Finite Group Databasebeta

← #10777934

Subgroups — #10777934

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#81933C211yesmaximal, characteristic, complemented
2#81045C212yesmaximal, complemented
3#79517C211yesmaximal, characteristic, complemented
4C3×#1695C211yesmaximal, characteristic, complemented
5#71380C211yesmaximal, characteristic, complemented
6#71138C212yesmaximal, complemented
7#70868C211yesmaximal, characteristic, complemented
8#68550C211yesmaximal, characteristic, complemented
9#68527C211yesmaximal, characteristic, complemented
10#68514C212yesmaximal, complemented
11#68482C212yesmaximal, complemented
12#19160—31nomaximal, Sylow
13#5725C2212yes—
14C4×#237C2211yescharacteristic
15C3×#500C2211yescharacteristic
16C3×#476C2212yes—
17C3×#436C2211yescharacteristic
18#5127—21no—
19Dic3×M16—21no—
20#5091C2212yes—
21#5088C2212yes—
22#5015C2212yes—
23#5012C2211yescharacteristic
24#4883—21no—
25#4883—21no—
26#4872—21no—
27#4872—21no—
28#4866—22no—
29#4852—22no—
30#4848C2212yes—
31#4833C2211yescharacteristic
32#4833C2212yes—
33#4831C2211yescharacteristic
34#4781—22no—
35#4756—22no—
36#4728—21no—
37#4718—21no—
38#4695C2211yescharacteristic
39C4×D3×C8C2211yescharacteristic
40#4679C2211yescharacteristic
41C3×#334C2211yescharacteristic
42C3×#333C2212yes—
43#4546C2212yes—
44#4485C2212yes—
45#4485C2212yes—
46#4479C2212yes—
47#4457C2211yescharacteristic
48#3169—31no—
49#3106—32no—
50#2965—31no—
51#1698—32no—
52#1697—32no—
53#1696—31no—
54#1695D311yescharacteristic, complemented
55#1688—31no—
56#1687—31no—
57#1564—31no—
58#1543—32no—
59C2×#239—22no—
60C2×#237D411yescharacteristic
61C2×C3×M16—21no—
62C3×C4⋊Q8C2312yes—
63C3×C4×D4C2311yescharacteristic
64#947—22no—
65D4×Dic3D411yescharacteristic
66C2×#213—22no—
67C4×#67—21no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C2×D24—21no—
71C2×D24—21no—
72C2×#204—22no—
73C2×D3×C8D411yescharacteristic
74C2×D3×C8D411yescharacteristic
75#895—22no—
76#895—22no—
77#893—22no—
78#893—22no—
79#892D411yescharacteristic
80#887D411yescharacteristic
81#67⋊C4—21no—
82#877—21no—
83#874C2312yes—
84C42×D3C2311yescharacteristic
85C4×Dic6C2311yescharacteristic
86C3×#107—21no—
87C3×#107—21no—
88C3×#106—22no—
89C3×C4⋊C8C2312yes—
90C3×C4⋊C8C2311yescharacteristic
91C3×#102—22no—
92C3×C22⋊C8—21no—
93C3×C4×C8C2311yescharacteristic
94#830—21no—
95#821—22no—
96(C2×D3)⋊C8—22no—
97(C2×D3)⋊C8—21no—
98#815—21no—
99#815—21no—
100Dic3⋊C8—21no—
101Dic3⋊C8—21no—
102C8×Dic3—21no—
103C8×Dic3—22no—
104C8×Dic3D411yescharacteristic
105C8×Dic3D411yescharacteristic
106#810—22no—
107#809—22no—
108#809—22no—
109#809—22no—
110#807—22no—
111C4⋊(C3⋊C8)C2311yescharacteristic
112C4⋊(C3⋊C8)C2312yes—
113C4×C3⋊C8C2311yescharacteristic
114#532—32no—
115C4×#41—31no—
116#500C2×D311yescharacteristic
117#493—31no—
118#486—32no—
119#482—62no—
120#477—32no—
121#476C2×D312yes—
122#466—61no—
123#466—61no—
124#460—32no—
125#459—32no—
126#449—62no—
127#446—31no—
128#446—31no—
129C4×SD16—62no—
130C4×D8—61no—
131C4×D8—61no—
132C4×D8—62no—
133#436—31no—
134#436C2×D311yescharacteristic
135#435—61no—
136#435—61no—
137#431—31no—
138#431—32no—
139#431—61no—
140C4×M16—61no—
141C2×C4×C8—31no—
142#335—32no—
143#334—31no—
144#334C2×D311yescharacteristic
145#333—32no—
146#333C2×D312yes—
147#331—32no—
148#331—32no—
149C2×C3×Q8—22no—
150C2×C3×D4C2×D411yescharacteristic
151C22×C3×C4—21no—
152C2×#67—21no—
153C22×Dic3—21no—
154C22×Dic3—21no—
155#239—84no—
156#237—42no—
157C2×D12—22no—
158C2×D12—22no—
159C2×D12—22no—
160C2×D12—42no—
161C2×C4×D3—42no—
162C2×C4×D3C2×D411yescharacteristic
163C2×C4×D3C2×D411yescharacteristic
164C2×C4×D3#4111yescharacteristic
165C2×Dic6C2×D411yescharacteristic
166C3×M16—42no—
167C2×C3×C8—21no—
168C2×C3×C8—22no—
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C3×C4⋊C4—22no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4C2×D411yescharacteristic
174C3×C22⋊C4—21no—
175C3×C42C2411yescharacteristic
176#217—21no—
177#213—44no—
178(C2×D3)⋊C4—21no—
179(C2×D3)⋊C4—42no—
180(C2×D3)⋊C4—42no—
181C4⋊Dic3C2×D411yescharacteristic
182Dic3⋊C4—21no—
183Dic3⋊C4—21no—
184C4×Dic3—21no—
185C4×Dic3—21no—
186C4×Dic3—21no—
187C4×Dic3C2×D411yescharacteristic
188C4×Dic3C2×D411yescharacteristic
189C4×Dic3#4111yescharacteristic
190#208—42no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—21no—
193C2×C3⋊C8C2×D411yescharacteristic
194C2×C3⋊C8C2×D411yescharacteristic
195D24—42no—
196D24—42no—
197#204—44no—
198D3×C8—22no—
199D3×C8—41no—
200C2×#41—31no—
201C2×#41—62no—
202C2×SD16—62no—
203C2×D8—62no—
204C2×D8—61no—
205C2×D8—61no—
206C2×M16—21no—
207C2×M16—61no—
208C22×C8—31no—
209C22×C8—31no—
210C4⋊Q8C22×D312yes—
211#127—32no—
212#124—62no—
213#121—62no—
214#121—62no—
215C4×Q8—31no—
216C4×Q8—31no—
217C4×D4—31no—
218C4×D4—62no—
219C4×D4—62no—
220C4×D4—61no—
221C4×D4—61no—
222C4×D4C22×D311yescharacteristic
223#117—31no—
224#117—61no—
225C2×C42—31no—
226C2×C42—61no—
227#107—21no—
228#107—21no—
229#107—62no—
230#106—22no—
231C4⋊C8—31no—
232C4⋊C8—32no—
233C4⋊C8—61no—
234C4⋊C8—61no—
235C4⋊C8C22×D311yescharacteristic
236C4⋊C8C22×D312yes—
237#103—62no—
238#102—22no—
239#102—62no—
240#102—62no—
241#102—62no—
242#102—62no—
243C22⋊C8—21no—
244C22⋊C8—61no—
245C22⋊C8—62no—
246C22⋊C8—61no—
247#97—61no—
248#97—61no—
249C4×C8—31no—
250C4×C8—31no—
251C4×C8—31no—
252C4×C8—61no—
253C4×C8—62no—
254C4×C8C22×D311yescharacteristic
255C23×C3—21no—
256C22×D3—42no—
257C22×D3#12111yescharacteristic
258C3×Q8—82no—
259C3×D4—22no—
260C2×C3×C4—21no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267#67—42no—
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3#12111yescharacteristic
273C2×Dic3#12111yescharacteristic
274C2×Dic3#12111yescharacteristic
275D12—42no—
276D12—42no—
277D12—42no—
278D12—42no—
279D12—82no—
280D12—84no—
281C4×D3—21no—
282C4×D3—21no—
283C4×D3—22no—
284C4×D3—22no—
285C4×D3—84no—
286Dic6—22no—
287C3×C8—21no—
288C3×C8—41no—
289C3×C8—42no—
290C3×C8#13512yes—
291C3⋊C8—21no—
292C3⋊C8—21no—
293C3⋊C8—41no—
294C3⋊C8—42no—
295#41—122no—
296#41—244no—
297C2×Q8—22no—
298C2×Q8—31no—
299C2×D4—61no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—122no—
304C2×D4D3×D411yescharacteristic
305C22×C4—21no—
306C22×C4—31no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—122no—
312SD16—124no—
313D8—122no—
314D8—124no—
315D8—122no—
316M16—42no—
317M16—122no—
318C2×C8—21no—
319C2×C8—22no—
320C2×C8—31no—
321C2×C8—31no—
322C2×C8—62no—
323C2×C8—61no—
324C2×C8—62no—
325C2×C8—121no—
326C2×C8D3×D411yescharacteristic
327C2×C8D3×D411yescharacteristic
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—31no—
331C4⋊C4—61no—
332C4⋊C4—61no—
333C4⋊C4D3×D411yescharacteristic
334C22⋊C4—21no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C22⋊C4—122no—
338C22⋊C4—122no—
339C42—31no—
340C42—31no—
341C42—31no—
342C42—61no—
343C42—61no—
344C42—61no—
345C42C23×D311yescharacteristic
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3C2×#12111yescharacteristic
350C2×D3—21no—
351C2×D3—21no—
352C2×D3—21no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—21no—
357C3×C4—41no—
358C3×C4—82no—
359C3×C4C2×#13511yescharacteristic
360C3×C4#57511yescharacteristic
361C3×C4#57512yes—
362Dic3—21no—
363Dic3—21no—
364Dic3—41no—
365Dic3—41no—
366Dic3#47212yes—
367C23—21no—
368C23—31no—
369C23—122no—
370Q8—62no—
371Q8—82no—
372D4—22no—
373D4—122no—
374D4—122no—
375D4—122no—
376D4—122no—
377D4—122no—
378D4—242no—
379D4—244no—
380C2×C4—21no—
381C2×C4—31no—
382C2×C4—31no—
383C2×C4—31no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—42no—
387C2×C4—62no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—62no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—244no—
396C2×C4C2×D3×D411yescharacteristic
397C2×C4C2×D3×D411yescharacteristic
398C2×C4D3×#4111yescharacteristic
399C8—21no—
400C8—41no—
401C8—42no—
402C8—61no—
403C8—61no—
404C8—121no—
405C8—122no—
406C8#91612yes—
407C2×C3—81no—
408C2×C3#303511yescharacteristic
409C2×C3#305411yescharacteristic
410C2×C3C2×#47211yescharacteristic
411D3—41no—
412D3—162no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—61no—
417C22—61no—
418C22—61no—
419C22—242no—
420C22—242no—
421C22—242no—
422C22D3×#12111yescharacteristic
423C4—21no—
424C4—32no—
425C4—41no—
426C4—61no—
427C4—61no—
428C4—82no—
429C4—121no—
430C4—121no—
431C4C2×#91611yescharacteristic
432C4#577712yes—
433C4#577711yescharacteristic
434C3#1916011yescharacteristic, complemented, Sylow
435C2—81no—
436C2—121no—
437C2—482no—
438C2#8042711yescharacteristic
439C2#8090311yescharacteristic
440C2D3×#47211yescharacteristic
subgroupG/H
centerC22D3×#121
commutator, cosocleC3×C42C24
FrattiniC42C23×D3
FittingC3×#1695C2
socleC22×C3C2×#121
hypercenter#1695D3
hypocenterC3#19160
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C42 ◁ #1695

Complements

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

NH ≅ G/Ntype
1. #81933C2semidirect
2. #81045C2semidirect
3. #79517C2semidirect
4. C3×#1695C2semidirect
5. #71380C2semidirect
6. #71138C2semidirect
7. #70868C2semidirect
8. #68550C2semidirect
9. #68527C2semidirect
10. #68514C2semidirect
11. #68482C2semidirect
54. #1695D3semidirect
434. C3#19160semidirect

Maximal subgroups

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

subgroupindex
#819332
#810452
#795172
C3×#16952
#713802
#711382
#708682
#685502
#685272
#685142
#684822
#191603