Finite Group Databasebeta

← #10798370

Subgroups — #10798370

Order 768.

890 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82454C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#73751C211yesmaximal, characteristic
4C3×#1763C211yesmaximal, characteristic, complemented
5#70276C212yesmaximal
6#70241C212yesmaximal, complemented
7#69812C211yesmaximal, characteristic
8#69806C211yesmaximal, characteristic, complemented
9C4×#808C211yesmaximal, characteristic, complemented
10#68895C212yesmaximal
11#68817C212yesmaximal, complemented
12#21133—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2211yescharacteristic
15C2×#893C2212yes—
16C2×C4×D12C2211yescharacteristic
17C2×C3×#103C2212yes—
18C2×C3×#97C2211yescharacteristic
19C3×C4×C4⋊C4C2211yescharacteristic
20#5229C2212yes—
21C2×#835C2211yescharacteristic
22C2×#821C2212yes—
23C4×#215C418yescomplemented
24#4984C2212yes—
25#4980—22no—
26Dic3×C4⋊C4C2212yes—
27C2×#809C2211yescharacteristic
28C2×#808C2212yes—
29C2×#808C2211yescharacteristic
30C4×(C2×D3)⋊C4—22no—
31#4937—21no—
32#4936—21no—
33C4×C4⋊Dic3C2211yescharacteristic
34#4932C2211yescharacteristic
35C2×C4×C3⋊C8C2211yescharacteristic
36#4666—21no—
37D3×#95—21no—
38C3×#340C2212yes—
39#4551C2212yes—
40#4550C2212yes—
41#4550C2212yes—
42#4529C2212yes—
43C2×C4×SD16—31no—
44#2320—31no—
45#2275—31no—
46#1927—32no—
47#1919—32no—
48#1877—32no—
49#1864—32no—
50C4×#106—31no—
51#1763D311yescharacteristic, complemented
52#1762—31no—
53#1746—31no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—21no—
57C22×C4×D3—22no—
58C22×C3×C8C2312yes—
59C3×C4×Q8—22no—
60C3×C4×Q8C2×C414yescomplemented
61C2×C3×C4⋊C4—21no—
62C2×C3×C4⋊C4C2311yescharacteristic
63C2×C3×C4⋊C4C2312yes—
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×#215—24no—
68C2×#215C2×C414yes—
69C2×(C2×D3)⋊C4—21no—
70C2×(C2×D3)⋊C4—22no—
71C2×C4⋊Dic3—22no—
72C2×C4⋊Dic3C2312yes—
73C2×C4⋊Dic3C2311yescharacteristic
74C2×C4×Dic3—21no—
75C2×C4×Dic3—22no—
76C2×C4×Dic3C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79#893—24no—
80#893—24no—
81C4×D12—22no—
82C4×D12C2×C414yes—
83C3×#103—24no—
84C3×#97—22no—
85C3×#95—21no—
86#835C2×C414yescomplemented
87C3⋊#95—21no—
88C3⋊#95—22no—
89C3⋊#95—22no—
90C3⋊#95—22no—
91C3⋊#95—22no—
92C3⋊#95D411yescharacteristic
93C3⋊#95D411yescharacteristic
94C3⋊#95D412yes—
95#821—24no—
96#809C2×C414yes—
97#808—24no—
98#808C2×C414yescomplemented
99C4×C3⋊C8C2×C414yescomplemented
100C22×SD16—31no—
101C2×C4×Q8C2×D311yescharacteristic
102C2×C4×D4—32no—
103C2×C4×D4—31no—
104C4×SD16—38no—
105C2×#106—31no—
106C2×#106—32no—
107C2×#103—31no—
108C2×#103C2×D312yes—
109C2×#102—31no—
110C2×#102—32no—
111C2×#97C2×D311yescharacteristic
112C2×C4×C8—31no—
113#391—32no—
114#389—32no—
115#387—62no—
116#383—61no—
117#382—61no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4—31no—
120C4×C4⋊C4C2×D311yescharacteristic
121C4×C22⋊C4—61no—
122C4×C22⋊C4—62no—
123#376—31no—
124C2×#95—61no—
125#340—32no—
126#340—32no—
127#340C2×D312yes—
128#336—32no—
129#336—32no—
130C23×D3—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8C22×C412yes—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—22no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2411yescharacteristic
141C22×Dic3—21no—
142C22×Dic3—22no—
143C22×Dic3C2×D412yes—
144C22×Dic3C2×D412yes—
145C2×D12—21no—
146C2×D12—22no—
147C2×D12C22×C412yes—
148C2×D12#4114yes—
149C2×C4×D3—28no—
150C2×C4×D3—24no—
151C2×C4×D3—82no—
152C2×C4×D3—82no—
153C2×C4×D3—82no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C3×C4⋊C4—24no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4C22×C412yes—
161C3×C4⋊C4#4114yes—
162C3×C42—24no—
163C3×C42—81no—
164C3×C42C22×C412yes—
165C3×C42#4112yes—
166#215—28no—
167(C2×D3)⋊C4—28no—
168(C2×D3)⋊C4—24no—
169C4⋊Dic3—21no—
170C4⋊Dic3—22no—
171C4⋊Dic3—22no—
172C4⋊Dic3—82no—
173C4⋊Dic3C22×C412yes—
174C4×Dic3—22no—
175C4×Dic3—22no—
176C4×Dic3—82no—
177C4×Dic3—81no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8C22×C412yes—
181C2×C3⋊C8C22×C412yes—
182C22×Q8C22×D311yescharacteristic
183C22×D4—31no—
184C23×C4—62no—
185C23×C4—61no—
186C2×SD16—34no—
187C2×SD16—64no—
188C22×C8—31no—
189C22×C8—31no—
190C22×C8C22×D312yes—
191C4×Q8—22no—
192C4×Q8C4×D314yescomplemented
193C4×D4—34no—
194C4×D4—64no—
195C4×D4—64no—
196C4×D4—62no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—32no—
200C2×C4⋊C4—62no—
201C2×C4⋊C4C22×D311yescharacteristic
202C2×C4⋊C4C22×D312yes—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—61no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—32no—
208C2×C42—62no—
209C2×C42—61no—
210C2×C42C22×D311yescharacteristic
211#106—34no—
212#106—64no—
213#103—24no—
214#103—34no—
215#102—34no—
216#102—64no—
217#97—22no—
218C4×C8—34no—
219#95—21no—
220#95—31no—
221#95—31no—
222#95—32no—
223#95—62no—
224#95—62no—
225#95—62no—
226#95—62no—
227#95—61no—
228C23×C3C22×D411yescharacteristic
229C22×D3—22no—
230C22×D3—24no—
231C22×D3—41no—
232C22×D3—42no—
233C22×D3—41no—
234C3×Q8—22no—
235C3×Q8—22no—
236C3×Q8—42no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—22no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—42no—
247C2×C3×C4—41no—
248C2×C3×C4C4×D412yes—
249C2×C3×C4C4×D412yes—
250C2×C3×C4C23×C411yescharacteristic
251C2×C3×C4C2×#4112yes—
252C2×C3×C4C2×#4111yescharacteristic
253C2×C3×C4C2×#4112yes—
254C2×Dic3—22no—
255C2×Dic3—22no—
256C2×Dic3—22no—
257C2×Dic3—22no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—22no—
261C2×Dic3—42no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265D12—22no—
266D12—24no—
267D12#11714yes—
268C4×D3—84no—
269C4×D3—88no—
270C3×C8—42no—
271C3⋊C8—22no—
272C3⋊C8—22no—
273C24—61no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8C2×C4×D312yes—
278C2×D4—32no—
279C2×D4—34no—
280C2×D4—62no—
281C2×D4—61no—
282C22×C4—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—32no—
286C22×C4—32no—
287C22×C4—68no—
288C22×C4—61no—
289C22×C4—64no—
290C22×C4—62no—
291C22×C4—242no—
292C22×C4—242no—
293C22×C4—242no—
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296C22×C4C23×D311yescharacteristic
297SD16—68no—
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—32no—
302C2×C8—32no—
303C2×C8—62no—
304C2×C8—62no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—32no—
309C4⋊C4—61no—
310C4⋊C4—62no—
311C4⋊C4—62no—
312C4⋊C4—242no—
313C4⋊C4C2×C4×D312yes—
314C4⋊C4#23514yes—
315C22⋊C4—68no—
316C22⋊C4—64no—
317C42—24no—
318C42—62no—
319C42—62no—
320C42—81no—
321C42—241no—
322C42—242no—
323C42C2×C4×D312yes—
324C42#23712yes—
325C22×C3C2×C4×D411yescharacteristic
326C22×C3#52512yes—
327C22×C3#57711yescharacteristic
328C22×C3#57712yes—
329C22×C3#57711yescharacteristic
330C2×D3—24no—
331C2×D3—44no—
332C2×D3—44no—
333C2×D3—42no—
334C2×D3—42no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—44no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4C2×#11711yescharacteristic
341C3×C4C4×#4111yescharacteristic
342C3×C4#52912yes—
343Dic3—42no—
344Dic3—42no—
345Dic3—42no—
346Dic3—82no—
347C23—62no—
348C23—64no—
349C23—121no—
350C23—122no—
351C23—121no—
352C23C2×D3×D411yescharacteristic
353Q8—22no—
354Q8—22no—
355Q8—42no—
356D4—34no—
357D4—62no—
358D4—64no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—24no—
365C2×C4—42no—
366C2×C4—41no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—62no—
373C2×C4—62no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—62no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—244no—
382C2×C4—248no—
383C2×C4#67⋊C412yes—
384C2×C4#67⋊C412yes—
385C2×C4C22×C4×D311yescharacteristic
386C2×C4C2×#23512yes—
387C2×C4C2×#23711yescharacteristic
388C2×C4D3×#4112yes—
389C8—42no—
390C8—62no—
391C8—62no—
392C2×C3#229011yescharacteristic
393C2×C3#294911yescharacteristic
394C2×C3#295211yescharacteristic
395C2×C3#312312yes—
396C2×C3#312512yes—
397D3—44no—
398C22—64no—
399C22—122no—
400C22—122no—
401C22—124no—
402C22—124no—
403C22C2×#67⋊C411yescharacteristic
404C22#554712yes—
405C22#577511yescharacteristic
406C22#577811yescharacteristic
407C22#577812yes—
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—44no—
413C4—122no—
414C4—122no—
415C4—122no—
416C4—242no—
417C4C4×#23511yescharacteristic
418C4D3×#11711yescharacteristic
419C4#555712yes—
420C3#2113311yescharacteristic, complemented, Sylow
421C2—124no—
422C2#7356211yescharacteristic
423C2#7886911yescharacteristic
424C2#7894711yescharacteristic
425C2#7978712yes—
426C2#7985012yes—
subgroupG/H
centerC23C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1763C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#1763D3
hypocenterC3#21133
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1763

Complements

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

NH ≅ G/Ntype
1. #82454C2semidirect
4. C3×#1763C2semidirect
6. #70241C2semidirect
8. #69806C2semidirect
9. C4×#808C2semidirect
11. #68817C2semidirect
23. C4×#215C4semidirect
51. #1763D3semidirect
60. C3×C4×Q8C2×C4semidirect
86. #835C2×C4semidirect
98. #808C2×C4semidirect
99. C4×C3⋊C8C2×C4semidirect
192. C4×Q8C4×D3semidirect
420. C3#21133semidirect

Maximal subgroups

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

subgroupindex
#824542
C2×C4×#2152
#737512
C3×#17632
#702762
#702412
#698122
#698062
C4×#8082
#688952
#688172
#211333