Finite Group Databasebeta

← #10818086

Subgroups — #10818086

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
1C3×#2948C211yesmaximal, characteristic, complemented
2C2×#5160C211yesmaximal, characteristic
3#82537C212yesmaximal, complemented
4#79179C211yesmaximal, characteristic, complemented
5#72038C211yesmaximal, characteristic, complemented
6#71647C212yesmaximal, complemented
7#70659C211yesmaximal, characteristic, complemented
8#70328C212yesmaximal, complemented
9#70318C212yesmaximal, complemented
10#69894C211yesmaximal, characteristic, complemented
11#69887C211yesmaximal, characteristic, complemented
12#23046—31nomaximal, Sylow
13C22×C3×D8C2211yescharacteristic
14C3×#518C2212yes—
15C2×#932C2212yes—
16C2×#901C2211yescharacteristic
17#5523C2212yes—
18C3×#442—22no—
19C3×C4×D8—21no—
20C3×C4×D8—21no—
21C3×#427C2211yescharacteristic
22C3×#418C2212yes—
23C3×#405C2211yescharacteristic
24#5221C2212yes—
25C2×#833C2212yes—
26C2×#832C2212yes—
27#5160C418yescomplemented
28C2×#825C2211yescharacteristic
29C2×#819C2211yescharacteristic
30C2×#818C2211yescharacteristic
31C2×C8×Dic3C2211yescharacteristic
32#5018—22no—
33C4×#214—22no—
34#5014—22no—
35C4×#213—22no—
36#5002C2211yescharacteristic
37#4997C2212yes—
38#4995C2212yes—
39#4995C2212yes—
40#4880—21no—
41D3×#107—21no—
42#4861—22no—
43#4558C2211yescharacteristic
44#4554C2212yes—
45#4532C2212yes—
46#4532C2211yescharacteristic
47C2×#472—31no—
48#2948D311yescharacteristic, complemented
49#2946—32no—
50#2913—31no—
51#1970—31no—
52#1952—31no—
53#1931—32no—
54#1883—32no—
55#1856—32no—
56#1772—31no—
57#1764—31no—
58C22×C3×D4C2312yes—
59C2×#235C2311yescharacteristic
60C2×C3×D8—21no—
61C2×C3×D8—21no—
62C2×C3×D8—22no—
63C2×C3×D8C2×C414yescomplemented
64C2×C3×M16C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C3×C4×D4—22no—
67C3×C4×D4—22no—
68C3×C4×D4—22no—
69C3×C4×D4—22no—
70C3×#117C2311yescharacteristic
71C3×#117C2312yes—
72C2×C3×C22⋊C4—42no—
73C2×#217—42no—
74C2×#214—22no—
75#932—24no—
76#932C2×C418yescomplemented
77C2×#213—22no—
78C2×C4⋊Dic3C2312yes—
79C2×C4×Dic3C2311yescharacteristic
80C2×#208C2311yescharacteristic
81C2×#208C2312yes—
82C22×C3⋊C8C2311yescharacteristic
83#901—22no—
84#901C2×C414yescomplemented
85C2×ΓL(1,𝔽25)—21no—
86C2×D3×C8—21no—
87#892—22no—
88D3×C4⋊C4—22no—
89#67⋊C4—42no—
90C4×D12—22no—
91C4×Dic6—22no—
92C3×#107D411yescharacteristic
93C3×#107D411yescharacteristic
94C3×#106D412yes—
95C3×#102—22no—
96C3×#102—22no—
97C3×#97—21no—
98C3×C4×C8—21no—
99#833C2×C418yescomplemented
100#832—22no—
101#832—22no—
102#825—22no—
103#819C2×C414yescomplemented
104#818—21no—
105#818—21no—
106C8×Dic3—22no—
107#810D411yescharacteristic
108#810D411yescharacteristic
109#809D411yescharacteristic
110#809D411yescharacteristic
111#808—22no—
112#808—22no—
113#807—22no—
114#807—22no—
115#803—22no—
116C4×C3⋊C8—22no—
117C2×AGL(1,ℤ/8)—32no—
118C22×D8C2×D311yescharacteristic
119C2×#131—31no—
120#520—32no—
121#518C2×D312yes—
122#472—38no—
123#442—22no—
124#442—62no—
125#440—62no—
126C4×SD16—62no—
127C4×D8—21no—
128C4×D8—21no—
129C4×D8—62no—
130C2×#108—31no—
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#418C2×D312yes—
140#416—31no—
141C2×#102—32no—
142C2×#100—32no—
143#405—32no—
144#405C2×D311yescharacteristic
145C2×C4×C8—31no—
146#390—32no—
147#343—32no—
148#341—31no—
149#339—31no—
150#339—32no—
151C24×C3—22no—
152C2×C3×D4—22no—
153C2×C3×D4—22no—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—22no—
157C2×C3×D4C22×C414yes—
158C22×C3×C4—42no—
159C22×C3×C4—42no—
160C22×C3×C4C2411yescharacteristic
161C2×#67—21no—
162C22×Dic3—21no—
163C22×Dic3—22no—
164#235—21no—
165#235—21no—
166#235C22×C412yes—
167#235#4112yes—
168C2×D12C2×D411yescharacteristic
169C2×C4×D3—21no—
170C2×C4×D3—42no—
171C2×Dic6C2×D411yescharacteristic
172C3×D8—24no—
173C3×D8—24no—
174C3×M16—21no—
175C3×M16—21no—
176C2×C3×C8—21no—
177C2×C3×C8C22×C412yes—
178C2×C3×C8C2×D411yescharacteristic
179C2×C3×C8C2×D411yescharacteristic
180C2×C3×C8#4112yes—
181C3×C4⋊C4C2×D411yescharacteristic
182C3×C4⋊C4C2×D411yescharacteristic
183C3×C4⋊C4C2×D412yes—
184C3×C4⋊C4C2×D412yes—
185C3×C22⋊C4—22no—
186C3×C22⋊C4—21no—
187C3×C22⋊C4—42no—
188C3×C22⋊C4—42no—
189C3×C42—22no—
190C3×C42—21no—
191#217—84no—
192#214—28no—
193#213—28no—
194(C2×D3)⋊C4—42no—
195C4⋊Dic3—22no—
196C4⋊Dic3—22no—
197Dic3⋊C4—42no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200C4×Dic3—42no—
201#208—21no—
202#208—22no—
203#208C22×C414yes—
204#208C22×C412yes—
205C2×C3⋊C8—22no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8C2×D411yescharacteristic
210C2×C3⋊C8C2×D411yescharacteristic
211ΓL(1,𝔽25)—24no—
212D3×C8—24no—
213C2×#41—31no—
214C22×D4C22×D312yes—
215AGL(1,ℤ/8)—38no—
216AGL(1,ℤ/8)—64no—
217C2×SD16—62no—
218C2×D8—22no—
219C2×D8—21no—
220C2×D8—21no—
221C2×D8—62no—
222C2×D8C4×D314yescomplemented
223#131—34no—
224#131—62no—
225C2×M16—31no—
226C2×M16—32no—
227C2×M16—61no—
228C2×M16C22×D311yescharacteristic
229C22×C8—31no—
230C22×C8—61no—
231C22×C8C22×D311yescharacteristic
232C4×Q8—62no—
233C4×D4—22no—
234C4×D4—22no—
235C4×D4—22no—
236C4×D4—22no—
237C4×D4—62no—
238C4×D4—122no—
239#117—62no—
240#117C22×D311yescharacteristic
241#117C22×D312yes—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—62no—
244C2×C22⋊C4—42no—
245C2×C22⋊C4—122no—
246C2×C42—31no—
247#108—34no—
248#107—61no—
249#107—61no—
250#107—62no—
251#107—62no—
252#107#6711yescharacteristic
253#107#6711yescharacteristic
254#106—62no—
255#106—62no—
256#106#6712yes—
257#104—62no—
258#103—31no—
259#103—31no—
260#102—22no—
261#102—22no—
262#102—31no—
263#102—31no—
264#102—62no—
265#102—62no—
266#100—38no—
267#97—21no—
268#97—62no—
269C4×C8—21no—
270C4×C8—62no—
271C4×C8—62no—
272C23×C3—24no—
273C23×C3—42no—
274C23×C3—42no—
275C23×C3—82no—
276C23×C3—82no—
277C23×C3H(5,𝔽2)11yescharacteristic
278C22×D3—21no—
279C3×D4—24no—
280C3×D4—24no—
281C3×D4—42no—
282C3×D4—42no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—162no—
288C2×C3×C4—162no—
289C2×C3×C4C23×C411yescharacteristic
290C2×C3×C4C22×D411yescharacteristic
291C2×C3×C4C22×D411yescharacteristic
292C2×C3×C4C2×#4111yescharacteristic
293C2×C3×C4C2×#4111yescharacteristic
294C2×C3×C42−1+411yescharacteristic
295#67—22no—
296#67—22no—
297C2×Dic3—21no—
298C2×Dic3—22no—
299C2×Dic3—41no—
300C2×Dic3—41no—
301C2×Dic3—42no—
302C2×Dic3—82no—
303D12—21no—
304D12C4×D412yes—
305C4×D3—22no—
306C4×D3—22no—
307C4×D3—162no—
308Dic6—21no—
309Dic6C4×D412yes—
310C3×C8—41no—
311C3×C8C4×D412yes—
312C3×C8C4×D412yes—
313C3⋊C8—22no—
314C3⋊C8—24no—
315C3⋊C8—21no—
316C3⋊C8—21no—
317C24—22no—
318#41—32no—
319#41—32no—
320#41—61no—
321#41—61no—
322C2×Q8—31no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—22no—
326C2×D4—22no—
327C2×D4—22no—
328C2×D4—31no—
329C2×D4—61no—
330C2×D4C2×C4×D314yes—
331C22×C4—42no—
332C22×C4—42no—
333C22×C4—61no—
334C22×C4—61no—
335C22×C4—62no—
336C22×C4—122no—
337C22×C4C23×D311yescharacteristic
338SD16—68no—
339D8—24no—
340D8—24no—
341D8—68no—
342M16—21no—
343M16—21no—
344M16—32no—
345M16—34no—
346M16—61no—
347M16—62no—
348M16—64no—
349C2×C8—21no—
350C2×C8—31no—
351C2×C8—31no—
352C2×C8—61no—
353C2×C8—62no—
354C2×C8—61no—
355C2×C8—61no—
356C2×C8—64no—
357C2×C8C2×C4×D312yes—
358C2×C8#23512yes—
359C2×C8C2×#6711yescharacteristic
360C2×C8C2×#6711yescharacteristic
361C4⋊C4—62no—
362C4⋊C4—62no—
363C4⋊C4—122no—
364C4⋊C4D3×D411yescharacteristic
365C4⋊C4D3×D411yescharacteristic
366C4⋊C4C2×#6712yes—
367C4⋊C4C2×#6712yes—
368C22⋊C4—21no—
369C22⋊C4—22no—
370C22⋊C4—42no—
371C22⋊C4—42no—
372C22⋊C4—122no—
373C22⋊C4—244no—
374C42—22no—
375C42—21no—
376C42—61no—
377C42—61no—
378C42—122no—
379C22×C3—21no—
380C22×C3—21no—
381C22×C3—44no—
382C22×C3—84no—
383C22×C3—82no—
384C22×C3—82no—
385C22×C3—162no—
386C22×C3#52011yescharacteristic
387C22×C3D4211yescharacteristic
388C22×C3#55311yescharacteristic
389C2×D3—22no—
390C2×D3—41no—
391C2×D3—41no—
392C3×C4—42no—
393C3×C4—41no—
394C3×C4—42no—
395C3×C4—41no—
396C3×C4C2×C4×D411yescharacteristic
397C3×C4C2×C4×D411yescharacteristic
398C3×C4#54711yescharacteristic
399C3×C4#54811yescharacteristic
400Dic3—22no—
401Dic3—42no—
402Dic3—82no—
403C23—24no—
404C23—42no—
405C23—42no—
406C23—61no—
407C23—82no—
408C23—82no—
409C23#100411yescharacteristic
410Q8—32no—
411Q8—61no—
412D4—24no—
413D4—24no—
414D4—32no—
415D4—42no—
416D4—42no—
417D4—62no—
418D4—62no—
419D4—61no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—122no—
431C2×C4—162no—
432C2×C4—162no—
433C2×C4—242no—
434C2×C4—482no—
435C2×C4C22×C4×D311yescharacteristic
436C2×C4C2×#23511yescharacteristic
437C2×C4C2×D3×D411yescharacteristic
438C2×C4D3×#4111yescharacteristic
439C2×C4#101011yescharacteristic
440C2×C4C22×#6711yescharacteristic
441C8—41no—
442C8—62no—
443C8—64no—
444C8—61no—
445C8—61no—
446C8C4×#6712yes—
447C8C4×#6712yes—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—84no—
451C2×C3#236111yescharacteristic
452C2×C3#329011yescharacteristic
453C2×C3#329211yescharacteristic
454D3—42no—
455C22—21no—
456C22—21no—
457C22—44no—
458C22—62no—
459C22—84no—
460C22—82no—
461C22—82no—
462C22—121no—
463C22—121no—
464C22—162no—
465C22#554611yescharacteristic
466C22#555811yescharacteristic
467C22D4×#6711yescharacteristic
468C4—41no—
469C4—42no—
470C4—41no—
471C4—42no—
472C4—62no—
473C4—122no—
474C4—242no—
475C4C4×D3×D411yescharacteristic
476C4#555111yescharacteristic
477C4C2×C4×#6711yescharacteristic
478C4#583111yescharacteristic
479C3#2304611yescharacteristic, complemented, Sylow
480C2—21no—
481C2—21no—
482C2—84no—
483C2—122no—
484C2#8239911yescharacteristic
485C2#8401011yescharacteristic
486C2#8401111yescharacteristic
subgroupG/H
centerC22D4×#67
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#2948C2
socleC22×C3D42
prime radicalC2×#5160C2
cosocleC22×C3×C4C24
hypercenter#2948D3
hypocenterC3#23046
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ #2948

Complements

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

NH ≅ G/Ntype
1. C3×#2948C2semidirect
3. #82537C2semidirect
4. #79179C2semidirect
5. #72038C2semidirect
6. #71647C2semidirect
7. #70659C2semidirect
8. #70328C2semidirect
9. #70318C2semidirect
10. #69894C2semidirect
11. #69887C2semidirect
27. #5160C4semidirect
48. #2948D3semidirect
63. C2×C3×D8C2×C4semidirect
76. #932C2×C4semidirect
84. #901C2×C4semidirect
99. #833C2×C4semidirect
103. #819C2×C4semidirect
222. C2×D8C4×D3semidirect
479. C3#23046semidirect

Maximal subgroups

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

subgroupindex
C3×#29482
C2×#51602
#825372
#791792
#720382
#716472
#706592
#703282
#703182
#698942
#698872
#230463