Finite Group Databasebeta

← #10801922

Subgroups — #10801922

Order 768.

1066 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4865C211yesmaximal, characteristic
2C2×#4869C211yesmaximal, characteristic, complemented
3C2×#4689C211yesmaximal, characteristic, complemented
4#76198C212yesmaximal, complemented
5C3×#1941C211yesmaximal, characteristic, complemented
6#70578C211yesmaximal, characteristic, complemented
7#70553C211yesmaximal, characteristic, complemented
8#69776C211yesmaximal, characteristic, complemented
9#69617C212yesmaximal, complemented
10#68883C212yesmaximal, complemented
11#68780C212yesmaximal, complemented
12#21541—31nomaximal, Sylow
13C22×D24C2211yescharacteristic
14C22×ΓL(1,𝔽25)C2211yescharacteristic
15C22×D3×C8C2211yescharacteristic
16C2×#895C2212yes—
17C2×#893C2212yes—
18C2×#872C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22C3×#389C2212yes—
23C2×#821C2212yes—
24C2×#818C2211yescharacteristic
25C2×#815C2211yescharacteristic
26C2×C8×Dic3C2211yescharacteristic
27#4989—22no—
28#4988—22no—
29#4987C2212yes—
30Dic3×C4⋊C4C2212yes—
31C2×#809C2212yes—
32C2×#809C2212yes—
33#4939C2212yes—
34C2×#803C2211yescharacteristic
35#4869—24no—
36#4865C418yescomplemented
37#4689—24no—
38#4669—22no—
39#4665—22no—
40C3×#340C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×#469—31no—
44C2×#442—31no—
45C2×#405—31no—
46#2324—32no—
47#1970—31no—
48#1941D311yescharacteristic, complemented
49#1936—32no—
50#1893—32no—
51#1809—31no—
52#1776—31no—
53C4×#102—32no—
54C22×D12—22no—
55C22×D12C2312yes—
56C22×C4×D3—22no—
57C22×C4×D3C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4C2312yes—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C42C2311yescharacteristic
64C2×(C2×D3)⋊C4—22no—
65C2×(C2×D3)⋊C4—22no—
66C2×(C2×D3)⋊C4—22no—
67C2×(C2×D3)⋊C4—22no—
68C2×(C2×D3)⋊C4—22no—
69C2×(C2×D3)⋊C4D412yes—
70C2×C4⋊Dic3C2312yes—
71C2×Dic3⋊C4D412yes—
72C2×C4×Dic3—22no—
73C2×C4×Dic3C2311yescharacteristic
74C22×C3⋊C8C2311yescharacteristic
75C22×C3⋊C8C2311yescharacteristic
76C2×D24—22no—
77C2×D24—22no—
78C2×D24C2×C414yes—
79C2×ΓL(1,𝔽25)—22no—
80C2×ΓL(1,𝔽25)—22no—
81C2×ΓL(1,𝔽25)C2×C414yescomplemented
82C2×D3×C8—22no—
83C2×D3×C8—22no—
84C2×D3×C8—22no—
85#895—24no—
86#895C2×C418yescomplemented
87#893—24no—
88#893—24no—
89#872—22no—
90#872—22no—
91C3×#106—22no—
92C3×#106C2×C414yescomplemented
93C3×C4×C8—22no—
94C3×#95—22no—
95C3⋊#95—22no—
96C3⋊#95—22no—
97#821—24no—
98#818—22no—
99#815D414yescomplemented
100C8×Dic3—22no—
101#809—24no—
102#809C2×C418yescomplemented
103#803—22no—
104C22×D8—31no—
105C22×M16—31no—
106C23×C8—31no—
107C2×#121—32no—
108C2×C4×D4—32no—
109C2×#117—31no—
110#469—38no—
111#442—64no—
112C2×#107—31no—
113C2×#106C2×D311yescharacteristic
114C2×#106C2×D311yescharacteristic
115C2×#102—32no—
116C2×#102—32no—
117C2×#102—32no—
118#405—64no—
119C2×#97—31no—
120C2×#97—31no—
121C2×C4×C8—31no—
122C2×C4×C8C2×D311yescharacteristic
123#390—32no—
124#390—62no—
125#389C2×D312yes—
126#388—62no—
127#386—62no—
128#385—62no—
129#380—32no—
130C4×C4⋊C4—32no—
131#340—32no—
132#340—32no—
133#340C2×D312yes—
134C23×D3—22no—
135C23×D3C2×D411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D412yes—
139C22×C3×C4C2411yescharacteristic
140C22×Dic3—22no—
141C22×Dic3C2×D411yescharacteristic
142C22×Dic3C2×D412yes—
143C2×D12—22no—
144C2×D12—22no—
145C2×D12—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—28no—
149C2×D12—82no—
150C2×D12—82no—
151C2×D12C22×C414yes—
152C2×C4×D3—21no—
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3—21no—
156C2×C4×D3—44no—
157C2×C4×D3—82no—
158C2×C4×D3—82no—
159C2×C4×D3C22×C412yes—
160C2×C4×D3#4112yes—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8C22×C412yes—
164C2×C3×C8C2×D412yes—
165C2×C3×C8#4112yes—
166C2×C3×C8#4112yes—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—22no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—82no—
171C3×C4⋊C4C22×C414yes—
172C3×C42—21no—
173C3×C42—21no—
174(C2×D3)⋊C4—28no—
175(C2×D3)⋊C4—24no—
176(C2×D3)⋊C4—44no—
177(C2×D3)⋊C4—82no—
178(C2×D3)⋊C4—82no—
179(C2×D3)⋊C4—82no—
180C4⋊Dic3—22no—
181C4⋊Dic3—22no—
182Dic3⋊C4—24no—
183C4×Dic3—82no—
184C4×Dic3C2×D412yes—
185C4×Dic3#4112yes—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8C22×C412yes—
191C2×C3⋊C8C2×D412yes—
192D24—28no—
193ΓL(1,𝔽25)—28no—
194D3×C8—28no—
195C22×D4—32no—
196C22×D4—62no—
197C23×C4—31no—
198C23×C4—62no—
199C2×D8—34no—
200C2×D8—62no—
201C2×D8—62no—
202C2×M16—34no—
203C2×M16—62no—
204C2×M16—62no—
205C22×C8—31no—
206C22×C8—31no—
207C22×C8—62no—
208C22×C8—62no—
209C22×C8—62no—
210C22×C8C22×D311yescharacteristic
211C22×C8C22×D311yescharacteristic
212#121—38no—
213#121—64no—
214C4×D4—64no—
215C4×D4—64no—
216#117—62no—
217#117—62no—
218C2×C4⋊C4—22no—
219C2×C4⋊C4—32no—
220C2×C4⋊C4—32no—
221C2×C4⋊C4C22×D312yes—
222C2×C4⋊C4C22×D312yes—
223C2×C22⋊C4—32no—
224C2×C22⋊C4—62no—
225C2×C22⋊C4—62no—
226C2×C22⋊C4—62no—
227C2×C22⋊C4—62no—
228C2×C22⋊C4—62no—
229C2×C42—31no—
230C2×C42—62no—
231C2×C42C22×D311yescharacteristic
232#107—62no—
233#106—22no—
234#106C4×D314yescomplemented
235#102—38no—
236#102—64no—
237#102—64no—
238#97—34no—
239#97—62no—
240C4×C8—22no—
241C4×C8—62no—
242#95—22no—
243#95—62no—
244#95—62no—
245C23×C3C22×D411yescharacteristic
246C22×D3—21no—
247C22×D3—21no—
248C22×D3—22no—
249C22×D3—22no—
250C22×D3—24no—
251C22×D3—42no—
252C22×D3—42no—
253C22×D3—42no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—42no—
257C22×D3C4×D412yes—
258C2×C3×C4—24no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4C23×C411yescharacteristic
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—22no—
276C2×Dic3—22no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—42no—
280C2×Dic3C4×D412yes—
281C2×Dic3#12114yes—
282D12—24no—
283D12—24no—
284D12—42no—
285D12—42no—
286D12—88no—
287C4×D3—22no—
288C4×D3—22no—
289C4×D3—22no—
290C4×D3—22no—
291C4×D3—88no—
292C3×C8—22no—
293C3×C8C4×D414yes—
294C3⋊C8—22no—
295C3⋊C8—22no—
296C24—31no—
297C24—62no—
298C2×D4—34no—
299C2×D4—68no—
300C2×D4—62no—
301C2×D4—62no—
302C2×D4—62no—
303C2×D4—62no—
304C2×D4—62no—
305C2×D4—242no—
306C2×D4—242no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—31no—
310C22×C4—32no—
311C22×C4—32no—
312C22×C4—32no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—62no—
318C22×C4—124no—
319C22×C4—242no—
320C22×C4—242no—
321C22×C4D3×D412yes—
322C22×C4C23×D311yescharacteristic
323D8—68no—
324M16—68no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—32no—
328C2×C8—32no—
329C2×C8—68no—
330C2×C8—61no—
331C2×C8—61no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8C2×C4×D312yes—
335C2×C8D3×D412yes—
336C2×C8#23912yes—
337C2×C8#23912yes—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—64no—
342C4⋊C4—62no—
343C4⋊C4—62no—
344C4⋊C4—82no—
345C4⋊C4C2×C4×D314yes—
346C22⋊C4—64no—
347C22⋊C4—68no—
348C22⋊C4—124no—
349C22⋊C4—242no—
350C22⋊C4—242no—
351C22⋊C4—242no—
352C42—21no—
353C42—21no—
354C42—32no—
355C42—32no—
356C42—242no—
357C22×C3C2×C4×D411yescharacteristic
358C22×C3C2×#12111yescharacteristic
359C22×C3#53211yescharacteristic
360C22×C3#57612yes—
361C22×C3#57712yes—
362C2×D3—24no—
363C2×D3—22no—
364C2×D3—41no—
365C2×D3—41no—
366C2×D3—42no—
367C2×D3—44no—
368C2×D3—44no—
369C2×D3—44no—
370C2×D3—82no—
371C2×D3—82no—
372C2×D3—82no—
373C2×D3—82no—
374C3×C4—44no—
375C3×C4—42no—
376C3×C4—82no—
377C3×C4C2×C4×D411yescharacteristic
378C3×C4C4×#4111yescharacteristic
379C3×C4#52511yescharacteristic
380C3×C4#52911yescharacteristic
381Dic3—24no—
382Dic3—22no—
383Dic3—82no—
384C23—32no—
385C23—61no—
386C23—62no—
387C23—62no—
388C23—61no—
389C23—64no—
390C23—122no—
391C23—122no—
392C23—122no—
393C23—122no—
394C23—122no—
395C23—122no—
396C23C2×D3×D411yescharacteristic
397D4—64no—
398D4—64no—
399D4—122no—
400D4—122no—
401D4—248no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—24no—
405C2×C4—22no—
406C2×C4—32no—
407C2×C4—34no—
408C2×C4—42no—
409C2×C4—42no—
410C2×C4—42no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—61no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—61no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—122no—
422C2×C4—122no—
423C2×C4—122no—
424C2×C4—248no—
425C2×C4C22×C4×D311yescharacteristic
426C2×C4C2×D3×D411yescharacteristic
427C2×C4C2×#23911yescharacteristic
428C2×C4C2×#23911yescharacteristic
429C2×C4D3×#4111yescharacteristic
430C2×C4D3×#4111yescharacteristic
431C8—22no—
432C8—62no—
433C8—62no—
434C8#89314yes—
435C2×C3C4×#12111yescharacteristic
436C2×C3#294612yes—
437C2×C3#306212yes—
438C2×C3#317311yescharacteristic
439C2×C3#317611yescharacteristic
440D3—42no—
441D3—84no—
442C22—64no—
443C22—62no—
444C22—122no—
445C22—121no—
446C22—121no—
447C22—124no—
448C22—124no—
449C22—124no—
450C22—242no—
451C22—242no—
452C22—242no—
453C22—242no—
454C22C4×D3×D411yescharacteristic
455C22D3×#12111yescharacteristic
456C22#572511yescharacteristic
457C22#576212yes—
458C22#576912yes—
459C4—42no—
460C4—44no—
461C4—64no—
462C4—62no—
463C4—82no—
464C4—242no—
465C4C2×#89311yescharacteristic
466C4C4×#23911yescharacteristic
467C4#562711yescharacteristic
468C4#569111yescharacteristic
469C3#2154111yescharacteristic, complemented, Sylow
470C2—122no—
471C2—244no—
472C2#7596411yescharacteristic
473C2#7964712yes—
474C2#8066112yes—
475C2#8188711yescharacteristic
476C2#8190911yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1941C2
socleC23×C3C22×D4
prime radicalC2×#4865C2
cosocleC22×C3×C4C24
hypercenter#1941D3
hypocenterC3#21541
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1941

Complements

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

NH ≅ G/Ntype
2. C2×#4869C2semidirect
3. C2×#4689C2semidirect
4. #76198C2semidirect
5. C3×#1941C2semidirect
6. #70578C2semidirect
7. #70553C2semidirect
8. #69776C2semidirect
9. #69617C2semidirect
10. #68883C2semidirect
11. #68780C2semidirect
36. #4865C4semidirect
48. #1941D3semidirect
81. C2×ΓL(1,𝔽25)C2×C4semidirect
86. #895C2×C4semidirect
92. C3×#106C2×C4semidirect
99. #815D4semidirect
102. #809C2×C4semidirect
234. #106C4×D3semidirect
469. C3#21541semidirect

Maximal subgroups

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

subgroupindex
C2×#48652
C2×#48692
C2×#46892
#761982
C3×#19412
#705782
#705532
#697762
#696172
#688832
#687802
#215413