Finite Group Databasebeta

← #10818092

Subgroups — #10818092

Order 768.

896 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4897C211yesmaximal, characteristic
2#79179C211yesmaximal, characteristic, complemented
3#79167C211yesmaximal, characteristic, complemented
4#78560C212yesmaximal, complemented
5C3×#1772C211yesmaximal, characteristic, complemented
6#70701C211yesmaximal, characteristic, complemented
7#69892C212yesmaximal, complemented
8#69789C211yesmaximal, characteristic, complemented
9#69776C211yesmaximal, characteristic, complemented
10#68776C212yesmaximal, complemented
11#68758C212yesmaximal, complemented
12#23046—31nomaximal, Sylow
13C2×#908C2212yes—
14C22×D24C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5523C2212yes—
17#5521C2212yes—
18C2×C3×#108C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#405C2212yes—
21C2×C3×C4×C8C2211yescharacteristic
22C2×#823C2212yes—
23#5121C2212yes—
24#5113C2211yescharacteristic
25C2×#821C2212yes—
26C2×#818C2211yescharacteristic
27#5101C2211yescharacteristic
28#5001C2212yes—
29#4995C2212yes—
30#4939C2212yes—
31C2×#805C2211yescharacteristic
32#4897C418yescomplemented
33#4880—21no—
34D3×#107—21no—
35#4872—21no—
36#4872—21no—
37#4869—22no—
38#4861—22no—
39#4711—22no—
40#4710—22no—
41C4×D24—22no—
42C4×#204—22no—
43C3×#339C2212yes—
44#4554C2212yes—
45#4552C2211yescharacteristic
46#4523C2211yescharacteristic
47C2×#472—31no—
48#2948—31no—
49#2946—32no—
50#2913—31no—
51#1970—31no—
52#1952—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1772D311yescharacteristic, complemented
57#1764—31no—
58C2×#235C2311yescharacteristic
59C22×D12C2312yes—
60C2×C3×M16C2312yes—
61C22×C3×C8C2311yescharacteristic
62C22×C3×C8C2311yescharacteristic
63C3×#117C2312yes—
64C2×C3×C42C2311yescharacteristic
65C2×(C2×D3)⋊C4—42no—
66#926C2311yescharacteristic
67C2×C4⋊Dic3C2312yes—
68C2×#208C2311yescharacteristic
69C2×#208C2311yescharacteristic
70#908—24no—
71#908C2×C418yescomplemented
72C2×D24—22no—
73C2×D24—21no—
74C2×D24—21no—
75C2×D24—22no—
76C2×D24C2×C414yescomplemented
77C2×#204—22no—
78#901—22no—
79#901C2×C414yescomplemented
80C2×ΓL(1,𝔽25)—21no—
81C2×D3×C8—21no—
82#893—22no—
83#893—22no—
84#892—22no—
85D3×C4⋊C4—22no—
86#67⋊C4—42no—
87D3×C22⋊C4—42no—
88C4×D12—22no—
89C4×D12—22no—
90C4×D12—22no—
91C4×Dic6—22no—
92C3×#108C2×C414yescomplemented
93C3×#107D411yescharacteristic
94C3×#107D411yescharacteristic
95C3×#106D412yes—
96C3×#97—22no—
97C3×C4×C8—22no—
98C3×C4×C8—22no—
99#823C2×C418yescomplemented
100#821—22no—
101#821—22no—
102#821D411yescharacteristic
103#821D411yescharacteristic
104#818—21no—
105#818—21no—
106#818—22no—
107#817—22no—
108#816D411yescharacteristic
109#816D411yescharacteristic
110#815—21no—
111C8×Dic3—21no—
112#809—22no—
113#809—22no—
114#808—22no—
115#807—22no—
116#805—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8—31no—
119C2×#131—31no—
120#520—32no—
121#518—32no—
122#472—38no—
123#442—62no—
124#442—62no—
125#440—62no—
126C4×SD16—62no—
127C4×D8—62no—
128C4×D8—61no—
129C4×D8—61no—
130C2×#108C2×D311yescharacteristic
131#428—32no—
132#428—32no—
133#428—62no—
134#427—61no—
135#427C2×D311yescharacteristic
136C2×#107—31no—
137C2×#107—61no—
138C2×#104—31no—
139#418—32no—
140#416—31no—
141C2×#102—32no—
142C2×#100—32no—
143#405—31no—
144#405C2×D312yes—
145C2×C4×C8C2×D311yescharacteristic
146#390—32no—
147#343—32no—
148#341—31no—
149#339—31no—
150#339C2×D312yes—
151C23×D3—22no—
152C22×C3×C4—21no—
153C22×C3×C4C2411yescharacteristic
154C2×#67—21no—
155C22×Dic3—22no—
156#235—21no—
157#235—21no—
158#235C22×C412yes—
159#235#4112yes—
160C2×D12—22no—
161C2×D12—22no—
162C2×D12—22no—
163C2×D12—22no—
164C2×D12—22no—
165C2×D12C22×C414yes—
166C2×D12C2×D411yescharacteristic
167C2×C4×D3—21no—
168C2×C4×D3—42no—
169C2×C4×D3—42no—
170C2×C4×D3—42no—
171C2×Dic6C2×D411yescharacteristic
172C3×M16—22no—
173C3×M16C22×C414yes—
174C2×C3×C8—22no—
175C2×C3×C8—21no—
176C2×C3×C8—21no—
177C2×C3×C8C22×C412yes—
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8C2×D411yescharacteristic
181C2×C3×C8C2×D411yescharacteristic
182C2×C3×C8#4112yes—
183C3×C4⋊C4C2×D412yes—
184C3×C4⋊C4C2×D412yes—
185C3×C22⋊C4—22no—
186C3×C42—22no—
187C3×C42—21no—
188C3×C42—21no—
189#217—21no—
190(C2×D3)⋊C4—42no—
191(C2×D3)⋊C4—42no—
192(C2×D3)⋊C4—42no—
193(C2×D3)⋊C4—84no—
194C4⋊Dic3—22no—
195C4⋊Dic3—22no—
196C4⋊Dic3C2×D411yescharacteristic
197C4⋊Dic3C2×D411yescharacteristic
198Dic3⋊C4—42no—
199C4×Dic3—21no—
200C4×Dic3—42no—
201#208—21no—
202#208—21no—
203#208—21no—
204#208C22×C412yes—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207D24—28no—
208D24—24no—
209D24—24no—
210#204—28no—
211ΓL(1,𝔽25)—24no—
212D3×C8—24no—
213C2×#41—31no—
214C22×D4—32no—
215AGL(1,ℤ/8)—38no—
216AGL(1,ℤ/8)—64no—
217C2×SD16—62no—
218C2×D8—34no—
219C2×D8—62no—
220C2×D8—61no—
221C2×D8—61no—
222C2×D8—62no—
223#131—34no—
224#131—62no—
225C2×M16—31no—
226C2×M16—31no—
227C2×M16—61no—
228C2×M16C22×D312yes—
229C22×C8—61no—
230C22×C8C22×D311yescharacteristic
231C22×C8C22×D311yescharacteristic
232C4×Q8—62no—
233C4×D4—62no—
234C4×D4—62no—
235C4×D4—62no—
236C4×D4—62no—
237C4×D4—62no—
238C4×D4—122no—
239#117—31no—
240#117—62no—
241#117C22×D312yes—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C22⋊C4—122no—
245C2×C22⋊C4—122no—
246C2×C42C22×D311yescharacteristic
247#108C4×D314yescomplemented
248#107—62no—
249#107—61no—
250#107—61no—
251#107—62no—
252#107D1211yescharacteristic
253#107D1211yescharacteristic
254#106—62no—
255#106—62no—
256#106D1212yes—
257#104—62no—
258#103—31no—
259#103—31no—
260#102—31no—
261#102—31no—
262#102—62no—
263#102—62no—
264#102—62no—
265#102—62no—
266#100—38no—
267#97—22no—
268#97—61no—
269C4×C8—22no—
270C4×C8—22no—
271C4×C8—61no—
272C23×C3H(5,𝔽2)11yescharacteristic
273C22×D3—21no—
274C22×D3—24no—
275C22×D3—42no—
276C22×D3—42no—
277C22×D3—82no—
278C22×D3—82no—
279C2×C3×C4—22no—
280C2×C3×C4—22no—
281C2×C3×C4—22no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4C23×C411yescharacteristic
285C2×C3×C4C22×D411yescharacteristic
286C2×C3×C4C22×D411yescharacteristic
287C2×C3×C4C2×#4111yescharacteristic
288C2×C3×C4C2×#4111yescharacteristic
289C2×C3×C42−1+411yescharacteristic
290#67—22no—
291#67—22no—
292C2×Dic3—21no—
293C2×Dic3—21no—
294C2×Dic3—21no—
295C2×Dic3—42no—
296C2×Dic3—82no—
297D12—24no—
298D12—24no—
299D12—21no—
300D12—42no—
301D12—42no—
302D12C4×D412yes—
303C4×D3—22no—
304C4×D3—22no—
305C4×D3—162no—
306C4×D3—162no—
307C4×D3—162no—
308Dic6—21no—
309Dic6C4×D412yes—
310C3×C8—24no—
311C3×C8—21no—
312C3×C8—21no—
313C3×C8C4×D412yes—
314C3×C8C4×D412yes—
315C3⋊C8—22no—
316C3⋊C8—41no—
317C24—62no—
318#41—32no—
319#41—32no—
320#41—61no—
321#41—61no—
322C2×Q8—31no—
323C2×D4—34no—
324C2×D4—31no—
325C2×D4—61no—
326C2×D4—62no—
327C2×D4—62no—
328C2×D4—62no—
329C2×D4—62no—
330C2×D4—62no—
331C22×C4—21no—
332C22×C4—61no—
333C22×C4—62no—
334C22×C4—122no—
335C22×C4—122no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—68no—
340D8—64no—
341D8—64no—
342M16—22no—
343M16—32no—
344M16—61no—
345M16—64no—
346M16—61no—
347M16—61no—
348M16C2×C4×D314yes—
349C2×C8—21no—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—61no—
353C2×C8—64no—
354C2×C8—61no—
355C2×C8C2×C4×D312yes—
356C2×C8C2×D1211yescharacteristic
357C2×C8C2×D1211yescharacteristic
358C2×C8#23512yes—
359C2×C8D3×D411yescharacteristic
360C2×C8D3×D411yescharacteristic
361C4⋊C4—31no—
362C4⋊C4—31no—
363C4⋊C4—62no—
364C4⋊C4—62no—
365C4⋊C4—122no—
366C4⋊C4C2×D1212yes—
367C4⋊C4C2×D1212yes—
368C22⋊C4—22no—
369C22⋊C4—61no—
370C22⋊C4—122no—
371C22⋊C4—122no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—22no—
375C42—21no—
376C42—21no—
377C42—61no—
378C42—122no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3#52011yescharacteristic
382C22×C3D4211yescharacteristic
383C22×C3#55311yescharacteristic
384C2×D3—22no—
385C2×D3—41no—
386C2×D3—41no—
387C2×D3—44no—
388C2×D3—82no—
389C2×D3—82no—
390C2×D3—84no—
391C2×D3—162no—
392C3×C4—42no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4C2×C4×D411yescharacteristic
396C3×C4C2×C4×D411yescharacteristic
397C3×C4#54711yescharacteristic
398C3×C4#54811yescharacteristic
399Dic3—22no—
400Dic3—41no—
401Dic3—41no—
402Dic3—82no—
403C23—61no—
404C23—64no—
405C23—122no—
406C23—122no—
407C23—242no—
408C23—242no—
409C23#100911yescharacteristic
410Q8—32no—
411Q8—61no—
412D4—32no—
413D4—62no—
414D4—62no—
415D4—64no—
416D4—61no—
417D4—64no—
418D4—122no—
419D4—122no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—61no—
429C2×C4—61no—
430C2×C4—122no—
431C2×C4—242no—
432C2×C4—482no—
433C2×C4—482no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C22×D1211yescharacteristic
437C2×C4C2×#23511yescharacteristic
438C2×C4C2×D3×D411yescharacteristic
439C2×C4#100711yescharacteristic
440C2×C4D3×#4111yescharacteristic
441C8—21no—
442C8—21no—
443C8—24no—
444C8—62no—
445C8—121no—
446C8C4×D1212yes—
447C8C4×D1212yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3#236111yescharacteristic
451C2×C3#329011yescharacteristic
452C2×C3#329211yescharacteristic
453D3—42no—
454D3—84no—
455C22—21no—
456C22—21no—
457C22—62no—
458C22—121no—
459C22—121no—
460C22—124no—
461C22—244no—
462C22—242no—
463C22—242no—
464C22—482no—
465C22#553011yescharacteristic
466C22D4×D1211yescharacteristic
467C22#557411yescharacteristic
468C4—42no—
469C4—42no—
470C4—42no—
471C4—62no—
472C4—121no—
473C4—121no—
474C4—242no—
475C4C2×C4×D1211yescharacteristic
476C4#550711yescharacteristic
477C4C4×D3×D411yescharacteristic
478C4#557611yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—122no—
483C2—244no—
484C2#7377411yescharacteristic
485C2#7989211yescharacteristic
486C2#7989411yescharacteristic
subgroupG/H
centerC22D4×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1772C2
socleC22×C3D42
prime radicalC2×#4897C2
cosocleC22×C3×C4C24
hypercenter#1772D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1772

Complements

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

NH ≅ G/Ntype
2. #79179C2semidirect
3. #79167C2semidirect
4. #78560C2semidirect
5. C3×#1772C2semidirect
6. #70701C2semidirect
7. #69892C2semidirect
8. #69789C2semidirect
9. #69776C2semidirect
10. #68776C2semidirect
11. #68758C2semidirect
32. #4897C4semidirect
56. #1772D3semidirect
71. #908C2×C4semidirect
76. C2×D24C2×C4semidirect
79. #901C2×C4semidirect
92. C3×#108C2×C4semidirect
99. #823C2×C4semidirect
247. #108C4×D3semidirect
479. C3#23046semidirect

Maximal subgroups

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

subgroupindex
C2×#48972
#791792
#791672
#785602
C3×#17722
#707012
#698922
#697892
#697762
#687762
#687582
#230463