Finite Group Databasebeta

← #10853308

Subgroups — #10853308

Order 768.

786 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2354C211yesmaximal, characteristic
2#73715C211yesmaximal, characteristic, complemented
3C4×#832C211yesmaximal, characteristic
4#71604C211yesmaximal, characteristic
5#69652C212yesmaximal
6#69638C212yesmaximal
7#69574C211yesmaximal, characteristic, complemented
8#69568C211yesmaximal, characteristic, complemented
9#69489C212yesmaximal, complemented
10#69484C211yesmaximal, characteristic, complemented
11#69460C212yesmaximal, complemented
12#26024—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C2×C3×#117—21no—
15C2×#888—21no—
16C2×#869C2212yes—
17C3×#390C2212yes—
18C3×#388—22no—
19C3×#384C2211yescharacteristic
20C3×#380—21no—
21C3×C4×C4⋊C4C2212yes—
22C3×C4×C4⋊C4C2211yescharacteristic
23C2×#832C2211yescharacteristic
24C2×#832C2211yescharacteristic
25C2×#832C2212yes—
26#4984—21no—
27C2×#808C2212yes—
28#4935C2211yescharacteristic
29C4×C4⋊Dic3C2211yescharacteristic
30#4928C2212yes—
31C2×C4⋊(C3⋊C8)C2211yescharacteristic
32C2×C4⋊(C3⋊C8)C2212yes—
33C2×C4×C3⋊C8C2211yescharacteristic
34#4656—22no—
35#4529C2212yes—
36#4529C2212yes—
37#4529C2212yes—
38#4529C2211yescharacteristic
39#4529C2212yes—
40#4529C2211yescharacteristic
41#4524C2212yes—
42#2551—31no—
43#2354D311yescharacteristic
44#2065—32no—
45#2023—32no—
46#1987—32no—
47#1948—31no—
48#1939—31no—
49#1920—32no—
50#1834—31no—
51#1802—31no—
52C4×#102—31no—
53C22×C3×D4C2311yescharacteristic
54C23×C3×C4—21no—
55C22×Dic6—22no—
56C3×C4×D4—22no—
57C3×C4×D4Q814yescomplemented
58C3×#117—44no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4D411yescharacteristic
62C2×C3×C4⋊C4D411yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C4⋊C4C2311yescharacteristic
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C22⋊C4—22no—
68C2×C3×C42—22no—
69C2×C3×C42C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3—21no—
73C2×C4⋊Dic3—21no—
74C2×C4⋊Dic3C2312yes—
75C2×C4⋊Dic3C2311yescharacteristic
76C2×C4⋊Dic3C2311yescharacteristic
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C22×C3⋊C8C2312yes—
82#888—44no—
83#869—24no—
84#869—42no—
85C3×#95—22no—
86C3×#95D412yes—
87#832—22no—
88#832—22no—
89#832—42no—
90C3⋊#95—21no—
91C3⋊#95—22no—
92#808—24no—
93C4⋊(C3⋊C8)—22no—
94C4⋊(C3⋊C8)—42no—
95C4×C3⋊C8—22no—
96C2×C4⋊Q8—32no—
97C2×C4⋊Q8—61no—
98C2×C4×D4C2×D311yescharacteristic
99C2×#117—21no—
100C2×#106—32no—
101C2×C4⋊C8—31no—
102C2×C4⋊C8—32no—
103C2×#102—31no—
104C2×#102—31no—
105C2×#102—32no—
106C2×C4×C8—31no—
107#395—62no—
108#390C2×D312yes—
109#389—32no—
110#389—61no—
111#388—22no—
112#384—31no—
113#384C2×D311yescharacteristic
114#380—21no—
115C4×C4⋊C4—31no—
116C4×C4⋊C4C2×D312yes—
117C4×C4⋊C4C2×D311yescharacteristic
118#340—32no—
119#340—32no—
120#340—32no—
121#340—31no—
122#340—32no—
123#340—31no—
124#336—32no—
125C24×C3—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4C2×Q812yes—
132C22×C3×C4—21no—
133C22×C3×C4—24no—
134C22×C3×C4—22no—
135C22×C3×C4—42no—
136C22×C3×C4—42no—
137C22×C3×C4C2×D412yes—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2411yescharacteristic
142C22×Dic3—21no—
143C22×Dic3—21no—
144C22×Dic3—22no—
145C2×Dic6—44no—
146C2×Dic6—44no—
147C2×Dic6—82no—
148C3×C4⋊C4—22no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—22no—
151C3×C4⋊C4—22no—
152C3×C4⋊C4—42no—
153C3×C4⋊C4—81no—
154C3×C4⋊C4SD1614yes—
155C3×C4⋊C4C2×Q812yes—
156C3×C22⋊C4—24no—
157C3×C22⋊C4—44no—
158C3×C22⋊C4—82no—
159C3×C42—22no—
160C3×C42—22no—
161C3×C42—44no—
162C3×C42C2×Q812yes—
163C3×C42#4112yes—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—22no—
167C4⋊Dic3—22no—
168C4⋊Dic3—44no—
169C4⋊Dic3—42no—
170C4⋊Dic3—81no—
171C4⋊Dic3#4112yes—
172C4⋊Dic3#4112yes—
173Dic3⋊C4—44no—
174C4×Dic3—42no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8—21no—
184C22×Q8—62no—
185C22×D4C22×D311yescharacteristic
186C23×C4—21no—
187C22×C8—31no—
188C22×C8—32no—
189C22×C8—31no—
190C4⋊Q8—64no—
191C4⋊Q8—122no—
192C4⋊Q8—124no—
193C4×D4—22no—
194C4×D4Dic614yescomplemented
195#117—44no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4—62no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—62no—
204C2×C4⋊C4—61no—
205C2×C4⋊C4#6711yescharacteristic
206C2×C4⋊C4#6711yescharacteristic
207C2×C4⋊C4C22×D312yes—
208C2×C4⋊C4C22×D311yescharacteristic
209C2×C22⋊C4—21no—
210C2×C22⋊C4—22no—
211C2×C22⋊C4—22no—
212C2×C42—22no—
213C2×C42—61no—
214C2×C42C22×D312yes—
215C2×C42C22×D311yescharacteristic
216#106—64no—
217C4⋊C8—62no—
218C4⋊C8—122no—
219#102—62no—
220#102—62no—
221#102—122no—
222C4×C8—62no—
223#95—22no—
224#95—61no—
225#95—62no—
226#95#6712yes—
227C23×C3—22no—
228C23×C3—41no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—41no—
234C23×C3C22×D411yescharacteristic
235C3×D4—22no—
236C3×D4—22no—
237C3×D4—41no—
238C3×D4—41no—
239C2×C3×C4—22no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—48no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4C22⋊Q812yes—
254C2×C3×C4C22⋊Q812yes—
255C2×C3×C4C2×SD1614yes—
256C2×C3×C4C2×SD1612yes—
257C2×C3×C4AGL(1,ℤ/8)12yes—
258C2×C3×C4C22×Q811yescharacteristic
259C2×C3×C4C2×#4111yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4H(5,𝔽2)11yescharacteristic
263C2×C3×C42−1+411yescharacteristic
264C2×Dic3—22no—
265C2×Dic3—22no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—41no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273Dic6—88no—
274C3⋊C8—41no—
275C3⋊C8—42no—
276C3⋊C8—41no—
277C24—21no—
278C2×Q8—124no—
279C2×Q8—124no—
280C2×Q8—242no—
281C2×D4—21no—
282C2×D4—21no—
283C2×D4—21no—
284C2×D4—21no—
285C2×D4—21no—
286C2×D4C2×Dic612yes—
287C22×C4—21no—
288C22×C4—24no—
289C22×C4—22no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—61no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4C2×#6712yes—
296C22×C4C2×#6711yescharacteristic
297C22×C4C2×#6712yes—
298C22×C4C2×#6711yescharacteristic
299C22×C4C23×D311yescharacteristic
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—62no—
303C2×C8—61no—
304C2×C8—61no—
305C2×C8—62no—
306C2×C8—62no—
307C2×C8—61no—
308C2×C8—61no—
309C4⋊C4—22no—
310C4⋊C4—21no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4—32no—
314C4⋊C4—32no—
315C4⋊C4—42no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—62no—
319C4⋊C4—62no—
320C4⋊C4—81no—
321C4⋊C4—124no—
322C4⋊C4—124no—
323C4⋊C4—122no—
324C4⋊C4—241no—
325C4⋊C4#21414yes—
326C4⋊C4C2×Dic612yes—
327C22⋊C4—24no—
328C22⋊C4—44no—
329C22⋊C4—82no—
330C42—22no—
331C42—22no—
332C42—44no—
333C42—122no—
334C42C2×Dic612yes—
335C42#23512yes—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3—81no—
340C22×C3—81no—
341C22×C3—81no—
342C22×C3—81no—
343C22×C3C2×C22⋊Q811yescharacteristic
344C22×C3#54611yescharacteristic
345C22×C3#54811yescharacteristic
346C22×C3C22×SD1611yescharacteristic
347C22×C3C2×AGL(1,ℤ/8)11yescharacteristic
348C22×C3#57611yescharacteristic
349C22×C3#57811yescharacteristic
350C3×C4—22no—
351C3×C4—24no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—82no—
355C3×C4#47614yes—
356C3×C4#53311yescharacteristic
357C3×C4#54411yescharacteristic
358C3×C4#55511yescharacteristic
359C3×C4#55611yescharacteristic
360Dic3—42no—
361Dic3—42no—
362Dic3—82no—
363C23—22no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—41no—
370C23C22×#6711yescharacteristic
371Q8—248no—
372D4—22no—
373D4—22no—
374D4—41no—
375D4—41no—
376C2×C4—22no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—48no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—122no—
397C2×C4—122no—
398C2×C4—122no—
399C2×C4#92412yes—
400C2×C4#92412yes—
401C2×C4C2×#21414yes—
402C2×C4C2×#21412yes—
403C2×C4#94912yes—
404C2×C4C22×Dic611yescharacteristic
405C2×C4C2×#23511yescharacteristic
406C2×C4D3×#4111yescharacteristic
407C2×C4D3×#4111yescharacteristic
408C2×C4#100911yescharacteristic
409C2×C4#101011yescharacteristic
410C8—121no—
411C8—122no—
412C8—121no—
413C2×C3—82no—
414C2×C3#243911yescharacteristic
415C2×C3C2×#47611yescharacteristic
416C2×C3#308311yescharacteristic
417C2×C3#329811yescharacteristic
418C2×C3#332511yescharacteristic
419C2×C3#336211yescharacteristic
420C2×C3#336711yescharacteristic
421C22—42no—
422C22—42no—
423C22—42no—
424C22—81no—
425C22—81no—
426C22—81no—
427C22—81no—
428C22C2×#92411yescharacteristic
429C22C22×#21411yescharacteristic
430C22C2×#94911yescharacteristic
431C22#582911yescharacteristic
432C22#583111yescharacteristic
433C22#583511yescharacteristic
434C22#583811yescharacteristic
435C4—24no—
436C4—22no—
437C4—42no—
438C4—42no—
439C4—82no—
440C4—122no—
441C4—122no—
442C4—242no—
443C4#500814yes—
444C4#551711yescharacteristic
445C4#551911yescharacteristic
446C4#553211yescharacteristic
447C4#553311yescharacteristic
448C3#2602411yescharacteristic, complemented, Sylow
449C2—82no—
450C2#8224811yescharacteristic
451C2C2×#500811yescharacteristic
452C2#8255911yescharacteristic
453C2#8353111yescharacteristic
454C2#8358611yescharacteristic
455C2#8365411yescharacteristic
456C2#8366811yescharacteristic
subgroupG/H
centerC23C22×#67
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2354C2
socleC23×C3C22×D4
prime radicalC22×C3×D4C23
hypercenter#2354D3
hypocenterC3#26024
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2354

Complements

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

NH ≅ G/Ntype
2. #73715C2semidirect
7. #69574C2semidirect
8. #69568C2semidirect
9. #69489C2semidirect
10. #69484C2semidirect
11. #69460C2semidirect
57. C3×C4×D4Q8semidirect
194. C4×D4Dic6semidirect
448. C3#26024semidirect

Maximal subgroups

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

subgroupindex
C3×#23542
#737152
C4×#8322
#716042
#696522
#696382
#695742
#695682
#694892
#694842
#694602
#260243