Finite Group Databasebeta

← #10876307

Subgroups — #10876307

Order 768.

1124 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#380C212yesmaximal
2C3×#2075C211yesmaximal, characteristic, complemented
3#70632C212yesmaximal, complemented
4#69955C211yesmaximal, characteristic
5#69938C211yesmaximal, characteristic, complemented
6#69518C212yesmaximal, complemented
7#68874C212yesmaximal, complemented
8#68871C212yesmaximal, complemented
9#68871C212yesmaximal, complemented
10#27600—31nomaximal, Sylow
11C23×C4×D3C2211yescharacteristic
12C2×D3×C4⋊C4C2212yes—
13C2×D3×C4⋊C4C2212yes—
14C2×D3×C22⋊C4—22no—
15C2×C3×#107C2211yescharacteristic, complemented
16C2×C3×C22⋊C8C2211yescharacteristic
17C3×#380C2212yes—
18#5212C2212yes—
19C2×#830C2211yescharacteristic
20C2×(C2×D3)⋊C8C2212yes—
21C2×(C2×D3)⋊C8C2212yes—
22C2×#818C2211yescharacteristic, complemented
23#4987C2212yes—
24#4987C2212yes—
25C2×#807C2212yescomplemented
26#4942—22no—
27#4667—22no—
28D3×#95—22no—
29C3×#340C2211yescharacteristic, complemented
30C3×#340C2212yescomplemented
31#4551C2211yescharacteristic, complemented
32#4551C2212yescomplemented
33#4529C2212yescomplemented
34#4529C2212yescomplemented
35#4529C2212yescomplemented
36C2×#380—32no—
37#2075—31no—
38#2075—32no—
39#2075D311yescharacteristic, complemented
40#1852—32no—
41#1852—32no—
42#1823—32no—
43#1823—32no—
44#1779—31no—
45C24×D3D411yescharacteristic
46C23×C3×C4C2311yescharacteristic
47C23×Dic3D411yescharacteristic
48C22×C4×D3—24no—
49C22×C4×D3—24no—
50C22×C4×D3—22no—
51C22×C4×D3—22no—
52C22×C4×D3—41no—
53C22×C4×D3—41no—
54C22×C4×D3D412yes—
55C22×C4×D3C2312yes—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C2×C3×C4⋊C4C2312yes—
59C2×C3×C4⋊C4C2312yes—
60C2×C3×C22⋊C4—22no—
61C2×#217—22no—
62C2×(C2×D3)⋊C4—22no—
63C2×(C2×D3)⋊C4—22no—
64C2×C4⋊Dic3C2312yes—
65C2×C4⋊Dic3C2312yes—
66C2×Dic3⋊C4—22no—
67C2×Dic3⋊C4—22no—
68C22×C3⋊C8C2312yes—
69D3×C4⋊C4—24no—
70D3×C4⋊C4—24no—
71D3×C4⋊C4—42no—
72D3×C4⋊C4—42no—
73D3×C22⋊C4—48no—
74C3×#107—41no—
75C3×C22⋊C8—41no—
76C3×#95—22no—
77C3⋊#95—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—41no—
81(C2×D3)⋊C8—24no—
82(C2×D3)⋊C8—24no—
83#818—41no—
84#807—42no—
85C24×C4—31no—
86C22×C4⋊C4—32no—
87C22×C4⋊C4—32no—
88C22×C22⋊C4—62no—
89C2×#107—31no—
90C2×#107—32no—
91C2×#107C2×D311yescharacteristic, complemented
92C2×C22⋊C8—32no—
93C2×C22⋊C8—32no—
94C2×C22⋊C8—31no—
95C2×C22⋊C8C2×D311yescharacteristic
96#380—32no—
97#380—32no—
98#380—32no—
99#380—62no—
100#380—62no—
101#380C2×D312yes—
102C2×#95—62no—
103#340—31no—
104#340—32no—
105#340—32no—
106#340—32no—
107#340—32no—
108#340C2×D311yescharacteristic, complemented
109#340C2×D312yescomplemented
110C24×C3C2×D411yescharacteristic
111C23×D3—24no—
112C23×D3—24no—
113C23×D3—41no—
114C23×D3—41no—
115C23×D3D814yes—
116C23×D3C2×D412yes—
117C22×C3×C4—22no—
118C22×C3×C4—22no—
119C22×C3×C4—21no—
120C22×C3×C4—21no—
121C22×C3×C4—22no—
122C22×C3×C4—22no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2411yescharacteristic
125C22×Dic3—24no—
126C22×Dic3—22no—
127C22×Dic3—22no—
128C22×Dic3—41no—
129C22×Dic3C2×D412yes—
130C2×C4×D3—22no—
131C2×C4×D3—22no—
132C2×C4×D3—22no—
133C2×C4×D3—22no—
134C2×C4×D3—24no—
135C2×C4×D3—24no—
136C2×C4×D3—44no—
137C2×C4×D3—44no—
138C2×C4×D3—48no—
139C2×C4×D3—42no—
140C2×C4×D3—44no—
141C2×C4×D3—44no—
142C2×C4×D3—44no—
143C2×C4×D3—44no—
144C2×C4×D3—44no—
145C2×C4×D3—44no—
146C2×C4×D3#4114yes—
147C2×C4×D3#4114yes—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C22⋊C4—44no—
157#217—44no—
158(C2×D3)⋊C4—44no—
159(C2×D3)⋊C4—44no—
160C4⋊Dic3—22no—
161C4⋊Dic3—22no—
162C4⋊Dic3—22no—
163C4⋊Dic3—22no—
164Dic3⋊C4—44no—
165Dic3⋊C4—44no—
166C2×C3⋊C8—22no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C25—31no—
170C23×C4—32no—
171C23×C4—31no—
172C23×C4—32no—
173C23×C4—64no—
174C23×C4—64no—
175C23×C4—62no—
176C23×C4—62no—
177C23×C4—121no—
178C23×C4—121no—
179C23×C4C22×D311yescharacteristic
180C22×C8—32no—
181C22×C8C22×D311yescharacteristic
182C22×C8C22×D311yescharacteristic
183C2×C4⋊C4—32no—
184C2×C4⋊C4—32no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—64no—
187C2×C4⋊C4—62no—
188C2×C4⋊C4—62no—
189C2×C4⋊C4—122no—
190C2×C4⋊C4—122no—
191C2×C4⋊C4C22×D312yes—
192C2×C4⋊C4C22×D312yes—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—62no—
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—128no—
198#107—41no—
199#107—122no—
200#107—121no—
201C22⋊C8—41no—
202C22⋊C8—64no—
203C22⋊C8—64no—
204C22⋊C8—121no—
205#95—22no—
206#95—62no—
207#95—62no—
208#95—62no—
209C23×C3—22no—
210C23×C3—21no—
211C23×C3—22no—
212C23×C3—21no—
213C23×C3C2×D812yes—
214C23×C3C22×D411yescharacteristic
215C22×D3—22no—
216C22×D3—22no—
217C22×D3—22no—
218C22×D3—22no—
219C22×D3—216no—
220C22×D3—44no—
221C22×D3—44no—
222C22×D3—48no—
223C22×D3—44no—
224C22×D3#12314yes—
225C22×D3C2×D814yes—
226C22×D3#13714yes—
227C2×C3×C4—22no—
228C2×C3×C4—21no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—42no—
235C2×C3×C4—42no—
236C2×C3×C4—42no—
237C2×C3×C4—42no—
238C2×C3×C4C2×#4112yes—
239C2×C3×C4C2×#4112yes—
240C2×C3×C4H(5,𝔽2)11yescharacteristic
241C2×C3×C4H(5,𝔽2)11yescharacteristic
242C2×Dic3—22no—
243C2×Dic3—22no—
244C2×Dic3—44no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—42no—
248C2×Dic3—42no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3#12314yes—
252C4×D3—24no—
253C4×D3—24no—
254C4×D3—24no—
255C4×D3—24no—
256C4×D3—48no—
257C4×D3—88no—
258C4×D3—88no—
259C3×C8—41no—
260C3×C8—41no—
261C3⋊C8—42no—
262C24—32no—
263C24—34no—
264C24—64no—
265C24—64no—
266C24—121no—
267C24—121no—
268C24D3×D411yescharacteristic
269C22×C4—21no—
270C22×C4—22no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—32no—
276C22×C4—34no—
277C22×C4—34no—
278C22×C4—64no—
279C22×C4—64no—
280C22×C4—64no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—62no—
287C22×C4—122no—
288C22×C4—124no—
289C22×C4—124no—
290C22×C4—128no—
291C22×C4—124no—
292C22×C4—124no—
293C22×C4—124no—
294C22×C4—124no—
295C22×C4—124no—
296C22×C4—124no—
297C22×C4—121no—
298C22×C4D3×D411yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—21no—
301C2×C8—21no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—62no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—62no—
312C4⋊C4—62no—
313C4⋊C4—62no—
314C4⋊C4—62no—
315C4⋊C4—124no—
316C4⋊C4—124no—
317C22⋊C4—44no—
318C22⋊C4—124no—
319C22⋊C4—124no—
320C22⋊C4—124no—
321C22×C3—24no—
322C22×C3—22no—
323C22×C3—42no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C2×#12312yes—
327C22×C3#53411yescharacteristic
328C22×C3C22×D811yescharacteristic
329C22×C3C2×#13712yes—
330C22×C3#57511yescharacteristic
331C2×D3—24no—
332C2×D3—24no—
333C2×D3—24no—
334C2×D3—24no—
335C2×D3—24no—
336C2×D3—24no—
337C2×D3—416no—
338C2×D3#48018yes—
339C3×C4—41no—
340C3×C4—82no—
341C3×C4—82no—
342C3×C4#55111yescharacteristic
343C3×C4#55111yescharacteristic
344C3×C4#55112yes—
345Dic3—24no—
346Dic3—82no—
347Dic3—82no—
348C23—22no—
349C23—21no—
350C23—22no—
351C23—21no—
352C23—34no—
353C23—34no—
354C23—34no—
355C23—616no—
356C23—62no—
357C23—62no—
358C23—62no—
359C23—62no—
360C23—124no—
361C23—124no—
362C23—124no—
363C23—128no—
364C23D3×D812yes—
365C23C2×D3×D411yescharacteristic
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—34no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—64no—
381C2×C4—64no—
382C2×C4—64no—
383C2×C4—64no—
384C2×C4—128no—
385C2×C4—124no—
386C2×C4—122no—
387C2×C4—122no—
388C2×C4—122no—
389C2×C4—122no—
390C2×C4—122no—
391C2×C4—122no—
392C2×C4—248no—
393C2×C4—248no—
394C2×C4#100411yescharacteristic
395C2×C4D3×#4112yes—
396C2×C4D3×#4112yes—
397C2×C4#100911yescharacteristic
398C8—41no—
399C8—41no—
400C8—122no—
401C2×C3—42no—
402C2×C3#248311yescharacteristic
403C2×C3C2×#48012yes—
404C2×C3#309512yes—
405C2×C3#318611yescharacteristic
406C2×C3#319511yescharacteristic
407D3—28no—
408C22—22no—
409C22—24no—
410C22—38no—
411C22—41no—
412C22—41no—
413C22—42no—
414C22—64no—
415C22—64no—
416C22—64no—
417C22—64no—
418C22—64no—
419C22—64no—
420C22—1216no—
421C22#560511yescharacteristic
422C22D3×#12312yes—
423C22C2×D3×D811yescharacteristic
424C22#576811yescharacteristic
425C22D3×#13712yes—
426C4—41no—
427C4—64no—
428C4—82no—
429C4—82no—
430C4—242no—
431C4—242no—
432C4#565911yescharacteristic
433C4#567911yescharacteristic
434C4#570512yes—
435C3#2760011yescharacteristic, complemented, Sylow
436C2—42no—
437C2—68no—
438C2#7767711yescharacteristic
439C2#8071911yescharacteristic
440C2#8090811yescharacteristic
441C2D3×#48012yes—
442C2#8151712yes—
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2075C2
socleC23×C3C22×D4
prime radicalC24×D3D4
hypercenter#2075D3
hypocenterC3#27600
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2075

Complements

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

NH ≅ G/Ntype
2. C3×#2075C2semidirect
3. #70632C2semidirect
5. #69938C2semidirect
6. #69518C2semidirect
7. #68874C2semidirect
8. #68871C2semidirect
9. #68871C2semidirect
15. C2×C3×#107C22semidirect
22. C2×#818C22semidirect
25. C2×#807C22semidirect
29. C3×#340C22semidirect
30. C3×#340C22semidirect
31. #4551C22semidirect
32. #4551C22semidirect
33. #4529C22semidirect
34. #4529C22semidirect
35. #4529C22semidirect
39. #2075D3semidirect
91. C2×#107C2×D3semidirect
108. #340C2×D3semidirect
109. #340C2×D3semidirect
435. C3#27600semidirect

Maximal subgroups

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

subgroupindex
D3×#3802
C3×#20752
#706322
#699552
#699382
#695182
#688742
#688712
#276003