Finite Group Databasebeta

← #10876250

Subgroups — #10876250

Order 768.

954 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82269C211yesmaximal, characteristic
2#82231C211yesmaximal, characteristic, complemented
3C3×#1818C211yesmaximal, characteristic, complemented
4#69959C212yesmaximal
5#69929C212yesmaximal, complemented
6#68682C211yesmaximal, characteristic, complemented
7#68681C211yesmaximal, characteristic, complemented
8#68637C211yesmaximal, characteristic, complemented
9#68637C211yesmaximal, characteristic, complemented
10#68626C212yesmaximal, complemented
11#68625C212yesmaximal, complemented
12#27594—31nomaximal, Sylow
13C23×D12C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C2×#926C2211yescharacteristic
16C2×C3×C22⋊C8C2212yes—
17#5212C2212yes—
18C2×#821C2212yes—
19C2×#821C2212yes—
20C2×#821C2212yes—
21C2×#821C2212yes—
22C2×#818C2212yescomplemented
23C2×#817C2212yescomplemented
24#4988—22no—
25#4980—22no—
26#4955—22no—
27#4951—22no—
28#4939C2211yescharacteristic
29#4939C2211yescharacteristic
30#4939C2211yescharacteristic
31#4939C2211yescharacteristic
32#4936C2211yescharacteristic, complemented
33#4935C2211yescharacteristic, complemented
34C4×C4⋊Dic3C2212yescomplemented
35C3×#336C2212yescomplemented
36C3×#336C2211yescharacteristic, complemented
37C3×#336C2211yescharacteristic, complemented
38#4551C2212yescomplemented
39#4551C2212yescomplemented
40#2314—31no—
41#2309—31no—
42#2094—31no—
43#2093—31no—
44#2073—31no—
45#2073—31no—
46#1926—32no—
47#1925—32no—
48#1854—32no—
49#1818D311yescharacteristic, complemented
50#1786—32no—
51C24×D3—21no—
52C23×C3×C4C2311yescharacteristic
53C22×D12—21no—
54C22×D12—22no—
55C22×D12—22no—
56C22×D12—24no—
57C22×D12—42no—
58C22×D12C2311yescharacteristic
59C22×D12C2311yescharacteristic
60C22×C3×C8C2312yes—
61C22×C3×C8C2312yes—
62C3×#117—24no—
63C2×C3×C4⋊C4D412yes—
64C2×C3×C22⋊C4D412yes—
65C2×C3×C42C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67C2×#217—21no—
68C2×#217—22no—
69C2×(C2×D3)⋊C4—22no—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73#926—42no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2311yescharacteristic
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4×Dic3—21no—
80C2×C4×Dic3—22no—
81C3×C22⋊C8—24no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95—22no—
85#821—24no—
86#821—24no—
87#821—42no—
88#821—42no—
89#818—42no—
90#817—42no—
91C23×D4—31no—
92C2×#117—31no—
93C2×#117C2×D311yescharacteristic
94C2×#107—32no—
95C2×#106—32no—
96C2×#102—32no—
97C2×#102—32no—
98C2×#102—32no—
99C2×#102—32no—
100C2×C22⋊C8C2×D312yes—
101#390—31no—
102#390—31no—
103#390—31no—
104#390—31no—
105#390—62no—
106#388—62no—
107#387—62no—
108#384—31no—
109#382—31no—
110#380—32no—
111#379—62no—
112C4×C4⋊C4—32no—
113#340—32no—
114#340—32no—
115#336C2×D311yescharacteristic, complemented
116#336C2×D311yescharacteristic, complemented
117#336C2×D312yescomplemented
118C24×C3C2×D411yescharacteristic
119C23×D3—21no—
120C23×D3—21no—
121C23×D3—24no—
122C23×D3—42no—
123C23×D3—42no—
124C23×D3—82no—
125C23×D3—82no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—22no—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4114yes—
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—21no—
137C22×Dic3—21no—
138C2×D12—22no—
139C2×D12—22no—
140C2×D12—22no—
141C2×D12—22no—
142C2×D12—24no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—48no—
146C2×D12—48no—
147C2×D12—42no—
148C2×D12—42no—
149C2×D12#4112yes—
150C2×D12#4112yes—
151C2×C3×C8—22no—
152C2×C3×C8—22no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C3×C4⋊C4—24no—
158C3×C22⋊C4—24no—
159C3×C42—22no—
160C3×C42—22no—
161#217—42no—
162#217—82no—
163(C2×D3)⋊C4—82no—
164(C2×D3)⋊C4—82no—
165(C2×D3)⋊C4—82no—
166(C2×D3)⋊C4—82no—
167C4⋊Dic3—22no—
168C4⋊Dic3—22no—
169C4⋊Dic3—22no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—22no—
173C4⋊Dic3—82no—
174C4×Dic3—42no—
175C4×Dic3—82no—
176C25—61no—
177C22×D4—31no—
178C22×D4—31no—
179C22×D4—62no—
180C22×D4—62no—
181C22×D4—61no—
182C22×D4—64no—
183C22×D4—122no—
184C23×C4C22×D311yescharacteristic
185C22×C8C22×D312yes—
186C22×C8C22×D312yes—
187#117—24no—
188#117—122no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—31no—
191C2×C4⋊C4—32no—
192C2×C4⋊C4—32no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4D1212yes—
195C2×C22⋊C4—61no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—62no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4D1212yes—
202C2×C42—61no—
203C2×C42—62no—
204C2×C42C22×D311yescharacteristic
205C2×C42C22×D311yescharacteristic
206#107—122no—
207#106—122no—
208#102—64no—
209#102—64no—
210#102—122no—
211#102—122no—
212C22⋊C8—24no—
213#95—62no—
214#95—62no—
215#95—62no—
216C23×C3—21no—
217C23×C3—21no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3#12312yes—
221C23×C3C22×D411yescharacteristic
222C22×D3—22no—
223C22×D3—22no—
224C22×D3—48no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—42no—
230C22×D3—42no—
231C22×D3—42no—
232C22×D3—42no—
233C22×D3—88no—
234C22×D3—88no—
235C22×D3—84no—
236C22×D3—84no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—24no—
240C2×C3×C4—24no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4#12312yes—
248C2×C3×C4C2×#4112yes—
249C2×C3×C4C2×#4111yescharacteristic
250C2×C3×C4C2×#4111yescharacteristic
251C2×C3×C4H(5,𝔽2)12yes—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C2×Dic3—42no—
257C2×Dic3—42no—
258C2×Dic3—42no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—42no—
262D12—24no—
263D12—24no—
264D12—48no—
265D12—42no—
266D12—42no—
267C3×C8—42no—
268C3×C8—42no—
269C24—64no—
270C24—61no—
271C24—61no—
272C24—122no—
273C24—122no—
274C24—242no—
275C24—242no—
276C24C2×D1211yescharacteristic
277C2×D4—32no—
278C2×D4—32no—
279C2×D4—64no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—61no—
283C2×D4—61no—
284C2×D4—62no—
285C2×D4—62no—
286C2×D4—128no—
287C2×D4—128no—
288C2×D4—122no—
289C2×D4—122no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—22no—
293C22×C4—62no—
294C22×C4—62no—
295C22×C4—61no—
296C22×C4—61no—
297C22×C4C2×D1212yes—
298C22×C4C2×D1212yes—
299C22×C4C2×D1211yescharacteristic
300C22×C4#23714yes—
301C22×C4C23×D311yescharacteristic
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C2×C8—22no—
308C4⋊C4—24no—
309C4⋊C4—62no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—242no—
316C22⋊C4—24no—
317C22⋊C4—122no—
318C22⋊C4—242no—
319C22⋊C4—242no—
320C22⋊C4—242no—
321C22⋊C4—242no—
322C22⋊C4—242no—
323C42—22no—
324C42—22no—
325C42—122no—
326C42—242no—
327C22×C3—24no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3C2×#12311yescharacteristic
333C22×C3#52511yescharacteristic
334C22×C3#53511yescharacteristic
335C22×C3#57612yes—
336C22×C3#57712yes—
337C2×D3—44no—
338C2×D3—44no—
339C2×D3—42no—
340C2×D3—42no—
341C2×D3—816no—
342C2×D3—82no—
343C2×D3—82no—
344C2×D3—81no—
345C2×D3—81no—
346C2×D3—81no—
347C2×D3—81no—
348C3×C4—22no—
349C3×C4—42no—
350C3×C4—42no—
351C3×C4#55112yes—
352C3×C4#55112yes—
353Dic3—82no—
354Dic3—81no—
355Dic3—82no—
356Dic3—81no—
357C23—22no—
358C23—22no—
359C23—21no—
360C23—21no—
361C23—62no—
362C23—62no—
363C23—128no—
364C23—121no—
365C23—121no—
366C23—122no—
367C23—122no—
368C23—122no—
369C23—122no—
370C23—121no—
371C23—121no—
372C23—248no—
373C23—248no—
374C23—244no—
375C23—244no—
376C23#88612yes—
377C23C22×D1211yescharacteristic
378D4—64no—
379D4—64no—
380D4—128no—
381D4—122no—
382D4—122no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—24no—
386C2×C4—24no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4—122no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—122no—
403C2×C4#88612yes—
404C2×C4C2×#23712yes—
405C2×C4D3×#4111yescharacteristic
406C2×C4D3×#4111yescharacteristic
407C2×C4#100912yes—
408C8—42no—
409C8—42no—
410C2×C3—42no—
411C2×C3#248211yescharacteristic
412C2×C3#309612yes—
413C2×C3#311812yes—
414C2×C3#322011yescharacteristic
415C2×C3#322211yescharacteristic
416D3—82no—
417D3—82no—
418C22—24no—
419C22—22no—
420C22—42no—
421C22—41no—
422C22—41no—
423C22—124no—
424C22—124no—
425C22—122no—
426C22—122no—
427C22—2416no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22—241no—
432C22—242no—
433C22—242no—
434C22C2×#88611yescharacteristic
435C22#552411yescharacteristic
436C22#552511yescharacteristic
437C22#575112yes—
438C22#575212yes—
439C4—22no—
440C4—42no—
441C4—42no—
442C4—242no—
443C4—242no—
444C4—241no—
445C4—241no—
446C4#565812yes—
447C4#565812yes—
448C3#2759411yescharacteristic, complemented, Sylow
449C2—42no—
450C2—242no—
451C2—242no—
452C2#7412911yescharacteristic
453C2#7865112yes—
454C2#7938312yes—
455C2#7939011yescharacteristic
456C2#7939111yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1818C2
socleC23×C3C22×D4
prime radicalC23×D12C22
hypercenter#1818D3
hypocenterC3#27594
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ #1818

Complements

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

NH ≅ G/Ntype
2. #82231C2semidirect
3. C3×#1818C2semidirect
5. #69929C2semidirect
6. #68682C2semidirect
7. #68681C2semidirect
8. #68637C2semidirect
9. #68637C2semidirect
10. #68626C2semidirect
11. #68625C2semidirect
22. C2×#818C22semidirect
23. C2×#817C22semidirect
32. #4936C22semidirect
33. #4935C22semidirect
34. C4×C4⋊Dic3C22semidirect
35. C3×#336C22semidirect
36. C3×#336C22semidirect
37. C3×#336C22semidirect
38. #4551C22semidirect
39. #4551C22semidirect
49. #1818D3semidirect
115. #336C2×D3semidirect
116. #336C2×D3semidirect
117. #336C2×D3semidirect
448. C3#27594semidirect

Maximal subgroups

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

subgroupindex
#822692
#822312
C3×#18182
#699592
#699292
#686822
#686812
#686372
#686262
#686252
#275943