Finite Group Databasebeta

← #10853321

Subgroups — #10853321

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×#2552C211yesmaximal, characteristic, complemented
2#73638C211yesmaximal, characteristic
3#71743C212yesmaximal
4#71738C212yesmaximal, complemented
5#69630C212yesmaximal
6#69603C212yesmaximal, complemented
7#69584C211yesmaximal, characteristic
8#69574C211yesmaximal, characteristic, complemented
9#69484C211yesmaximal, characteristic, complemented
10C4×#809C211yesmaximal, characteristic
11#69466C211yesmaximal, characteristic, complemented
12#26025—31nomaximal, Sylow
13C2×C3×#125C2212yes—
14C2×#892—21no—
15C2×C4×D12C2211yescharacteristic
16C3×#398—22no—
17C3×#389—21no—
18C3×#389C2212yes—
19C3×#384—21no—
20C3×#384C2211yescharacteristic
21C3×C4×C4⋊C4C2211yescharacteristic
22C3×C4×C4⋊C4C2211yescharacteristic
23#4987—21no—
24C2×#809C2211yescharacteristic
25C2×#809C2211yescharacteristic
26C2×#809C2212yes—
27C2×#807C2212yes—
28#4939C2212yes—
29#4935C2211yescharacteristic
30C4×C4⋊Dic3C2212yes—
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4670—22no—
35#4529C2211yescharacteristic
36#4529C2212yes—
37#4529C2212yes—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4524C2212yes—
42#2552D311yescharacteristic, complemented
43#2354—31no—
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×D12C2311yescharacteristic
54C22×C4×D3—21no—
55C3×#125—24no—
56C3×#125—42no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—22no—
59C2×C3×C4⋊C4—22no—
60C2×C3×C4⋊C4—22no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C4⋊C4C2311yescharacteristic
67C2×C3×C42—21no—
68C2×C3×C42C2312yes—
69C2×C3×C42C2311yescharacteristic
70C2×(C2×D3)⋊C4—21no—
71C2×(C2×D3)⋊C4—22no—
72C2×(C2×D3)⋊C4—22no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3—21no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×C4×Dic3—22no—
78C22×C3⋊C8C2311yescharacteristic
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2312yes—
81#892—44no—
82C4×D12—22no—
83C4×D12Q814yes—
84C3×#95—21no—
85C3×#95—22no—
86C3×#95D412yes—
87C3⋊#95—22no—
88#809—22no—
89#809—22no—
90#809—42no—
91#807—24no—
92C4⋊(C3⋊C8)—22no—
93C4⋊(C3⋊C8)—42no—
94C4×C3⋊C8—22no—
95C2×#125C2×D312yes—
96C2×C4×D4—31no—
97C2×#117—61no—
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—22no—
106#390—32no—
107#389—21no—
108#389C2×D312yes—
109#388—62no—
110#384—21no—
111#384—31no—
112#384C2×D311yescharacteristic
113#380—61no—
114C4×C4⋊C4—32no—
115C4×C4⋊C4C2×D311yescharacteristic
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—
124C23×D3—21no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—22no—
135C2×D12—21no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×D12C2×Q812yes—
141C2×C4×D3—24no—
142C2×C4×D3—42no—
143C2×C4×D3—42no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—22no—
150C3×C4⋊C4—44no—
151C3×C4⋊C4—44no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—81no—
154C3×C4⋊C4—82no—
155C3×C4⋊C4#4112yes—
156C3×C4⋊C4#4112yes—
157C3×C42—22no—
158C3×C42—22no—
159C3×C42—42no—
160C3×C42C2×Q812yes—
161C3×C42#4112yes—
162(C2×D3)⋊C4—24no—
163(C2×D3)⋊C4—44no—
164(C2×D3)⋊C4—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—42no—
169C4⋊Dic3—81no—
170C4⋊Dic3C2×Q812yes—
171C4×Dic3—44no—
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×D4—31no—
182C23×C4—61no—
183C22×C8—31no—
184C22×C8—32no—
185C22×C8—31no—
186#125—24no—
187#125—42no—
188C4×D4—34no—
189C4×D4—62no—
190#117—124no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—22no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—31no—
197C2×C4⋊C4—32no—
198C2×C4⋊C4—61no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4#6711yescharacteristic
201C2×C4⋊C4#6711yescharacteristic
202C2×C4⋊C4C22×D312yes—
203C2×C4⋊C4C22×D311yescharacteristic
204C2×C4⋊C4C22×D311yescharacteristic
205C2×C22⋊C4—61no—
206C2×C22⋊C4—62no—
207C2×C22⋊C4—62no—
208C2×C42—21no—
209C2×C42—62no—
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—21no—
220#95—22no—
221#95—62no—
222#95#6712yes—
223C23×C3C22×D411yescharacteristic
224C22×D3—22no—
225C22×D3—41no—
226C22×D3—41no—
227C22×D3—41no—
228C22×D3—41no—
229C22×D3—41no—
230C22×D3—41no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—22no—
234C2×C3×C4—22no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—22no—
238C2×C3×C4—22no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—42no—
248C2×C3×C4C22⋊Q812yes—
249C2×C3×C4C22⋊Q812yes—
250C2×C3×C4#13512yes—
251C2×C3×C4AGL(1,ℤ/8)12yes—
252C2×C3×C4C22×Q811yescharacteristic
253C2×C3×C4C2×#4111yescharacteristic
254C2×C3×C4C2×#4111yescharacteristic
255C2×C3×C4C2×#4111yescharacteristic
256C2×C3×C4H(5,𝔽2)11yescharacteristic
257C2×C3×C42−1+411yescharacteristic
258C2×Dic3—22no—
259C2×Dic3—41no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264D12—22no—
265D12—22no—
266D12—41no—
267D12—41no—
268C4×D3—48no—
269C3⋊C8—41no—
270C3⋊C8—42no—
271C3⋊C8—41no—
272C24—61no—
273C2×D4—32no—
274C2×D4—61no—
275C2×D4—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—61no—
283C22×C4—64no—
284C22×C4—62no—
285C22×C4—122no—
286C22×C4—122no—
287C22×C4C2×#6712yes—
288C22×C4C2×#6712yes—
289C22×C4C2×#6711yescharacteristic
290C22×C4C2×#6711yescharacteristic
291C22×C4C23×D311yescharacteristic
292C2×C8—61no—
293C2×C8—62no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—61no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—22no—
306C4⋊C4—22no—
307C4⋊C4—32no—
308C4⋊C4—44no—
309C4⋊C4—44no—
310C4⋊C4—42no—
311C4⋊C4—61no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—81no—
315C4⋊C4—82no—
316C4⋊C4—122no—
317C4⋊C4—241no—
318C4⋊C4#23712yes—
319C4⋊C4#23912yes—
320C22⋊C4—64no—
321C22⋊C4—124no—
322C22⋊C4—242no—
323C42—22no—
324C42—22no—
325C42—42no—
326C42—124no—
327C42D3×Q812yes—
328C42#23912yes—
329C22×C3C2×C22⋊Q811yescharacteristic
330C22×C3#54611yescharacteristic
331C22×C3#54811yescharacteristic
332C22×C3C2×#13511yescharacteristic
333C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
334C22×C3#57611yescharacteristic
335C22×C3#57811yescharacteristic
336C2×D3—42no—
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—81no—
340C2×D3—81no—
341C2×D3—81no—
342C2×D3—81no—
343C3×C4—22no—
344C3×C4—42no—
345C3×C4—42no—
346C3×C4—42no—
347C3×C4—42no—
348C3×C4—82no—
349C3×C4#47814yes—
350C3×C4#53311yescharacteristic
351C3×C4#54411yescharacteristic
352C3×C4#55511yescharacteristic
353C3×C4#55611yescharacteristic
354Dic3—42no—
355Dic3—82no—
356C23—62no—
357C23—121no—
358C23—121no—
359C23—121no—
360C23—121no—
361C23—121no—
362C23—121no—
363C23C22×#6711yescharacteristic
364D4—62no—
365D4—62no—
366D4—121no—
367D4—121no—
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—22no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—62no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—128no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4#94612yes—
393C2×C4#94612yes—
394C2×C4#94912yes—
395C2×C4#95012yes—
396C2×C4C2×#23711yescharacteristic
397C2×C4#100411yescharacteristic
398C2×C4C2×D3×Q811yescharacteristic
399C2×C4C2×#23911yescharacteristic
400C2×C4C2×#23911yescharacteristic
401C2×C4#100711yescharacteristic
402C8—121no—
403C8—122no—
404C8—121no—
405C2×C3#243911yescharacteristic
406C2×C3C2×#47811yescharacteristic
407C2×C3#308311yescharacteristic
408C2×C3#330511yescharacteristic
409C2×C3#333011yescharacteristic
410C2×C3#336411yescharacteristic
411C2×C3#336611yescharacteristic
412D3—82no—
413C22—122no—
414C22—122no—
415C22—122no—
416C22—241no—
417C22—241no—
418C22—241no—
419C22—241no—
420C22#580611yescharacteristic
421C22C2×#94611yescharacteristic
422C22#581711yescharacteristic
423C22C2×#94911yescharacteristic
424C22C2×#95011yescharacteristic
425C22#583511yescharacteristic
426C22#583811yescharacteristic
427C4—22no—
428C4—42no—
429C4—42no—
430C4—42no—
431C4—42no—
432C4—82no—
433C4—122no—
434C4—242no—
435C4#506614yes—
436C4#572811yescharacteristic
437C4#572911yescharacteristic
438C4#573211yescharacteristic
439C4#573411yescharacteristic
440C3#2602511yescharacteristic, complemented, Sylow
441C2—242no—
442C2C2×#506611yescharacteristic
443C2#8325611yescharacteristic
444C2#8347511yescharacteristic
445C2#8351411yescharacteristic
446C2#8357111yescharacteristic
447C2#8363811yescharacteristic
448C2#8368911yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2552C2
socleC23×C3C22×D4
prime radicalC22×D12C23
hypercenter#2552D3
hypocenterC3#26025
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2552

Complements

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

NH ≅ G/Ntype
1. C3×#2552C2semidirect
4. #71738C2semidirect
6. #69603C2semidirect
8. #69574C2semidirect
9. #69484C2semidirect
11. #69466C2semidirect
42. #2552D3semidirect
440. C3#26025semidirect

Maximal subgroups

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

subgroupindex
C3×#25522
#736382
#717432
#717382
#696302
#696032
#695842
#695742
#694842
C4×#8092
#694662
#260253