—
125#417—32no—
126C2×#102—32no—
127C2×#100—32no—
128C2×#100C2×D312yes—
129#408—31no—
130#408—31no—
131C4×M16—31no—
132C4×M16—32no—
133C2×C4×C8—31no—
134#343—32no—
135#343—32no—
136#341—31no—
137#341C2×D311yescharacteristic
138C24×C3—22no—
139C2×C3×D4—22no—
140C2×C3×D4—22no—
141C2×C3×D4—22no—
142C2×C3×D4—22no—
143C2×C3×D4—22no—
144C2×C3×D4#4114yes—
145C22×C3×C4C2411yescharacteristic
146C2×#67—21no—
147C22×Dic3—22no—
148C22×Dic3—42no—
149C22×Dic3—42no—
150C22×Dic3—84no—
151C22×Dic3#4111yescharacteristic
152C22×Dic3#4111yescharacteristic
153#235—22no—
154#235—22no—
155C2×D12C2×D411yescharacteristic
156C2×C4×D3—21no—
157C2×C4×D3—21no—
158C2×C4×D3—21no—
159C2×Dic6C2×D411yescharacteristic
160C3×D8—28no—
161C3×M16—22no—
162C3×M16—22no—
163C3×M16—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—21no—
167C2×C3×C8C2×D412yes—
168C3×C4⋊C4C2×D411yescharacteristic
169C3×C4⋊C4C2×D411yescharacteristic
170C3×C22⋊C4—21no—
171C3×C42—21no—
172#217—42no—
173#217—42no—
174#214—28no—
175#213—28no—
176(C2×D3)⋊C4—21no—
177(C2×D3)⋊C4—21no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180Dic3⋊C4—21no—
181Dic3⋊C4—21no—
182C4×Dic3—21no—
183C4×Dic3—21no—
184C4×Dic3C2×D411yescharacteristic
185C4×Dic3C2×D411yescharacteristic
186C4×Dic3C2×D411yescharacteristic
187C4×Dic3C2×D411yescharacteristic
188#208—22no—
189#208—22no—
190#208#4114yes—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8C2×D412yes—
195C2×#41—31no—
196C22×D4C22×D312yes—
197C23×C4—62no—
198AGL(1,ℤ/8)—64no—
199AGL(1,ℤ/8)—64no—
200C2×SD16—62no—
201C2×D8—24no—
202C2×D8—22no—
203C2×D8—62no—
204C2×M16—32no—
205C2×M16—31no—
206C2×M16C22×D312yes—
207C2×M16C22×D311yescharacteristic
208C22×C8—31no—
209C22×C8C22×D311yescharacteristic
210C4×Q8—31no—
211C4×Q8—31no—
212C4×D4—31no—
213C4×D4—31no—
214C4×D4—62no—
215C4×D4—62no—
216C4×D4—62no—
217C4×D4—62no—
218C4×D4—61no—
219C4×D4—61no—
220#117—61no—
221#117C22×D311yescharacteristic
222C2×C4⋊C4—32no—
223C2×C22⋊C4—62no—
224C2×C42—31no—
225C2×C42—61no—
226#108—64no—
227#107—62no—
228#107—62no—
229#106—62no—
230C4⋊C8—22no—
231C4⋊C8—62no—
232#103—62no—
233#102—22no—
234#102—22no—
235#102—62no—
236#102—64no—
237#100—24no—
238#100—64no—
239C22⋊C8—62no—
240C22⋊C8—62no—
241#97—61no—
242#97—62no—
243C4×C8—32no—
244C4×C8—32no—
245C4×C8—61no—
246C4×C8—62no—
247C23×C3—24no—
248C23×C3—42no—
249C23×C3—42no—
250C23×C3—82no—
251C23×C3—82no—
252C23×C3C2×#4111yescharacteristic
253C22×D3—21no—
254C3×D4—24no—
255C3×D4—24no—
256C3×D4—42no—
257C3×D4—42no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C22×D411yescharacteristic
262C2×C3×C4C2×#4112yes—
263C2×C3×C4C2×#4112yes—
264#67—42no—
265C2×Dic3—22no—
266C2×Dic3—21no—
267C2×Dic3—41no—
268C2×Dic3—42no—
269C2×Dic3—84no—
270C2×Dic3—84no—
271C2×Dic3—82no—
272C2×Dic3#12111yescharacteristic
273C2×Dic3#12111yescharacteristic
274C2×Dic3#12111yescharacteristic
275C2×Dic3#12111yescharacteristic
276D12—22no—
277C4×D3—42no—
278C4×D3—41no—
279C4×D3—41no—
280C4×D3—42no—
281Dic6—22no—
282C3×C8—22no—
283C3×C8—41no—
284C3×C8—42no—
285C3⋊C8—24no—
286C3⋊C8—22no—
287C3⋊C8—41no—
288C24—22no—
289#41—62no—
290#41—62no—
291C2×Q8—31no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—31no—
298C2×D4—61no—
299C2×D4#23714yes—
300C22×C4—31no—
301C22×C4—31no—
302C22×C4—61no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4—122no—
307C22×C4—122no—
308C22×C4—244no—
309C22×C4C23×D311yescharacteristic
310SD16—68no—
311D8—28no—
312D8—68no—
313M16—22no—
314M16—22no—
315M16—22no—
316M16—34no—
317M16—62no—
318M16—62no—
319C2×C8—22no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—32no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—61no—
326C2×C8D3×D412yes—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—62no—
330C4⋊C4—62no—
331C4⋊C4D3×D411yescharacteristic
332C4⋊C4D3×D411yescharacteristic
333C22⋊C4—21no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—122no—
337C22⋊C4—122no—
338C42—21no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—31no—
343C42—61no—
344C42—61no—
345C22×C3—21no—
346C22×C3—21no—
347C22×C3—44no—
348C22×C3—84no—
349C22×C3—84no—
350C22×C3—82no—
351C22×C3—82no—
352C22×C3C2×#12111yescharacteristic
353C22×C3#52912yes—
354C2×D3—41no—
355C2×D3—41no—
356C2×D3—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4#52511yescharacteristic
360C3×C4#52512yes—
361C3×C4#53211yescharacteristic
362Dic3—21no—
363Dic3—21no—
364Dic3—21no—
365Dic3—41no—
366Dic3—82no—
367Dic3#47212yes—
368C23—24no—
369C23—42no—
370C23—42no—
371C23—61no—
372C23—82no—
373C23—82no—
374C23D3×#4111yescharacteristic
375Q8—62no—
376D4—24no—
377D4—24no—
378D4—42no—
379D4—42no—
380D4—62no—
381D4—122no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—31no—
385C2×C4—31no—
386C2×C4—31no—
387C2×C4—31no—
388C2×C4—62no—
389C2×C4—61no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—122no—
396C2×C4—244no—
397C2×C4—244no—
398C2×C4—242no—
399C2×C4C2×D3×D411yescharacteristic
400C2×C4C2×D3×D411yescharacteristic
401C2×C4C2×#23712yes—
402C2×C4D3×#4112yes—
403C8—22no—
404C8—41no—
405C8—42no—
406C8—62no—
407C8—64no—
408C8—121no—
409C2×C3—22no—
410C2×C3—84no—
411C2×C3#239211yescharacteristic
412C2×C3C2×#47211yescharacteristic
413C2×C3#306711yescharacteristic
414D3—81no—
415C22—21no—
416C22—21no—
417C22—44no—
418C22—84no—
419C22—84no—
420C22—82no—
421C22—82no—
422C22—121no—
423C22—121no—
424C22—121no—
425C22D3×#12111yescharacteristic
426C22#564912yes—
427C4—32no—
428C4—41no—
429C4—41no—
430C4—61no—
431C4—61no—
432C4—61no—
433C4—121no—
434C4—242no—
435C4#558611yescharacteristic
436C4#560612yes—
437C4#566811yescharacteristic
438C3#2371611yescharacteristic, complemented, Sylow
439C2—22no—
440C2—84no—
441C2—241no—
442C2#7689111yescharacteristic
443C2D3×#47211yescharacteristic
444C2#8093711yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2045C2
socleC22×C3C2×#121
prime radical#72046C2
hypercenter#2045D3
hypocenterC3#23716
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2045

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×D8C2semidirect
2. #79070C2semidirect
3. #78914C2semidirect
4. C3×#2045C2semidirect
6. #70532C2semidirect
7. #70317C2semidirect
8. #70092C2semidirect
9. Dic3×#100C2semidirect
10. #68921C2semidirect
11. #68916C2semidirect
51. #2045D3semidirect
438. C3#23716semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
#790702
#789142
C3×#20452
#720462
#705322
#703172
#700922
Dic3×#1002
#689212
#689162
#237163