Finite Group Databasebeta

← #10798317

Subgroups — #10798317

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2319C211yesmaximal, characteristic, complemented
2C2×C4×#213C211yesmaximal, characteristic
3#73412C211yesmaximal, characteristic, complemented
4#71609C211yesmaximal, characteristic, complemented
5#69657C212yesmaximal
6#69636C212yesmaximal, complemented
7#69625C212yesmaximal, complemented
8#69603C212yesmaximal, complemented
9C4×#807C211yesmaximal, characteristic, complemented
10#69474C211yesmaximal, characteristic, complemented
11#69435C211yesmaximal, characteristic, complemented
12#21129—31nomaximal, Sylow
13C2×C3×C4×D4C2211yescharacteristic
14C22×#213C2211yescharacteristic
15C2×C4×D12C2212yes—
16C2×C4×D12C2211yescharacteristic
17C3×#390C2212yes—
18C3×#389C2212yes—
19C3×#388—22no—
20C3×#381—21no—
21C3×#380—21no—
22C3×C4×C4⋊C4C2211yescharacteristic
23C3×#376C2211yescharacteristic
24C2×#832C2212yes—
25C2×#832C2211yescharacteristic
26C4×#213C418yescomplemented
27C2×#809C2212yes—
28C2×#809C2211yescharacteristic
29C2×#807C2212yes—
30C2×#807C2211yescharacteristic
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×D8—31no—
44#2319D311yescharacteristic, complemented
45#2275—31no—
46#1925—32no—
47#1921—32no—
48#1879—32no—
49#1862—32no—
50C4×#107—31no—
51#1762—31no—
52#1762—31no—
53#1746—31no—
54C22×C3×D4C2311yescharacteristic
55C23×C3×C4—21no—
56C22×D12C2311yescharacteristic
57C22×C4×D3—22no—
58C22×C4×D3—21no—
59C3×C4×D4—22no—
60C3×C4×D4C2×C414yescomplemented
61C2×C3×C4⋊C4—22no—
62C2×C3×C4⋊C4C2312yes—
63C2×C3×C4⋊C4C2311yescharacteristic
64C2×C3×C22⋊C4—21no—
65C2×C3×C22⋊C4—22no—
66C2×C3×C22⋊C4—22no—
67C2×C3×C42—21no—
68C2×C3×C42C2312yes—
69C2×C3×C42C2311yescharacteristic
70C2×#213—24no—
71C2×#213C2×C414yes—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—21no—
74C2×C4⋊Dic3C2311yescharacteristic
75C2×C4⋊Dic3C2312yes—
76C2×C4×Dic3—22no—
77C2×C4×Dic3—21no—
78C22×C3⋊C8C2312yes—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C4×D12—24no—
82C4×D12—22no—
83C4×D12—24no—
84C4×D12C2×C414yes—
85C3×#95—21no—
86C3×#95—22no—
87C3×#95D411yescharacteristic
88C3×#95D411yescharacteristic
89C3×#95D412yes—
90#832—24no—
91#832C2×C414yescomplemented
92C3⋊#95—22no—
93C3⋊#95—21no—
94#809—24no—
95#809C2×C414yes—
96#807—24no—
97#807C2×C414yescomplemented
98#803—22no—
99C4×C3⋊C8C2×C414yescomplemented
100C22×D8—31no—
101C2×C4×D4—31no—
102C2×C4×D4—32no—
103C2×C4×D4C2×D311yescharacteristic
104C4×D8—38no—
105C2×#107—31no—
106C2×#107—32no—
107C2×#102—31no—
108C2×#102—31no—
109C2×#102—32no—
110C2×#102—32no—
111C2×#97—31no—
112C2×C4×C8—31no—
113#390C2×D312yes—
114#389C2×D312yes—
115#388—22no—
116#381—21no—
117#380—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—
130C24×C3—21no—
131C23×D3—21no—
132C2×C3×D4—22no—
133C2×C3×D4—22no—
134C2×C3×D4—21no—
135C2×C3×D4C22×C412yes—
136C22×C3×C4—21no—
137C22×C3×C4—24no—
138C22×C3×C4—22no—
139C22×C3×C4—82no—
140C22×C3×C4C2×D412yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4C2411yescharacteristic
145C22×Dic3—21no—
146C22×Dic3—22no—
147C2×D12—21no—
148C2×D12—22no—
149C2×D12C22×C412yes—
150C2×D12#4114yes—
151C2×C4×D3—28no—
152C2×C4×D3—24no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C2×C4×D3—82no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—22no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—82no—
160C3×C4⋊C4C22×C412yes—
161C3×C22⋊C4—24no—
162C3×C22⋊C4—82no—
163C3×C22⋊C4—82no—
164C3×C42—22no—
165C3×C42—22no—
166C3×C42—81no—
167C3×C42C22×C412yes—
168C3×C42#4112yes—
169#213—28no—
170(C2×D3)⋊C4—28no—
171(C2×D3)⋊C4—24no—
172C4⋊Dic3—21no—
173C4⋊Dic3—22no—
174C4⋊Dic3C22×C412yes—
175C4⋊Dic3#4114yes—
176C4×Dic3—81no—
177C4×Dic3—82no—
178C2×C3⋊C8—22no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—22no—
183C2×C3⋊C8C22×C412yes—
184C2×C3⋊C8C22×C412yes—
185C22×D4—31no—
186C22×D4C22×D311yescharacteristic
187C23×C4—21no—
188C23×C4—62no—
189C23×C4—61no—
190C2×D8—34no—
191C2×D8—64no—
192C22×C8—31no—
193C22×C8—31no—
194C22×C8—32no—
195C4×D4—22no—
196C4×D4—34no—
197C4×D4—64no—
198C4×D4—64no—
199C4×D4—62no—
200C4×D4C4×D314yescomplemented
201C2×C4⋊C4—22no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4C22×D312yes—
205C2×C4⋊C4C22×D311yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—22no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C42—21no—
212C2×C42—62no—
213C2×C42—61no—
214C2×C42C22×D312yes—
215C2×C42C22×D311yescharacteristic
216#107—34no—
217#107—64no—
218#102—34no—
219#102—34no—
220#102—64no—
221#102—64no—
222#97—62no—
223C4×C8—34no—
224#95—21no—
225#95—22no—
226#95—62no—
227#95—61no—
228#95#6711yescharacteristic
229#95#6711yescharacteristic
230#95#6712yes—
231C23×C3—22no—
232C23×C3—42no—
233C23×C3—41no—
234C23×C3—42no—
235C23×C3—41no—
236C23×C3C22×D411yescharacteristic
237C22×D3—22no—
238C22×D3—24no—
239C22×D3—41no—
240C22×D3—42no—
241C22×D3—41no—
242C3×D4—22no—
243C3×D4—22no—
244C3×D4—42no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—84no—
260C2×C3×C4C4×D412yes—
261C2×C3×C4C4×D412yes—
262C2×C3×C4C23×C411yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264C2×C3×C4C2×#4112yes—
265C2×C3×C4C2×#4112yes—
266C2×Dic3—22no—
267C2×Dic3—24no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271D12—22no—
272D12—24no—
273D12#11714yes—
274C4×D3—84no—
275C4×D3—88no—
276C3⋊C8—22no—
277C3⋊C8—22no—
278C3⋊C8—42no—
279C24—21no—
280C24—61no—
281C2×D4—21no—
282C2×D4—22no—
283C2×D4—22no—
284C2×D4—32no—
285C2×D4—34no—
286C2×D4—61no—
287C2×D4—62no—
288C2×D4C2×C4×D312yes—
289C22×C4—21no—
290C22×C4—24no—
291C22×C4—22no—
292C22×C4—68no—
293C22×C4—61no—
294C22×C4—62no—
295C22×C4—64no—
296C22×C4—82no—
297C22×C4—242no—
298C22×C4—242no—
299C22×C4—242no—
300C22×C4C2×#6712yes—
301C22×C4C2×#6712yes—
302C22×C4C2×#6711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304C22×C4C23×D311yescharacteristic
305D8—68no—
306C2×C8—32no—
307C2×C8—32no—
308C2×C8—62no—
309C2×C8—62no—
310C2×C8—62no—
311C2×C8—62no—
312C2×C8—62no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—32no—
317C4⋊C4—34no—
318C4⋊C4—61no—
319C4⋊C4—62no—
320C4⋊C4—82no—
321C4⋊C4C2×C4×D312yes—
322C22⋊C4—24no—
323C22⋊C4—68no—
324C22⋊C4—64no—
325C22⋊C4—82no—
326C22⋊C4—82no—
327C42—22no—
328C42—22no—
329C42—81no—
330C42—241no—
331C42—242no—
332C42C2×C4×D312yes—
333C42#23512yes—
334C22×C3—42no—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—82no—
338C22×C3—82no—
339C22×C3C2×C4×D411yescharacteristic
340C22×C3#52512yes—
341C22×C3#57611yescharacteristic
342C22×C3#57612yes—
343C22×C3#57811yescharacteristic
344C2×D3—24no—
345C2×D3—44no—
346C2×D3—44no—
347C2×D3—42no—
348C2×D3—42no—
349C3×C4—22no—
350C3×C4—22no—
351C3×C4—42no—
352C3×C4—42no—
353C3×C4—42no—
354C3×C4—82no—
355C3×C4C2×#11711yescharacteristic
356C3×C4C4×#4111yescharacteristic
357C3×C4#52912yes—
358Dic3—44no—
359Dic3—42no—
360C23—22no—
361C23—42no—
362C23—41no—
363C23—42no—
364C23—41no—
365C23—62no—
366C23—64no—
367C23—122no—
368C23—121no—
369C23—121no—
370C23C22×#6711yescharacteristic
371D4—22no—
372D4—22no—
373D4—34no—
374D4—42no—
375D4—62no—
376D4—64no—
377C2×C4—21no—
378C2×C4—22no—
379C2×C4—21no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—62no—
392C2×C4—64no—
393C2×C4—84no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4—244no—
398C2×C4—248no—
399C2×C4C4×#6712yes—
400C2×C4C4×#6712yes—
401C2×C4C22×C4×D311yescharacteristic
402C2×C4C2×#23511yescharacteristic
403C2×C4D3×#4112yes—
404C2×C4D3×#4112yes—
405C8—62no—
406C8—62no—
407C8—122no—
408C2×C3—82no—
409C2×C3#229011yescharacteristic
410C2×C3#294811yescharacteristic
411C2×C3#295111yescharacteristic
412C2×C3#312212yes—
413C2×C3#312412yes—
414D3—44no—
415C22—42no—
416C22—42no—
417C22—42no—
418C22—64no—
419C22—82no—
420C22—82no—
421C22—124no—
422C22—122no—
423C22—122no—
424C22—124no—
425C22C2×C4×#6711yescharacteristic
426C22#582812yes—
427C22#583512yes—
428C22#583511yescharacteristic
429C22#583811yescharacteristic
430C4—22no—
431C4—22no—
432C4—42no—
433C4—42no—
434C4—42no—
435C4—82no—
436C4—122no—
437C4—124no—
438C4D3×#11711yescharacteristic
439C4#552211yescharacteristic
440C4#552812yes—
441C3#2112911yescharacteristic, complemented, Sylow
442C2—82no—
443C2—124no—
444C2#8226811yescharacteristic
445C2#8253811yescharacteristic
446C2#8255711yescharacteristic
447C2#8343512yes—
448C2#8346112yes—
subgroupG/H
centerC23C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2319C2
socleC23×C3C22×D4
prime radicalC22×#213C22
cosocleC22×C3×C4C24
hypercenter#2319D3
hypocenterC3#21129
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ #376 ◁ #2319

Complements

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

NH ≅ G/Ntype
1. C3×#2319C2semidirect
3. #73412C2semidirect
4. #71609C2semidirect
6. #69636C2semidirect
7. #69625C2semidirect
8. #69603C2semidirect
9. C4×#807C2semidirect
10. #69474C2semidirect
11. #69435C2semidirect
26. C4×#213C4semidirect
44. #2319D3semidirect
60. C3×C4×D4C2×C4semidirect
91. #832C2×C4semidirect
97. #807C2×C4semidirect
99. C4×C3⋊C8C2×C4semidirect
200. C4×D4C4×D3semidirect
441. C3#21129semidirect

Maximal subgroups

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

subgroupindex
C3×#23192
C2×C4×#2132
#734122
#716092
#696572
#696362
#696252
#696032
C4×#8072
#694742
#694352
#211293