Finite Group Databasebeta

← #10877227

Subgroups — #10877227

Order 768.

886 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#384C211yesmaximal, characteristic
2C4×#924C211yesmaximal, characteristic
3#71559C212yesmaximal
4#71555C212yesmaximal
5#71530C211yesmaximal, characteristic
6#69590C212yesmaximal, complemented
7#69588C212yesmaximal, complemented
8#69572C212yesmaximal, complemented
9#69567C212yesmaximal, complemented
10#27681—31nomaximal, Sylow
11C22×C3×C4⋊C4—21no—
12C22×C3×C4⋊C4—21no—
13C22×C3×C4⋊C4C2212yes—
14C22×C3×C42C2211yescharacteristic
15C2×#924C2211yescharacteristic
16C2×C4×Dic6C2211yescharacteristic
17C3×#384—21no—
18C3×#384—21no—
19C3×#384C2212yes—
20C3×#384C2212yes—
21#5212C2211yescharacteristic
22#5210—21no—
23C4×#217—21no—
24C2×#830C2212yes—
25C2×#810C2214yes—
26C2×#807C2214yes—
27C4×C4⋊Dic3C2211yescharacteristic
28#4933—21no—
29C4×Dic3⋊C4—21no—
30#4929C2211yescharacteristic
31#4928C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33#4529C2214yes—
34#4529C2214yes—
35#4524C2212yes—
36C4×C22⋊Q8—31no—
37C2×#384D311yescharacteristic
38#2100—32no—
39#2095—32no—
40#1947—32no—
41#1872—32no—
42#1852—32no—
43#1803—32no—
44#1797—31no—
45C23×C3×C4—22no—
46C23×C3×C4D411yescharacteristic
47C23×C3×C4D411yescharacteristic
48C23×C3×C4C2311yescharacteristic
49C22×Dic6C2311yescharacteristic
50C2×C3×C4⋊C4—22no—
51C2×C3×C4⋊C4—22no—
52C2×C3×C4⋊C4—22no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—44no—
56C2×C3×C4⋊C4—44no—
57C2×C3×C4⋊C4—44no—
58C2×C3×C4⋊C4C2314yes—
59C2×C3×C42—24no—
60C2×C3×C42D411yescharacteristic
61C2×C3×C42D411yescharacteristic
62C2×C3×C42C2311yescharacteristic
63C2×C3×C42C2311yescharacteristic
64C2×#217—21no—
65C2×#217—21no—
66C2×C4⋊Dic3C2311yescharacteristic
67C2×C4⋊Dic3C2311yescharacteristic
68C2×C4⋊Dic3C2311yescharacteristic
69#924—22no—
70#924Q814yes—
71C2×Dic3⋊C4—21no—
72C2×Dic3⋊C4—21no—
73C2×Dic3⋊C4—21no—
74C2×C4×Dic3—21no—
75C2×C4×Dic3—21no—
76C22×C3⋊C8C2314yes—
77C4×Dic6—24no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82C3⋊#95—21no—
83#830—24no—
84#810—44no—
85#807—44no—
86C4⋊(C3⋊C8)—24no—
87C2×C22⋊Q8—31no—
88C2×C4×Q8—31no—
89C22×C4⋊C4—21no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×D312yes—
92C22×C42C2×D311yescharacteristic
93C2×#107—34no—
94C2×C4⋊C8—32no—
95C2×#103—34no—
96C2×C22⋊C8—32no—
97#391—31no—
98#389—31no—
99#387—61no—
100#385—61no—
101#384—21no—
102#384—21no—
103#384C2×D312yes—
104#384C2×D312yes—
105#380—31no—
106C4×C4⋊C4—31no—
107C4×C4⋊C4—61no—
108C4×C22⋊C4—61no—
109#340—34no—
110#340—34no—
111#336—32no—
112C24×C3C2×D411yescharacteristic
113C22×C3×C4—22no—
114C22×C3×C4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—24no—
121C22×C3×C4—44no—
122C22×C3×C4—44no—
123C22×C3×C4—44no—
124C22×C3×C4Q1614yes—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4C2×D411yescharacteristic
130C22×C3×C4C2×Q812yes—
131C22×C3×C4#4112yes—
132C22×C3×C4C2411yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—21no—
135C22×Dic3—21no—
136C22×Dic3—21no—
137C2×Dic6—22no—
138C2×Dic6—22no—
139C2×Dic6—21no—
140C2×Dic6C2×Q812yes—
141C3×C4⋊C4—24no—
142C3×C4⋊C4—24no—
143C3×C4⋊C4—44no—
144C3×C4⋊C4—44no—
145C3×C4⋊C4—44no—
146C3×C4⋊C4—44no—
147C3×C4⋊C4—44no—
148C3×C42—22no—
149C3×C42—22no—
150C3×C42—22no—
151C3×C42#4114yes—
152#217—24no—
153#217—42no—
154C4⋊Dic3—21no—
155C4⋊Dic3—22no—
156C4⋊Dic3—22no—
157C4⋊Dic3C2×Q812yes—
158Dic3⋊C4—24no—
159Dic3⋊C4—42no—
160Dic3⋊C4—42no—
161C4×Dic3—42no—
162C4×Dic3—42no—
163C2×C3⋊C8—24no—
164C2×C3⋊C8—24no—
165C2×C3⋊C8—24no—
166C22×Q8—31no—
167C23×C4—22no—
168C23×C4#6711yescharacteristic
169C23×C4#6711yescharacteristic
170C23×C4C22×D311yescharacteristic
171C22×C8—34no—
172C22⋊Q8—34no—
173C22⋊Q8—62no—
174C4×Q8—64no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—22no—
180C2×C4⋊C4—31no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—31no—
183C2×C4⋊C4—44no—
184C2×C4⋊C4—44no—
185C2×C4⋊C4—44no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4—61no—
188C2×C4⋊C4—61no—
189C2×C4⋊C4C22×D314yes—
190C2×C22⋊C4—61no—
191C2×C22⋊C4—61no—
192C2×C42—24no—
193C2×C42—61no—
194C2×C42—61no—
195C2×C42#6711yescharacteristic
196C2×C42#6711yescharacteristic
197C2×C42C22×D311yescharacteristic
198C2×C42C22×D311yescharacteristic
199#107—124no—
200C4⋊C8—64no—
201#103—124no—
202C22⋊C8—64no—
203#95—61no—
204#95—61no—
205#95—61no—
206#95—61no—
207#95—61no—
208C23×C3—21no—
209C23×C3—21no—
210C23×C3—21no—
211C23×C3—21no—
212C23×C3C22⋊Q812yes—
213C23×C3C2×Q1612yes—
214C23×C3AGL(1,ℤ/8)12yes—
215C23×C3C22×D411yescharacteristic
216C2×C3×C4—28no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—24no—
226C2×C3×C4—24no—
227C2×C3×C4—28no—
228C2×C3×C4—44no—
229C2×C3×C4—44no—
230C2×C3×C4—44no—
231C2×C3×C4—88no—
232C2×C3×C4C22⋊Q812yes—
233C2×C3×C4#12312yes—
234C2×C3×C4#12312yes—
235C2×C3×C4C2×Q1612yes—
236C2×C3×C4C2×Q1612yes—
237C2×C3×C4AGL(1,ℤ/8)12yes—
238C2×C3×C4#13712yes—
239C2×C3×C4C22×Q811yescharacteristic
240C2×C3×C4C2×#4112yes—
241C2×C3×C4C2×#4111yescharacteristic
242C2×C3×C4H(5,𝔽2)12yes—
243C2×Dic3—22no—
244C2×Dic3—22no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—41no—
252C2×Dic3—41no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255Dic6—24no—
256Dic6—42no—
257C3⋊C8—44no—
258C24C2×#6711yescharacteristic
259C2×Q8—32no—
260C2×Q8—61no—
261C2×Q8—62no—
262C2×Q8—62no—
263C22×C4—22no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—24no—
271C22×C4—44no—
272C22×C4—44no—
273C22×C4—44no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4#21614yes—
279C22×C4D3×Q812yes—
280C22×C4#23912yes—
281C22×C4C2×#6711yescharacteristic
282C22×C4C2×#6711yescharacteristic
283C22×C4C2×#6711yescharacteristic
284C22×C4C2×#6711yescharacteristic
285C22×C4C2×#6711yescharacteristic
286C22×C4C23×D311yescharacteristic
287C2×C8—64no—
288C2×C8—64no—
289C2×C8—64no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—32no—
293C4⋊C4—44no—
294C4⋊C4—44no—
295C4⋊C4—44no—
296C4⋊C4—44no—
297C4⋊C4—44no—
298C4⋊C4—64no—
299C4⋊C4—61no—
300C4⋊C4—62no—
301C4⋊C4—62no—
302C4⋊C4—122no—
303C4⋊C4—122no—
304C22⋊C4—64no—
305C22⋊C4—122no—
306C42—22no—
307C42—22no—
308C42—22no—
309C42—122no—
310C42—122no—
311C42#23514yes—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—22no—
316C22×C3—22no—
317C22×C3—22no—
318C22×C3#47714yes—
319C22×C3C2×C22⋊Q811yescharacteristic
320C22×C3C2×#12311yescharacteristic
321C22×C3#53511yescharacteristic
322C22×C3C22×Q1611yescharacteristic
323C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
324C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
325C22×C3C2×#13711yescharacteristic
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—84no—
331C3×C4#48414yes—
332C3×C4#55312yes—
333C3×C4#55412yes—
334Dic3—42no—
335Dic3—42no—
336Dic3—81no—
337Dic3—81no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23#93212yes—
343C23C2×#21612yes—
344C23#94612yes—
345C23C22×#6711yescharacteristic
346Q8—64no—
347Q8—122no—
348C2×C4—28no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—28no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—44no—
361C2×C4—44no—
362C2×C4—44no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—88no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—121no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4#92912yes—
377C2×C4#92912yes—
378C2×C4#93212yes—
379C2×C4#94212yes—
380C2×C4C2×#21612yes—
381C2×C4C2×#21612yes—
382C2×C4#94612yes—
383C2×C4C2×#23512yes—
384C2×C4#100412yes—
385C2×C4C2×D3×Q811yescharacteristic
386C2×C4C2×#23911yescharacteristic
387C8—124no—
388C2×C3—24no—
389C2×C3#248711yescharacteristic
390C2×C3C2×#47711yescharacteristic
391C2×C3#307911yescharacteristic
392C2×C3C2×#48411yescharacteristic
393C2×C3#309211yescharacteristic
394C2×C3#320911yescharacteristic
395C2×C3#321711yescharacteristic
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—22no—
402C22#509114yes—
403C22C2×#92911yescharacteristic
404C22C2×#93211yescharacteristic
405C22C2×#93211yescharacteristic
406C22#580511yescharacteristic
407C22C2×#94211yescharacteristic
408C22C22×#21611yescharacteristic
409C22C2×#94611yescharacteristic
410C4—22no—
411C4—22no—
412C4—22no—
413C4—22no—
414C4—84no—
415C4—122no—
416C4—122no—
417C4—241no—
418C4—241no—
419C4#497014yes—
420C4#551112yes—
421C4#551412yes—
422C3#2768111yescharacteristic, complemented, Sylow
423C2—24no—
424C2#8213911yescharacteristic
425C2C2×#497011yescharacteristic
426C2#8219711yescharacteristic
427C2#8316711yescharacteristic
428C2#8321211yescharacteristic
429C2C2×#509111yescharacteristic
430C2#8322211yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#384C2
socleC23×C3C22×D4
prime radicalC24×C3C2×D4
hypercenterC2×#384D3
hypocenterC3#27681
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#384

Complements

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

NH ≅ G/Ntype
6. #69590C2semidirect
7. #69588C2semidirect
8. #69572C2semidirect
9. #69567C2semidirect
422. C3#27681semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3842
C4×#9242
#715592
#715552
#715302
#695902
#695882
#695722
#695672
#276813