Finite Group Databasebeta

← #10802091

Subgroups — #10802091

Order 768.

938 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4897C211yesmaximal, characteristic
2C4×#908C212yesmaximal, complemented
3C2×#4711C211yesmaximal, characteristic, complemented
4#78486C211yesmaximal, characteristic, complemented
5C3×#1778C211yesmaximal, characteristic, complemented
6#70720C211yesmaximal, characteristic, complemented
7#70709C211yesmaximal, characteristic, complemented
8#70700C211yesmaximal, characteristic, complemented
9C4×#823C212yesmaximal, complemented
10#68948C212yesmaximal, complemented
11#68931C212yesmaximal, complemented
12#21555—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15C2×#901C2211yescharacteristic
16C2×#901C2211yescharacteristic
17C4×#235C2211yescharacteristic
18C2×C4×D12C2212yes—
19C2×C3×#108C2211yescharacteristic
20C2×C3×#108C2211yescharacteristic
21C3×C4×M16C2212yes—
22C2×C3×#97C2211yescharacteristic
23#5133C2212yes—
24#5131C2212yes—
25C2×#823C2212yes—
26C2×#823C2212yes—
27#5121C2212yes—
28C2×#821C2212yes—
29C2×#817C2211yescharacteristic
30#5101C2211yescharacteristic
31#5101C2211yescharacteristic
32C4×#208C2211yescharacteristic
33#4897C418yescomplemented
34#4711—22no—
35#4711—21no—
36#4711—21no—
37#4711—21no—
38#4711—21no—
39#4710—22no—
40#4708—21no—
41#4708—21no—
42#4707—21no—
43D3×#97—21no—
44C4×D24—22no—
45C4×#204—22no—
46C3×#341C2212yes—
47#4554C2212yes—
48#4554C2212yes—
49C2×#472—31no—
50C4×AGL(1,ℤ/8)—32no—
51C2×#442—31no—
52#2875—31no—
53#1978—31no—
54#1951—31no—
55#1931—32no—
56#1907—32no—
57#1815—31no—
58#1778D311yescharacteristic, complemented
59C4×#100—32no—
60C2×#235C2311yescharacteristic
61C22×D12C2312yes—
62C22×C4×D3—22no—
63C2×C3×M16C2312yes—
64C2×C3×M16C2312yes—
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68C4×#67—22no—
69C2×(C2×D3)⋊C4—22no—
70#926C2311yescharacteristic
71C2×C4⋊Dic3C2312yes—
72C2×#208C2311yescharacteristic
73C2×#208C2311yescharacteristic
74#908—24no—
75#908C2×C418yescomplemented
76C2×D24—22no—
77C2×D24—21no—
78C2×D24—21no—
79C2×D24—22no—
80C2×D24C2×C414yescomplemented
81C2×#204—22no—
82#901—22no—
83#901—22no—
84#901—22no—
85#901C2×C414yescomplemented
86C2×ΓL(1,𝔽25)—21no—
87C2×ΓL(1,𝔽25)—21no—
88C2×D3×C8—21no—
89C2×D3×C8—21no—
90C4×D12—22no—
91C4×D12—22no—
92C4×D12—22no—
93C4×D12—22no—
94C4×D12D412yes—
95#872—21no—
96C42×D3—21no—
97C4×Dic6D412yes—
98C3×#108—22no—
99C3×#108C2×C414yescomplemented
100C3×#97—22no—
101C3×#97D411yescharacteristic
102C3×#97D411yescharacteristic
103C3×#97D411yescharacteristic
104C3×#97D411yescharacteristic
105C3×C4×C8—22no—
106#823—24no—
107#823C2×C418yescomplemented
108#821—22no—
109#821—22no—
110#821—22no—
111#821—22no—
112#821—22no—
113#818—22no—
114#817—21no—
115#817—21no—
116#817—22no—
117#816—22no—
118#815—21no—
119#815—21no—
120C8×Dic3—21no—
121C8×Dic3—21no—
122#803—21no—
123C4×C3⋊C8—21no—
124C2×AGL(1,ℤ/8)—32no—
125C22×D8—31no—
126C2×#131—31no—
127C2×#131—31no—
128C4×#41—31no—
129C2×C4×D4—32no—
130#472—38no—
131#442—62no—
132#442—61no—
133#442—61no—
134#442—61no—
135#442—61no—
136#440—62no—
137C4×SD16—62no—
138C4×D8—62no—
139C2×#108C2×D311yescharacteristic
140C2×#108C2×D311yescharacteristic
141#428—32no—
142C2×#106—31no—
143#418—32no—
144#417—32no—
145C2×#102—32no—
146C2×#100—32no—
147C2×#100—32no—
148#405—31no—
149#405—31no—
150#405—61no—
151#405—61no—
152C4×M16—31no—
153C4×M16—61no—
154C4×M16C2×D312yes—
155C2×#97—61no—
156C2×#97C2×D311yescharacteristic
157#343—32no—
158#343—32no—
159#341C2×D312yes—
160C23×D3—22no—
161C22×C3×C4#4112yes—
162C22×C3×C4C2411yescharacteristic
163C2×#67—21no—
164C22×Dic3—22no—
165#235—21no—
166#235—21no—
167#235C22×C412yes—
168#235#4112yes—
169C2×D12—22no—
170C2×D12—22no—
171C2×D12—22no—
172C2×D12—22no—
173C2×D12—22no—
174C2×D12C22×C414yes—
175C2×D12C2×D411yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×C4×D3—42no—
179C2×C4×D3—42no—
180C2×C4×D3—84no—
181C2×Dic6C2×D411yescharacteristic
182C3×M16—22no—
183C3×M16—22no—
184C3×M16—22no—
185C3×M16C22×C414yes—
186C2×C3×C8—22no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—22no—
190C2×C3×C8C22×C412yes—
191C2×C3×C8C2×D411yescharacteristic
192C2×C3×C8C2×D411yescharacteristic
193C2×C3×C8C2×D411yescharacteristic
194C2×C3×C8C2×D411yescharacteristic
195C2×C3×C8#4112yes—
196C3×C42C2×D411yescharacteristic
197C3×C42C2×D411yescharacteristic
198C3×C42C2×D412yes—
199#217—21no—
200(C2×D3)⋊C4—22no—
201(C2×D3)⋊C4—42no—
202(C2×D3)⋊C4—42no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3C2×D412yes—
206Dic3⋊C4—22no—
207C4×Dic3—22no—
208C4×Dic3—21no—
209#208—21no—
210#208—21no—
211#208C22×C412yes—
212#208#4112yes—
213C2×C3⋊C8—21no—
214C2×C3⋊C8—21no—
215D24—28no—
216D24—24no—
217D24—24no—
218#204—28no—
219ΓL(1,𝔽25)—24no—
220ΓL(1,𝔽25)—24no—
221D3×C8—24no—
222D3×C8—24no—
223C2×#41—31no—
224C22×D4—32no—
225C23×C4—62no—
226AGL(1,ℤ/8)—38no—
227AGL(1,ℤ/8)—64no—
228C2×SD16—62no—
229C2×D8—34no—
230C2×D8—62no—
231C2×D8—61no—
232C2×D8—61no—
233C2×D8—62no—
234#131—34no—
235#131—62no—
236#131—62no—
237#131—62no—
238C2×M16—31no—
239C2×M16—31no—
240C2×M16—61no—
241C2×M16—61no—
242C2×M16C22×D312yes—
243C2×M16C22×D312yes—
244C22×C8—61no—
245C22×C8—61no—
246C22×C8C22×D311yescharacteristic
247C22×C8C22×D311yescharacteristic
248C4×Q8—32no—
249C4×D4—32no—
250C4×D4—62no—
251C4×D4—62no—
252C4×D4—62no—
253C4×D4—62no—
254C4×D4—62no—
255#117—31no—
256#117—61no—
257C2×C4⋊C4—32no—
258C2×C22⋊C4—62no—
259C2×C42—61no—
260C2×C42C22×D311yescharacteristic
261#108—22no—
262#108C4×D314yescomplemented
263#107—62no—
264#106—62no—
265#106—61no—
266#106—61no—
267#103—62no—
268#102—62no—
269#102—62no—
270#102—62no—
271#102—62no—
272#102—62no—
273#100—38no—
274#100—64no—
275#97—22no—
276#97—61no—
277#97—61no—
278#97—61no—
279#97D1211yescharacteristic
280#97D1211yescharacteristic
281#97D1211yescharacteristic
282#97D1211yescharacteristic
283C4×C8—22no—
284C4×C8—61no—
285C4×C8—61no—
286C4×C8—61no—
287C23×C3C2×#4111yescharacteristic
288C22×D3—21no—
289C22×D3—24no—
290C22×D3—42no—
291C22×D3—42no—
292C22×D3—82no—
293C22×D3—82no—
294C2×C3×C4—42no—
295C2×C3×C4#12112yes—
296C2×C3×C4#12112yes—
297C2×C3×C4C23×C411yescharacteristic
298C2×C3×C4C22×D411yescharacteristic
299C2×C3×C4C22×D411yescharacteristic
300C2×C3×C4C2×#4111yescharacteristic
301C2×C3×C4C2×#4111yescharacteristic
302C2×C3×C4C2×#4111yescharacteristic
303#67—22no—
304#67—22no—
305C2×Dic3—21no—
306C2×Dic3—22no—
307C2×Dic3—42no—
308C2×Dic3—82no—
309D12—24no—
310D12—24no—
311D12—21no—
312D12—42no—
313D12—42no—
314D12C4×D412yes—
315C4×D3—22no—
316C4×D3—22no—
317C4×D3—82no—
318C4×D3—84no—
319C4×D3—84no—
320Dic6—21no—
321Dic6C4×D412yes—
322C3×C8—24no—
323C3×C8—21no—
324C3×C8—21no—
325C3×C8—42no—
326C3×C8C4×D412yes—
327C3×C8C4×D412yes—
328C3⋊C8—22no—
329C3⋊C8—22no—
330C24—62no—
331#41—32no—
332#41—32no—
333#41—61no—
334#41—61no—
335C2×Q8—31no—
336C2×D4—34no—
337C2×D4—31no—
338C2×D4—61no—
339C2×D4—62no—
340C2×D4—62no—
341C2×D4—62no—
342C2×D4—62no—
343C2×D4—62no—
344C22×C4—61no—
345C22×C4—62no—
346C22×C4—62no—
347C22×C4—122no—
348C22×C4—122no—
349C22×C4—244no—
350C22×C4#23912yes—
351C22×C4C23×D311yescharacteristic
352SD16—68no—
353D8—68no—
354D8—64no—
355D8—64no—
356M16—22no—
357M16—22no—
358M16—22no—
359M16—32no—
360M16—32no—
361M16—64no—
362M16—61no—
363M16—61no—
364M16—64no—
365M16C2×C4×D314yes—
366C2×C8—22no—
367C2×C8—21no—
368C2×C8—21no—
369C2×C8—22no—
370C2×C8—64no—
371C2×C8—61no—
372C2×C8—61no—
373C2×C8—64no—
374C2×C8C2×C4×D312yes—
375C2×C8C2×D1211yescharacteristic
376C2×C8C2×D1211yescharacteristic
377C2×C8C2×D1211yescharacteristic
378C2×C8C2×D1211yescharacteristic
379C2×C8#23512yes—
380C4⋊C4—32no—
381C4⋊C4—62no—
382C4⋊C4—62no—
383C4⋊C4—62no—
384C22⋊C4—61no—
385C22⋊C4—62no—
386C22⋊C4—122no—
387C22⋊C4—122no—
388C42—62no—
389C42—61no—
390C42C2×D1211yescharacteristic
391C42C2×D1211yescharacteristic
392C42D3×D412yes—
393C22×C3—22no—
394C22×C3C4×#4111yescharacteristic
395C22×C3C2×#12111yescharacteristic
396C22×C3#52911yescharacteristic
397C2×D3—22no—
398C2×D3—42no—
399C2×D3—44no—
400C2×D3—84no—
401C2×D3—82no—
402C2×D3—82no—
403C2×D3—84no—
404C3×C4—22no—
405C3×C4—22no—
406C3×C4C2×C4×D411yescharacteristic
407C3×C4C2×C4×D411yescharacteristic
408C3×C4#52511yescharacteristic
409C3×C4#53211yescharacteristic
410Dic3—22no—
411Dic3—42no—
412Dic3—82no—
413C23—61no—
414C23—64no—
415C23—122no—
416C23—122no—
417C23—242no—
418C23—242no—
419C23C2×#23911yescharacteristic
420Q8—32no—
421Q8—61no—
422D4—32no—
423D4—62no—
424D4—62no—
425D4—64no—
426D4—64no—
427D4—61no—
428D4—122no—
429D4—122no—
430C2×C4—42no—
431C2×C4—61no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—122no—
436C2×C4—242no—
437C2×C4—244no—
438C2×C4—244no—
439C2×C4—242no—
440C2×C4#89512yes—
441C2×C4#89512yes—
442C2×C4C22×C4×D311yescharacteristic
443C2×C4C22×D1211yescharacteristic
444C2×C4C2×#23511yescharacteristic
445C2×C4C2×D3×D411yescharacteristic
446C2×C4D3×#4111yescharacteristic
447C2×C4D3×#4111yescharacteristic
448C8—21no—
449C8—21no—
450C8—24no—
451C8—42no—
452C8—62no—
453C8—62no—
454C8C4×D1212yes—
455C8C4×D1212yes—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3C4×#12111yescharacteristic
459C2×C3#306812yes—
460D3—42no—
461D3—84no—
462C22—22no—
463C22—62no—
464C22—122no—
465C22—124no—
466C22—244no—
467C22—244no—
468C22—242no—
469C22—242no—
470C22C2×#89511yescharacteristic
471C22C4×#23911yescharacteristic
472C22#558011yescharacteristic
473C4—22no—
474C4—22no—
475C4—62no—
476C4—122no—
477C4—242no—
478C4C2×C4×D1211yescharacteristic
479C4#552411yescharacteristic
480C4C4×D3×D411yescharacteristic
481C4#554811yescharacteristic
482C3#2155511yescharacteristic, complemented, Sylow
483C2—21no—
484C2—21no—
485C2—122no—
486C2—244no—
487C2C4×#89511yescharacteristic
488C2#7926112yes—
subgroupG/H
centerC22C2×#895
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1778C2
socleC22×C3C2×#121
prime radicalC2×#4897C2
cosocleC22×C3×C4C24
hypercenter#1778D3
hypocenterC3#21555
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1778

Complements

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

NH ≅ G/Ntype
2. C4×#908C2semidirect
3. C2×#4711C2semidirect
4. #78486C2semidirect
5. C3×#1778C2semidirect
6. #70720C2semidirect
7. #70709C2semidirect
8. #70700C2semidirect
9. C4×#823C2semidirect
10. #68948C2semidirect
11. #68931C2semidirect
33. #4897C4semidirect
58. #1778D3semidirect
75. #908C2×C4semidirect
80. C2×D24C2×C4semidirect
85. #901C2×C4semidirect
99. C3×#108C2×C4semidirect
107. #823C2×C4semidirect
262. #108C4×D3semidirect
482. C3#21555semidirect

Maximal subgroups

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

subgroupindex
C2×#48972
C4×#9082
C2×#47112
#784862
C3×#17782
#707202
#707092
#707002
C4×#8232
#689482
#689312
#215553