Finite Group Databasebeta

← #10800517

Subgroups — #10800517

Order 768.

1122 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#85738C211yesmaximal, characteristic
2C3×#2921C211yesmaximal, characteristic, complemented
3Dic3×#117C211yesmaximal, characteristic, complemented
4#70515C211yesmaximal, characteristic, complemented
5#70514C211yesmaximal, characteristic, complemented
6#70513C211yesmaximal, characteristic, complemented
7#70512C211yesmaximal, characteristic, complemented
8#69966C212yesmaximal, complemented
9#69965C212yesmaximal, complemented
10#69866C212yesmaximal, complemented
11C4×#825C212yesmaximal, complemented
12#21419—31nomaximal, Sylow
13C2×C3×#117C2211yescharacteristic
14#5791C2212yes—
15C2×#926C2211yescharacteristic
16C22×C4×Dic3C2211yescharacteristic
17#5535C412yescomplemented
18#5534C414yescomplemented
19#5528—24no—
20#5518C412yescomplemented
21#5515—22no—
22#5494—22no—
23#5477C2211yescharacteristic
24C4×#235C2212yes—
25#5470C2211yescharacteristic
26C3×#433—24no—
27C3×#423C2211yescharacteristic
28C3×#423C2211yescharacteristic
29C3×#407C2212yes—
30C3×C4×M16C2212yes—
31C2×#825C2212yes—
32C2×#825C2212yes—
33C2×#825C2212yes—
34C2×#825C2212yes—
35Dic3×C4⋊C4—22no—
36Dic3×C22⋊C4—22no—
37#4932C2211yescharacteristic
38#4932C2211yescharacteristic
39C42×Dic3C2211yescharacteristic
40C42×Dic3C2211yescharacteristic
41#4555C2212yes—
42#4555C2212yes—
43#4555C2212yes—
44#4555C2212yes—
45#3455—31no—
46#2921D311yescharacteristic, complemented
47C4×#117—31no—
48#1995—31no—
49#1994—31no—
50#1963—31no—
51#1962—31no—
52#1897—32no—
53#1896—32no—
54#1759—32no—
55C4×#104—32no—
56C23×C3×C4C2311yescharacteristic
57C23×Dic3—21no—
58C2×#235C2312yes—
59C2×C3×M16C2312yes—
60C2×C3×M16C2312yes—
61C3×#117C2×C412yes—
62C3×#117C2×C412yes—
63C3×#117D414yes—
64C2×C3×C4⋊C4D412yes—
65C2×C3×C22⋊C4D412yes—
66C2×C3×C42C2311yescharacteristic
67C2×C3×C42C2311yescharacteristic
68#953—22no—
69C2×#217—21no—
70#930—22no—
71#930C2×C414yes—
72#929—22no—
73C4×#67—24no—
74C4×#67—24no—
75#926—22no—
76#926—22no—
77#926C2312yes—
78C2×C4⋊Dic3—21no—
79#924—22no—
80#924C2×C414yes—
81C2×C4×Dic3—21no—
82C2×C4×Dic3—22no—
83C2×C4×Dic3—22no—
84C2×C4×Dic3—22no—
85C2×C4×Dic3—22no—
86C2×C4×Dic3C2311yescharacteristic
87C2×C4×Dic3C2311yescharacteristic
88C2×C4×Dic3C2311yescharacteristic
89C2×C4×Dic3C2311yescharacteristic
90#898—24no—
91#896—24no—
92#894—24no—
93#889—24no—
94#885—24no—
95#884—24no—
96#883—24no—
97#879—24no—
98#875C2×C412yescomplemented
99#874C2×C411yescharacteristic, complemented
100C4×D12—22no—
101C4×D12C2×C414yescomplemented
102#872—22no—
103C42×D3—22no—
104#870C2×C412yescomplemented
105#869C2×C411yescharacteristic, complemented
106#869C2×C412yescomplemented
107C4×Dic6—22no—
108C4×Dic6C2×C414yescomplemented
109C3×C4⋊C8—22no—
110C3×C4⋊C8—22no—
111C3×C22⋊C8—24no—
112C3×#97—22no—
113C3×C4×C8—22no—
114C3⋊#95—22no—
115C3⋊#95—22no—
116#825—24no—
117#825—24no—
118#825—24no—
119#825—24no—
120#542—34no—
121#541—32no—
122#540—32no—
123#539—62no—
124#538—62no—
125#533—31no—
126#532—31no—
127#529—64no—
128#525—32no—
129C4×#41—32no—
130C2×#117—31no—
131C2×#117C2×D311yescharacteristic
132C22×C42—31no—
133#433—24no—
134#423C2×D311yescharacteristic
135#423C2×D311yescharacteristic
136C2×#104—32no—
137C2×#104—32no—
138C2×#104—32no—
139C2×#104—32no—
140#407C2×D312yes—
141C4×M16C2×D312yes—
142C4×C4⋊C4—62no—
143C4×C22⋊C4—62no—
144#376—31no—
145#376—31no—
146C43—31no—
147C43—31no—
148#339—32no—
149#339—32no—
150#339—32no—
151#339—32no—
152C24×C3C2×D411yescharacteristic
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4—21no—
157C22×C3×C4C22×C412yes—
158C22×C3×C4C2×D411yescharacteristic
159C22×C3×C4C2×D412yes—
160C22×C3×C4C2×D412yes—
161C22×C3×C4C2×D412yes—
162C22×C3×C4C2411yescharacteristic
163C2×#67—22no—
164C2×#67—24no—
165C22×Dic3—21no—
166C22×Dic3—24no—
167C22×Dic3—21no—
168C22×Dic3—21no—
169C22×Dic3—21no—
170C22×Dic3—44no—
171#235—22no—
172#235—22no—
173#235—22no—
174#235—22no—
175C2×D12—22no—
176C2×D12C22×C412yes—
177C2×C4×D3—22no—
178C2×C4×D3—24no—
179C2×Dic6—22no—
180C2×Dic6C22×C412yes—
181C3×M16—22no—
182C3×M16—22no—
183C3×M16—22no—
184C3×M16—22no—
185C2×C3×C8—22no—
186C2×C3×C8—22no—
187C3×C4⋊C4C22⋊C414yes—
188C3×C4⋊C4C22⋊C414yes—
189C3×C22⋊C4C22⋊C414yes—
190C3×C22⋊C4C22⋊C414yes—
191C3×C42C22×C412yes—
192C3×C42C22×C411yescharacteristic
193C3×C42C22×C411yescharacteristic
194C3×C42C2×D412yes—
195C3×C42C2×D412yes—
196#217—24no—
197#217—24no—
198#217—22no—
199(C2×D3)⋊C4—24no—
200(C2×D3)⋊C4—24no—
201(C2×D3)⋊C4—24no—
202C4⋊Dic3—24no—
203C4⋊Dic3—22no—
204C4⋊Dic3—22no—
205C4⋊Dic3C22×C414yes—
206Dic3⋊C4—24no—
207Dic3⋊C4—24no—
208Dic3⋊C4—24no—
209C4×Dic3—24no—
210C4×Dic3—24no—
211C4×Dic3—22no—
212C4×Dic3—21no—
213C4×Dic3—21no—
214C4×Dic3—21no—
215C4×Dic3—21no—
216C4×Dic3—44no—
217C4×Dic3—44no—
218C4×Dic3#4112yes—
219C4×Dic3#4112yes—
220C4×Dic3#4112yes—
221C4×Dic3#4112yes—
222C2×#41—32no—
223C23×C4—61no—
224C23×C4C22×D311yescharacteristic
225C2×M16C22×D312yes—
226C2×M16C22×D312yes—
227C4⋊Q8—32no—
228C4⋊Q8—31no—
229#127—31no—
230#126—64no—
231#126—64no—
232#125—32no—
233#125—64no—
234#124—32no—
235#124—64no—
236#123—62no—
237#123—64no—
238#123—64no—
239C22⋊Q8—34no—
240C22⋊Q8—62no—
241C22⋊Q8—64no—
242#121—34no—
243#121—62no—
244#121—64no—
245#120—62no—
246C4×Q8—34no—
247C4×Q8—62no—
248C4×D4—34no—
249C4×D4—64no—
250C4×D4—64no—
251C4×D4—62no—
252#117—32no—
253#117—62no—
254#117—62no—
255#117—62no—
256#117C4×D312yes—
257#117C4×D312yes—
258#117#6714yes—
259C2×C4⋊C4—61no—
260C2×C4⋊C4D1212yes—
261C2×C22⋊C4—61no—
262C2×C22⋊C4D1212yes—
263C2×C42—31no—
264C2×C42—31no—
265C2×C42—31no—
266C2×C42—31no—
267C2×C42—62no—
268C2×C42—62no—
269C2×C42—61no—
270C2×C42—62no—
271C2×C42—62no—
272C2×C42—62no—
273C2×C42C22×D311yescharacteristic
274C2×C42C22×D311yescharacteristic
275C4⋊C8—22no—
276C4⋊C8—22no—
277#104—64no—
278#104—64no—
279#104—64no—
280#104—64no—
281C22⋊C8—24no—
282#97—22no—
283C4×C8—22no—
284#95—62no—
285#95—62no—
286C23×C3—21no—
287C23×C3—21no—
288C23×C3—42no—
289C23×C3C2×C22⋊C412yes—
290C23×C3C22×D411yescharacteristic
291C22×D3—22no—
292C2×C3×C4—21no—
293C2×C3×C4—21no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—42no—
299C2×C3×C4—42no—
300C2×C3×C4C2×C22⋊C414yes—
301C2×C3×C4C2×C22⋊C412yes—
302C2×C3×C4C2×C22⋊C412yes—
303C2×C3×C4C2×C22⋊C412yes—
304C2×C3×C4C23×C411yescharacteristic
305C2×C3×C4C22×D411yescharacteristic
306C2×C3×C4C2×#4111yescharacteristic
307C2×C3×C4C2×#4111yescharacteristic
308C2×C3×C4C2×#4111yescharacteristic
309C2×C3×C4C2×#4111yescharacteristic
310#67—42no—
311#67—42no—
312#67—84no—
313C2×Dic3—24no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—24no—
317C2×Dic3—22no—
318C2×Dic3—22no—
319C2×Dic3—22no—
320C2×Dic3—48no—
321C2×Dic3—42no—
322C2×Dic3—42no—
323C2×Dic3—42no—
324D12—22no—
325D12—22no—
326D12—82no—
327C4×D3—42no—
328C4×D3—42no—
329C4×D3—84no—
330Dic6—22no—
331Dic6—22no—
332Dic6—82no—
333C3×C8—42no—
334C3×C8—42no—
335C24C2×D1211yescharacteristic
336#41—62no—
337#41—62no—
338#41—62no—
339#41—62no—
340C2×Q8—32no—
341C2×Q8—62no—
342C2×D4—32no—
343C2×D4—62no—
344C2×D4—64no—
345C2×D4—62no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—61no—
351C22×C4—64no—
352C22×C4—61no—
353C22×C4—61no—
354C22×C4—61no—
355C22×C4—62no—
356C22×C4—64no—
357C22×C4—124no—
358C22×C4C2×C4×D312yes—
359C22×C4C2×D1211yescharacteristic
360C22×C4C2×D1212yes—
361C22×C4C2×D1212yes—
362C22×C4C2×#6712yes—
363C22×C4C23×D311yescharacteristic
364M16—22no—
365M16—22no—
366M16—22no—
367M16—22no—
368C2×C8—22no—
369C2×C8—22no—
370C4⋊C4—34no—
371C4⋊C4—64no—
372C4⋊C4—64no—
373C4⋊C4—62no—
374C4⋊C4—64no—
375C4⋊C4—64no—
376C4⋊C4—62no—
377C4⋊C4(C2×D3)⋊C414yes—
378C4⋊C4(C2×D3)⋊C414yes—
379C22⋊C4—64no—
380C22⋊C4—64no—
381C22⋊C4—62no—
382C22⋊C4—64no—
383C22⋊C4—64no—
384C22⋊C4—64no—
385C22⋊C4(C2×D3)⋊C414yes—
386C22⋊C4(C2×D3)⋊C414yes—
387C42—32no—
388C42—32no—
389C42—32no—
390C42—32no—
391C42—64no—
392C42—64no—
393C42—61no—
394C42—61no—
395C42—61no—
396C42—61no—
397C42—62no—
398C42—124no—
399C42—124no—
400C42C2×C4×D312yes—
401C42C2×C4×D311yescharacteristic
402C42C2×C4×D311yescharacteristic
403C42C2×#6712yes—
404C42C2×#6712yes—
405C22×C3—22no—
406C22×C3—24no—
407C22×C3—22no—
408C22×C3—42no—
409C22×C3—42no—
410C22×C3C22×C22⋊C411yescharacteristic
411C22×C3#52511yescharacteristic
412C22×C3#52511yescharacteristic
413C2×D3—44no—
414C2×D3—42no—
415C3×C4—22no—
416C3×C4—24no—
417C3×C4—22no—
418C3×C4—22no—
419C3×C4C4×#4111yescharacteristic
420C3×C4C4×#4111yescharacteristic
421C3×C4#53211yescharacteristic
422C3×C4#53211yescharacteristic
423Dic3—24no—
424Dic3—42no—
425Dic3—42no—
426Dic3—42no—
427Dic3—44no—
428Dic3—42no—
429C23—21no—
430C23—21no—
431C23—42no—
432C23—62no—
433C23C2×(C2×D3)⋊C412yes—
434C23C22×D1211yescharacteristic
435Q8—62no—
436Q8—62no—
437Q8—242no—
438D4—62no—
439D4—62no—
440D4—122no—
441D4—122no—
442D4—242no—
443D4—244no—
444C2×C4—21no—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—64no—
453C2×C4—62no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—64no—
458C2×C4—62no—
459C2×C4—128no—
460C2×C4—122no—
461C2×C4—122no—
462C2×C4—122no—
463C2×C4—122no—
464C2×C4—122no—
465C2×C4—244no—
466C2×C4C2×(C2×D3)⋊C414yes—
467C2×C4C2×(C2×D3)⋊C412yes—
468C2×C4C2×(C2×D3)⋊C412yes—
469C2×C4C2×(C2×D3)⋊C412yes—
470C2×C4C22×C4×D311yescharacteristic
471C2×C4D3×#4111yescharacteristic
472C2×C4D3×#4111yescharacteristic
473C2×C4D3×#4111yescharacteristic
474C2×C4D3×#4111yescharacteristic
475C2×C4C22×#6711yescharacteristic
476C8—42no—
477C8—42no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—42no—
481C2×C3#229711yescharacteristic
482C2×C3#290112yes—
483D3—82no—
484C22—22no—
485C22—22no—
486C22—24no—
487C22—42no—
488C22—42no—
489C22—122no—
490C22—124no—
491C22#552411yescharacteristic
492C22#552411yescharacteristic
493C22C22×(C2×D3)⋊C411yescharacteristic
494C4—22no—
495C4—24no—
496C4—22no—
497C4—22no—
498C4—64no—
499C4—122no—
500C4—122no—
501C4—122no—
502C4—122no—
503C4—124no—
504C4#552211yescharacteristic
505C4#552211yescharacteristic
506C4#583211yescharacteristic
507C4#583211yescharacteristic
508C3#2141911yescharacteristic, complemented, Sylow
509C2—21no—
510C2—21no—
511C2—42no—
512C2—242no—
513C2#8222811yescharacteristic
514C2#8381712yes—
subgroupG/H
centerC2×C4C22×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2921C2
socleC22×C3C22×C22⋊C4
prime radical#85738C2
cosocleC22×C3×C4C24
hypercenter#2921D3
hypocenterC3#21419
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×#117 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. C3×#2921C2semidirect
3. Dic3×#117C2semidirect
4. #70515C2semidirect
5. #70514C2semidirect
6. #70513C2semidirect
7. #70512C2semidirect
8. #69966C2semidirect
9. #69965C2semidirect
10. #69866C2semidirect
11. C4×#825C2semidirect
17. #5535C4semidirect
18. #5534C4semidirect
20. #5518C4semidirect
46. #2921D3semidirect
98. #875C2×C4semidirect
99. #874C2×C4semidirect
101. C4×D12C2×C4semidirect
104. #870C2×C4semidirect
105. #869C2×C4semidirect
106. #869C2×C4semidirect
108. C4×Dic6C2×C4semidirect
508. C3#21419semidirect

Maximal subgroups

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

subgroupindex
#857382
C3×#29212
Dic3×#1172
#705152
#705142
#705132
#705122
#699662
#699652
#698662
C4×#8252
#214193