Finite Group Databasebeta

← #10814354

Subgroups — #10814354

Order 768.

874 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2354C211yesmaximal, characteristic
2C2×C4×#214C211yesmaximal, characteristic
3C4⋊C4×Dic6C211yesmaximal, characteristic, complemented
4#71604C211yesmaximal, characteristic
5#69660C212yesmaximal
6#69645C212yesmaximal
7#69588C211yesmaximal, characteristic, complemented
8#69574C211yesmaximal, characteristic, complemented
9#69477C211yesmaximal, characteristic, complemented
10#69475C212yesmaximal, complemented
11#69459C212yesmaximal, complemented
12#22759—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×#117—21no—
15C22×#214C2211yescharacteristic
16C2×C4×Dic6C2212yes—
17C2×C4×Dic6C2211yescharacteristic
18C3×#390C2212yes—
19C3×#388—22no—
20C3×#384C2211yescharacteristic
21C3×#380—21no—
22C3×C4×C4⋊C4C2212yes—
23C3×C4×C4⋊C4C2211yescharacteristic
24C2×#832C2212yes—
25C2×#832C2211yescharacteristic
26C4×#214C418yescomplemented
27#4975—21no—
28Dic3×C4⋊C4—21no—
29C2×#810C2212yes—
30C2×#810C2211yescharacteristic
31C2×#808C2211yescharacteristic
32#4935C2211yescharacteristic
33#4928C2212yes—
34C2×C4⋊(C3⋊C8)C2211yescharacteristic
35C2×#803C2212yes—
36C2×C4×C3⋊C8C2211yescharacteristic
37#4647—22no—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43C2×C4×SD16—31no—
44#2354D311yescharacteristic
45Q8×C4⋊C4—31no—
46#1936—32no—
47#1919—32no—
48#1880—32no—
49#1834—31no—
50#1803—31no—
51#1802—31no—
52#1770—31no—
53#1763—32no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×Dic6C2311yescharacteristic
57C3×C4×D4—22no—
58C3×C4×D4C2×C414yes—
59C3×#117—44no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4C2312yes—
65C2×C3×C4⋊C4C2311yescharacteristic
66C2×C3×C22⋊C4—21no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—22no—
69C2×C3×C42—22no—
70C2×C3×C42C2312yes—
71C2×C3×C42C2311yescharacteristic
72C2×#214—22no—
73C2×#214—22no—
74C2×#214C2×C414yes—
75C2×C4⋊Dic3—21no—
76C2×C4⋊Dic3—21no—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×Dic3⋊C4—21no—
80C2×Dic3⋊C4—22no—
81C2×Dic3⋊C4—22no—
82C2×C4×Dic3—21no—
83C2×C4×Dic3—22no—
84C22×C3⋊C8C2312yes—
85C22×C3⋊C8C2311yescharacteristic
86C22×C3⋊C8C2311yescharacteristic
87C4×Dic6—22no—
88C4×Dic6—24no—
89C4×Dic6—42no—
90C4×Dic6C2×C414yes—
91C3×#95—22no—
92C3×#95D412yes—
93#832—42no—
94#832C2×C414yes—
95C3⋊#95—22no—
96#810—24no—
97#810C2×C414yes—
98#808C2×C414yes—
99C4⋊(C3⋊C8)—22no—
100#803—42no—
101C4×C3⋊C8C2×C414yes—
102C22×SD16—31no—
103C2×C4×Q8—31no—
104C2×C4×Q8—32no—
105C2×C4×D4C2×D311yescharacteristic
106C2×#117—21no—
107C4×SD16—38no—
108C2×#106—31no—
109C2×C4⋊C8—31no—
110C2×#103—31no—
111C2×#103—32no—
112C2×#102—31no—
113C2×#102—32no—
114C2×#97—32no—
115C2×C4×C8—31no—
116#390C2×D312yes—
117#389—32no—
118#389—62no—
119#388—22no—
120#384—31no—
121#384—61no—
122#384C2×D311yescharacteristic
123#380—21no—
124C4×C4⋊C4—61no—
125C4×C4⋊C4C2×D312yes—
126C4×C4⋊C4C2×D311yescharacteristic
127#340—32no—
128#340—32no—
129#340—32no—
130#340—32no—
131#336—32no—
132C24×C3—21no—
133C2×C3×D4—21no—
134C2×C3×D4—21no—
135C2×C3×D4—21no—
136C2×C3×D4—21no—
137C2×C3×D4—21no—
138C2×C3×D4C22×C412yes—
139C22×C3×C4—21no—
140C22×C3×C4—24no—
141C22×C3×C4—22no—
142C22×C3×C4—42no—
143C22×C3×C4—42no—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—22no—
152C2×Dic6—21no—
153C2×Dic6—21no—
154C2×Dic6—21no—
155C2×Dic6C22×C412yes—
156C2×Dic6#4112yes—
157C2×Dic6#4112yes—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—42no—
164C3×C4⋊C4—81no—
165C3×C4⋊C4C22×C412yes—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—82no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42—44no—
172C3×C42C22×C412yes—
173C3×C42#4112yes—
174#214—28no—
175C4⋊Dic3—21no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—42no—
179C4⋊Dic3—81no—
180C4⋊Dic3C22×C412yes—
181Dic3⋊C4—24no—
182Dic3⋊C4—44no—
183Dic3⋊C4—82no—
184C4×Dic3—24no—
185C4×Dic3—44no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190C2×C3⋊C8—22no—
191C2×C3⋊C8—22no—
192C2×C3⋊C8—22no—
193C2×C3⋊C8C22×C412yes—
194C2×C3⋊C8C22×C412yes—
195C22×Q8—31no—
196C22×D4C22×D311yescharacteristic
197C23×C4—21no—
198C2×SD16—34no—
199C2×SD16—62no—
200C2×SD16—62no—
201C22×C8—31no—
202C22×C8—31no—
203C22×C8—32no—
204C4×Q8—34no—
205C4×Q8—64no—
206C4×Q8—62no—
207C4×Q8—122no—
208C4×D4—22no—
209C4×D4C4×D314yes—
210#117—44no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—31no—
214C2×C4⋊C4—32no—
215C2×C4⋊C4—61no—
216C2×C4⋊C4—62no—
217C2×C4⋊C4—61no—
218C2×C4⋊C4—61no—
219C2×C4⋊C4—62no—
220C2×C4⋊C4#6711yescharacteristic
221C2×C4⋊C4#6711yescharacteristic
222C2×C4⋊C4C22×D312yes—
223C2×C4⋊C4C22×D311yescharacteristic
224C2×C22⋊C4—21no—
225C2×C22⋊C4—22no—
226C2×C22⋊C4—22no—
227C2×C42—22no—
228C2×C42—61no—
229C2×C42—62no—
230C2×C42C22×D312yes—
231C2×C42C22×D311yescharacteristic
232#106—34no—
233C4⋊C8—62no—
234#103—34no—
235#103—64no—
236#102—34no—
237#102—122no—
238#97—122no—
239C4×C8—34no—
240#95—22no—
241#95—62no—
242#95#6712yes—
243C23×C3—22no—
244C23×C3—41no—
245C23×C3—41no—
246C23×C3—41no—
247C23×C3—41no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3C22×D411yescharacteristic
251C3×D4—22no—
252C3×D4—22no—
253C3×D4—41no—
254C3×D4—41no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—48no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4C4×D412yes—
271C2×C3×C4C4×D412yes—
272C2×C3×C4AGL(1,ℤ/8)12yes—
273C2×C3×C4#13712yes—
274C2×C3×C4C23×C411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4H(5,𝔽2)11yescharacteristic
279C2×C3×C42−1+411yescharacteristic
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—22no—
283C2×Dic3—22no—
284C2×Dic3—41no—
285C2×Dic3—41no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C2×Dic3—42no—
289Dic6—22no—
290Dic6—22no—
291Dic6—22no—
292Dic6#11714yescomplemented
293C3⋊C8—22no—
294C3⋊C8—22no—
295C3⋊C8—42no—
296C24—21no—
297C2×Q8—32no—
298C2×Q8—32no—
299C2×Q8—32no—
300C2×Q8—61no—
301C2×Q8—61no—
302C2×Q8—61no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4C2×C4×D312yes—
309C22×C4—21no—
310C22×C4—24no—
311C22×C4—22no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—62no—
317C22×C4C2×#6712yes—
318C22×C4C2×#6712yes—
319C22×C4C2×#6711yescharacteristic
320C22×C4C2×#6711yescharacteristic
321C22×C4C23×D311yescharacteristic
322SD16—68no—
323C2×C8—32no—
324C2×C8—32no—
325C2×C8—61no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—61no—
329C2×C8—62no—
330C2×C8—62no—
331C2×C8—62no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—32no—
338C4⋊C4—42no—
339C4⋊C4—64no—
340C4⋊C4—61no—
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—81no—
344C4⋊C4—124no—
345C4⋊C4—122no—
346C4⋊C4—242no—
347C4⋊C4—241no—
348C4⋊C4C2×C4×D312yes—
349C22⋊C4—24no—
350C22⋊C4—44no—
351C22⋊C4—82no—
352C42—22no—
353C42—22no—
354C42—44no—
355C42—64no—
356C42—124no—
357C42C2×C4×D312yes—
358C42#23512yes—
359C22×C3—42no—
360C22×C3—42no—
361C22×C3—42no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—81no—
365C22×C3—81no—
366C22×C3C2×C4×D411yescharacteristic
367C22×C3#54611yescharacteristic
368C22×C3#54811yescharacteristic
369C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
370C22×C3C2×#13711yescharacteristic
371C22×C3#57711yescharacteristic
372C22×C3#57711yescharacteristic
373C3×C4—22no—
374C3×C4—42no—
375C3×C4—42no—
376C3×C4—42no—
377C3×C4—82no—
378C3×C4#44014yes—
379C3×C4C2×#11711yescharacteristic
380C3×C4#52011yescharacteristic
381C3×C4#55211yescharacteristic
382C3×C4#55311yescharacteristic
383Dic3—24no—
384Dic3—42no—
385Dic3—82no—
386C23—22no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23C22×#6711yescharacteristic
394Q8—34no—
395Q8—62no—
396Q8—62no—
397Q8—62no—
398D4—22no—
399D4—22no—
400D4—41no—
401D4—41no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—22no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—48no—
414C2×C4—42no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4C4×#6712yes—
427C2×C4C4×#6712yes—
428C2×C4#93212yes—
429C2×C4#95112yes—
430C2×C4C22×C4×D311yescharacteristic
431C2×C4C2×#23511yescharacteristic
432C2×C4D3×#4111yescharacteristic
433C2×C4D3×#4111yescharacteristic
434C2×C4#100911yescharacteristic
435C2×C4#101011yescharacteristic
436C8—62no—
437C8—62no—
438C8—122no—
439C2×C3—82no—
440C2×C3#235211yescharacteristic
441C2×C3C2×#44011yescharacteristic
442C2×C3#295211yescharacteristic
443C2×C3#332911yescharacteristic
444C2×C3#333211yescharacteristic
445C2×C3#336811yescharacteristic
446C2×C3#336911yescharacteristic
447C22—42no—
448C22—42no—
449C22—42no—
450C22—81no—
451C22—81no—
452C22—81no—
453C22—81no—
454C22C2×C4×#6711yescharacteristic
455C22C2×#93211yescharacteristic
456C22C2×#95111yescharacteristic
457C22#582911yescharacteristic
458C22#583111yescharacteristic
459C22#583611yescharacteristic
460C22#583711yescharacteristic
461C4—22no—
462C4—42no—
463C4—42no—
464C4—42no—
465C4—64no—
466C4—82no—
467C4—122no—
468C4—242no—
469C4#501814yes—
470C4D3×#11711yescharacteristic
471C4#552311yescharacteristic
472C4#552911yescharacteristic
473C4#553011yescharacteristic
474C3#2275911yescharacteristic, complemented, Sylow
475C2—82no—
476C2#8227211yescharacteristic
477C2C2×#501811yescharacteristic
478C2#8255811yescharacteristic
479C2#8353311yescharacteristic
480C2#8358711yescharacteristic
481C2#8365711yescharacteristic
482C2#8367711yescharacteristic
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
cosocleC22×C3×C4C24
hypercenter#2354D3
hypocenterC3#22759
LCSG ▷ C2×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
3. C4⋊C4×Dic6C2semidirect
7. #69588C2semidirect
8. #69574C2semidirect
9. #69477C2semidirect
10. #69475C2semidirect
11. #69459C2semidirect
26. C4×#214C4semidirect
292. Dic6#117semidirect
474. C3#22759semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
C2×C4×#2142
C4⋊C4×Dic62
#716042
#696602
#696452
#695882
#695742
#694772
#694752
#694592
#227593