Finite Group Databasebeta

← #10710913

Subgroups — #10710913

Order 768.

934 conjugacy classes of subgroups.

The subgroup lattice has 486 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5752C211yesmaximal, characteristic
2C2×C3×#350C211yesmaximal, characteristic, complemented
3#72016C211yesmaximal, characteristic, complemented
4C2×#4516C211yesmaximal, characteristic, complemented
5#71439C211yesmaximal, characteristic, complemented
6C2×#4510C211yesmaximal, characteristic, complemented
7C2×#4509C211yesmaximal, characteristic, complemented
8#12748—31nomaximal, Sylow
9C2×#1009—21no—
10C2×C3×#131C2211yescharacteristic
11#5752—22no—
12#5752—22no—
13#5752—22no—
14#5752C414yescomplemented
15C2×#908—21no—
16C2×#908—22no—
17C2×#904—21no—
18C2×#904C2211yescharacteristic
19C22×D24—21no—
20C22×D24C2211yescharacteristic
21C2×C3×M32C2211yescharacteristic, complemented
22C22×C3×C16C2211yescharacteristic, complemented
23#5232—21no—
24C2×#819C2211yescharacteristic, complemented
25C2×#818C2211yescharacteristic, complemented
26C3×#350—22no—
27#4554—21no—
28#4516—22no—
29#4510—22no—
30#4509—21no—
31#4509—22no—
32#4503—21no—
33C2×#576—31no—
34C2×#361—31no—
35C2×#359—31no—
36#2139—31no—
37C2×#357—31no—
38C2×#350D311yescharacteristic, complemented
39#1816—31no—
40C2×C3×#41D411yescharacteristic
41#1009—24no—
42#1009—24no—
43#1009—42no—
44#1009—42no—
45C2×#239—41no—
46C2×D3×D4—42no—
47C2×D3×D4—41no—
48C2×#235—22no—
49C2×#235—21no—
50C22×D12—22no—
51C22×D12—21no—
52C3×#131—21no—
53C3×#131—21no—
54C3×#131C2×C412yes—
55C3×#131D412yes—
56C2×C3×M16—21no—
57C2×C3×M16D411yescharacteristic
58C22×C3×C8—21no—
59C22×C3×C8C2311yescharacteristic
60C2×C4⋊Dic3—21no—
61C2×#208—21no—
62#908—24no—
63#908—28no—
64#908—24no—
65#908—28no—
66C2×Dic12D411yescharacteristic
67#904—24no—
68#904—21no—
69#904—21no—
70#904—24no—
71#904C2×C412yes—
72#904D412yes—
73C2×D24—22no—
74C2×D24—24no—
75C2×D24—21no—
76C2×D24—21no—
77C2×D24—21no—
78C2×D24—24no—
79C2×D24C2×C412yes—
80C2×D24D411yescharacteristic
81C2×D24D412yes—
82C2×D24D412yes—
83C2×#204—22no—
84C2×#204—21no—
85C3×M32—21no—
86C3×M32—21no—
87C2×C3×C16—21no—
88C2×C3×C16—21no—
89C2×C3×C16—21no—
90C2×C3×C16—21no—
91#835—41no—
92#832—41no—
93#819—21no—
94#819—21no—
95#818—21no—
96#818—21no—
97C2×H(5,𝔽2)—61no—
98#576—34no—
99#576—62no—
100#576—62no—
101#576—62no—
102C2×AGL(1,ℤ/8)—62no—
103C2×AGL(1,ℤ/8)—61no—
104C2×#135—31no—
105C2×#135—61no—
106C22×D8—31no—
107C22×D8—61no—
108C2×#131C2×D311yescharacteristic
109C2×M32C2×D311yescharacteristic, complemented
110C22×C16C2×D311yescharacteristic, complemented
111C2×#108—31no—
112C2×#107—31no—
113#417—61no—
114#361—62no—
115#359—62no—
116#358—61no—
117#357—62no—
118#357—61no—
119#350—22no—
120#343—61no—
121C23×D3—42no—
122C23×D3—41no—
123C3×#41—21no—
124C3×#41—21no—
125C3×#41C22⋊C412yes—
126C3×#41SD1612yes—
127C2×C3×Q8D811yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4D811yescharacteristic
130C22×C3×C4—21no—
131C22×C3×C4C2×D411yescharacteristic
132C2×#67—42no—
133C2×#67—41no—
134C22×Dic3—41no—
135#239—44no—
136#239—44no—
137D3×D4—48no—
138D3×D4—48no—
139D3×D4—44no—
140D3×D4—44no—
141#235—28no—
142#235—22no—
143#235—22no—
144#235—44no—
145#235—41no—
146#235—41no—
147C2×D12—28no—
148C2×D12—22no—
149C2×D12—22no—
150C2×D12—21no—
151C2×D12—22no—
152C2×D12—42no—
153C2×D12—44no—
154C2×D12—41no—
155C2×D12—41no—
156C2×D12—41no—
157C2×C4×D3—42no—
158C2×C4×D3—41no—
159C2×Dic6—21no—
160C3×M16—21no—
161C3×M16—22no—
162C3×M16—22no—
163C3×M16C22⋊C412yes—
164C2×C3×C8—21no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C22×C411yescharacteristic
168C2×C3×C8C2×D411yescharacteristic
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8C2×D411yescharacteristic
173C4⋊Dic3—41no—
174C4⋊Dic3—41no—
175#208—41no—
176#208—41no—
177C2×C3⋊C8—41no—
178Dic12—21no—
179Dic12C22⋊C412yes—
180D24—28no—
181D24—22no—
182D24—21no—
183D24—22no—
184D24—22no—
185D24C22⋊C412yes—
186D24C22⋊C414yes—
187#204—28no—
188#204—22no—
189#204—22no—
190C3×C16—41no—
191C3×C16—41no—
192H(5,𝔽2)—64no—
193H(5,𝔽2)—64no—
194H(5,𝔽2)—122no—
195H(5,𝔽2)—122no—
196C2×#41—62no—
197C2×#41—61no—
198C2×#41—121no—
199C2×#41D1211yescharacteristic
200C22×D4—62no—
201C22×D4—61no—
202C22×D4—122no—
203C22×D4—121no—
204AGL(1,ℤ/8)—68no—
205AGL(1,ℤ/8)—64no—
206AGL(1,ℤ/8)—68no—
207AGL(1,ℤ/8)—64no—
208#135—32no—
209#135—32no—
210#135—64no—
211#135—64no—
212#135—61no—
213#135—61no—
214C2×Q16—31no—
215C2×SD16—62no—
216C2×SD16—61no—
217C2×D8—32no—
218C2×D8—32no—
219C2×D8—32no—
220C2×D8—31no—
221C2×D8—64no—
222C2×D8—62no—
223C2×D8—64no—
224C2×D8—61no—
225C2×D8—61no—
226C2×D8—61no—
227#131—21no—
228#131—21no—
229#131C4×D312yes—
230#131#6712yes—
231C2×M16—21no—
232C2×M16—61no—
233C2×M16D1211yescharacteristic
234C22×C8—21no—
235C22×C8C22×D311yescharacteristic
236C2×C4⋊C4—61no—
237M32—21no—
238M32—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—21no—
242C2×C16—21no—
243#108—61no—
244#108—61no—
245#107—61no—
246#107—61no—
247#103—121no—
248#102—121no—
249C23×C3—21no—
250C23×C3AGL(1,ℤ/8)11yescharacteristic
251C22×D3—42no—
252C22×D3—41no—
253C22×D3—48no—
254C22×D3—42no—
255C22×D3—42no—
256C22×D3—161no—
257C22×D3—161no—
258C22×D3—162no—
259C22×D3—162no—
260C3×Q8—21no—
261C3×Q8#10212yes—
262C3×D4—22no—
263C3×D4—22no—
264C3×D4—21no—
265C3×D4#10212yes—
266C2×C3×C4—21no—
267C2×C3×C4—22no—
268C2×C3×C4—22no—
269C2×C3×C4C2×C22⋊C411yescharacteristic
270C2×C3×C4#12011yescharacteristic
271C2×C3×C4#12011yescharacteristic
272C2×C3×C4C2×D811yescharacteristic
273C2×C3×C4C2×SD1611yescharacteristic
274C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
275#67—48no—
276#67—42no—
277#67—42no—
278C2×Dic3—41no—
279C2×Dic3—81no—
280C2×Dic3—161no—
281D12—24no—
282D12—22no—
283D12—48no—
284D12—42no—
285D12—42no—
286D12—41no—
287D12—42no—
288C4×D3—48no—
289C4×D3—42no—
290C4×D3—42no—
291Dic6—22no—
292Dic6—41no—
293C3×C8—22no—
294C3×C8C2×C22⋊C411yescharacteristic
295C3×C8C2×C22⋊C411yescharacteristic
296C3×C8#12011yescharacteristic
297C3×C8#12011yescharacteristic
298C3⋊C8—81no—
299C24—122no—
300C24—121no—
301#41—21no—
302#41—21no—
303#41—68no—
304#41—62no—
305#41—62no—
306#41—124no—
307#41—124no—
308#41—124no—
309#41—121no—
310#41—121no—
311#41#20412yes—
312#41(C2×D3)⋊C412yes—
313C2×Q8—61no—
314C2×Q8D2411yescharacteristic
315C2×D4—21no—
316C2×D4—68no—
317C2×D4—62no—
318C2×D4—61no—
319C2×D4—62no—
320C2×D4—62no—
321C2×D4—122no—
322C2×D4—124no—
323C2×D4—128no—
324C2×D4—128no—
325C2×D4—124no—
326C2×D4—122no—
327C2×D4—121no—
328C2×D4—124no—
329C2×D4—121no—
330C2×D4—121no—
331C2×D4—121no—
332C2×D4D2411yescharacteristic
333C22×C4—21no—
334C22×C4—122no—
335C22×C4—121no—
336C22×C4—121no—
337C22×C4C2×D1211yescharacteristic
338Q16—32no—
339Q16—61no—
340SD16—68no—
341SD16—62no—
342SD16—62no—
343D8—34no—
344D8—32no—
345D8—68no—
346D8—62no—
347D8—62no—
348D8—61no—
349D8—62no—
350M16—21no—
351M16—22no—
352M16—22no—
353M16—121no—
354M16—121no—
355M16(C2×D3)⋊C412yes—
356C2×C8—21no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—121no—
360C2×C8C2×C4×D311yescharacteristic
361C2×C8D3×D411yescharacteristic
362C2×C8D3×D411yescharacteristic
363C2×C8D3×D411yescharacteristic
364C2×C8D3×D411yescharacteristic
365C2×C8C2×#6711yescharacteristic
366C4⋊C4—121no—
367C4⋊C4—121no—
368C16—41no—
369C16—41no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—22no—
373C22×C3—41no—
374C22×C3—41no—
375C22×C3#41811yescharacteristic
376C22×C3#44711yescharacteristic
377C22×C3#45011yescharacteristic
378C2×D3—44no—
379C2×D3—42no—
380C2×D3—82no—
381C2×D3—81no—
382C2×D3—82no—
383C2×D3—81no—
384C2×D3—162no—
385C2×D3—162no—
386C2×D3—168no—
387C3×C4—22no—
388C3×C4#37911yescharacteristic
389C3×C4C2×#10211yescharacteristic
390C3×C4#44711yescharacteristic
391C3×C4#45011yescharacteristic
392Dic3—42no—
393Dic3—161no—
394C23—21no—
395C23—128no—
396C23—122no—
397C23—121no—
398C23—122no—
399C23—122no—
400C23—481no—
401C23—481no—
402C23—482no—
403C23—482no—
404C23#90811yescharacteristic
405Q8—21no—
406Q8—62no—
407Q8—121no—
408Q8#82112yes—
409D4—22no—
410D4—22no—
411D4—21no—
412D4—64no—
413D4—62no—
414D4—128no—
415D4—128no—
416D4—122no—
417D4—122no—
418D4—121no—
419D4—122no—
420D4—122no—
421D4—122no—
422D4#82112yes—
423C2×C4—21no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—128no—
427C2×C4—121no—
428C2×C4—122no—
429C2×C4—122no—
430C2×C4—241no—
431C2×C4—481no—
432C2×C4#88211yescharacteristic
433C2×C4#88211yescharacteristic
434C2×C4C2×#20411yescharacteristic
435C2×C4C2×D2411yescharacteristic
436C2×C4#90811yescharacteristic
437C2×C4C2×(C2×D3)⋊C411yescharacteristic
438C8—22no—
439C8—241no—
440C8D3×C22⋊C411yescharacteristic
441C8D3×C22⋊C411yescharacteristic
442C8#93711yescharacteristic
443C8#93711yescharacteristic
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—42no—
447C2×C3#178611yescharacteristic
448C2×C3#219211yescharacteristic
449C2×C3#219411yescharacteristic
450D3—84no—
451D3—82no—
452C22—21no—
453C22—21no—
454C22—22no—
455C22—41no—
456C22—41no—
457C22—124no—
458C22—122no—
459C22—242no—
460C22—241no—
461C22—242no—
462C22—241no—
463C22—482no—
464C22—482no—
465C22—488no—
466C22#473111yescharacteristic
467C22#473211yescharacteristic
468C22#513111yescharacteristic
469C4—22no—
470C4—122no—
471C4—481no—
472C4#473111yescharacteristic
473C4#473211yescharacteristic
474C4#495511yescharacteristic
475C4C2×#82111yescharacteristic
476C3#1274811yescharacteristic, complemented, Sylow
477C2—21no—
478C2—21no—
479C2—42no—
480C2—244no—
481C2—242no—
482C2#6912611yescharacteristic
483C2#6912811yescharacteristic
484C2#6995911yescharacteristic
subgroupG/H
centerC22#4732
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC2×C3×#350C2
socleC22×C3#447
prime radicalC2×#5752C2
cosocleC22×C3×C8C23
hypercenterC2×#350D3
hypocenterC3#12748
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
2. C2×C3×#350C2semidirect
3. #72016C2semidirect
4. C2×#4516C2semidirect
5. #71439C2semidirect
6. C2×#4510C2semidirect
7. C2×#4509C2semidirect
14. #5752C4semidirect
21. C2×C3×M32C22semidirect
22. C22×C3×C16C22semidirect
24. C2×#819C22semidirect
25. C2×#818C22semidirect
38. C2×#350D3semidirect
109. C2×M32C2×D3semidirect
110. C22×C16C2×D3semidirect
476. C3#12748semidirect

Maximal subgroups

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

subgroupindex
C2×#57522
C2×C3×#3502
#720162
C2×#45162
#714392
C2×#45102
C2×#45092
#127483