Finite Group Databasebeta

← #10764046

Subgroups — #10764046

Order 768.

990 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#86190C211yesmaximal, characteristic
2#79340C212yesmaximal, complemented
3C3×#1555C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#70820C212yesmaximal, complemented
6#68388C212yesmaximal, complemented
7#68386C212yesmaximal, complemented
8#68384C212yesmaximal, complemented
9#68381C212yesmaximal, complemented
10#17771—31nomaximal, Sylow
11C2×C3×#127C2211yescharacteristic
12#5806—22no—
13#5806—22no—
14#5805C412yes—
15#5798C412yes—
16#5719C414yes—
17#5718—22no—
18#5716C2212yes—
19#5716C2212yes—
20#5713—22no—
21C4×#235C2211yescharacteristic
22#5470C2211yescharacteristic
23C3×#432C2212yes—
24C2×C4⋊(C3⋊C8)C2212yes—
25#4837C2212yescomplemented
26#4835C2212yescomplemented
27#4833C2212yescomplemented
28D3×C4⋊C8C2212yescomplemented
29#4730—22no—
30#67⋊C8—22no—
31#4706—22no—
32#4701—22no—
33C3×#331C2212yes—
34C3×#331C2211yescharacteristic
35C3×#331C2211yescharacteristic
36#4544C2211yescharacteristic
37#4544C2212yes—
38#4544C2211yescharacteristic
39#4486C2212yescomplemented
40#4486C2212yescomplemented
41#4482C2212yescomplemented
42#4482C2212yescomplemented
43#3508—31no—
44#2990—32no—
45#1557—32no—
46#1555D311yescharacteristic, complemented
47#1553—32no—
48#1552—32no—
49#1547—32no—
50#1540—32no—
51C2×#331—31no—
52C22×C3×D4—22no—
53C22×C3×D4C2×C412yes—
54C2×#237—22no—
55C2×#237—22no—
56C2×#235C2×C411yescharacteristic
57C3×#127—22no—
58C3×#127C2×C412yes—
59C3×#127C2311yescharacteristic
60C3×#127C2311yescharacteristic
61C2×C3×C42C2311yescharacteristic
62#953—24no—
63C2×#217—22no—
64C2×#217—22no—
65#940—22no—
66#939—24no—
67#939—24no—
68#938—22no—
69#938—22no—
70#938C2×C414yes—
71#938C2×C414yes—
72#936—22no—
73#936—22no—
74#936—22no—
75#935—24no—
76#935—24no—
77D4×Dic3—24no—
78D4×Dic3—22no—
79D4×Dic3—22no—
80D4×Dic3C2×C414yes—
81D4×Dic3C2×C414yes—
82C4×#67D412yes—
83C4×#67D412yes—
84#926C2×C412yes—
85#926C2×C411yescharacteristic
86#924—22no—
87C22×C3⋊C8—22no—
88C2×ΓL(1,𝔽25)—22no—
89C2×D3×C8—22no—
90C4×D12D412yes—
91#872C2×C412yes—
92C42×D3C2312yes—
93#870C2×C412yes—
94#869C2312yes—
95C4×Dic6—22no—
96C4×Dic6D412yes—
97C3×C4⋊C8C2312yes—
98C3×C4⋊C8C2312yes—
99C3×C22⋊C8—22no—
100C3×#97—22no—
101(C2×D3)⋊C8—22no—
102(C2×D3)⋊C8—22no—
103#815—22no—
104Dic3⋊C8—22no—
105Dic3⋊C8—22no—
106C8×Dic3—22no—
107C4⋊(C3⋊C8)—22no—
108C4⋊(C3⋊C8)C2312yes—
109C4⋊(C3⋊C8)C2312yes—
110#559—62no—
111#550—62no—
112#546—62no—
113#546—62no—
114#540—34no—
115#535—32no—
116#533—31no—
117#532—32no—
118#532—32no—
119C2×#127C2×D311yescharacteristic
120#518—32no—
121C4×#41—31no—
122#446—62no—
123#435—62no—
124#435—62no—
125C8×D4—62no—
126#432—32no—
127#432C2×D312yes—
128#431—32no—
129#423—32no—
130C2×C4⋊C8—32no—
131C2×C4⋊C8—32no—
132#332—32no—
133#332—32no—
134#332—32no—
135#332—32no—
136#331—31no—
137#331—32no—
138#331—31no—
139#331C2×D312yes—
140#331C2×D311yescharacteristic
141#331C2×D311yescharacteristic
142C24×C3—22no—
143C2×C3×D4—22no—
144C2×C3×D4—24no—
145C2×C3×D4—22no—
146C2×C3×D4—22no—
147C2×C3×D4—44no—
148C2×C3×D4—82no—
149C2×C3×D4—82no—
150C2×C3×D4C22×C412yes—
151C2×C3×D4C22×C412yes—
152C22×C3×C4C22×C411yescharacteristic
153C22×C3×C4C2×D412yes—
154C2×#67—24no—
155C2×#67—24no—
156C2×#67C22⋊C412yes—
157C22×Dic3—24no—
158C22×Dic3—24no—
159#237—82no—
160#237—82no—
161#237—82no—
162#237—82no—
163#235—42no—
164C2×D12C22⋊C411yescharacteristic
165C2×C4×D3C22×C412yes—
166C2×C4×D3C2×D412yes—
167C2×C4×D3C2×D412yes—
168C2×Dic6—22no—
169C2×Dic6C22⋊C411yescharacteristic
170C2×C3×C8—22no—
171C2×C3×C8—22no—
172C3×C42C22×C411yescharacteristic
173C3×C42C2×D412yes—
174C3×C42C2411yescharacteristic
175#217—22no—
176#217—24no—
177#217—24no—
178#217—24no—
179#217—24no—
180#217C22⋊C411yescharacteristic
181#217C22⋊C411yescharacteristic
182(C2×D3)⋊C4C22⋊C412yes—
183(C2×D3)⋊C4C22⋊C412yes—
184C4⋊Dic3—22no—
185C4⋊Dic3C22⋊C412yes—
186C4⋊Dic3C22×C414yes—
187Dic3⋊C4—24no—
188Dic3⋊C4C22⋊C412yes—
189Dic3⋊C4C22⋊C412yes—
190C4×Dic3—22no—
191C4×Dic3C22×C412yes—
192C4×Dic3C2×D412yes—
193C4×Dic3C2×D412yes—
194C2×C3⋊C8—22no—
195C2×C3⋊C8—22no—
196C2×C3⋊C8—82no—
197ΓL(1,𝔽25)—82no—
198D3×C8—82no—
199C2×#41—31no—
200C2×#41—62no—
201C2×#41—62no—
202C22×D4—22no—
203C22×D4C4×D312yes—
204C2×M16—62no—
205C22×C8—62no—
206C22×C8—62no—
207C4⋊Q8—32no—
208#127—22no—
209#127—62no—
210#127C4×D312yes—
211#127C22×D311yescharacteristic
212#127C22×D311yescharacteristic
213#125—32no—
214#124—62no—
215#124—62no—
216#124—62no—
217#123—64no—
218#123—64no—
219C22⋊Q8—62no—
220#121—34no—
221#121—34no—
222#121—64no—
223#121—64no—
224#121—62no—
225#121—62no—
226#120—64no—
227C4×Q8—32no—
228C4×Q8—62no—
229C4×D4—32no—
230C4×D4—32no—
231C4×D4—34no—
232C4×D4—34no—
233C4×D4—32no—
234C4×D4—62no—
235C4×D4—62no—
236C4×D4—64no—
237#117—31no—
238#117—32no—
239#117—32no—
240C2×C22⋊C4—62no—
241C2×C22⋊C4—62no—
242C2×C42—32no—
243C2×C42C22×D311yescharacteristic
244C4⋊C8—32no—
245C4⋊C8—32no—
246C4⋊C8—62no—
247C4⋊C8—62no—
248C4⋊C8—62no—
249C4⋊C8C22×D312yes—
250C4⋊C8C22×D312yes—
251C22⋊C8—22no—
252C22⋊C8—62no—
253C22⋊C8—62no—
254#97—22no—
255#97—62no—
256C4×C8—62no—
257C23×C3—22no—
258C23×C3—22no—
259C23×C3—44no—
260C23×C3—84no—
261C23×C3C2×C22⋊C411yescharacteristic
262C22×D3C2×C22⋊C412yes—
263C3×D4—42no—
264C3×D4—42no—
265C3×D4—44no—
266C3×D4—42no—
267C3×D4—42no—
268C3×D4—42no—
269C3×D4—42no—
270C3×D4—84no—
271C2×C3×C4—22no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4C2×C22⋊C411yescharacteristic
275C2×C3×C4C2×C22⋊C412yes—
276C2×C3×C4C23×C411yescharacteristic
277C2×C3×C4C22×D412yes—
278#67—24no—
279#67—84no—
280#67—84no—
281C2×Dic3—24no—
282C2×Dic3—84no—
283C2×Dic3—84no—
284C2×Dic3C2×C22⋊C412yes—
285C2×Dic3C2×C22⋊C412yes—
286C2×Dic3C2×C22⋊C412yes—
287D12—41no—
288C4×D3—22no—
289C4×D3—22no—
290C4×D3—22no—
291C4×D3—22no—
292C4×D3—42no—
293Dic6—41no—
294Dic6—42no—
295Dic6—42no—
296C3×C8—42no—
297C3×C8—42no—
298C3⋊C8—42no—
299C3⋊C8—42no—
300C24—22no—
301#41—122no—
302#41—242no—
303#41—242no—
304#41—242no—
305#41—242no—
306C2×Q8—31no—
307C2×Q8—62no—
308C2×D4—22no—
309C2×D4—24no—
310C2×D4—22no—
311C2×D4—22no—
312C2×D4—32no—
313C2×D4—31no—
314C2×D4—44no—
315C2×D4—64no—
316C2×D4—64no—
317C2×D4—82no—
318C2×D4—82no—
319C2×D4C2×C4×D312yes—
320C2×D4C2×C4×D312yes—
321C22×C4—32no—
322C22×C4—32no—
323C22×C4—32no—
324C22×C4—64no—
325C22×C4—64no—
326C22×C4C2×C4×D311yescharacteristic
327C22×C4D3×D412yes—
328M16—242no—
329C2×C8—22no—
330C2×C8—22no—
331C2×C8—62no—
332C2×C8—62no—
333C2×C8—242no—
334C2×C8—242no—
335C4⋊C4—32no—
336C4⋊C4—32no—
337C4⋊C4—32no—
338C4⋊C4—34no—
339C4⋊C4—64no—
340C4⋊C4—62no—
341C22⋊C4—31no—
342C22⋊C4—31no—
343C22⋊C4—32no—
344C22⋊C4—32no—
345C22⋊C4—64no—
346C22⋊C4—64no—
347C22⋊C4—64no—
348C22⋊C4—64no—
349C22⋊C4—62no—
350C42—32no—
351C42—32no—
352C42—32no—
353C42—62no—
354C42C2×C4×D311yescharacteristic
355C42D3×D412yes—
356C42C23×D311yescharacteristic
357C22×C3—22no—
358C22×C3—42no—
359C22×C3—42no—
360C22×C3—44no—
361C22×C3—44no—
362C22×C3—164no—
363C22×C3#41312yes—
364C22×C3C22×C22⋊C411yescharacteristic
365C2×D3—22no—
366C2×D3—22no—
367C2×D3—22no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—22no—
371C3×C4#57612yes—
372C3×C4#57712yes—
373Dic3—22no—
374Dic3—22no—
375Dic3—22no—
376Dic3—44no—
377C23—22no—
378C23—22no—
379C23—32no—
380C23—44no—
381C23—84no—
382C23D3×C22⋊C411yescharacteristic
383Q8—121no—
384Q8—122no—
385Q8—122no—
386D4—42no—
387D4—42no—
388D4—44no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—42no—
393D4—64no—
394D4—84no—
395D4—121no—
396D4—244no—
397D4—244no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—32no—
404C2×C4—62no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—64no—
409C2×C4—122no—
410C2×C4—244no—
411C2×C4—244no—
412C2×C4D3×C22⋊C411yescharacteristic
413C2×C4D3×C22⋊C412yes—
414C2×C4C22×C4×D311yescharacteristic
415C2×C4C2×D3×D412yes—
416C8—42no—
417C8—42no—
418C8—122no—
419C8—122no—
420C2×C3—22no—
421C2×C3—82no—
422C2×C3—82no—
423C2×C3C2×#41311yescharacteristic
424C2×C3#289712yes—
425D3—42no—
426C22—22no—
427C22—44no—
428C22—44no—
429C22—42no—
430C22—42no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—164no—
435C22#474512yes—
436C22C2×D3×C22⋊C411yescharacteristic
437C4—22no—
438C4—21no—
439C4—21no—
440C4—62no—
441C4—62no—
442C4—62no—
443C4—124no—
444C4#577412yes—
445C4#577512yes—
446C3#1777111yescharacteristic, complemented, Sylow
447C2—22no—
448C2—82no—
449C2—82no—
450C2—122no—
451C2C2×#474511yescharacteristic
452C2#7892512yes—
subgroupG/H
centerC22C2×D3×C22⋊C4
commutatorC2×C3×C4C23×C4
FrattiniC42C23×D3
FittingC3×#1555C2
socleC22×C3C22×C22⋊C4
prime radical#86190C2
cosocleC3×C42C24
hypercenter#1555D3
hypocenterC3#17771
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1555

Complements

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

NH ≅ G/Ntype
2. #79340C2semidirect
3. C3×#1555C2semidirect
4. C2×#4544C2semidirect
5. #70820C2semidirect
6. #68388C2semidirect
7. #68386C2semidirect
8. #68384C2semidirect
9. #68381C2semidirect
25. #4837C22semidirect
26. #4835C22semidirect
27. #4833C22semidirect
28. D3×C4⋊C8C22semidirect
39. #4486C22semidirect
40. #4486C22semidirect
41. #4482C22semidirect
42. #4482C22semidirect
46. #1555D3semidirect
446. C3#17771semidirect

Maximal subgroups

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

subgroupindex
#861902
#793402
C3×#15552
C2×#45442
#708202
#683882
#683862
#683842
#683812
#177713