Finite Group Databasebeta

← #10798328

Subgroups — #10798328

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
1C3×#2320C211yesmaximal, characteristic, complemented
2C2×C4×#215C211yesmaximal, characteristic
3#73412C211yesmaximal, characteristic
4#71631C211yesmaximal, characteristic, complemented
5#69687C212yesmaximal
6#69667C212yesmaximal, complemented
7#69625C212yesmaximal
8#69602C212yesmaximal, complemented
9C4×#807C211yesmaximal, characteristic, complemented
10#69474C211yesmaximal, characteristic
11#69435C211yesmaximal, characteristic, complemented
12#21130—31nomaximal, Sylow
13C2×C3×C4×Q8C2211yescharacteristic
14C22×#215C2211yescharacteristic
15C2×C4×D12C2212yes—
16C2×C4×D12C2211yescharacteristic
17C3×#391C2212yes—
18C3×#389C2212yes—
19C3×#387—22no—
20C3×#383—21no—
21C3×#382—21no—
22C3×C4×C4⋊C4C2211yescharacteristic
23C3×#376C2211yescharacteristic
24C2×#835C2212yes—
25C2×#835C2211yescharacteristic
26C4×#215C418yescomplemented
27C2×#809C2212yes—
28C2×#809C2211yescharacteristic
29C2×#808C2211yescharacteristic
30C2×#807C2212yes—
31C4×C4⋊Dic3C2212yes—
32C4×C4⋊Dic3C2211yescharacteristic
33C2×#803C2211yescharacteristic
34C2×C4×C3⋊C8C2211yescharacteristic
35#4666—21no—
36#4666—22no—
37D3×#95—21no—
38#4529C2212yes—
39#4529C2212yes—
40#4529C2212yes—
41#4524C2212yes—
42#4524C2212yes—
43C2×C4×SD16—31no—
44#2320D311yescharacteristic, complemented
45#2275—31no—
46#1928—32no—
47#1920—32no—
48#1877—32no—
49#1862—32no—
50C4×#107—31no—
51#1763—31no—
52#1762—31no—
53#1746—31no—
54C22×C3×Q8C2311yescharacteristic
55C22×D12C2311yescharacteristic
56C22×C4×D3—22no—
57C22×C4×D3—21no—
58C3×C4×Q8—22no—
59C3×C4×Q8C2×C414yescomplemented
60C2×C3×C4⋊C4—21no—
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C42—21no—
65C2×C3×C42—21no—
66C2×C3×C42C2312yes—
67C2×C3×C42C2311yescharacteristic
68C2×#215—24no—
69C2×#215C2×C414yes—
70C2×(C2×D3)⋊C4—22no—
71C2×(C2×D3)⋊C4—21no—
72C2×C4⋊Dic3C2311yescharacteristic
73C2×C4⋊Dic3C2312yes—
74C2×C4×Dic3—22no—
75C2×C4×Dic3—21no—
76C22×C3⋊C8C2312yes—
77C22×C3⋊C8C2311yescharacteristic
78C22×C3⋊C8C2311yescharacteristic
79C4×D12—24no—
80C4×D12—22no—
81C4×D12—24no—
82C4×D12C2×C414yes—
83C3×#95—21no—
84C3×#95—22no—
85C3×#95—22no—
86C3×#95—22no—
87C3×#95D411yescharacteristic
88C3×#95D411yescharacteristic
89C3×#95D412yes—
90#835—24no—
91#835C2×C414yescomplemented
92C3⋊#95—22no—
93C3⋊#95—21no—
94#809—24no—
95#809C2×C414yes—
96#808C2×C414yescomplemented
97#807—24no—
98#803—22no—
99C4×C3⋊C8C2×C414yescomplemented
100C22×SD16—31no—
101C2×C4×Q8C2×D311yescharacteristic
102C2×C4×D4—31no—
103C2×C4×D4—32no—
104C4×SD16—38no—
105C2×#107—32no—
106C2×#106—31no—
107C2×#103—31no—
108C2×#103—32no—
109C2×#102—31no—
110C2×#102—32no—
111C2×#97—31no—
112C2×C4×C8—31no—
113#391C2×D312yes—
114#389C2×D312yes—
115#387—22no—
116#383—21no—
117#382—21no—
118C4×C4⋊C4—32no—
119C4×C4⋊C4—31no—
120C4×C4⋊C4C2×D311yescharacteristic
121C4×C22⋊C4—61no—
122C4×C22⋊C4—62no—
123#376C2×D311yescharacteristic
124C2×#95—61no—
125#340—32no—
126#340—32no—
127#340—32no—
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×D412yes—
139C22×C3×C4C2×D412yes—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2411yescharacteristic
143C22×Dic3—21no—
144C22×Dic3—22no—
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—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—82no—
159C3×C4⋊C4C22×C412yes—
160C3×C42—24no—
161C3×C42—22no—
162C3×C42—22no—
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⋊Dic3C22×C412yes—
172C4⋊Dic3#4114yes—
173C4×Dic3—81no—
174C4×Dic3—82no—
175C2×C3⋊C8—22no—
176C2×C3⋊C8—22no—
177C2×C3⋊C8—22no—
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×C8—32no—
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—22no—
199C2×C4⋊C4—31no—
200C2×C4⋊C4—32no—
201C2×C4⋊C4C22×D312yes—
202C2×C4⋊C4C22×D311yescharacteristic
203C2×C22⋊C4—62no—
204C2×C22⋊C4—61no—
205C2×C42—21no—
206C2×C42—21no—
207C2×C42—62no—
208C2×C42—61no—
209C2×C42C22×D312yes—
210C2×C42C22×D311yescharacteristic
211#107—64no—
212#106—34no—
213#103—34no—
214#103—64no—
215#102—34no—
216#102—64no—
217#97—62no—
218C4×C8—34no—
219#95—21no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—62no—
224#95—61no—
225#95#6711yescharacteristic
226#95#6711yescharacteristic
227#95#6712yes—
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—22no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—42no—
253C2×C3×C4—42no—
254C2×C3×C4C4×D412yes—
255C2×C3×C4C4×D412yes—
256C2×C3×C4C23×C411yescharacteristic
257C2×C3×C4C2×#4111yescharacteristic
258C2×C3×C4C2×#4112yes—
259C2×C3×C4C2×#4112yes—
260C2×Dic3—22no—
261C2×Dic3—24no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265D12—22no—
266D12—24no—
267D12#11714yes—
268C4×D3—84no—
269C4×D3—88no—
270C3⋊C8—22no—
271C3⋊C8—22no—
272C3⋊C8—42no—
273C24—61no—
274C2×Q8—21no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8C2×C4×D312yes—
278C2×D4—32no—
279C2×D4—34no—
280C2×D4—61no—
281C2×D4—62no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—22no—
285C22×C4—68no—
286C22×C4—61no—
287C22×C4—62no—
288C22×C4—64no—
289C22×C4—242no—
290C22×C4—242no—
291C22×C4—242no—
292C22×C4C2×#6712yes—
293C22×C4C2×#6712yes—
294C22×C4C2×#6711yescharacteristic
295C22×C4C2×#6711yescharacteristic
296C22×C4C23×D311yescharacteristic
297SD16—68no—
298C2×C8—32no—
299C2×C8—32no—
300C2×C8—62no—
301C2×C8—62no—
302C2×C8—62no—
303C2×C8—62no—
304C2×C8—62no—
305C4⋊C4—24no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—32no—
310C4⋊C4—34no—
311C4⋊C4—61no—
312C4⋊C4—62no—
313C4⋊C4—82no—
314C4⋊C4C2×C4×D312yes—
315C22⋊C4—68no—
316C22⋊C4—64no—
317C42—24no—
318C42—22no—
319C42—22no—
320C42—81no—
321C42—241no—
322C42—242no—
323C42C2×C4×D312yes—
324C42#23512yes—
325C22×C3C2×C4×D411yescharacteristic
326C22×C3#52512yes—
327C22×C3#57611yescharacteristic
328C22×C3#57712yes—
329C22×C3#57811yescharacteristic
330C2×D3—24no—
331C2×D3—44no—
332C2×D3—44no—
333C2×D3—42no—
334C2×D3—42no—
335C3×C4—22no—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—42no—
340C3×C4—42no—
341C3×C4—82no—
342C3×C4C2×#11711yescharacteristic
343C3×C4C4×#4111yescharacteristic
344C3×C4#52912yes—
345Dic3—44no—
346Dic3—42no—
347C23—62no—
348C23—64no—
349C23—121no—
350C23—122no—
351C23—121no—
352C23C22×#6711yescharacteristic
353Q8—22no—
354Q8—22no—
355Q8—42no—
356D4—34no—
357D4—62no—
358D4—64no—
359C2×C4—21no—
360C2×C4—22no—
361C2×C4—21no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—62no—
377C2×C4—64no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—122no—
381C2×C4—244no—
382C2×C4—248no—
383C2×C4C4×#6712yes—
384C2×C4C4×#6712yes—
385C2×C4C22×C4×D311yescharacteristic
386C2×C4C2×#23511yescharacteristic
387C2×C4D3×#4112yes—
388C2×C4D3×#4112yes—
389C8—62no—
390C8—62no—
391C8—122no—
392C2×C3#229011yescharacteristic
393C2×C3#294911yescharacteristic
394C2×C3#295111yescharacteristic
395C2×C3#311812yes—
396C2×C3#312012yes—
397D3—44no—
398C22—64no—
399C22—124no—
400C22—122no—
401C22—122no—
402C22—124no—
403C22C2×C4×#6711yescharacteristic
404C22#582812yes—
405C22#583511yescharacteristic
406C22#583712yes—
407C22#583811yescharacteristic
408C4—22no—
409C4—22no—
410C4—42no—
411C4—42no—
412C4—42no—
413C4—42no—
414C4—82no—
415C4—124no—
416C4—122no—
417C4D3×#11711yescharacteristic
418C4#552211yescharacteristic
419C4#552812yes—
420C3#2113011yescharacteristic, complemented, Sylow
421C2—124no—
422C2#8226811yescharacteristic
423C2#8254811yescharacteristic
424C2#8255711yescharacteristic
425C2#8344612yes—
426C2#8346012yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2320C2
socleC23×C3C22×D4
prime radicalC22×D12C23
cosocleC22×C3×C4C24
hypercenter#2320D3
hypocenterC3#21130
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2320

Complements

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

NH ≅ G/Ntype
1. C3×#2320C2semidirect
4. #71631C2semidirect
6. #69667C2semidirect
8. #69602C2semidirect
9. C4×#807C2semidirect
11. #69435C2semidirect
26. C4×#215C4semidirect
44. #2320D3semidirect
59. C3×C4×Q8C2×C4semidirect
91. #835C2×C4semidirect
96. #808C2×C4semidirect
99. C4×C3⋊C8C2×C4semidirect
192. C4×Q8C4×D3semidirect
420. C3#21130semidirect

Maximal subgroups

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

subgroupindex
C3×#23202
C2×C4×#2152
#734122
#716312
#696872
#696672
#696252
#696022
C4×#8072
#694742
#694352
#211303