Finite Group Databasebeta

← #10688564

Subgroups — #10688564

Order 768.

940 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#380C211yesmaximal, characteristic
2#72127C211yesmaximal, characteristic
3#72126C211yesmaximal, characteristic
4#67624C214yesmaximal, complemented
5#8713—31nomaximal, Sylow
6C24×C3×C4C2211yescharacteristic
7C22×C3×C4⋊C4C411yescharacteristic, complemented
8C22×C3×C4⋊C4C411yescharacteristic, complemented
9C22×C3×C22⋊C4—21no—
10C22×#217—21no—
11C3×#380—21no—
12C3×#380—21no—
13C3×#380C412yescomplemented
14C3×#380C2212yes—
15C2×C3×#95—21no—
16C2×C3⋊#95—21no—
17C2×C3⋊#95C2212yes—
18C2×#830—21no—
19C2×#830—21no—
20C2×#830C2212yes—
21C2×#380D311yescharacteristic
22#1779—31no—
23C22⋊#95—31no—
24#1294—34no—
25C25×C3D411yescharacteristic
26C23×C3×C4—24no—
27C23×C3×C4—21no—
28C23×C3×C4—21no—
29C23×C3×C4—42no—
30C23×C3×C4—42no—
31C23×C3×C4C2×C411yescharacteristic
32C23×C3×C4C2×C411yescharacteristic
33C23×C3×C4D411yescharacteristic
34C23×C3×C4D411yescharacteristic
35C23×C3×C4D411yescharacteristic
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—22no—
38C2×C3×C4⋊C4—21no—
39C2×C3×C4⋊C4—21no—
40C2×C3×C4⋊C4—42no—
41C2×C3×C4⋊C4—42no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C4⋊C4C2×C412yes—
44C2×C3×C22⋊C4—22no—
45C2×C3×C22⋊C4—22no—
46C2×C3×C22⋊C4—44no—
47C2×#217—24no—
48C2×#217—42no—
49C2×#217—42no—
50C22×C3⋊C8—22no—
51C3×#95—22no—
52C3×#95—22no—
53C3⋊#95—24no—
54C3⋊#95—42no—
55#830—24no—
56#830—42no—
57#830—42no—
58C24×C4C2×D311yescharacteristic
59C22×C4⋊C4Dic311yescharacteristic, complemented
60C22×C4⋊C4Dic311yescharacteristic, complemented
61C22×C22⋊C4—21no—
62C22×C22⋊C4—61no—
63C2×C22⋊C8—32no—
64C2×C22⋊C8—61no—
65C2×C22⋊C8—61no—
66#380—21no—
67#380—21no—
68#380Dic312yescomplemented
69#380C2×D312yes—
70C2×#95—21no—
71C2×#95—32no—
72C2×#95—61no—
73C24×C3—22no—
74C24×C3—22no—
75C24×C3—22no—
76C24×C3—22no—
77C24×C3—41no—
78C24×C3—41no—
79C24×C3C22⋊C411yescharacteristic
80C24×C3C22⋊C411yescharacteristic
81C24×C3SD1612yes—
82C24×C3Q1612yes—
83C24×C3C2×D411yescharacteristic
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—22no—
87C22×C3×C4—22no—
88C22×C3×C4—22no—
89C22×C3×C4—22no—
90C22×C3×C4—22no—
91C22×C3×C4—22no—
92C22×C3×C4—22no—
93C22×C3×C4—22no—
94C22×C3×C4—22no—
95C22×C3×C4—22no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4—41no—
99C22×C3×C4—41no—
100C22×C3×C4—41no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4—416no—
104C22×C3×C4—42no—
105C22×C3×C4—42no—
106C22×C3×C4—42no—
107C22×C3×C4—42no—
108C22×C3×C4—42no—
109C22×C3×C4—42no—
110C22×C3×C4C22⋊C411yescharacteristic
111C22×C3×C4C22⋊C411yescharacteristic
112C22×C3×C4C22×C411yescharacteristic
113C22×C3×C4C2×D411yescharacteristic
114C22×C3×C4#4114yes—
115C22×Dic3—24no—
116C22×Dic3—42no—
117C22×Dic3—42no—
118C22×Dic3—42no—
119C22×Dic3—42no—
120C22×Dic3—42no—
121C22×Dic3—42no—
122C3×C4⋊C4—42no—
123C3×C4⋊C4—42no—
124C3×C4⋊C4—42no—
125C3×C4⋊C4—42no—
126C3×C22⋊C4—44no—
127C3×C22⋊C4—44no—
128#217—48no—
129C2×C3⋊C8—42no—
130C2×C3⋊C8—42no—
131C2×C3⋊C8—42no—
132C25#6711yescharacteristic
133C23×C4—24no—
134C23×C4—21no—
135C23×C4—21no—
136C23×C4—42no—
137C23×C4—42no—
138C23×C4—62no—
139C23×C4C2×Dic311yescharacteristic
140C23×C4C2×Dic311yescharacteristic
141C23×C4#6711yescharacteristic
142C23×C4#6711yescharacteristic
143C23×C4#6711yescharacteristic
144C23×C4C22×D311yescharacteristic
145C22×C8—62no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—42no—
149C2×C4⋊C4—42no—
150C2×C4⋊C4C2×Dic312yes—
151C2×C4⋊C4C2×Dic312yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—22no—
154C2×C22⋊C4—44no—
155C2×C22⋊C4—64no—
156C2×C22⋊C4—122no—
157C2×C22⋊C4—122no—
158C22⋊C8—64no—
159C22⋊C8—122no—
160C22⋊C8—122no—
161#95—22no—
162#95—22no—
163#95—64no—
164#95—122no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—24no—
174C23×C3—24no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—24no—
182C23×C3—24no—
183C23×C3—42no—
184C23×C3—42no—
185C23×C3—44no—
186C23×C3—44no—
187C23×C3—44no—
188C23×C3—44no—
189C23×C3#9912yes—
190C23×C3#10312yes—
191C23×C3#10312yes—
192C23×C3#10412yes—
193C23×C3C2×C22⋊C411yescharacteristic
194C23×C3#12312yes—
195C23×C3C2×SD1611yescharacteristic
196C23×C3C2×Q1611yescharacteristic
197C23×C3AGL(1,ℤ/8)12yes—
198C2×C3×C4—28no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—44no—
211C2×C3×C4—44no—
212C2×C3×C4—44no—
213C2×C3×C4—42no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—84no—
220C2×C3×C4—84no—
221C2×C3×C4#9912yes—
222C2×C3×C4#11712yes—
223C2×C3×C4#12312yes—
224C2×Dic3—44no—
225C2×Dic3—44no—
226C2×Dic3—44no—
227C2×Dic3—82no—
228C2×Dic3—82no—
229C2×Dic3—82no—
230C2×Dic3—82no—
231C3⋊C8—82no—
232C24—22no—
233C24—22no—
234C24—22no—
235C24—22no—
236C24—41no—
237C24—41no—
238C24#21512yes—
239C24#21612yes—
240C24#21711yescharacteristic
241C24#21711yescharacteristic
242C24C2×#6711yescharacteristic
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—41no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—41no—
262C22×C4—416no—
263C22×C4—42no—
264C22×C4—42no—
265C22×C4—42no—
266C22×C4—42no—
267C22×C4—42no—
268C22×C4—42no—
269C22×C4—64no—
270C22×C4—122no—
271C22×C4—122no—
272C22×C4—122no—
273C22×C4—122no—
274C22×C4—122no—
275C22×C4—122no—
276C22×C4#21711yescharacteristic
277C22×C4#21711yescharacteristic
278C22×C4#23514yes—
279C22×C4C22×Dic311yescharacteristic
280C22×C4C2×#6711yescharacteristic
281C2×C8—122no—
282C2×C8—122no—
283C2×C8—122no—
284C4⋊C4—42no—
285C4⋊C4—42no—
286C4⋊C4—42no—
287C4⋊C4—42no—
288C22⋊C4—44no—
289C22⋊C4—44no—
290C22⋊C4—128no—
291C22×C3—22no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—22no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—21no—
303C22×C3—22no—
304C22×C3—21no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—48no—
309C22×C3—48no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3#32814yes—
315C22×C3#38111yescharacteristic
316C22×C3C2×#9911yescharacteristic
317C22×C3C2×#9911yescharacteristic
318C22×C3C2×#10311yescharacteristic
319C22×C3#41811yescharacteristic
320C22×C3C2×#10411yescharacteristic
321C22×C3#42111yescharacteristic
322C22×C3#48112yes—
323C22×C3#48412yes—
324C3×C4—22no—
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—41no—
328C3×C4—82no—
329C3×C4—82no—
330Dic3—84no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—24no—
344C23—24no—
345C23—21no—
346C23—21no—
347C23—24no—
348C23—24no—
349C23—42no—
350C23—42no—
351C23—44no—
352C23—44no—
353C23—44no—
354C23—44no—
355C23#83412yes—
356C23#83512yes—
357C23#83512yes—
358C23#83712yes—
359C23#92912yes—
360C23#93212yes—
361C23C2×#21511yescharacteristic
362C23C2×#21611yescharacteristic
363C23C2×#21711yescharacteristic
364C2×C4—28no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—44no—
375C2×C4—44no—
376C2×C4—44no—
377C2×C4—44no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—84no—
386C2×C4—84no—
387C2×C4—124no—
388C2×C4—124no—
389C2×C4—124no—
390C2×C4—242no—
391C2×C4—242no—
392C2×C4—242no—
393C2×C4—242no—
394C2×C4#83412yes—
395C2×C4#92612yes—
396C2×C4#92912yes—
397C8—242no—
398C2×C3—22no—
399C2×C3—24no—
400C2×C3—22no—
401C2×C3—42no—
402C2×C3C2×#32811yescharacteristic
403C2×C3#150711yescharacteristic
404C2×C3#151011yescharacteristic
405C2×C3#152011yescharacteristic
406C2×C3#181911yescharacteristic
407C2×C3#182411yescharacteristic
408C2×C3#182611yescharacteristic
409C22—21no—
410C22—21no—
411C22—21no—
412C22—22no—
413C22—21no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—48no—
427C22—48no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22#453814yes—
433C22#496812yes—
434C22#497012yes—
435C22#521111yescharacteristic
436C22#521511yescharacteristic
437C22C2×#83411yescharacteristic
438C22C2×#83411yescharacteristic
439C22C2×#83511yescharacteristic
440C22C2×#83711yescharacteristic
441C22#523611yescharacteristic
442C4—22no—
443C4—22no—
444C4—22no—
445C4—41no—
446C4—82no—
447C4—82no—
448C4—244no—
449C3#871311yescharacteristic, complemented, Sylow
450C2—22no—
451C2—22no—
452C2—24no—
453C2—42no—
454C2#7154211yescharacteristic
455C2#7155411yescharacteristic
456C2#7159011yescharacteristic
457C2#7167311yescharacteristic
458C2#7167411yescharacteristic
459C2C2×#453811yescharacteristic
460C2#7169011yescharacteristic
subgroupG/H
centerC23C2×#217
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC2×C3×#380C2
socleC23×C3C2×C22⋊C4
prime radicalC25×C3D4
cosocleC23×C3×C4C23
hypercenterC2×#380D3
hypocenterC3#8713
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
4. #67624C2semidirect
7. C22×C3×C4⋊C4C4semidirect
8. C22×C3×C4⋊C4C4semidirect
13. C3×#380C4semidirect
59. C22×C4⋊C4Dic3semidirect
60. C22×C4⋊C4Dic3semidirect
68. #380Dic3semidirect
449. C3#8713semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3802
#721272
#721262
#676242
#87133