Finite Group Databasebeta

← #10694332

Subgroups — #10694332

Order 768.

650 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
1C3×#2850C211yesmaximal, characteristic
2C2×#4538C212yesmaximal, complemented
3C2×#4537C212yesmaximal
4#71534C211yesmaximal, characteristic, complemented
5#71530C211yesmaximal, characteristic, complemented
6#9716—31nomaximal, Sylow
7C2×C3×#124C411yescharacteristic, complemented
8C2×C3×C22⋊Q8C2211yescharacteristic
9C2×C3×#121C2211yescharacteristic
10C22×C3×C42C2211yescharacteristic
11C3×#391C411yescharacteristic, complemented
12C3×#390C411yescharacteristic, complemented
13C3×#388—21no—
14C3×#388—21no—
15C3×#386—21no—
16C3×#386—21no—
17C3×#382C411yescharacteristic, complemented
18C2×C3⋊#95—21no—
19C2×C3⋊#95—21no—
20C2×C3⋊#95C2212yes—
21C2×#830C2212yes—
22C2×C4⋊(C3⋊C8)—21no—
23#4538—24no—
24#4537—24no—
25#4524—21no—
26#2850D311yescharacteristic
27#1797—31no—
28C2×#328—32no—
29C2×#327—32no—
30#1434—31no—
31C22×C3×Q8C2×C411yescharacteristic
32C22×C3×D4—21no—
33C22×C3×D4C2×C411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—22no—
38C3×#124—42no—
39C3×C22⋊Q8—22no—
40C3×C22⋊Q8—22no—
41C3×#121—22no—
42C3×#121—22no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4C2×C411yescharacteristic
45C2×C3×C4⋊C4C2×C411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C22⋊C4—21no—
48C2×C3×C22⋊C4—21no—
49C2×C3×C22⋊C4—21no—
50C2×C3×C22⋊C4—21no—
51C2×C3×C22⋊C4—21no—
52C2×C3×C42—42no—
53C2×C3×C42C2×C411yescharacteristic
54C2×C3×C42C2×C411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42D411yescharacteristic
57C22×C3⋊C8—22no—
58C3×#95—21no—
59C3×#95—21no—
60C3×#95—21no—
61C3×#95—21no—
62C3⋊#95—22no—
63C3⋊#95—42no—
64C3⋊#95—42no—
65C3⋊#95D414yes—
66#830—22no—
67#830—22no—
68C4⋊(C3⋊C8)—42no—
69C2×#124Dic311yescharacteristic, complemented
70C2×C22⋊Q8C2×D311yescharacteristic
71C2×#121C2×D311yescharacteristic
72C22×C42C2×D311yescharacteristic
73C2×C4⋊C8—61no—
74C2×C22⋊C8—32no—
75#391Dic311yescharacteristic, complemented
76#390Dic311yescharacteristic, complemented
77#388—21no—
78#388—21no—
79#386—21no—
80#386—21no—
81#382Dic311yescharacteristic, complemented
82C2×#95—32no—
83C2×#95—61no—
84C2×#95—61no—
85#336—61no—
86#328—64no—
87#327—64no—
88C24×C3—21no—
89C24×C3C2×D411yescharacteristic
90C2×C3×Q8—22no—
91C2×C3×Q8—41no—
92C2×C3×Q8—41no—
93C2×C3×D4—22no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4—42no—
97C2×C3×D4—42no—
98C2×C3×D4—41no—
99C2×C3×D4—41no—
100C22×C3×C4—22no—
101C22×C3×C4—21no—
102C22×C3×C4—21no—
103C22×C3×C4—22no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—41no—
108C22×C3×C4—41no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C22⋊C411yescharacteristic
113C22×C3×C4C22⋊C411yescharacteristic
114C22×C3×C4C22⋊C411yescharacteristic
115C22×C3×C4C22⋊C411yescharacteristic
116C22×C3×C4C22×C411yescharacteristic
117C22×C3×C4C2×D412yes—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4112yes—
120C22×Dic3—24no—
121C22×Dic3—42no—
122C22×Dic3—42no—
123C22×Dic3—42no—
124C22×Dic3—42no—
125C22×Dic3—42no—
126C22×Dic3—42no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—41no—
130C3×C4⋊C4—41no—
131C3×C22⋊C4—42no—
132C3×C22⋊C4—42no—
133C3×C22⋊C4—42no—
134C3×C22⋊C4—42no—
135C3×C22⋊C4—42no—
136C3×C22⋊C4—41no—
137C3×C22⋊C4—41no—
138C3×C42—22no—
139C3×C42—22no—
140C3×C42—41no—
141C3×C42—41no—
142C2×C3⋊C8—42no—
143C2×C3⋊C8—42no—
144C2×C3⋊C8—42no—
145C22×Q8C2×Dic311yescharacteristic
146C22×D4—21no—
147C22×D4C2×Dic311yescharacteristic
148C23×C4—62no—
149C23×C4#6711yescharacteristic
150C23×C4#6711yescharacteristic
151C23×C4C22×D311yescharacteristic
152C22×C8—62no—
153#124—42no—
154C22⋊Q8—22no—
155C22⋊Q8—22no—
156#121—22no—
157#121—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4C2×Dic311yescharacteristic
160C2×C4⋊C4C2×Dic311yescharacteristic
161C2×C22⋊C4—21no—
162C2×C22⋊C4—21no—
163C2×C22⋊C4—21no—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—21no—
166C2×C22⋊C4—21no—
167C2×C42—42no—
168C2×C42C2×Dic311yescharacteristic
169C2×C42C2×Dic311yescharacteristic
170C2×C42#6711yescharacteristic
171C2×C42#6711yescharacteristic
172C4⋊C8—122no—
173C22⋊C8—62no—
174C22⋊C8—62no—
175#95—21no—
176#95—21no—
177#95—21no—
178#95—21no—
179#95—34no—
180#95—62no—
181#95—122no—
182#95—122no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—41no—
188C23×C3—41no—
189C23×C3—41no—
190C23×C3—41no—
191C23×C3—41no—
192C23×C3—41no—
193C23×C3—41no—
194C23×C3C2×C22⋊C411yescharacteristic
195C23×C3#12112yes—
196C23×C3#13512yes—
197C23×C3AGL(1,ℤ/8)11yescharacteristic
198C23×C3#13711yescharacteristic
199C3×Q8—82no—
200C3×D4—82no—
201C3×D4—82no—
202C3×D4—82no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—42no—
219C2×C3×C4—42no—
220C2×C3×C4—44no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—41no—
232C2×C3×C4#9912yes—
233C2×C3×C4C4×D412yes—
234C2×C3×C4#12411yescharacteristic
235C2×C3×C4#12711yescharacteristic
236C2×Dic3—44no—
237C2×Dic3—44no—
238C2×Dic3—44no—
239C2×Dic3—82no—
240C2×Dic3—82no—
241C2×Dic3—82no—
242C2×Dic3—82no—
243C3⋊C8—82no—
244C24—21no—
245C24C2×#6711yescharacteristic
246C2×Q8—22no—
247C2×Q8—41no—
248C2×Q8—41no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—42no—
253C2×D4—41no—
254C2×D4—42no—
255C2×D4—41no—
256C22×C4—21no—
257C22×C4—22no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—41no—
264C22×C4—41no—
265C22×C4—41no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—64no—
269C22×C4—122no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4—122no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4#21711yescharacteristic
276C22×C4#21711yescharacteristic
277C22×C4#21711yescharacteristic
278C22×C4#21711yescharacteristic
279C22×C4D3×D412yes—
280C22×C4#23712yes—
281C22×C4C22×Dic311yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C2×C8—122no—
284C2×C8—122no—
285C2×C8—122no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C22⋊C4—42no—
291C22⋊C4—42no—
292C22⋊C4—42no—
293C22⋊C4—42no—
294C22⋊C4—42no—
295C22⋊C4—41no—
296C22⋊C4—41no—
297C42—22no—
298C42—22no—
299C42—41no—
300C42—41no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—81no—
310C22×C3—81no—
311C22×C3—81no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—81no—
316C22×C3#39011yescharacteristic
317C22×C3C2×#9911yescharacteristic
318C22×C3#41011yescharacteristic
319C22×C3#41611yescharacteristic
320C22×C3#41711yescharacteristic
321C22×C3#42111yescharacteristic
322C22×C3#42111yescharacteristic
323C22×C3#46312yes—
324C22×C3#46412yes—
325C3×C4—21no—
326C3×C4—22no—
327C3×C4—21no—
328C3×C4—22no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—81no—
335Dic3—84no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23—41no—
346C23—41no—
347C23#93812yes—
348C23#94911yescharacteristic
349C23#95012yes—
350C23#95111yescharacteristic
351C23C2×#21711yescharacteristic
352Q8—82no—
353D4—82no—
354D4—82no—
355D4—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—44no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—124no—
388C2×C4—242no—
389C2×C4—242no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4#83412yes—
393C2×C4D4×Dic312yes—
394C2×C4#93611yescharacteristic
395C2×C4#94011yescharacteristic
396C8—242no—
397C2×C3—22no—
398C2×C3—22no—
399C2×C3—161no—
400C2×C3#151211yescharacteristic
401C2×C3#151311yescharacteristic
402C2×C3#151611yescharacteristic
403C2×C3#152111yescharacteristic
404C2×C3#196311yescharacteristic
405C2×C3#196411yescharacteristic
406C2×C3#198311yescharacteristic
407C22—22no—
408C22—21no—
409C22—21no—
410C22—22no—
411C22—21no—
412C22—21no—
413C22—22no—
414C22—22no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—81no—
422C22#505412yes—
423C22#505712yes—
424C22C2×#83411yescharacteristic
425C22#522111yescharacteristic
426C22#523211yescharacteristic
427C22#523311yescharacteristic
428C22#523611yescharacteristic
429C22#523611yescharacteristic
430C22#524011yescharacteristic
431C4—21no—
432C4—22no—
433C4—21no—
434C4—22no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—81no—
439C4—81no—
440C4—81no—
441C4—244no—
442C3#971611yescharacteristic, complemented, Sylow
443C2—22no—
444C2—22no—
445C2—161no—
446C2#7168611yescharacteristic
447C2#7170911yescharacteristic
448C2#7171011yescharacteristic
449C2#7172111yescharacteristic
450C2#7175611yescharacteristic
451C2#7175711yescharacteristic
452C2#7181611yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#2850C2
socleC23×C3C2×C22⋊C4
prime radicalC2×C3×#121C22
cosocleC23×C3×C4C23
hypercenter#2850D3
hypocenterC3#9716
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2850

Complements

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

NH ≅ G/Ntype
2. C2×#4538C2semidirect
4. #71534C2semidirect
5. #71530C2semidirect
7. C2×C3×#124C4semidirect
11. C3×#391C4semidirect
12. C3×#390C4semidirect
17. C3×#382C4semidirect
69. C2×#124Dic3semidirect
75. #391Dic3semidirect
76. #390Dic3semidirect
81. #382Dic3semidirect
442. C3#9716semidirect

Maximal subgroups

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

subgroupindex
C3×#28502
C2×#45382
C2×#45372
#715342
#715302
#97163