Finite Group Databasebeta

← #10812401

Subgroups — #10812401

Order 768.

978 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2341C211yesmaximal, characteristic
2C2×#5233C211yesmaximal, characteristic
3#73460C211yesmaximal, characteristic, complemented
4#72135C211yesmaximal, characteristic
5C4×#835C212yesmaximal, complemented
6#71629C212yesmaximal, complemented
7#69644C212yesmaximal, complemented
8#69637C212yesmaximal, complemented
9#69511C211yesmaximal, characteristic, complemented
10#69503C211yesmaximal, characteristic, complemented
11C22⋊C4×C3⋊C8C211yesmaximal, characteristic, complemented
12#22577—31nomaximal, Sylow
13C22×C3×#41C2211yescharacteristic
14C2×C3×C4×Q8C2212yes—
15C2×#926C2211yescharacteristic
16C23×C3⋊C8C2211yescharacteristic
17C3×#390—22no—
18C3×#390C2212yes—
19C3×#380—21no—
20C3×#380C2211yescharacteristic
21C3×C4×C22⋊C4—21no—
22C3×C4×C22⋊C4C2211yescharacteristic
23#5233C418yescomplemented
24C2×#835C2212yes—
25C2×#835C2211yescharacteristic
26C2×#835C2211yescharacteristic
27C2×#832C2212yes—
28C2×#832C2211yescharacteristic
29C2×#832C2211yescharacteristic
30#5212C2211yescharacteristic
31C2×#830C2211yescharacteristic
32C2×#808C2212yes—
33#4945—22no—
34C4×C4⋊Dic3C2212yes—
35#4928C2212yes—
36C2×C4×C3⋊C8C2212yes—
37#4651—21no—
38#4649—21no—
39#4529C2212yes—
40#4529C2212yes—
41#4529C2212yes—
42#4524C2212yes—
43C2×#416—31no—
44#2347—31no—
45#2341D311yescharacteristic
46#1934—32no—
47#1926—32no—
48#1865—32no—
49#1825—31no—
50#1807—31no—
51#1788—31no—
52C4×#103—32no—
53C8×C22⋊C4—31no—
54C2×C3×#41—22no—
55C2×C3×#41—24no—
56C2×C3×#41—42no—
57C2×C3×#41C2×C414yes—
58C22×C3×Q8C2311yescharacteristic
59C22×C3×D4—21no—
60C22×C3×D4C2311yescharacteristic
61C23×C3×C4—21no—
62C23×C3×C4C2311yescharacteristic
63C3×C4×Q8—24no—
64C3×C4×Q8—42no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4C2312yes—
67C2×C3×C22⋊C4—22no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C22⋊C4—22no—
70C2×C3×C22⋊C4D412yes—
71C2×C3×C42—22no—
72C2×C3×C42C2312yes—
73C2×#217—21no—
74C2×#217—21no—
75#926—22no—
76#926C2×C414yes—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2311yescharacteristic
79C2×C4⋊Dic3C2311yescharacteristic
80C2×Dic3⋊C4—22no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—22no—
83C22×C3⋊C8—42no—
84C22×C3⋊C8C2×C414yes—
85C22×C3⋊C8C2312yes—
86C22×C3⋊C8C2311yescharacteristic
87C22×C3⋊C8C2311yescharacteristic
88C3×#95—21no—
89C3×#95D411yescharacteristic
90C3×#95D411yescharacteristic
91#835—24no—
92#835C2×C414yes—
93#835C2×C414yes—
94#832—42no—
95#832C2×C414yes—
96#832C2×C414yes—
97C3⋊#95—21no—
98C3⋊#95—22no—
99C3⋊#95—22no—
100#830—22no—
101#808—42no—
102C4×C3⋊C8—42no—
103C22×#41C2×D311yescharacteristic
104C23×C8—31no—
105C2×C4×Q8C2×D312yes—
106C2×#117—31no—
107C2×#106—32no—
108#416—38no—
109C2×#103—31no—
110C2×#103—31no—
111C2×#103—32no—
112C2×#102—31no—
113C2×#102—31no—
114C2×#102—32no—
115C2×C22⋊C8—31no—
116C2×C4×C8—32no—
117#390—22no—
118#390C2×D312yes—
119#389—32no—
120#388—62no—
121#383—61no—
122#380—21no—
123#380—31no—
124#380C2×D311yescharacteristic
125C4×C4⋊C4—32no—
126C4×C4⋊C4—61no—
127C4×C22⋊C4—21no—
128C4×C22⋊C4C2×D311yescharacteristic
129#340—32no—
130#340—32no—
131#340—32no—
132#336—32no—
133C24×C3—21no—
134C24×C3C2×D411yescharacteristic
135C3×#41—28no—
136C3×#41—44no—
137C3×#41—48no—
138C2×C3×Q8—21no—
139C2×C3×Q8—22no—
140C2×C3×Q8C22×C412yes—
141C2×C3×Q8#4114yes—
142C2×C3×D4—24no—
143C2×C3×D4—22no—
144C2×C3×D4—21no—
145C2×C3×D4—22no—
146C2×C3×D4—44no—
147C2×C3×D4C22×C412yes—
148C22×C3×C4—21no—
149C22×C3×C4—22no—
150C22×C3×C4—22no—
151C22×C3×C4—24no—
152C22×C3×C4—22no—
153C22×C3×C4—44no—
154C22×C3×C4C22×C412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D412yes—
158C22×C3×C4#4112yes—
159C22×C3×C4C2411yescharacteristic
160C22×Dic3—21no—
161C22×Dic3—21no—
162C22×Dic3—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—44no—
166C3×C22⋊C4—24no—
167C3×C22⋊C4—44no—
168C3×C22⋊C4—82no—
169C3×C22⋊C4—81no—
170C3×C42—22no—
171C3×C42—22no—
172C3×C42—44no—
173#217—24no—
174#217—81no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3C22×C412yes—
180C4⋊Dic3C22×C412yes—
181Dic3⋊C4—44no—
182C4×Dic3—24no—
183C4×Dic3—82no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—22no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—22no—
189C2×C3⋊C8—44no—
190C2×C3⋊C8C22×C412yes—
191C2×C3⋊C8C22×C412yes—
192C2×#41—22no—
193C2×#41—24no—
194C2×#41—42no—
195C2×#41C4×D314yes—
196C22×Q8C22×D311yescharacteristic
197C22×D4—21no—
198C22×D4C22×D311yescharacteristic
199C23×C4—21no—
200C23×C4C22×D311yescharacteristic
201C22×C8—31no—
202C22×C8—31no—
203C22×C8—34no—
204C22×C8—32no—
205C22×C8—122no—
206C4×Q8—24no—
207C4×Q8—42no—
208#117—34no—
209#117—62no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—31no—
213C2×C4⋊C4—32no—
214C2×C4⋊C4—62no—
215C2×C4⋊C4C22×D312yes—
216C2×C22⋊C4—22no—
217C2×C22⋊C4—21no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—61no—
220C2×C22⋊C4—61no—
221C2×C22⋊C4#6712yes—
222C2×C42—22no—
223C2×C42—61no—
224C2×C42—62no—
225C2×C42C22×D312yes—
226#106—122no—
227#103—34no—
228#103—34no—
229#103—64no—
230#102—34no—
231#102—34no—
232#102—122no—
233C22⋊C8—62no—
234C4×C8—122no—
235#95—21no—
236#95—62no—
237#95—61no—
238#95—62no—
239#95#6711yescharacteristic
240#95#6711yescharacteristic
241C23×C3—22no—
242C23×C3—22no—
243C23×C3—21no—
244C23×C3—21no—
245C23×C3—22no—
246C23×C3—42no—
247C23×C3—42no—
248C23×C3—41no—
249C23×C3—41no—
250C23×C3C4×D412yes—
251C23×C3C22×D411yescharacteristic
252C3×Q8—22no—
253C3×Q8—24no—
254C3×Q8#11714yes—
255C3×D4—22no—
256C3×D4—22no—
257C3×D4—48no—
258C3×D4—42no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—24no—
262C2×C3×C4—22no—
263C2×C3×C4—22no—
264C2×C3×C4—22no—
265C2×C3×C4—22no—
266C2×C3×C4—24no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4—22no—
270C2×C3×C4—48no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4C4×D412yes—
275C2×C3×C4#13514yes—
276C2×C3×C4C23×C411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4112yes—
279C2×C3×C4H(5,𝔽2)12yes—
280C2×Dic3—22no—
281C2×Dic3—22no—
282C2×Dic3—42no—
283C2×Dic3—41no—
284C2×Dic3—41no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—42no—
288C3⋊C8—22no—
289C3⋊C8—22no—
290C3⋊C8—42no—
291C24—21no—
292C24C2×#6711yescharacteristic
293#41—28no—
294#41—44no—
295#41—48no—
296C2×Q8—21no—
297C2×Q8—22no—
298C2×Q8C2×C4×D312yes—
299C2×Q8#23714yes—
300C2×D4—24no—
301C2×D4—21no—
302C2×D4—22no—
303C2×D4—22no—
304C2×D4—44no—
305C2×D4C2×C4×D312yes—
306C22×C4—21no—
307C22×C4—22no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4—22no—
311C22×C4—44no—
312C22×C4—61no—
313C22×C4—61no—
314C22×C4—62no—
315C22×C4C2×C4×D312yes—
316C22×C4#23512yes—
317C22×C4C2×#6712yes—
318C22×C4C2×#6711yescharacteristic
319C22×C4C2×#6712yes—
320C22×C4C23×D311yescharacteristic
321C2×C8—32no—
322C2×C8—32no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—62no—
328C2×C8—124no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—32no—
332C4⋊C4—32no—
333C4⋊C4—44no—
334C4⋊C4—61no—
335C4⋊C4—61no—
336C4⋊C4—62no—
337C4⋊C4—62no—
338C4⋊C4—124no—
339C22⋊C4—24no—
340C22⋊C4—44no—
341C22⋊C4—64no—
342C22⋊C4—82no—
343C22⋊C4—81no—
344C22⋊C4—241no—
345C42—22no—
346C42—22no—
347C42—44no—
348C42—64no—
349C42—242no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3—42no—
356C22×C3—42no—
357C22×C3—42no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3C2×C4×D411yescharacteristic
361C22×C3#54612yes—
362C22×C3C2×#13512yes—
363C22×C3#57711yescharacteristic
364C22×C3#57711yescharacteristic
365C3×C4—22no—
366C3×C4—24no—
367C3×C4—24no—
368C3×C4—42no—
369C3×C4—82no—
370C3×C4C2×#11711yescharacteristic
371C3×C4#51811yescharacteristic
372C3×C4#55112yes—
373Dic3—42no—
374Dic3—42no—
375Dic3—82no—
376C23—22no—
377C23—22no—
378C23—21no—
379C23—21no—
380C23—22no—
381C23—42no—
382C23—42no—
383C23—41no—
384C23—41no—
385C23C4×#6712yes—
386C23C22×#6711yescharacteristic
387Q8—22no—
388Q8—24no—
389Q8#87714yes—
390D4—22no—
391D4—22no—
392D4—48no—
393D4—42no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—22no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—24no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—48no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—122no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4C4×#6712yes—
418C2×C4#95014yes—
419C2×C4C22×C4×D311yescharacteristic
420C2×C4C2×#23511yescharacteristic
421C2×C4C2×#23712yes—
422C2×C4#100412yes—
423C8—62no—
424C8—62no—
425C8—122no—
426C2×C3—42no—
427C2×C3—82no—
428C2×C3#234211yescharacteristic
429C2×C3C4×#13511yescharacteristic
430C2×C3#295211yescharacteristic
431C2×C3#330112yes—
432C2×C3#330412yes—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—42no—
437C22—42no—
438C22—42no—
439C22—42no—
440C22—42no—
441C22—82no—
442C22—82no—
443C22C2×C4×#6711yescharacteristic
444C22#580612yes—
445C22C2×#95012yes—
446C22#583611yescharacteristic
447C22#583711yescharacteristic
448C4—24no—
449C4—22no—
450C4—24no—
451C4—42no—
452C4—82no—
453C4—122no—
454C4—122no—
455C4—242no—
456C4C2×#87711yescharacteristic
457C4#548611yescharacteristic
458C4#549512yes—
459C3#2257711yescharacteristic, complemented, Sylow
460C2—42no—
461C2—82no—
462C2#8209811yescharacteristic
463C2C4×#95011yescharacteristic
464C2#8255811yescharacteristic
465C2#8353512yes—
466C2#8357012yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2341C2
socleC23×C3C22×D4
prime radicalC22×C3×#41C22
cosocleC22×C3×C4C24
hypercenter#2341D3
hypocenterC3#22577
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ #2341

Complements

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

NH ≅ G/Ntype
3. #73460C2semidirect
5. C4×#835C2semidirect
6. #71629C2semidirect
7. #69644C2semidirect
8. #69637C2semidirect
9. #69511C2semidirect
10. #69503C2semidirect
11. C22⋊C4×C3⋊C8C2semidirect
23. #5233C4semidirect
459. C3#22577semidirect

Maximal subgroups

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

subgroupindex
C3×#23412
C2×#52332
#734602
#721352
C4×#8352
#716292
#696442
#696372
#695112
#695032
C22⋊C4×C3⋊C82
#225773