Finite Group Databasebeta

← #10824590

Subgroups — #10824590

Order 768.

1062 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Dic3×D8C211yesmaximal, characteristic, complemented
2Dic3×#121C212yesmaximal, complemented
3C2×#4718C211yesmaximal, characteristic, complemented
4C3×#1999C211yesmaximal, characteristic
5#72040C211yesmaximal, characteristic, complemented
6#71679C212yesmaximal, complemented
7#70194C212yesmaximal, complemented
8Dic3×#102C212yesmaximal, complemented
9#69938C211yesmaximal, characteristic, complemented
10#68838C211yesmaximal, characteristic, complemented
11#68785C211yesmaximal, characteristic, complemented
12#23696—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C2×C3×#121C2212yes—
15C2×D4×Dic3—22no—
16C2×D4×Dic3C2212yes—
17C22×C4×Dic3C2211yescharacteristic
18C2×C3×#102C2212yes—
19C2×C3×C22⋊C8C2211yescharacteristic
20#5221C2212yes—
21#5220—22no—
22C2×#832C2212yes—
23C2×#832C2212yes—
24#5212C2212yes—
25C2×#830C2211yescharacteristic
26Dic3×D8—24no—
27C2×#818C2211yescharacteristic
28C2×Dic3⋊C8C2211yescharacteristic
29C2×Dic3⋊C8C2211yescharacteristic
30C2×C8×Dic3C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#4984C2212yes—
33Dic3×C4⋊C4C2212yes—
34C2×#807C2212yes—
35#4951—22no—
36Dic3×C22⋊C4—22no—
37#4718—24no—
38C3×#340C2211yescharacteristic
39#4551C2212yes—
40#4551C2211yescharacteristic
41#4529C2212yes—
42C2×C4×D8—31no—
43C2×#431—31no—
44C4×#121—32no—
45#2086—31no—
46#2075—31no—
47#2067—32no—
48#2054—31no—
49#2033—32no—
50#1999D311yescharacteristic
51C4×#107—31no—
52C4×#102—32no—
53C22×C3×D4—22no—
54C22×C3×D4C2312yes—
55C23×C3×C4C2311yescharacteristic
56C23×Dic3—22no—
57C23×Dic3D411yescharacteristic
58C23×Dic3D411yescharacteristic
59C2×C3×D8—24no—
60C2×C3×D8—22no—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#121—24no—
64C3×#121—24no—
65C2×C3×C4⋊C4C2312yes—
66C2×C3×C22⋊C4—22no—
67C2×#217—22no—
68C2×#217—22no—
69C2×#217—22no—
70C2×#217—22no—
71D4×Dic3—24no—
72D4×Dic3—24no—
73D4×Dic3—48no—
74C2×C4⋊Dic3—22no—
75C2×C4⋊Dic3C2312yes—
76C2×C4⋊Dic3C2312yes—
77C2×C4×Dic3—22no—
78C2×C4×Dic3—22no—
79C2×C4×Dic3—22no—
80C2×C4×Dic3D411yescharacteristic
81C2×C4×Dic3D411yescharacteristic
82C2×C4×Dic3C2311yescharacteristic
83C2×C4×Dic3C2311yescharacteristic
84C22×C3⋊C8C2311yescharacteristic
85C22×C3⋊C8C2311yescharacteristic
86C3×#102—24no—
87C3×C22⋊C8—22no—
88#832—24no—
89#832—24no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95—22no—
93#830—22no—
94#818—22no—
95Dic3⋊C8—22no—
96Dic3⋊C8—22no—
97C8×Dic3—22no—
98C8×Dic3D414yescomplemented
99#807—24no—
100C22×D8C2×D311yescharacteristic
101C2×#121C2×D312yes—
102C2×C4×D4—32no—
103C2×C4×D4—62no—
104C22×C42—31no—
105C4×D8—64no—
106#431—64no—
107C2×#107—31no—
108C2×#107—32no—
109C2×C4⋊C8—31no—
110C2×C4⋊C8—31no—
111C2×#102—32no—
112C2×#102—32no—
113C2×#102C2×D312yes—
114C2×C22⋊C8—31no—
115C2×C22⋊C8C2×D311yescharacteristic
116C2×C4×C8—31no—
117C2×C4×C8—31no—
118#390—32no—
119#389—32no—
120#388—62no—
121#386—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124C4×C22⋊C4—62no—
125#340—31no—
126#340—32no—
127#340—32no—
128#340C2×D311yescharacteristic
129C24×C3—22no—
130C24×C3C2×D411yescharacteristic
131C2×C3×D4—28no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—22no—
135C2×C3×D4—22no—
136C2×C3×D4—22no—
137C2×C3×D4—44no—
138C2×C3×D4—44no—
139C2×C3×D4#4114yes—
140C22×C3×C4—22no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—22no—
144C22×C3×C4—22no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2411yescharacteristic
147C22×Dic3—22no—
148C22×Dic3—22no—
149C22×Dic3—22no—
150C22×Dic3—22no—
151C22×Dic3—22no—
152C22×Dic3—42no—
153C22×Dic3—44no—
154C22×Dic3—44no—
155C22×Dic3—44no—
156C22×Dic3D814yescomplemented
157C22×Dic3C2×D411yescharacteristic
158C22×Dic3C2×D411yescharacteristic
159C22×Dic3C2×D411yescharacteristic
160C22×Dic3C2×D411yescharacteristic
161C3×D8—28no—
162C2×C3×C8—22no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8C2×D412yes—
166C3×C4⋊C4—22no—
167C3×C4⋊C4#4114yes—
168C3×C22⋊C4—28no—
169#217—44no—
170#217—44no—
171#217—44no—
172#217—44no—
173C4⋊Dic3—22no—
174C4⋊Dic3—22no—
175C4⋊Dic3—22no—
176C4⋊Dic3—22no—
177C4⋊Dic3—44no—
178C4×Dic3—21no—
179C4×Dic3—21no—
180C4×Dic3—22no—
181C4×Dic3—22no—
182C4×Dic3—44no—
183C4×Dic3C2×D412yes—
184C4×Dic3#4112yes—
185C2×C3⋊C8—22no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8C2×D412yes—
189C22×D4—22no—
190C22×D4C22×D312yes—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—62no—
194C23×C4C22×D311yescharacteristic
195C2×D8—24no—
196C2×D8—22no—
197C22×C8—31no—
198C22×C8—31no—
199C22×C8C22×D311yescharacteristic
200C22×C8C22×D311yescharacteristic
201#121—24no—
202#121—24no—
203C4×D4—64no—
204C4×D4—64no—
205C4×D4—128no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—32no—
208C2×C4⋊C4—62no—
209C2×C4⋊C4C22×D312yes—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4—62no—
213C2×C22⋊C4—62no—
214C2×C22⋊C4—62no—
215C2×C42—31no—
216C2×C42—31no—
217C2×C42—31no—
218C2×C42—31no—
219C2×C42—62no—
220C2×C42—62no—
221C2×C42—62no—
222#107—62no—
223#107—64no—
224C4⋊C8—62no—
225C4⋊C8—62no—
226#102—24no—
227#102—64no—
228#102—64no—
229C22⋊C8—22no—
230C22⋊C8—62no—
231C4×C8—34no—
232C4×C8—62no—
233#95—62no—
234#95—62no—
235#95—62no—
236C23×C3—21no—
237C23×C3—22no—
238C23×C3—21no—
239C23×C3—22no—
240C23×C3—24no—
241C23×C3—42no—
242C23×C3—42no—
243C23×C3—42no—
244C23×C3—42no—
245C23×C3—42no—
246C23×C3—42no—
247C23×C3C2×D812yes—
248C23×C3C22×D411yescharacteristic
249C3×D4—24no—
250C3×D4—24no—
251C3×D4—416no—
252C3×D4—42no—
253C3×D4—42no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—24no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—41no—
262C2×C3×C4C22×D411yescharacteristic
263C2×C3×C4C2×#4112yes—
264C2×C3×C4C2×#4112yes—
265C2×C3×C4C2×#4111yescharacteristic
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—22no—
269C2×Dic3—28no—
270C2×Dic3—21no—
271C2×Dic3—22no—
272C2×Dic3—44no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278C2×Dic3—42no—
279C2×Dic3—88no—
280C2×Dic3#12112yes—
281C2×Dic3#12112yes—
282C2×Dic3C2×D812yes—
283C2×Dic3C2×D812yes—
284C2×Dic3#13512yes—
285C3×C8—22no—
286C3×C8—41no—
287C3⋊C8—22no—
288C3⋊C8—41no—
289C24—22no—
290C24D3×D411yescharacteristic
291C2×D4—28no—
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—22no—
295C2×D4—22no—
296C2×D4—22no—
297C2×D4—44no—
298C2×D4—44no—
299C2×D4#23714yes—
300C22×C4—22no—
301C22×C4—21no—
302C22×C4—22no—
303C22×C4—22no—
304C22×C4—21no—
305C22×C4—31no—
306C22×C4—34no—
307C22×C4—31no—
308C22×C4—31no—
309C22×C4—31no—
310C22×C4—62no—
311C22×C4—62no—
312C22×C4—62no—
313C22×C4—62no—
314C22×C4—62no—
315C22×C4—124no—
316C22×C4—124no—
317C22×C4—124no—
318C22×C4—122no—
319C22×C4D3×D411yescharacteristic
320C22×C4C23×D311yescharacteristic
321D8—28no—
322C2×C8—22no—
323C2×C8—21no—
324C2×C8—22no—
325C2×C8—32no—
326C2×C8—62no—
327C2×C8—61no—
328C2×C8—62no—
329C2×C8D3×D412yes—
330C4⋊C4—22no—
331C4⋊C4—62no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—124no—
336C4⋊C4#23714yes—
337C22⋊C4—28no—
338C22⋊C4—124no—
339C22⋊C4—124no—
340C22⋊C4—124no—
341C22⋊C4—124no—
342C42—32no—
343C42—32no—
344C42—62no—
345C42—62no—
346C42—61no—
347C42—61no—
348C42—124no—
349C22×C3—24no—
350C22×C3—22no—
351C22×C3—42no—
352C22×C3—41no—
353C22×C3—41no—
354C22×C3—44no—
355C22×C3—44no—
356C22×C3—44no—
357C22×C3—82no—
358C22×C3—82no—
359C22×C3—82no—
360C22×C3—82no—
361C22×C3C2×#12111yescharacteristic
362C22×C3#52512yes—
363C22×C3C22×D811yescharacteristic
364C22×C3C2×#13511yescharacteristic
365C22×C3#57512yes—
366C3×C4—44no—
367C3×C4—41no—
368C3×C4#52511yescharacteristic
369C3×C4#52912yes—
370C3×C4#53211yescharacteristic
371Dic3—22no—
372Dic3—22no—
373Dic3—41no—
374Dic3—82no—
375Dic3—82no—
376Dic3#46614yes—
377C23—21no—
378C23—22no—
379C23—22no—
380C23—21no—
381C23—24no—
382C23—42no—
383C23—42no—
384C23—42no—
385C23—42no—
386C23—42no—
387C23—42no—
388C23D3×D812yes—
389C23C2×D3×D411yescharacteristic
390D4—24no—
391D4—24no—
392D4—416no—
393D4—42no—
394D4—42no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—21no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—32no—
402C2×C4—32no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—41no—
406C2×C4—41no—
407C2×C4—42no—
408C2×C4—68no—
409C2×C4—61no—
410C2×C4—62no—
411C2×C4—61no—
412C2×C4—62no—
413C2×C4—61no—
414C2×C4—124no—
415C2×C4—122no—
416C2×C4—122no—
417C2×C4—122no—
418C2×C4—122no—
419C2×C4—122no—
420C2×C4—122no—
421C2×C4—248no—
422C2×C4C2×D3×D411yescharacteristic
423C2×C4C2×#23712yes—
424C2×C4C2×#23712yes—
425C2×C4D3×#4111yescharacteristic
426C8—22no—
427C8—41no—
428C8—62no—
429C8—121no—
430C2×C3—42no—
431C2×C3—84no—
432C2×C3#239211yescharacteristic
433C2×C3C2×#46611yescharacteristic
434C2×C3#305411yescharacteristic
435C2×C3#310012yes—
436C2×C3#310212yes—
437C22—22no—
438C22—24no—
439C22—41no—
440C22—42no—
441C22—41no—
442C22—44no—
443C22—44no—
444C22—44no—
445C22—82no—
446C22—82no—
447C22—82no—
448C22—82no—
449C22#558612yes—
450C22D3×#12111yescharacteristic
451C22C2×D3×D811yescharacteristic
452C22D3×#13511yescharacteristic
453C22#577312yes—
454C4—34no—
455C4—44no—
456C4—41no—
457C4—62no—
458C4—62no—
459C4—121no—
460C4—242no—
461C4—242no—
462C4#558611yescharacteristic
463C4#560012yes—
464C4#571611yescharacteristic
465C3#2369611yescharacteristic, complemented, Sylow
466C2—42no—
467C2—84no—
468C2#7666611yescharacteristic
469C2#7998812yes—
470C2#8006812yes—
471C2D3×#46611yescharacteristic
472C2#8086311yescharacteristic
subgroupG/H
centerC23C2×D3×D4
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
Fitting, prime radicalC3×#1999C2
socleC23×C3C22×D4
hypercenter#1999D3
hypocenterC3#23696
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1999

Complements

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

NH ≅ G/Ntype
1. C2×Dic3×D8C2semidirect
2. Dic3×#121C2semidirect
3. C2×#4718C2semidirect
5. #72040C2semidirect
6. #71679C2semidirect
7. #70194C2semidirect
8. Dic3×#102C2semidirect
9. #69938C2semidirect
10. #68838C2semidirect
11. #68785C2semidirect
98. C8×Dic3D4semidirect
156. C22×Dic3D8semidirect
465. C3#23696semidirect

Maximal subgroups

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

subgroupindex
C2×Dic3×D82
Dic3×#1212
C2×#47182
C3×#19992
#720402
#716792
#701942
Dic3×#1022
#699382
#688382
#687852
#236963