Finite Group Databasebeta

← #10853319

Subgroups — #10853319

Order 768.

734 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2354C211yesmaximal, characteristic
2#73716C211yesmaximal, characteristic, complemented
3C4×#832C211yesmaximal, characteristic
4#71604C211yesmaximal, characteristic
5#69653C212yesmaximal
6#69639C212yesmaximal
7#69574C211yesmaximal, characteristic, complemented
8#69567C211yesmaximal, characteristic, complemented
9#69491C212yesmaximal, complemented
10#69484C211yesmaximal, characteristic, complemented
11#69461C212yesmaximal, complemented
12#26025—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×#117—21no—
15C2×#870C2212yes—
16C3×#390C2212yes—
17C3×#388—22no—
18C3×#384C2211yescharacteristic
19C3×#380—21no—
20C3×C4×C4⋊C4C2212yes—
21C3×C4×C4⋊C4C2211yescharacteristic
22C2×#832C2211yescharacteristic
23C2×#832C2211yescharacteristic
24C2×#832C2212yes—
25#4975—21no—
26#4972—21no—
27C2×#807C2212yes—
28#4935C2211yescharacteristic
29C4×C4⋊Dic3C2211yescharacteristic
30#4928C2212yes—
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4657—22no—
35#4529C2212yes—
36#4529C2212yes—
37#4529C2212yes—
38#4529C2211yescharacteristic
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4524C2212yes—
42#2552—31no—
43#2354D311yescharacteristic
44#2067—32no—
45#2025—32no—
46#1989—32no—
47#1947—31no—
48#1939—31no—
49#1921—32no—
50#1834—31no—
51#1802—31no—
52C4×#102—31no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C3×C4×D4—22no—
56C3×C4×D4Q814yes—
57C3×#117—44no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—22no—
68C2×C3×C42C2312yes—
69C2×C3×C42C2311yescharacteristic
70C2×C4⋊Dic3—21no—
71C2×C4⋊Dic3—21no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4⋊Dic3C2311yescharacteristic
75C2×Dic3⋊C4—22no—
76C2×Dic3⋊C4—22no—
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C22×C3⋊C8C2312yes—
82#870—24no—
83#870—42no—
84C3×#95—22no—
85C3×#95D412yes—
86#832—22no—
87#832—22no—
88#832—42no—
89C3⋊#95—21no—
90C3⋊#95—22no—
91#807—24no—
92C4⋊(C3⋊C8)—22no—
93C4⋊(C3⋊C8)—42no—
94C4×C3⋊C8—22no—
95C2×#125—32no—
96C2×C4×D4C2×D311yescharacteristic
97C2×#117—21no—
98C2×#107—32no—
99C2×C4⋊C8—31no—
100C2×C4⋊C8—32no—
101C2×#102—31no—
102C2×#102—31no—
103C2×#102—32no—
104C2×C4×C8—31no—
105#398—62no—
106#390C2×D312yes—
107#389—32no—
108#389—61no—
109#388—22no—
110#384—31no—
111#384—61no—
112#384C2×D311yescharacteristic
113#380—21no—
114C4×C4⋊C4—31no—
115C4×C4⋊C4C2×D312yes—
116C4×C4⋊C4C2×D311yescharacteristic
117#340—32no—
118#340—32no—
119#340—32no—
120#340—31no—
121#340—32no—
122#340—31no—
123#336—32no—
124C24×C3—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4C2×Q812yes—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—22no—
134C22×C3×C4—42no—
135C22×C3×C4—42no—
136C22×C3×C4C2×D412yes—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—21no—
143C22×Dic3—22no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—22no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—42no—
150C3×C4⋊C4—81no—
151C3×C4⋊C4C2×Q812yes—
152C3×C22⋊C4—24no—
153C3×C22⋊C4—44no—
154C3×C22⋊C4—82no—
155C3×C42—22no—
156C3×C42—22no—
157C3×C42—44no—
158C3×C42C2×Q812yes—
159C3×C42#4112yes—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164C4⋊Dic3—42no—
165C4⋊Dic3—81no—
166C4⋊Dic3#4112yes—
167C4⋊Dic3#4112yes—
168Dic3⋊C4—44no—
169Dic3⋊C4—44no—
170Dic3⋊C4—82no—
171C4×Dic3—42no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—21no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—21no—
181C22×D4C22×D311yescharacteristic
182C23×C4—21no—
183C22×C8—31no—
184C22×C8—32no—
185C22×C8—31no—
186#125—64no—
187#125—122no—
188C4×D4—22no—
189C4×D4Dic614yes—
190#117—44no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—32no—
196C2×C4⋊C4—62no—
197C2×C4⋊C4—62no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4#6711yescharacteristic
202C2×C4⋊C4#6711yescharacteristic
203C2×C4⋊C4C22×D312yes—
204C2×C4⋊C4C22×D311yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C42—22no—
209C2×C42—61no—
210C2×C42C22×D312yes—
211C2×C42C22×D311yescharacteristic
212#107—64no—
213C4⋊C8—62no—
214C4⋊C8—122no—
215#102—62no—
216#102—62no—
217#102—122no—
218C4×C8—62no—
219#95—22no—
220#95—61no—
221#95—62no—
222#95#6712yes—
223C23×C3—22no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—41no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3C22×D411yescharacteristic
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—48no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—42no—
250C2×C3×C4C22⋊Q812yes—
251C2×C3×C4C22⋊Q812yes—
252C2×C3×C4#13512yes—
253C2×C3×C4AGL(1,ℤ/8)12yes—
254C2×C3×C4C22×Q811yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4C2×#4111yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4H(5,𝔽2)11yescharacteristic
259C2×C3×C42−1+411yescharacteristic
260C2×Dic3—22no—
261C2×Dic3—22no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C3⋊C8—41no—
270C3⋊C8—42no—
271C3⋊C8—41no—
272C24—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4C2×Dic612yes—
279C22×C4—21no—
280C22×C4—24no—
281C22×C4—22no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—61no—
285C22×C4—61no—
286C22×C4—62no—
287C22×C4C2×#6712yes—
288C22×C4C2×#6712yes—
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291C22×C4C23×D311yescharacteristic
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—62no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—62no—
298C2×C8—62no—
299C2×C8—61no—
300C2×C8—61no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—32no—
307C4⋊C4—32no—
308C4⋊C4—42no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—81no—
314C4⋊C4—124no—
315C4⋊C4—124no—
316C4⋊C4—122no—
317C4⋊C4—242no—
318C4⋊C4—241no—
319C4⋊C4C2×Dic612yes—
320C22⋊C4—24no—
321C22⋊C4—44no—
322C22⋊C4—82no—
323C42—22no—
324C42—22no—
325C42—44no—
326C42—122no—
327C42C2×Dic612yes—
328C42#23512yes—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—42no—
332C22×C3—81no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3C2×C22⋊Q811yescharacteristic
337C22×C3#54611yescharacteristic
338C22×C3#54811yescharacteristic
339C22×C3C2×#13511yescharacteristic
340C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
341C22×C3#57611yescharacteristic
342C22×C3#57811yescharacteristic
343C3×C4—22no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—82no—
348C3×C4#47814yes—
349C3×C4#53311yescharacteristic
350C3×C4#54411yescharacteristic
351C3×C4#55511yescharacteristic
352C3×C4#55611yescharacteristic
353Dic3—42no—
354Dic3—42no—
355Dic3—82no—
356C23—22no—
357C23—41no—
358C23—41no—
359C23—41no—
360C23—41no—
361C23—41no—
362C23—41no—
363C23C22×#6711yescharacteristic
364D4—22no—
365D4—22no—
366D4—41no—
367D4—41no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—48no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—62no—
384C2×C4—62no—
385C2×C4—121no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—121no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4#92412yes—
393C2×C4#92412yes—
394C2×C4#93212yes—
395C2×C4#95012yes—
396C2×C4C22×Dic611yescharacteristic
397C2×C4C2×#23511yescharacteristic
398C2×C4D3×#4111yescharacteristic
399C2×C4D3×#4111yescharacteristic
400C2×C4#100911yescharacteristic
401C2×C4#101011yescharacteristic
402C8—121no—
403C8—122no—
404C8—121no—
405C2×C3—82no—
406C2×C3#243911yescharacteristic
407C2×C3C2×#47811yescharacteristic
408C2×C3#308311yescharacteristic
409C2×C3#330511yescharacteristic
410C2×C3#333011yescharacteristic
411C2×C3#336411yescharacteristic
412C2×C3#336611yescharacteristic
413C22—42no—
414C22—42no—
415C22—42no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22C2×#92411yescharacteristic
421C22C2×#93211yescharacteristic
422C22C2×#95011yescharacteristic
423C22#582911yescharacteristic
424C22#583111yescharacteristic
425C22#583511yescharacteristic
426C22#583811yescharacteristic
427C4—22no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—82no—
432C4—122no—
433C4—122no—
434C4—242no—
435C4#500914yes—
436C4#551711yescharacteristic
437C4#551911yescharacteristic
438C4#553211yescharacteristic
439C4#553311yescharacteristic
440C3#2602511yescharacteristic, complemented, Sylow
441C2—82no—
442C2#8224811yescharacteristic
443C2C2×#500911yescharacteristic
444C2#8255911yescharacteristic
445C2#8352711yescharacteristic
446C2#8358111yescharacteristic
447C2#8365511yescharacteristic
448C2#8367611yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2354D3
hypocenterC3#26025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #73716C2semidirect
7. #69574C2semidirect
8. #69567C2semidirect
9. #69491C2semidirect
10. #69484C2semidirect
11. #69461C2semidirect
440. C3#26025semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
#737162
C4×#8322
#716042
#696532
#696392
#695742
#695672
#694912
#694842
#694612
#260253