Finite Group Databasebeta

← #10824665

Subgroups — #10824665

Order 768.

932 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#433C211yesmaximal, characteristic, complemented
2#82236C212yesmaximal, complemented
3C2×C4×D24C211yesmaximal, characteristic, complemented
4#70483C212yesmaximal, complemented
5#70481C211yesmaximal, characteristic, complemented
6#70462C211yesmaximal, characteristic, complemented
7#69961C211yesmaximal, characteristic
8#69938C211yesmaximal, characteristic, complemented
9#69935C212yesmaximal, complemented
10#69812C212yesmaximal, complemented
11C4×#817C211yesmaximal, characteristic, complemented
12#23702—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14C2×#930C2212yes—
15C22×D24C2211yescharacteristic
16C2×C4×D12C2212yes—
17C3×#433—24no—
18C2×C3×C4⋊C8C2211yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C22⋊C8C2211yescharacteristic
21C2×C3×C22⋊C8C2211yescharacteristic
22C2×C3×#97C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24#5212C2212yes—
25C2×#821C2212yes—
26C2×#821C2212yes—
27C2×#821C2212yes—
28C2×#818C2211yescharacteristic
29C2×#817C2212yes—
30#4990—22no—
31#4989—22no—
32#4957—22no—
33#4956—22no—
34#4939C2212yes—
35C4×C4⋊Dic3C2212yes—
36#4928C2212yes—
37C4×D24—24no—
38#4551C2212yes—
39#4551C2211yescharacteristic
40#4551C2212yes—
41#4551C2211yescharacteristic
42C2×C4×D8—31no—
43C2×#433D311yescharacteristic, complemented
44#2324—32no—
45#2086—31no—
46#2075—31no—
47#2065—32no—
48#2054—31no—
49#2034—32no—
50#1999—31no—
51C4×#106—31no—
52#1762—32no—
53C23×C3×C4C2311yescharacteristic
54C22×#67—22no—
55C22×D12C2312yes—
56C22×C4×D3—22no—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C3×#117—22no—
62C3×#117—22no—
63C2×C3×C4⋊C4D411yescharacteristic
64C2×C3×C4⋊C4D411yescharacteristic
65C2×C3×C22⋊C4D411yescharacteristic
66C2×C3×C22⋊C4D411yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C2×C3×C42C2311yescharacteristic
69C2×#217—22no—
70#930—24no—
71#930—24no—
72C2×(C2×D3)⋊C4—22no—
73C2×(C2×D3)⋊C4—22no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—22no—
76C2×C4⋊Dic3C2312yes—
77C2×C4⋊Dic3C2312yes—
78C2×C4⋊Dic3C2312yes—
79C2×Dic3⋊C4—22no—
80C2×C4×Dic3—22no—
81C2×D24—24no—
82C2×D24—22no—
83C4×D12—24no—
84C4×D12—24no—
85C3×C4⋊C8—22no—
86C3×C4⋊C8—22no—
87C3×C22⋊C8—22no—
88C3×C22⋊C8—22no—
89C3×#97—22no—
90C3×C4×C8D414yescomplemented
91C3⋊#95—22no—
92C3⋊#95—22no—
93C3⋊#95—22no—
94#821—24no—
95#821—24no—
96#821—24no—
97#818—22no—
98#817—24no—
99C22×D8—31no—
100C2×#121—32no—
101C2×C4×D4—32no—
102C2×#117C2×D311yescharacteristic
103C4×D8—64no—
104#433—24no—
105C2×#107—31no—
106C2×#106—32no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×C4⋊C8C2×D311yescharacteristic
109C2×#102—32no—
110C2×#102—32no—
111C2×#102—32no—
112C2×C22⋊C8C2×D311yescharacteristic
113C2×C22⋊C8C2×D311yescharacteristic
114C2×#97C2×D311yescharacteristic
115C2×C4×C8C2×D311yescharacteristic
116#390—32no—
117#390—62no—
118#389—32no—
119#388—62no—
120#386—62no—
121#385—62no—
122#380—32no—
123C4×C4⋊C4—32no—
124#340—31no—
125#340—32no—
126#340—31no—
127#340—32no—
128C24×C3C2×D411yescharacteristic
129C23×D3—22no—
130C22×C3×C4—21no—
131C22×C3×C4—22no—
132C22×C3×C4—22no—
133C22×C3×C4—21no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4C2411yescharacteristic
140C2×#67—28no—
141C2×#67—82no—
142C2×#67—82no—
143C22×Dic3—22no—
144C22×Dic3—22no—
145C22×Dic3—22no—
146C2×D12—22no—
147C2×D12—22no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—22no—
151C2×D12#4114yes—
152C2×C4×D3—44no—
153C2×C4×D3—82no—
154C2×C4×D3—82no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—21no—
158C2×C3×C8—21no—
159C2×C3×C8—22no—
160C2×C3×C8—22no—
161C2×C3×C8C2×D412yes—
162C2×C3×C8C2×D412yes—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C22⋊C4—22no—
166C3×C22⋊C4—22no—
167C3×C42—21no—
168C3×C42—21no—
169C3×C42C2×D412yes—
170C3×C42#4112yes—
171#217—82no—
172(C2×D3)⋊C4—28no—
173(C2×D3)⋊C4—44no—
174(C2×D3)⋊C4—82no—
175(C2×D3)⋊C4—82no—
176C4⋊Dic3—22no—
177C4⋊Dic3—22no—
178C4⋊Dic3—22no—
179C4⋊Dic3—22no—
180C4⋊Dic3—22no—
181C4⋊Dic3#4114yes—
182Dic3⋊C4—82no—
183C4×Dic3—82no—
184D24—28no—
185C22×D4—32no—
186C22×D4—62no—
187C23×C4—62no—
188C23×C4C22×D311yescharacteristic
189C2×D8—64no—
190C2×D8—62no—
191C22×C8C22×D311yescharacteristic
192C22×C8C22×D311yescharacteristic
193C22×C8C22×D311yescharacteristic
194C22×C8C22×D311yescharacteristic
195#121—64no—
196#121—64no—
197C4×D4—64no—
198C4×D4—64no—
199#117—22no—
200#117—22no—
201C2×C4⋊C4—32no—
202C2×C4⋊C4—32no—
203C2×C4⋊C4—32no—
204C2×C4⋊C4—62no—
205C2×C4⋊C4D1211yescharacteristic
206C2×C4⋊C4D1211yescharacteristic
207C2×C22⋊C4—62no—
208C2×C22⋊C4—62no—
209C2×C22⋊C4—62no—
210C2×C22⋊C4—62no—
211C2×C22⋊C4—62no—
212C2×C22⋊C4D1211yescharacteristic
213C2×C22⋊C4D1211yescharacteristic
214C2×C42—62no—
215C2×C42C22×D311yescharacteristic
216C2×C42C22×D311yescharacteristic
217#107—62no—
218#106—64no—
219C4⋊C8—22no—
220C4⋊C8—22no—
221#102—64no—
222#102—64no—
223#102—64no—
224C22⋊C8—22no—
225C22⋊C8—22no—
226#97—22no—
227C4×C8#6714yescomplemented
228#95—62no—
229#95—62no—
230#95—62no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3—21no—
234C23×C3—21no—
235C23×C3—22no—
236C23×C3C22×D411yescharacteristic
237C22×D3—24no—
238C22×D3—42no—
239C22×D3—42no—
240C22×D3—42no—
241C22×D3—42no—
242C22×D3—42no—
243C22×D3—42no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—22no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—41no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4#12112yes—
258C2×C3×C4#12112yes—
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C2×#4111yescharacteristic
261C2×C3×C4C2×#4112yes—
262C2×C3×C4C2×#4112yes—
263#67—88no—
264C2×Dic3—24no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—42no—
268C2×Dic3—42no—
269C2×Dic3—42no—
270C2×Dic3—42no—
271C2×Dic3—42no—
272C2×Dic3—42no—
273D12—24no—
274D12—24no—
275D12—42no—
276D12—42no—
277C4×D3—88no—
278C3×C8—22no—
279C3×C8—22no—
280C3×C8—41no—
281C3×C8—41no—
282C24—62no—
283C24C2×D1211yescharacteristic
284C2×D4—34no—
285C2×D4—68no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—62no—
289C2×D4—62no—
290C2×D4—62no—
291C2×D4—242no—
292C2×D4—242no—
293C22×C4—21no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4—62no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—124no—
301C22×C4—242no—
302C22×C4—242no—
303C22×C4C2×D1211yescharacteristic
304C22×C4C2×D1211yescharacteristic
305C22×C4C2×D1211yescharacteristic
306C22×C4C2×D1211yescharacteristic
307C22×C4C2×D1211yescharacteristic
308C22×C4C23×D311yescharacteristic
309D8—68no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—21no—
315C2×C8—21no—
316C2×C8C2×#6712yes—
317C2×C8C2×#6712yes—
318C4⋊C4—22no—
319C4⋊C4—22no—
320C4⋊C4—34no—
321C4⋊C4—62no—
322C4⋊C4—62no—
323C4⋊C4—62no—
324C4⋊C4—62no—
325C4⋊C4—62no—
326C4⋊C4—242no—
327C22⋊C4—22no—
328C22⋊C4—22no—
329C22⋊C4—68no—
330C22⋊C4—124no—
331C22⋊C4—242no—
332C22⋊C4—242no—
333C22⋊C4—242no—
334C42—21no—
335C42—21no—
336C42—242no—
337C42#23512yes—
338C42C2×#6712yes—
339C22×C3—42no—
340C22×C3—42no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3C2×#12111yescharacteristic
345C22×C3#52512yes—
346C22×C3#57611yescharacteristic
347C22×C3#57611yescharacteristic
348C22×C3#57712yes—
349C2×D3—44no—
350C2×D3—44no—
351C2×D3—44no—
352C2×D3—82no—
353C2×D3—82no—
354C2×D3—82no—
355C2×D3—82no—
356C3×C4—22no—
357C3×C4—22no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—41no—
361C3×C4#52511yescharacteristic
362C3×C4#52912yes—
363C3×C4#53211yescharacteristic
364Dic3—44no—
365Dic3—82no—
366Dic3—82no—
367C23—22no—
368C23—21no—
369C23—21no—
370C23—22no—
371C23—21no—
372C23—64no—
373C23—122no—
374C23—122no—
375C23—122no—
376C23—122no—
377C23—122no—
378C23—122no—
379C23C22×D1211yescharacteristic
380D4—64no—
381D4—64no—
382D4—122no—
383D4—122no—
384D4—248no—
385C2×C4—22no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—22no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—64no—
399C2×C4—122no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—248no—
408C2×C4#93012yes—
409C2×C4#93012yes—
410C2×C4C2×#23511yescharacteristic
411C2×C4D3×#4112yes—
412C2×C4D3×#4112yes—
413C2×C4C22×#6711yescharacteristic
414C8—22no—
415C8—22no—
416C8—41no—
417C8—41no—
418C2×C3—81no—
419C2×C3#239211yescharacteristic
420C2×C3#305811yescharacteristic
421C2×C3#306011yescharacteristic
422C2×C3#311812yes—
423C2×C3#311912yes—
424D3—84no—
425C22—42no—
426C22—42no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—124no—
431C22—124no—
432C22—124no—
433C22—242no—
434C22—242no—
435C22—242no—
436C22—242no—
437C22#552412yes—
438C22#575112yes—
439C22#575211yescharacteristic
440C22#575211yescharacteristic
441C22C2×#93011yescharacteristic
442C4—22no—
443C4—22no—
444C4—41no—
445C4—41no—
446C4—41no—
447C4—124no—
448C4—242no—
449C4—242no—
450C4#552812yes—
451C4#582811yescharacteristic
452C4#583211yescharacteristic
453C3#2370211yescharacteristic, complemented, Sylow
454C2—81no—
455C2—244no—
456C2#7938312yes—
457C2#7938412yes—
458C2#8227311yescharacteristic
459C2#8379911yescharacteristic
460C2#8380011yescharacteristic
subgroupG/H
centerC23C22×D12
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC2×C3×#433C2
socleC23×C3C22×D4
prime radical#69961C2
hypercenterC2×#433D3
hypocenterC3#23702
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#117 ◁ C2×#433

Complements

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

NH ≅ G/Ntype
1. C2×C3×#433C2semidirect
2. #82236C2semidirect
3. C2×C4×D24C2semidirect
4. #70483C2semidirect
5. #70481C2semidirect
6. #70462C2semidirect
8. #69938C2semidirect
9. #69935C2semidirect
10. #69812C2semidirect
11. C4×#817C2semidirect
43. C2×#433D3semidirect
90. C3×C4×C8D4semidirect
227. C4×C8#67semidirect
453. C3#23702semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4332
#822362
C2×C4×D242
#704832
#704812
#704622
#699612
#699382
#699352
#698122
C4×#8172
#237023