Finite Group Databasebeta

← #10798325

Subgroups — #10798325

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×D8C211yesmaximal, characteristic
2#82325C211yesmaximal, characteristic, complemented
3#82317C211yesmaximal, characteristic, complemented
4#72037C212yesmaximal
5#71609C211yesmaximal, characteristic, complemented
6#71609C211yesmaximal, characteristic, complemented
7#70610C212yesmaximal, complemented
8#70168C212yesmaximal, complemented
9#70155C212yesmaximal, complemented
10C4×#818C211yesmaximal, characteristic, complemented
11#69751C211yesmaximal, characteristic, complemented
12#21129—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×C4×D4C2211yescharacteristic
15C2×C3×C4×D4C2211yescharacteristic
16C2×D4×Dic3C2212yes—
17C3×C4×D8C418yescomplemented
18C2×C3×#107C2211yescharacteristic
19C2×C3×#102C2211yescharacteristic
20C2×C3×#102C2211yescharacteristic
21C2×C3×C4×C8C2211yescharacteristic
22#5221C2212yes—
23C2×#832C2212yes—
24C2×#832C2212yes—
25#5212—21no—
26#5211—21no—
27C4×#217—21no—
28C2×C3⋊#95—21no—
29C2×#818C2212yes—
30#4984C2212yes—
31Dic3×C4⋊C4C2212yes—
32#4951—22no—
33Dic3×C22⋊C4—22no—
34C4×C4⋊Dic3C2211yescharacteristic
35C4×C4⋊Dic3C2211yescharacteristic
36#4932C2211yescharacteristic
37C2×#803C2211yescharacteristic
38#4551C2212yes—
39#4550C2212yes—
40#4550C2212yes—
41#4529C2212yes—
42#4529C2212yes—
43C2×C4×D8D311yescharacteristic
44#2319—31no—
45#2275—31no—
46#1925—32no—
47#1921—32no—
48#1879—32no—
49#1862—32no—
50C4×#107—31no—
51#1762—31no—
52#1762—31no—
53#1746—31no—
54C22×C3×D4C2311yescharacteristic
55C22×C3×D4C2311yescharacteristic
56C23×C3×C4—21no—
57C23×C3×C4—21no—
58C23×Dic3—22no—
59C2×C3×D8—24no—
60C2×C3×D8C2×C414yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×C4×D4—22no—
64C3×C4×D4—22no—
65C3×C4×D4C2×C414yes—
66C3×C4×D4C2×C414yescomplemented
67C2×C3×C4⋊C4C2311yescharacteristic
68C2×C3×C4⋊C4C2311yescharacteristic
69C2×C3×C22⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×#217—22no—
73C2×#217—22no—
74C2×#217—22no—
75D4×Dic3—24no—
76D4×Dic3—24no—
77C2×C4⋊Dic3—22no—
78C2×C4⋊Dic3C2312yes—
79C2×C4⋊Dic3C2312yes—
80C2×C4×Dic3—21no—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3C2312yes—
84C22×C3⋊C8C2312yes—
85C3×#107C2×C414yescomplemented
86C3×#102C2×C414yes—
87C3×#102C2×C414yescomplemented
88C3×C4×C8C2×C414yescomplemented
89#832—24no—
90#832—24no—
91C3⋊#95—21no—
92C3⋊#95—21no—
93C3⋊#95—22no—
94C3⋊#95—22no—
95C3⋊#95D412yes—
96C3⋊#95D411yescharacteristic
97C3⋊#95D411yescharacteristic
98#818—24no—
99#803—22no—
100C22×D8C2×D311yescharacteristic
101C2×C4×D4—32no—
102C2×C4×D4C2×D311yescharacteristic
103C2×C4×D4C2×D311yescharacteristic
104C4×D8Dic318yescomplemented
105C2×#107—32no—
106C2×#107C2×D311yescharacteristic
107C2×#102—32no—
108C2×#102—32no—
109C2×#102C2×D311yescharacteristic
110C2×#102C2×D311yescharacteristic
111C2×#97—31no—
112C2×C4×C8C2×D311yescharacteristic
113#390—32no—
114#389—32no—
115#388—62no—
116#381—61no—
117#380—61no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4—31no—
120C4×C4⋊C4—31no—
121C4×C22⋊C4—62no—
122C4×C22⋊C4—61no—
123#376—31no—
124C2×#95—61no—
125#340—32no—
126#340—32no—
127#340—32no—
128#336—32no—
129#336—32no—
130C24×C3—21no—
131C24×C3—21no—
132C2×C3×D4—22no—
133C2×C3×D4—21no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—21no—
137C2×C3×D4C22×C412yes—
138C2×C3×D4C22×C412yes—
139C2×C3×D4#4114yes—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—24no—
143C22×C3×C4—24no—
144C22×C3×C4—82no—
145C22×C3×C4—82no—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2411yescharacteristic
149C22×Dic3—22no—
150C22×Dic3—28no—
151C22×Dic3—22no—
152C22×Dic3—82no—
153C22×Dic3—82no—
154C22×Dic3C2×D412yes—
155C22×Dic3C2×D412yes—
156C3×D8—28no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8C22×C412yes—
160C2×C3×C8C22×C412yes—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4C22×C412yes—
164C3×C4⋊C4C22×C412yes—
165C3×C22⋊C4—24no—
166C3×C22⋊C4—24no—
167C3×C42C22×C412yes—
168C3×C42#4112yes—
169#217—28no—
170#217—82no—
171#217—82no—
172C4⋊Dic3—22no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—82no—
176C4⋊Dic3#4114yes—
177C4×Dic3—22no—
178C4×Dic3—22no—
179C4×Dic3—82no—
180C4×Dic3—81no—
181C4×Dic3—81no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—22no—
185C22×D4C22×D311yescharacteristic
186C22×D4C22×D311yescharacteristic
187C23×C4—21no—
188C23×C4—21no—
189C23×C4—62no—
190C2×D8—24no—
191C2×D8C2×Dic314yes—
192C22×C8—32no—
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195C4×D4—22no—
196C4×D4—22no—
197C4×D4—64no—
198C4×D4—64no—
199C4×D4C2×Dic314yes—
200C4×D4C2×Dic314yescomplemented
201C2×C4⋊C4—32no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4C22×D311yescharacteristic
205C2×C4⋊C4C22×D311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C42—32no—
212C2×C42—62no—
213C2×C42—61no—
214C2×C42—61no—
215C2×C42C22×D311yescharacteristic
216#107—64no—
217#107C2×Dic314yescomplemented
218#102—64no—
219#102—64no—
220#102C2×Dic314yes—
221#102C2×Dic314yescomplemented
222#97—62no—
223C4×C8C2×Dic314yescomplemented
224#95—31no—
225#95—31no—
226#95—32no—
227#95—62no—
228#95—62no—
229#95—61no—
230#95—61no—
231C23×C3—22no—
232C23×C3—24no—
233C23×C3—22no—
234C23×C3—41no—
235C23×C3—42no—
236C23×C3—41no—
237C23×C3—42no—
238C23×C3—41no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3C22×D411yescharacteristic
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—22no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4#11714yes—
248C2×C3×C4—21no—
249C2×C3×C4—22no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—84no—
256C2×C3×C4—84no—
257C2×C3×C4C4×D412yes—
258C2×C3×C4C4×D412yes—
259C2×C3×C4C23×C411yescharacteristic
260C2×C3×C4C2×#4112yes—
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4112yes—
263C2×Dic3—22no—
264C2×Dic3—22no—
265C2×Dic3—24no—
266C2×Dic3—22no—
267C2×Dic3—22no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—88no—
276C3×C8—22no—
277C3×C8—22no—
278C3⋊C8—42no—
279C24—21no—
280C24—21no—
281C2×D4—22no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—22no—
285C2×D4—22no—
286C2×D4#23514yes—
287C2×D4C22×Dic312yes—
288C2×D4C22×Dic312yes—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—24no—
292C22×C4—24no—
293C22×C4—32no—
294C22×C4—32no—
295C22×C4—62no—
296C22×C4—68no—
297C22×C4—62no—
298C22×C4—82no—
299C22×C4—82no—
300C22×C4—242no—
301C22×C4—242no—
302C22×C4D3×D411yescharacteristic
303C22×C4D3×D411yescharacteristic
304C22×C4C23×D311yescharacteristic
305D8—28no—
306C2×C8—22no—
307C2×C8—22no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8C22×Dic312yes—
312C2×C8C22×Dic312yes—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—34no—
316C4⋊C4—62no—
317C4⋊C4—62no—
318C4⋊C4—62no—
319C4⋊C4—242no—
320C4⋊C4C22×Dic312yes—
321C4⋊C4C22×Dic312yes—
322C22⋊C4—24no—
323C22⋊C4—24no—
324C22⋊C4—68no—
325C22⋊C4—242no—
326C22⋊C4—242no—
327C42—62no—
328C42—62no—
329C42—241no—
330C42—241no—
331C42—242no—
332C42#23712yes—
333C42C22×Dic312yes—
334C22×C3—24no—
335C22×C3—44no—
336C22×C3—44no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—42no—
342C22×C3—82no—
343C22×C3—82no—
344C22×C3C2×C4×D411yescharacteristic
345C22×C3#52512yes—
346C22×C3#57612yes—
347C22×C3#57611yescharacteristic
348C22×C3#57811yescharacteristic
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4C2×#11711yescharacteristic
354C3×C4C4×#4111yescharacteristic
355C3×C4#52912yes—
356Dic3—42no—
357Dic3—44no—
358Dic3—42no—
359Dic3—82no—
360C23—22no—
361C23—24no—
362C23—22no—
363C23—42no—
364C23—41no—
365C23—41no—
366C23—42no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23C2×D3×D411yescharacteristic
371D4—22no—
372D4—24no—
373D4—22no—
374D4—22no—
375D4—42no—
376D4#92614yes—
377C2×C4—22no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—62no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—64no—
391C2×C4—84no—
392C2×C4—84no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—248no—
399C2×C4D4×Dic312yes—
400C2×C4D4×Dic312yes—
401C2×C4C2×#23512yes—
402C2×C4C2×#23711yescharacteristic
403C2×C4D3×#4112yes—
404C2×C4C23×Dic311yescharacteristic
405C8—22no—
406C8—22no—
407C8—122no—
408C2×C3—44no—
409C2×C3—82no—
410C2×C3#229011yescharacteristic
411C2×C3#294811yescharacteristic
412C2×C3#295111yescharacteristic
413C2×C3#312212yes—
414C2×C3#312412yes—
415C22—24no—
416C22—42no—
417C22—42no—
418C22—44no—
419C22—44no—
420C22—42no—
421C22—42no—
422C22—42no—
423C22—82no—
424C22—82no—
425C22#554712yes—
426C22#575712yes—
427C22#576911yescharacteristic
428C22#577111yescharacteristic
429C22C2×D4×Dic311yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—122no—
435C4—124no—
436C4—122no—
437C4—242no—
438C4#553812yes—
439C4C2×#92611yescharacteristic
440C4Dic3×#4111yescharacteristic
441C3#2112911yescharacteristic, complemented, Sylow
442C2—44no—
443C2—82no—
444C2#7959612yes—
445C2#7959712yes—
446C2#8231811yescharacteristic
447C2#8382311yescharacteristic
448C2#8389811yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×C4×D8C2
socleC23×C3C22×D4
prime radicalC22×C3×D8C22
cosocleC22×C3×C4C24
hypercenterC2×C4×D8D3
hypocenterC3#21129
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4×D8

Complements

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

NH ≅ G/Ntype
2. #82325C2semidirect
3. #82317C2semidirect
5. #71609C2semidirect
6. #71609C2semidirect
7. #70610C2semidirect
8. #70168C2semidirect
9. #70155C2semidirect
10. C4×#818C2semidirect
11. #69751C2semidirect
17. C3×C4×D8C4semidirect
66. C3×C4×D4C2×C4semidirect
85. C3×#107C2×C4semidirect
87. C3×#102C2×C4semidirect
88. C3×C4×C8C2×C4semidirect
104. C4×D8Dic3semidirect
200. C4×D4C2×Dic3semidirect
217. #107C2×Dic3semidirect
221. #102C2×Dic3semidirect
223. C4×C8C2×Dic3semidirect
441. C3#21129semidirect

Maximal subgroups

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

subgroupindex
C2×C3×C4×D82
#823252
#823172
#720372
#716092
#706102
#701682
#701552
C4×#8182
#697512
#211293